PATRON
Dr. S. Marriya
Regional Director
DAV Public School (U.P. & Panipat Zone)
Board Of Studies
Chairman
Ms.Ritu Dilbagi
Principal
DAV Public School Thermal Colony Panipat
Co-Ordinators
Ms. Renu Sharma
Principal
SF DAV Public School
Muzaffarnagar,
Mr. Vijay Chandel
Mr. O. P. Mishra
Principal
ND DAV Public School
Kumarganj
Ms.Anu Maheshwari
Principal
DAV Public School
Unchahar
PGT Maths
DAV Public School
Thermal Colony Panipat
Designer
Ruchi Chawla
C.S. Department
DAV Public School
Thermal Colony
Panipat
CONTENTS
Course Structure & Exam Pattern (2016-17)
1. Relations and Functions
1
2. Inverse Trigonometric Functions
13
3 & 4. Matrices and Determinants
24
5. Continuity and Differentiability
52
6. Applications of Derivatives
65
7. Integrals
82
8. Applications of Integrals
119
9. Differential Equations
130
10. Vectors
146
11. Three-Dimensional Geometry
167
12. Linear Programming
188
13. Probability
202
14. Design of Question Paper & Sample Question
Paper issued by CBSE
224
Course Structure & Exam Pattern (2016-17)
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Chapter 1 : Relations and Functions
Points to Remember
 A relation R between two non empty sets A and B is a subset of their Cartesian Product A  B.
If A = B then relation R on A is a subset of A A
 If (a, b) belongs to R, then a is related to b, and written as a R b If (a, b) does not belongs to
R and written as (a, b) R
 Let R be a relation from A to B.
o Then Domain of R  A and Range of R  B co domain is either set B or any of its
superset or subset containing range of R
 A relation R in a set A is called empty relation, if no element of A is related to any element of
A, i.e., R =  A × A.
 A relation R in a set A is called universal relation, if each element of A is related to every
element of A, i.e., R = A × A.
 A relation R in a set A is called
o
Reflexive, if (a, a)  R, for every a A,
o
Symmetric, if (a,b)  R implies that (b, a)  R, for all a, b  A.
o
Transitive, if (a, b)  R and (b, c)  R implies that (a, c) R, or all a, b, c  A.
 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and
transitive.
 The empty relation R on a non-empty set X (i.e. a R b is never true) is not an equivalence
relation, because although it is vacuously symmetric and transitive, it is not reflexive (except
when X is also empty)
 Given an arbitrary equivalence relation R in a set X, R divides X into mutually disjoint
subsets Si called partitions or subdivisions of X satisfying:
 All elements of Si are related to each other, for all i
 No element of Si is related to Sj ,if i j
n
 Sj=X and Si
Sj=, if ij
i1



The subsets Sj are called Equivalence classes. 
 A function from a non empty set A to another non empty set B is a correspondence or a rule
which associates every element of A to a unique element of B written as
f:A  B s.t f(x) = y for all xA, yB.
All functions are relations but converse is not true.
 If f: A  B is a function then set A is the domain, set B is co-domain and set {f(x):xA } is
1
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
the range of f. Range  codomain.
 f: A  B is one-to-one if
o For all x1, x2 A f(x1) = f(x2)  x1 = x2 or x1x2 f(x1)  f(x2)
o A one- one function is known as injection or an Injective Function. Otherwise, f is
called many-one.

f: A  B is an onto function ,if for each b Bthere is atleastone a A such that f(a) = b
o i.e if every element in B is the image of some element in A, f is onto.
o
For an onto function range = co-domain.
 A function which is both one-one and onto is called a bijective function or a bijection.
 A one – one function defined from a finite set to itself is always onto but if the set is
infinite then it is not the case.
Let f :AB and g : BC be two functions. Then the composition of f and g, denoted by gof is
defined as the function gof: A  C given by gof(x): A  C defined by gof(x) = g(f(x))  x A
Composition of f and g is written as gof and not fog
gof is defined if the range of f  domain of f and fog is defined if range of g  domain of f
 Composition of functions is not commutative in general fog(x) ≠ gof(x).Composition is
associative
o
If f: X  Y, g: Y  Z and h: Z  S are functions then ho(g o f)=(h o g)of
 If f:A B and g: B C are one-one and onto then gof: AC is alsoone-one and onto. But If
g o f is one –one then only f is one –one g may or may not be one-one. If g o f is onto then
g is onto f may or may not be onto.
 If f: RR is invertible, f(x)=y, then f1 (y)=x and (f-1)-1 is the function f itself.
 A function f: X  Y is defined to be invertible, if there exists a function
g : Y  X such that gof = IX and fog = IY. The function g is called the inverse of f and is
denoted by f –1
 If f is invertible, then f must be one-one and onto and conversely, if f is one- one and onto,
then f must be invertible.
 Let f: X  Y and g: Y Z betwo invertible functions. Then gof is also Invertible with (gof)–1
= f –1o g–1.
2
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
 A binary operation * on a set A is a function *: A X A A.
 Addition, subtraction and multiplication are binary operations on R, the set of real numbers.
Division is not binary on R, however, division is a binary operation on R-{0}, the set of nonzero real numbers
 A binary operation  on the set X is called commutative, if a  b= b  a, for every a,bX
 A binary operation  on the set X is called associative, if a  (b*c) =(a*b)*c, for every a, b,
cX
 An element eA is called an identityfor the operation *:A X A A, if for each aA,
a * e = a = e * a.
 The identity element if it exists, is unique.
 Given a binary operation  from A A A, with the identity element e in A, an element aA is
said to be invertible with respect to the operation  , if there exists an element b in A such
that a  b=e= b a, then b is called the inverse of a and is denoted by a-1.
Addition '+' and multiplication '·' on N, the set of natural numbers are binary operations But
subtraction ‘–‘ and division ‘÷’ are not since (4, 5) = 4 - 5 = -1N and 4/5 =.8 N
2
 Total number of binary operations on a set consisting of n elements is 𝑛𝑛

3
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
1. If A is the set of students of a school then write, which of following relations are
Universal, Empty or neither of the two.
i. R1= {(a, b) : a, b are ages of students and |a – b|0}
ii. R2= {(a, b) : a, b are weights of students, and |a – b| < 0}
iii. R3= {(a, b) : a, b are students studying in same class}
2. Is the relation R in the set A = {1, 2, 3, 4, 5} defined as
i. R = {(a, b) : b = a + 1} reflexive?
3. If R, is a relation in set N given by
i. R = {(a, b) : a = b – 3, b > 5},then does element (5, 7)  R?
4.
If f : {1, 3}  {1, 2, 5} and g : {1, 2, 5}  {1, 2, 3, 4} be given by f = {(1, 2), (3, 5)},
g = {(1, 3), (2, 3), (5, 1)},writegof.
5. Let g, f : RR be defined by 𝑔(𝑥)
6. If
f : RR defined by 𝑓(𝑥)
7. If 𝑓(𝑥)
=
𝑥
𝑥+1
=
=
𝑥+2
3
, 𝑓(𝑥) = 3𝑥 − 2write f o g(x)
2𝑥−1
5
be an invertible function, write f–1(x).
∀ 𝑥 ≠ −1, 𝑤𝑟𝑖𝑡𝑒 𝑓 𝑜 𝑓(𝑥).
8. Let * be a Binary operation defined on R, then if
a. a * b = a + b + ab, write 3 * 2
b. a * b =
(𝑎+𝑏)2
3
, write (2*3)*4
9. If n(A) = n(B) = 3, then how many bijective functions from A to B can be formed?
10. If f (x) = x + 1, g(x) = x – 1, then (gof) (3) = ?
11. Is f :NN given by f(x) = x2 one-one? Give reason.
12. If f : RA, given by f(x) = x2- 2x+2 is onto function, find set A
13. If f :AB is bijective function such that n (A) = 10, then n (B) = ?
14. R = {(a, b) : a, b  N, a  b and a divides b}. Is R reflexive? Give reason
15. Is f :RR, given by f(x) = |x – 1| one-one? Give reason
16. f : R  B given by f(x) = sin x is onto function, then write set B.
4
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
1+𝑥
2𝑥
17.If f(x) = log (
), show thatf(1+𝑥 2) = 2 f(x)
1−𝑥
18. If * is Binary operation on N defined by a*b = a + aba, bN, write the identity
element in N if it exists.
19. Check the following functions for one-one and onto.
3 x 1
f x 
(i) f : R – {2} R,
x 2
(ii) f : R [–1, 1], f(x) = sin2x
20. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a* b = H.C.F. ofaand b. Write the
operation table for the operation*.
−4
4𝑥
4
21. Let 𝑓: 𝑅 − ( 3 ) → 𝑅 − (3) be a function given by f(x) = 3𝑥+4 Show that f is invertible with 𝑓 −1 (𝑥) =
4𝑥
4−3𝑥
22. Let R be the relationon set A = {x : x ∈ Z, 0 ≤ x ≤ 10} given byR = {(a, b) : (a – b) is divisible by 4}.
Show that R is an equivalencerelation. Also, write all elements related to 4.
23. If * is a binary operation defined on R – {0} defined by a * b
=
2𝑎
, thencheck * for commutativity and
𝑏2
associativity.
24. If A = N × N and binary operation * is definedonA as (a, b) * (c, d) = (ac, bd).
a.
Check * for commutativity and associativity.
b.
Find the identity element for * in A (If it exists).
𝑎𝑏
25. Let * be a binary operation on set Q defined by a*b
a. 4 is the identity element in Q.
4
,show that
b. Every non zero element of Q is invertible with𝑎 −1
26. Show that f : R+→ R+ defined by f (x )=
1
2𝑥
=
16
𝑎
, 𝑎 ∈ 𝑄 − {0}
is bijective where R+ is the set of all non-zero positive real
numbers.
27. Let A = {1, 2, 3, ...., 12} and R be a relation in A × A defined by (a, b) R (c, d) if ad = bc∀ (a, b), (c, d), ∈ A ×
A. Prove that R is an equivalence relation. Also obtain the equivalence class [(3, 4)].
28. If f, g : R → R defined by f(x) = x2 – x and g(x) = x + 1 find (fog) (x) and (gof) (x). Are they equal?
29. If 𝑓: [1, ∞) → [2, ∞)𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓(𝑥) = 𝑥 +
1
𝑥
, 𝑓𝑖𝑛𝑑 𝑓 −1 (𝑥)
30. Let R be the equivalence relation in the set A={0,1,2,3,4,5} Given by R = {(𝑎, 𝑏): 2 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 (𝑎 −
𝑏)}write the equivalence class for [0].
31. Let A = { 1, 2, 3 } Then find the number of equivalence relation containing ( 1 , 2 )
5
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
32. Let f:{1, 3, 4} → {1, 2, 5} and g = {1, 2, 5} → {1 , 2 ,3}given by f = {(1 , 2), (3 , 5), (4 ,1) and g =
{(1, 3), (2 , 3), (5, 1)} Write down gof. }
33. Find the number of binary operation on the set { a, b }.
34. Let f : R→ R be defined as f(x) = 10x + 7. find the function g: R → R such that
gof = fog = IR .
35. Let us consider a set A = set of all roots of quadric equationx 2 + x − 6 = 0 , a function f: A →
B is defined such that f(x) = x 2 + 3 , find the range of set B.
36. If A ={a, b, c, d} and the function f = {(a, b), (b, d), (c, a), (d, c)} , then write f −1
37. Write fog , if f : R → R And g ∶ R → R are given by f(x) = |x| and g(x) = |5x − 2| .
38. Is{(1, 1), (2, 3), (3, 5), (4, 7)} a function ? if g is described by g(x) = αx +
β , then what value would be assigned to α and β. ?
39. Let f : X → Y be a function .Define a relation R on X given by R = { (a ,b ) : f(a) = f(b) }.
Show that R is an equivalence relation on X . Give an example of equivalence relation in a
family .
40. Using the definition , prove that the function f : A → B is invertible if and only if f is both one –
one and onto .
41. What is the range of the function f ( x ) =
42. If f : R → R be the function defined by
|x−1|
x−1
?.
f ( x ) = 4 x3 + 7 , show that f is a bijection .
43. If f( x ) = ex and g( x) = loge x ( x > 0 ) , then find
44. If f ( x )
=
3x−2
2x−3
, prove that f ( f ( x) ) = x
45. Let f ( x ) = | x | , show that
fog and gof . Is fog = g of ? .
for all x ∈ R - {3/2} .
fof = f .
46. Let A = R - {3} and B = R - { 1 } . Consider thefunction f : A → B defined by f ( x ) = ( x
-2 ) / ( x – 3 ). Show that f is one –one and onto and hence find f -1.
47. Given the relation R = { ( 1 ,2 ) , ( 2 ,3 ) } on the set A = {1 , 2 ,3 } , add minimum number
of ordered pairs so that the enlarged relation is reflexive ,symmetric and transitive .
6
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
48. Let S be the set of all points in a plane and R be the on S defined as
R = {(P, Q): distance between Pand Q is less than 2 units}.Show that R is reflexive and
symmetric but not transitive.
49. Show that the relation R defined in the set A of all triangles as R = {(T 1, T2): T1 is similar to T2},
is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides
5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?.
50. If set A has two elements and set B has three elements .find number of relation on AXB
51. Let Z be the set of all integers and R be the relation on Z defined as R={(a,b):a,b∈ Z, and a-b
is divisible by 5 } Prove that R is an equivalence relation
52. Show that the function f:R⟶{x∈R ,-1<x<1},defined by f(x)=x/(1+x) ,x∈R is one-one and onto
function
53. If f:R→R Is defined as f(x)=x2- 3x+2,find f(f(x)).
54.
3
2x
Let f: R − {− } → R be a function defined as f(x) =
5
5x+3
Find f −1 ∶
3
Range of f → R − {− }
5
55.
If f(x) = [ x ] and g (x) = │x │, then evaluate (f o g) (1/2) – (g o f) (−1/2)
7
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PROFICIENCY EXERCISE
1. Let f: N → N be a function defined as f(x) = 9x2 + 6x – 5. Show that f: N→ S, where S is the
range of f, is invertible. Find the inverse of f and hence find f-1(43) & f-1(163)
2. Show that the binary operation *on A =R – {-1} defined as a*b = a+b+ab,a,b∈A is
commutative and associative on A. Also find the identity element of * in A and prove that every
element of A is invertible.
3. Show that the relation R defined by (a,b) R (c,d) a+d = b+c on the AxA, where A = {
1,2,3,……10} is an equivalence relation. Hence write the equivalence class [(3,4)];a,b,c,d ∈ A
4. Let A = R x R and * be a binary operation on A defined by (a,b) * (c,d) = (a+c, b+d).
Show that * is commutative and associative. Find the identity element for * on A. Also find the
inverse of every element (a,b) ∈ A.
5. Let f: N →N be a function defined as f(x) = 4x2 + 12x + 15. Show that f: N→ S is invertible
(where S is range of f) Find the inverse of f and hence find f-1(31) & f-1(87)
6. If f,g: R→ R be two functions defined as f(x) = |x| + x and g(x) = |x | - x, x ∈ R. then find
fog and gof . Hence find fog(-3), fog(5) and gof(-2).
7. Let N denote the set of all natural numbers and R be the relation on NxN defined by
(a,b)R(c,d) is ad(b+c) = bc (a+d). Show that R is an equivalence relation.
8. Determine whether the relation R defined on the set R of all real numbers as R = {(a,b) :
a,b∈ R & a-b + √3∈ S, where S is the set of all irrational numbers}, is reflexive, symmetric
and transitive.
9. Consider f ; R+→ [-9, ∞) given by f(x) = 5x2 + 6x-9. Prove that f is invertible withf-1 (y)=
√54+5y − 3
5
10. Let A = {-1,0,1,2}, B = {-4, -2, 0,2}, and f,g : A → B be functions defined by f(x)=x2-x,
x∈A and g(x) = 2| x – ½ | - 1, x∈ A. Find (gof) (x) and hence show that f=g=gof.
8
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
11. Let A = QxQ , where Q is the set of all rational numbers and * be a binary operation on A
defined by (a,b) * (c,d) = (ac, b+ad) for (a,b), (c,d)∈ A . Then find
i)
ii)
The identity element of * in A.
Invertible elements of A, hence write the inverse of elements (5,3) and (1/2 , 4).
𝑛 − 1 , 𝑛 𝑖𝑠 𝑜𝑑𝑑
12. Let f: W→ W be defined as 𝑓(𝑥) = {
𝑛 + 1 , 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
Show that f is invertible and find the inverse of f. Here, W is the set of all whole numbers.
13. If the function f: R→ R be defined by f(x) = 2x-3 and g: R→ R by g(x) = x3+5, then find
the value of (fog)-1(x).
14. Show that the relation R in the set A={1,2,3,4,5} given by R = {(a,b): |a-b| is divisible by 2}
is an equivalence relation. Show that all the elements of {1,3,5} are related to each other and all
the elements of {2,4} are related to each other, but no element of {1,3,5} is related to any
element of {2,4}.
15. On the set {0,1,2,3,4,5,6} a binary operation * is defined as
𝑎+𝑏
𝑖𝑓 𝑎 + 𝑏 < 7
𝑎∗𝑏 = {
𝑎 + 𝑏 − 7 𝑖𝑓 𝑎 + 𝑏 ≥ 7
Write the operation table of the operation * and prove that zero is the identity element for this
operation and each element a ≠ 0 of the set is invertible with 7 –a being the inverse of a.
9
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
14.
10
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
15.
16.
18.
19.
(i)
(ii)
20.
22.
23.
24.
27.
28.
11
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
29.
30.
[0] = { 0 , ±2 , ±4 , ±6 ------------}
31.
32.gof = { ( 1 ,3) , (3 ,1 ) , (4, 3) }
33.
2
16
35.{ 7 , 12}
f-1 = { (b,a) ,(d, b),(a,c) ,(c,d) }
36.
37. | 5x-2 |
38.α = 2 , β = -1
41.{ -1 , 1 }
43. (fog) (x) = x
, (gof) (x) = x
,
gof ≠ fog
3𝑥−2
46.
f-1(x) =
47.
(1,1) , (2,2) , (3,3) , (1,3) , (2,1) , (3,2) , (3,1)
50.
64
54.
f-1(x) =
𝑥−1
53.x4 – 6x3 + 10x2 – 3x
3𝑥
55.
2−5𝑥
-1
ANSWERS OF PROFICIENCY EXERCISE
−1+√𝑦+6
1. f-1(y)=
3
, 2, 4
2.Identity = 0, Inverse =
−𝑎
1+𝑎
3. [(3,4)] = {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8), (8,9), (9,10)}
4. (0,0), (-a, -b)
5.f-1(y)=
√𝑦−6 − 3
2
, 1, 3
6. (gof) (x) = 0 ∀x∈ R(fog) (x) ={ 0
−4𝑥
(gof) (-2) = 0,
3
13. √
𝑥≥0
𝑥<0
(fog) (-3) = 12, (fog) (5) = 0
𝑥−7
2
12
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Chapter 2 :Inverse Trigonometric Functions
Points to Remember

Inverse trigonometric functions map real numbers back to angles.

Inverse of sine function denoted by sin-1 or arc sin(x) is defined on [-1,1]

and range could be any of the intervals
[
The branch of sin-1 function with range
[
sin-1: [-1,1]→ [

−3𝜋 −𝜋
,
2
2
],[
−𝜋 𝜋
2
𝜋 3𝜋
, ],[ ,
2
2
2
]
−𝜋 𝜋
2
, ]is the principal branch So
2
−𝜋 𝜋
, ]
2 2
The graph of sin-1 x is obtained from the graph of sine x by
interchanging the x and y axes

Graph of the inverse function is the mirror image (i.e reflection) of the
original function along the line y = x.

Inverse of cosine function denoted by cos-1 or arc cos(x) is defined in
[-1,1] and range could be any of the intervals [-,0], [0,],[,2].
So,cos-1: [-1,1] [0,].

−𝜋 𝜋
The branch of tan-1 function with range ( 2 , 2)is the principal branch
So
−𝜋 𝜋
tan-1: R→ ( 2 , 2 )

The branch of cosec-1 function with range
branch So
[
−𝜋 𝜋
2
, ] − {0}is
2
the principal
−𝜋 𝜋
cosec-1:R- (-1,1)→ [ 2 , 2 ]- {0}
𝜋
 The branch of sec-1 function with range [0,𝜋]-{
𝜋
branch So sec-1 : R- (-1,1)→ [0,𝜋]-{

2
2
}
}
is the principal
cot-1is defined as a function with domain R and range as any of
13
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
theintervals (-,0), (0,),(,2). The principal branch is (0,) So cot-1 :
R  (0,)

The value of an inverse trigonometric function which lies in the
range of principal branch is called the principal value of the inverse
trigonometric functions.

Inverse of a function is not equal to the reciprocal of the function.

Properties of inverse trigonometric functions are valid only on
theprincipal value branches of corresponding inverse functions or
wherever the functions are defined
14
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Holds for all other trigonometric ratios as well.
15
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
(iii)𝐬𝐢𝐧−𝟏 𝒙 + 𝐬𝐢𝐧−𝟏 𝒚 = 𝐬𝐢𝐧−𝟏 [𝒙√𝟏 − 𝒚𝟐 + 𝒚√𝟏 − 𝒙𝟐 ]
(iv)𝐬𝐢𝐧−𝟏 𝒙 − 𝐬𝐢𝐧−𝟏 𝒚 = 𝐬𝐢𝐧−𝟏 [𝒙√𝟏 − 𝒚𝟐 − 𝒚√𝟏 − 𝒙𝟐 ]
(v)𝐜𝐨𝐬 −𝟏 𝒙 + 𝐜𝐨𝐬 −𝟏 𝒚 = 𝐜𝐨𝐬 −𝟏 [𝒙𝒚 − √𝟏 − 𝒙𝟐 √𝟏 − 𝒚𝟐 ]
(vi)𝐜𝐨𝐬 −𝟏 𝒙 − 𝐜𝐨𝐬 −𝟏 𝒚 = 𝐜𝐨𝐬 −𝟏 [𝒙𝒚 + √𝟏 − 𝒙𝟐 √𝟏 − 𝒚𝟐 ]
Important substitutions to simplify trigonometrical expressions involving
inverse trigonometrical functions
√𝑎 2 + 𝑥 2 ,
𝑥 = 𝑎 𝑡𝑎𝑛𝜃 𝑜𝑟 𝑎 cot 𝜃
√𝑎 2 − 𝑥 2 ,
𝑥 = 𝑎 𝑠𝑖𝑛 𝜃 𝑜𝑟 𝑎 cos 𝜃
√𝑥 2 − 𝑎 2 ,
√
𝑥 = 𝑎 𝑠𝑒𝑐𝜃 𝑜𝑟 𝑎 cosec 𝜃
𝑎+𝑥
𝑎−𝑥
𝑜𝑟 √
,
𝑎−𝑥
𝑎+𝑥
𝑥 = 𝑎 𝑐𝑜𝑠2𝜃
16
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
17
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
18
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
23.If tan−1 x + tan−1 y =
4π
5
,then find the value of cot −1 x + cot −1 y .
24.Express the expression tan−1 [
x
a+ √a2 −𝑥 2
]in simplest form.
2
25.If sin ( sin-1 + cos-1 x ) = 1, then find the value of x.
3
33𝜋
26.Find the value of sin−1 [cos (
5
)].
27.Solve for x , sin−1 x + sin−1 2x =
1−𝑥 2
28.If 5cos −1 (
1+𝑥 2
π
3
.
2𝑥
2𝑥
) + 7 sin−1 (1+𝑥 2) - 4 tan−1 (1−𝑥 2) - tan−1 x = 5π ,then
find the value of x .
19
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
29.Find the real solutions of the equation
tan−1 √x(x + 1) + sin−1 √𝑥 2 + 𝑥 + 1 =
π
2
30.If α and β are the roots of the equation 2x2 – 7x + 3 = 0 , then find the
value of tan−1 α + tan−1 β .
1
1
31.If two geometric series are α = 1 + +
2 4
1
3
+
1
9
+
1
27
+
1
8
+ ⋯ … and
β=1+
+ ⋯ … … … … … , then find the value of the expression sec -1α
+ sec-1 β.
32. Prove that cot -1 7 + cot -1 8 + cot -1 18 = cot-13 .

tan  sin

33. Write the value of
x
y
2
3
1
3
3
 cot 1 
5
2
34.If cos −1 + cos −1 = α, then prove that 9x 2 − 12xy cos α + 4y 2 =
36 sin2 α
3
4
5
5
35.Simplify cos −1 { cos x + sin x}
36.Write the value of cos −1 (cos 6)
37. Prove that cos[tan−1 {sin(cot −1 𝑥)}] = √
𝑥 2 +1
𝑥 2 +2
𝑥
38. If 𝑦 = cot −1 √𝑐𝑜𝑠𝑥 − tan−1 √𝑐𝑜𝑠𝑥 , 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑠𝑖𝑛𝑦 = 𝑡𝑎𝑛2 ( )
2
3
8
5
17
39. Prove that sin−1 + sin−1
= cos −1
36
85
20
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PROFICIENCY EXERCISE
1. If sin [ cot-1 (x+1)]= cos (tan-1 x) , then find x.
2. If (tan-1 x)2 + (cot-1x)2=
5𝜋2
1
8
𝜋
5
4
,then find x.
3. Evaluate: tan { 2 tan-1 + }
4. Solve for x: tan-1(x+1) + tan-1(x-1)= tan-1
5. Prove that: cot
𝑥𝑦+1
-1
𝑥−𝑦
3
-1
6. Prove that: 2 sin
−1
7. If tan
−1
𝑥 + tan
+ cot
𝑦𝑧+1
-1
𝑦−𝑧
17
𝜋
-1
- tan
5
31
−1
𝑦 + tan
=
31
𝑧𝑥+1
-1
+ cot
4
8
𝑧−𝑥
=0 (0<xy,yz,zx<1)
𝜋
𝑧 = , x,y,z>0,then find the value of
2
xy+yz+zx.
𝑥−5
𝑥+5
8. If tan−1 (
) + tan−1 (𝑥+6) =
𝑥−6
𝜋
9. Solve for x:sin−1 (1 − 𝑥) − 2 sin
10. Prove that: 2tan−1 √
𝑎−𝑏
tan
𝑎+𝑏
−1
𝑥
2
, then find the value of x
4
−1
𝑥=
𝜋
2
= cos −1 (
𝑎 𝑐𝑜𝑠𝑥+𝑏
𝑎+𝑏 cos 𝑥
)
11. If 2tan−1 (cos 𝜃) = tan (2 cosec 𝜃), (𝜃 ≠ 0) , then find the value of 𝜃
12. If tan−1 (
1
1
1
) + tan−1 (1+2.3) + − − + tan−1 (1+𝑥(𝑥+1)) = tan−1 𝜃,
1+1.2
then find the value of 𝜃.
2−𝑥
13. Solve for x : tan−1 (
)=
2+𝑥
6𝑥−8𝑥 3
14. Prove that: tan−1 (
1−12𝑥 2
1
2
𝑥
tan−1 , x>0
2
4𝑥
) − tan−1 (1−4𝑥 2) = tan−1 2𝑥, |2x|<
√3
3
15. Solve for x: cos (tan−1 𝑥) = sin (cot −1 )
4
16. Prove that: cot −1 [
√1+sin 𝑥+√1−sin 𝑥
√1+sin 𝑥− √1−sin 𝑥
1
]=
21
𝑥
2
𝜋
, 0<x<
2
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
ANSWERS
22
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
𝜋
23.
5
25.
2/3
1
3
2
7
27. √
1
𝑥
2
𝑎
24. sin−1
26.
−𝜋
10
28.√3
30.π - tan−1 (7)
29.x = 0 , -1
𝜋
3
3
2
31. + sec −1 ( )
35.x -tan−1
33.
4
17
6
36. 2𝜋 − 6
3
ANSWERS OF PROFICIENCY EXERCISE
1. X = - ½
2. X= -1
3. 17 / 7
4.1 / 4
7. 1
8.X= ± 7√2/2
9. X= 0
11.
12. n / n+2
14. x=2/√3
𝜋
4
16. ¾
23
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Chapter 3 & 4: Matrices & Determinants
Points to Remember
MATRICES
 Matrix is an ordered rectangular array of numbers (real or complex) or
functions or names or any type of data. The numbers or functions are called
the elements or the entries of the matrix.
 The horizontal lines of elements are said to constitute, rows of the matrix
and the vertical lines of elements are said to constitute, columns of the
matrix.
 Order of a matrix gives the number of rows and columns present in the
matrix.
 If the matrix A has m rows and n columns then it is denoted by A=[aij]m x
th
th
naijis the element of i row and j column.
 A matrix is said to be a column matrix if it has only one column.
 A = [aij]m x 1 is a column matrix of order m x 1
 A matrix is said to be a row matrix if it has only one row. B = [bij]1 x n is a
row matrix of order 1 x n.
 A matrix in which the number of rows is equal to the number of columns is
said to be a square matrix. A matrix of order “m  n” is said to be a square
matrix if m = n and is known as a square matrix of order „n‟.
 A = [aij]m x m is a square matrix of order m.
 If A = [aij] is a square matrix of order n, then elements a11, a22, …, ann
are said to constitute the diagonal, of the matrix A
 A square matrix B = [bij]m x m is said to be a diagonal matrix if all its non
diagonal elements are zero, that is a matrix B = [bij]m x m is said to be a
diagonal matrix if bij = 0, when i ≠ j.
 A diagonal matrix is said to be a scalar matrix if its diagonal elements
are equal
 A square matrix in which all diagonal elements are 1and rest of the
elements arezero is called an identity matrix.
 A matrix is said to be zero matrix or null matrix if all its elements are zero.
 Two matrices of same order are comparable matrices and only these
Matrices can be added or subtracted
 Two matrices A = [aij] and B = [bij] are said to be equal if
24
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
o
They are of the same order
o
Each elements of A is equal to the corresponding element of B,
i.e.aij= bij for all i and j.
 If A = [aij] be an m x n matrix, then the matrix obtained by interchanging
the rows and columns of A is called the transpose of A.
 Transpose of the matrix A is denoted by A’ or (AT).
 If A=[aij]nxn is an nxn matrix such that AT= A, then A is called Symmetric
Matrix. In a symmetric Matrix aij=aji for all i and j
 If A=[aij]nxn is an nxn matrix such that AT= - A, then A is called Skew
Symmetric Matrix. In a skew symmetric Matrix aij= -aji for all i≠ j and
aij=0 for i=j
 For any square matrix A with real number entries, A+ATis a symmetric
matrix and A-ATis a skew symmetric matrix.
 Every square matrix can be expressed as the sum of a symmetric and skew
symmetric matrix i.e A = ½ (A+AT) + ½ (A-AT)for any square matrix A

Properties of matrix addition
 Matrix addition is commutative i.e A+B = B+A.
 Matrix addition is associative i.e (A + B) + C = A + (B + C).
 Existence of additive identity: Null matrix is the identity w.r.t addition of
matrices i.e.
Given a matrix A= [aij]m x n , there will be corresponding null matrix O
of the same order such that A+O=O+A=A
 The existence of additive inverse Let A = [aij]m x n be any matrix, then
there exists another matrix –A= -[aij]m x n such that A+(–A)=(–A)+A = O.
 A = [aij]m
x n
is a matrix and k is a scalar, then kA is another matrix which is
obtained by multiplying each element of A by the scalar k. Hence kA = [kaij]m x n
25
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT


Properties of Matrix Multiplication
 Commutative law does not hold in matrices, whereas the associative and
distributive laws hold for matrix multiplication
o In general AB  BA
 Matrix multiplication is associative A(BC)=(AB)C
 Distributive laws:
o A(B+C)=AB+BC;
o (A+B)C=AC+BC
 The multiplication of two non zero matrices can result in a null matrix.
 Properties of transpose of matrices
o
o
If A is a matrix, then (AT)T=A
(A + B) T = A T + B T,
o (kB) T = kB T, where k is any constant.
o (AB)T= BTAT
 A square matrix A is called an orthogonal matrix when AAT=ATA=I.
 A null matrix is both symmetric as well as skew symmetric.
 Multiplication of diagonal matrices of same order will be commutative.
 Let A and B be two square matrices of order n such that AB=BA=I.
ThenA is called inverse of B and is denoted by B=A-1.If B is the inverse of
A, then A is also the inverse of B.
 If A and B are two invertible matrices of same order, then (AB)-1 = B-1 A-1
 Elementary operations helps in transforming a square matrix to identity
matrix
26
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
 Inverse of a square matrix, if it exists is unique.
 Inverse of a matrix can be obtained by applying elementary row operations
on the matrix A= IA. In order to use column operations write A=AI
 There are 6 elementary operations on matrices.Three on rows and 3 on
columns.
 Row operations on matrices are
o First operation is interchanging the two rows i.eRiRj implies
theith row is interchanged with jth row. The two rows are
interchangedwitone another the rest of the matrix remains
same.
o Second operation on matrices is to multiply a row with a scalar or
a real number i.eRikRi that ith row of a matrix A is multiplied by
k.
o Third operation is the addition to the elements of any row, the
corresponding elements of any other row multiplied by any non zero
number i.eRiRikRj k multiples of jth row elements are added to
ith row elements
 Column operation on matrices are
o
Interchanging the two columns: CrCk indicates that rth column is
interchanged with kth column
o
Multiply a column with a non zero constant i.eCikCi
o
Addition of scalar multiple of any column to another column i.e
CiCikCj
 Either of the two operations namely row or column operations can be
applied. Both cannot be applied simultaneously
27
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
DETERMINANTS
 To every square matrix A =[aij] a unique number (real or complex) called
determinant of the square matrix A can be associated. Determinant of
matrix A is denoted by det(A) or |A| or .
 A determinant can be thought of as a function which associates each square
matrix to a unique number (real or complex).
 f:M  K is defined by f(A) = k where A  M the set of square matrices and
k K set of numbers(real or complex)
 Let A = [a] be the matrix of order 1, then determinant of A is defined to be
equal to a.
 Determinant of order 2
a11
If A=[a
21
a12
a11
a22 ] then , |A| = |a21
a12
a22 | = a11 a22− a12 a21
 Determinant of order 3
a11 a12 a13
If A=[a21 a22 a23 ]
a31 a32 a33
a11 a12 a13
a22
a
then , |A| = | 21 a22 a23 | = a11 |a
32
a31 a32 a33
a23
a21
a33 | − a12 |a31
a23
a21
a33 | + a13 |a31
a22
a32 |
 A determinant can be expanded along any of its row (or column). For easier
calculations it must be expanded along the row (or column) containing
maximum zeros.
 Minor of an element aij of a determinant is the determinant obtained by
deleting its ith row and jth column in which element aij lies. Minor of an
element aij is denoted by Mij.
 Minor of an element of a determinant of order n(n ≥ 2) is a determinant of
order n – 1.
 Cofactor of an element aij, denoted by Aij is defined by Aij = (-1)i+j.Mij
where Mij is the minor of aij.
28
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
 The adjoint of a square matrix A=[aij] is the transpose of the cofactor
matrix [Aij]nn.
 A square matrix A is said to be singular if |A| = 0
 A square matrix A is said to be non – singular if |A|≠ 0
 The determinant of the product of matrices is equal to product of the
respective determinants, that is, |AB| = |A| |B|, where A and B are square
matrices of the same order.


 A square matrix A is invertible i.e its inverse exists if and only if A is
nonsingular matrix. Inverse of matrix A if exists is given by𝐴−1 =
1
|𝐴|
(𝑎𝑑𝑗𝐴),
|𝐴| ≠ 0
 If A is a nonsingular matrix of order n then |𝑎𝑑𝑗𝐴| = |𝐴|𝑛−1
 If A=kB where A and B are square matrices of order n, then |A|=kn |B|
where n =1,2,3.
Properties of Determinants
 Property 1 Value of the determinant remains unchanged if its rowand
columns are interchanged. If A is a square matrix, the
det (A) = det (A’), where A’ = transpose of A.
 Property 2 If two rows or columns of a determinant are interchanged,
then the sign of the determinant is changed. Interchange of rows and
columns is written as RiRj or CiCj
 Property 3: If any two rows (or columns) of a determinant areidentical,
then value of determinant is zero.
29
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
 Property 4: If each element of a row (or a column) of a determinant
is multiplied by a constant k, then its value get multiplied by k.
If Δ1 is the determinant obtained by applying Ri→kRi or Ci→kCi to the
determinant Δ, then Δ1= kΔ. So if A is a square matrix of order n and k is a
scalar, then |kA|=kn|A|. This property enables taking out of common
factors from a given row or column.
 Property 5: If in a determinant, the elements in two rows or columnsare
proportional, then the value of the determinant is zero. For example.
 Property 6: If the elements of a row (or column) of a determinant
areexpressed as sum of two terms, then the determinant can be expressed
as sum of two determinants.
 Property 7: If to any row or column of a determinant, a multiple ofanother
row or column is added, the value of the determinant remains
the same i.e the value of the determinant remains same on applying the
operationRiRi + kRj or CiCi + k Cj
 If more than one operation like Ri→Ri + kRj is done in one step, care
should be taken to see that a row that is affected in one operation should
not be used in another operation. A similar remark applies to column
operations.

 Area of a Triangle with vertices (x1, y1), (x2,y2) & (x3,y3) is
1 x1 y1 1
∆= |x2 y2 1|
2 x y 1
3
3
 Since area is a positive quantity, so the absolute value of the determinant is
30
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
taken in case of finding the area of the triangle.
 If area is given, then both positive and negative values of the determinant
are used for calculation.
 If the area of the triangle is zero then three points are collinear
 Value of determinant of a matrix A is obtained by sum of product of
elements of a row (or a column) with corresponding cofactors. For example
|A|=
a11A11+a12A12+a13A13
 If elements of a row (or column) are multiplied with cofactors of any other
row (or column), then their sum is zero. For example
a11A21+a12A22+a13A23 = 0
 Determinants and matrices can be used to solve the system of linear
equations in two or three variables.

a1
can be written as A X = B, where𝐴 = [a2
a3

b1
b2
b3
c1
x
d1
c2 ] , X = [y] and B = [d2 ]
c3
d3
z
Then matrix X = A-1 B gives the unique solution of the system of equations,
if |A| is non zero and A-1 exists.
For a square matrix A in matrix equation AX = B,
i. |A|≠ 0, there exists unique solution.
ii. |A|= 0 and (adj A) B ≠ 0, then there exists no solution.
iii. |A| = 0, and (adj A) B = 0, then system may or may not be consistent.
 A system of equations is said to be consistent if its solution (one or more)
exists.
 A system of equations is said to be inconsistent if its solution does not exist.
31
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
32
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
33
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
34
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
35
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
36
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
37
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
38
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
39
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
40
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
87. If
𝑎+4
[
8
88. If [ 2𝑥
3𝑏
2𝑎 + 2 𝑏 + 2
] = [
] , write the value of a - 2b .
−6
8
𝑎 − 8𝑏
𝑥
−4] [ 8] =
O , find the positive value of x.
89. If A &B are symmetric matrices , then prove that
90. If matrix
0 𝑎
[2 𝑏
𝑐 1
AB + BA is a symmetric matrix .
3
−1] is a skew –symmetric matrix then find the values of a , b and c.
0
91. Let us define three matrices ,
1
A= [2
6
0 −1
1 −1 1
],
B
=
[
3 −4
3 2 −1]
1 3
3 1
3
1 −2 3
, C = [4 1 2 ]
0 1 3
.
Then prove that A , B and C are in A.P .
92. Express the matrix A as a sum of a symmetric and a skew- symmetric matrices ,
2 4 −6
where A = [ 7 3
5] .
1 −2 4
93. Using elementary operations ,find the inverse of the matrix A = [
41
9 5
].
7 4
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
−1 1
94. Using elementary operations ,find the inverse of the matrix A = [ 1 2
3 1
2
95. If A = [ 2
1
0 1
1 3] , then find the value of
−1 0
2
3].
1
A2 – 5A + 4I . & hence find a matric X
such that A2- 5A+ 4I+ X =0
96. Two shopkeepers A and B of a particular school have stock of books on moral education ,
non – violence and truth as given by
[
M. Edu
24
12
48
24
N.violence
Truth
36 𝑆ℎ𝑜𝑝𝐴
](
)
60 𝑆ℎ𝑜𝑝𝐵
If the selling prices of these books are respectively Rs 400 ,Rs 350 and Rs 300 per book .
Find the total amount received by each shopkeeper , if all the books are sold ,using matrices.
Out of these given values, which value(s) is / are used by Gandhi ji during freedom
struggle.
3  4
1  2n  4n 
97. If A = 
, then prove that An = 
, where n is any positive integer.

1  2n
 n
1  1
𝑙𝑜𝑔 256
98. Evaluate | 3
𝑙𝑜𝑔3 8
1 𝑥
99. If ∆= | 1 𝑦
1 𝑧
𝑥2
𝑦2|
𝑧2
𝑙𝑜𝑔4 3
|.
𝑙𝑜𝑔4 9
1
and ∆1 = |𝑦𝑧
𝑥
1
𝑧𝑥
𝑦
1
𝑥𝑦| , then show that
𝑧
∆ + ∆1 = 0
100.
101.
102.
𝑎2 + 1
𝑎𝑏
𝑎𝑐
2
Prove that
| 𝑎𝑏
b +1
𝑏𝑐 | = 1+ a2 + b2 + c2 .
𝑐𝑎
𝑐𝑏
c2 + 1
𝑎+𝑏 𝑏+𝑐 𝑐+𝑎
𝑎 𝑏 𝑐
Prove
| 𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏| = 2 |𝑏 𝑐 𝑎|by using the properties of
𝑐+𝑎 𝑎+𝑏 𝑏+𝑐
𝑐 𝑎 𝑏
determinants.
103.
104.
𝑥+𝑦
By using the properties of determinants , prove that | 5𝑥 + 4𝑦
10𝑥 + 8𝑦
If x = -9 is a root of
𝑥 3
| 2 𝑥
7 6
𝑥
4𝑥
8𝑥
7
2 | = 0 , then find the other two roots .
𝑥
42
𝑥
2𝑥 | = x3 .
3𝑥
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
𝑎2
𝑏2
𝑐2
𝑏𝑐
1 𝑎2
𝑐𝑎 | = | 1 𝑏 2
𝑎𝑏
1 𝑐2
𝑎3
𝑏3 | .
𝑐3
105.
𝑎
Without expanding the determinants prove that |𝑏
𝑐
106.
Two schools A and B want to award their selected students on the values of sincerity
, truthfulness and helpfulness. The school A wants to award Rs x each , Rs y each and Rs z
each for three respective values of 3 ,2 and 1 students respectively with a total award money
of Rs 1600.School B wants to spend Rs 2300 to award 4 ,1 ,3students on respective values (
by giving the same award money to three values ) . If the total amount for one prize on each
value is Rs 900, using matrices, find the award money for each value. Apart from these three
values, suggest one more value which should be considered for award.
107.
A school wants to award its students for the values of honesty, regularity and hard
work with a total cash award of Rs 6000. Three times the award money for hard work added
to that given for honesty amounts to Rs 11000. The award money given for honesty and hard
work together is double the one given for regularity. Represent the above situation
algebraically and find the award money for each value, using matrix method. Apart from these
values suggest one more value which the school must include for award.
108.
The management committee of a residential colony decided to award some of its
members ( say X) for honesty , some (say Y )for helping others and some ( say Z) for
supervising the workers to keep the colony neat and clean .The sum of all the awardees is 12
. Three times the sum of the awardees for cooperation and supervision added to two times
the number of awardees for honesty is 33 .If the sum of the number of awardees of honesty
and supervision is twice the number of awardees for helping others , using matrix method ,
find the number of awardees for each category .Apart from these three values suggest one
more value which the management of each colony must include for awards .
109.
𝑎
If a+b+c≠0 and | 𝑏
𝑐
𝑏
𝑐
𝑎
𝑐
𝑎 |
𝑏
= 0 ,then using properties of determinants, prove that
a=b=c
110.
𝑎
If ∆ = |𝑏
𝑐
𝑝
𝑞
𝑟
𝑥
𝑝+𝑥
𝑦| = 16 , then prove that , ∆1 = |𝑞 + 𝑦
𝑧
𝑟+𝑧
43
𝑎+𝑥
𝑏+𝑦
𝑐+𝑧
𝑎+𝑝
𝑏 + 𝑞 | = 32 .
𝑐+𝑟
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
111.
If the coordinates of vertices of an equilateral triangle with the sides of length a are (
𝑥1
|𝑥2
𝑥3
x1 ,y1 ),( x2 ,y2) &( x3 , y3 ) then prove that
112.
1
|𝑐𝑜𝑠𝐶
𝑐𝑜𝑠𝐵
113.
114.
115.
𝑦1
𝑦2
𝑦3
1 2
1 | = 3a4 /4 .
1
If A , B and C are angles of a triangle, then find the value of the determinant given by
𝑐𝑜𝑠𝐶
1
𝑐𝑜𝑠𝐴
𝑐𝑜𝑠𝐵
𝑐𝑜𝑠𝐴 |
1
𝑥 − 2 2𝑥 − 3
3𝑥 − 4
Solve for x :|𝑥 − 4 2𝑥 − 9 3𝑥 − 16| = o
𝑥 − 8 2𝑥 − 27 3𝑥 − 64
2 2 1
1 3 2
Suppose A = [ −2 1 2] and B = [1 1 1], verify that ( AB ) -1 =
1 −2 2
2 −3 1
For any 2 x 2 matrix, if A (adj A) =
10
0

0  , then
10

B-1A-1
find |A|
PROFICIENCY EXERCISE
1. If A is a square matrix such that A2=I, then find the simplified value of (A-I)3+(A+I)3-7A
2. The monthly income of Aryan and Babban are m the ration 3:4 and their monthly
expenditures are in the ratio 5:7. If each saves Rs. 15000 per month find their monthly
incomes using matrix method. This problem reflects which value?
3. A trust invested some money in two types of bonds. The first bond pays 10% interest second
bond pays 12% interest. The trust received Rs.2800 as interest. However if trust had
interchanged money in bonds, they would have got Rs100 less as interest. Using matrix
method, find the amount invested by the trust.
4. On her birthday Seema decided to donate some money to children of an orphanage home. If
there were 8 children less, everyone would have got Rs.10 more. However if there were 16
children more, everyone would have got Rs.10 less. Using matrix method, find the number of
children and the amount distributed by Seema. What values are reflected by Seema’s
decision?
5. A typist charges Rs.145 for typing 10 English and 3 Hindi pages, while charges for typing 3
English and 10 Hindi pages are Rs. 180. Using matrices, find the charges of typing one
English and one Hindi page separately. However typist charged only Rs. 2 per page from a
poor student. Shyam for 5 hindi pages. How much less was charged from this poor boy?
Which values are reflected in this problem?
𝑥
𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
6. If| −𝑠𝑖𝑛𝜃 −𝑥
1 |=8, write the value of x.
𝑐𝑜𝑠𝜃
1
𝑥
2 −1 3
7. Using elementary row operations, find the inverse of the matrix A=[ −5 3 1]
−3
2 3
44
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
8. Using properties of determinents, Prove that
(𝑏 + 𝑐)2
| (𝑐 + 𝑎)2
(𝑎 + 𝑏)2
𝑎2
𝑏2
𝑐2
𝑏𝑐
𝑐𝑎 |=(a-b)(b-c)(c-a)(a+b+c)(a2+b2+c2)
𝑎𝑏
9. If A is a square matrix such that |A|=5. Write the value of |AAT|
1 2
1 −4
10. If A=[
]and B=[
], find |AB|
3 −1
3 −2
11. For what value of k, the system of linear equations x+y+z=2, 2x+y – z=3, 3x+2y+kz=4 has a
unique solutions?
3 −3 4
12. Using elementary row operations find the inverse of Matrix A=[2 −3 4]and hence solve
0 −1 1
the following system of equations: 3 x - 3y+ 4z = 21, 2x-3y+4z = 20 , -y+z = 5
13. Using properties of determinants, show that ∆ABC is isosceles if:
1
| 1 + cos 𝐴
𝑐𝑜𝑠 2 𝐴 + cos 𝐴
1
1 + cos 𝐵
𝑐𝑜𝑠 2 𝐵 + cos 𝐵
1
1 + cos 𝐶 |
𝑐𝑜𝑠 2 𝐶 + cos 𝐶
14. Use elementary column operation C2 → C2+2C1 in the following matrix equation
[
2 1
3 1 1
] = [
][
2 0
2 0 −1
𝑦𝑧 − 𝑥 2
15. Prove that|𝑧𝑥 − 𝑦 2
𝑥𝑦 − 𝑧 2
𝑧𝑥 − 𝑦 2
𝑥𝑦 − 𝑧 2
𝑦𝑧 − 𝑥 2
0
]
1
𝑥𝑦 − 𝑧 2
𝑦𝑧 − 𝑥 2 |is divisible by (x+y+z), and hence find the quotient.
𝑧𝑥 − 𝑦 2
𝑎
−1 0
16. If f(x)=| 𝑎𝑥
𝑎 −1|using properties of determinants, find the value of f(2x)-f(x).
2
𝑎𝑥
𝑎𝑥 𝑎
17. There are 2 families A and B. there are 4 men, 6 women and 2 children in family B. the
recommended daily amount of calories is 2400 for men, 1900 for women, 1800 for children
and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent
the above information using matrices. Using matrix multiplication, calculate the total
requirement of calories and proteins for each of the 2 families. What awareness can you
create among people about the balanced diet from this question.
18. Using properties of determinants, prove that
𝑎3 2 𝑎
|𝑏 3 2 𝑏 | =2(a-b)(b-c)(c- a)(a+b+c)
𝑐3 2 𝑐
−1 −2 −2
19. Find the adjoint of the matrix A=[ 2
1 −2] and hence, show that A (adj A)= |A| I3
2 −2 1
45
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
20. Three schools A,B and C want to award their selected students for the values of Honesty,
Regularity and Hardwork. Each school decided to award a sum of Rs.2500, Rs.3100 and
Rs.5100 per student for the respective values. The number of students to be awarded by the
three schools is given below in the table:
Values
A
B
C
Honesty
3
4
6
Regularity 4
5
2
Hardwork 6
3
4
Find the total money given in awards by the three schools separately, using matrices.
1
21. If A=[2
2
2 2
1 2] , then show that A2-4A-5I=0 and hence find A-1
2 1
𝑥+2 𝑥+6 𝑥−1
22. Using the properties of determinants, solveforx: |𝑥 + 6 𝑥 − 1 𝑥 + 2| = 0
𝑥−1 𝑥+2 𝑥+6
𝑎 1
1 −1
23. If A= [
] and B= [
] and (A+B)2= A2+B2, then find the values of a and b.
𝑏 −1
2 −1
46
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
47
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
48
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
49
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
87. 0
90.
88.
4
a = -2, b = 0, c = -3
2
11/2 −5/2
3
3/2 ]&
92. symmetric matrix = [ 11/2
−5/2 3/2
4
0
−3/2 −7/2
0
7/2 ]
skew symmetric matrix = = [3/2
7/2 −7/2
0
4 −5
93. A = [
]
−7 9
-1
−1 −1
95. [−1 −3
−5 4
94.
1 −1 1
A == [−8 7 −5]
5
4 −3
-1
−3
1 1
3
−10], [1 3
10 ]
2
5 −4 −2.
96. shopkeeper A - Rs 37200 , shopkeeper B - Rs 31200
98.
13/2
103.
2,7
105.
200, 300, 400
106.
500, 2000, 3500
107.
3, 4, 5
111.
0
112.
x=4
114.
10
ANSWERS OF PROFICIENCY EXERCISE
1.
2.
3.
4.
5.
6.
A
Aryan- Rs. 90000 , Babban- Rs. 120000
First – Rs. 10000 , Second-Rs. 15000
32 , 30
Eng- 10 , Hindi – 15
x = -2
−7
−9 10
7. [−12 −15 17 ]
1
1
−1
9.25
50
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
10.
-70
11. k≠0
12.
1
[−2
−2
−1 0
3 −4]
3 −3
15.
(x+ y+ z)(xy + yz + zx –x2 –y2 –z2)2
, x = 1 , y = -2 , z = 3.
16. ax(2a+3x)
24000 576
17. ⌊
⌋
15800 332
−3 6
6
19. adj. A = [−6 3 −6]
−6 −6 3
20. Money awarded by School A = Rs. 50500
Money awarded by School B = Rs. 40800
Money awarded by School C = Rs. 41600
21.
22.
23.
−3 2
2
[ 2 −3 2 ]
5
2
2 −3
1
x= -7/3
a = 1 , b= 4
51
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Chapter 5: Continuity and Differentiability
POINTS TO REMEMBER






52
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT









53
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT








54
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT







55
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT



56
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT

57
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
58
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
59
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
60
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
61
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
1
𝑒 𝑥 −1
39. Show that the function f(x) = {
, 𝑥 ≠ 0is discontinuous at x= 0
1
𝑒 𝑥 +1
0
40. If
y = [√𝑥 + √𝑥 2 +
41. If y = √
42.
1−𝑥
𝑑𝑦
𝑎2 ]n , then prove that
, prove that ( 1 – x2 )
1+𝑥
𝑥=0
𝑑𝑦
𝑑𝑥
𝑑𝑥
𝑛𝑦
√𝑥 2 +
𝑎2
.
+y=0.
3𝑐𝑜𝑠𝑥−4𝑠𝑖𝑛𝑥
Find the derivative of 𝑐𝑜𝑠 −1 (
43. Differentiate
=
5
).
cosx cos2x cos3x with respect to x .
44. If y = x sinx + sinxcosx
, then find
𝑑𝑦
𝑑𝑥
.
45. If x = a ( cost + t sin t ) and y = a ( sin t – t cos t ) , then find
𝑑2𝑦
𝑑𝑥 2
at t = π /4 .
46. If y = ( tan-1x ) 2 , then show that ( x2 + 1 ) y2 + 2x ( x2 + 1 ) y1 = 2 .
47. If y = log [ x + √𝑥 2 + 1] , then show that
( x2 + 1)y2 + x y1= 0 .
48. Verify the Rolle’s theorem for the function f ( x ) = log( x2 + 2 ) - log3 , in the interval
[ -1 ,1 ]
49. Verify Mean Value theorem for the function f( x ) = x3- 2x2 – x + 3 in the interval [ 0 ,1 ]
50. Differentiate
y = logcosxsinx
with respect to x .
51. Let the geometric series in x be y = 1 + x + x2 + x3 + …….. ∞ , then determine
√1+𝑥 −
√1+𝑥 +
52. If y = tan -1(
√1−𝑥
)
√1−𝑥
, then find
𝑑𝑦
𝑑𝑥
.
53. If 𝑥√1 + 𝑦 + 𝑦√1 + 𝑥 = 0 , prove that (1+ x2 )
54. If xy = ex-y
, prove that
𝑑𝑦
𝑑𝑥
=
𝑙𝑜𝑔𝑥
( 1+𝑙𝑜𝑔𝑥)2
55. If x = a sec3θ , y = a tan3 θ , find
56. If y = 2 cos(logx) + 3 sin(logx)
𝑑𝑦
𝑑𝑥
at
𝑑𝑦
𝑑𝑥
+ 1 = 0.
.
θ = π/3 .
, prove that x2y2 + xy1 + y = 0 .
62
𝑑𝑦
𝑑𝑥
–𝑦 .
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PROFICIENCY EXERCISE
1. If x= a cos  b sin  , y= a sin   b cos , show that y2y2 – xy1+ y = 0
2. If x= a(cos 2t+ 2t sin 2t) and y=a (sin2t-2tcos2t), then Find
𝑑2𝑦
𝑑𝑥 2
3. Discuss the continuity and differentiability of the function f(x)=|x|+|x-1| in the interval (-1,2)
4. If (ax+b)e y/x=x, then show that x3y2=(xy1-y)2
5. Let f(x)=x-|x-x3|, x∈[-1,1], find the point of discontinuity, (if any), of this function on [-1,1].
6. If y=𝑥 𝑒
−𝑥2
, find
𝑑𝑦
𝑑𝑥
𝑥
𝑑𝑦
𝑦
𝑑𝑥
7. If log√𝑥 2 + 𝑦 2 =tan-1 , then show that
𝑦−𝑥
=
𝑦+𝑥
8. Verify mean value theorem for the function f(x)=(x-4)(x-6)(x-8) on the interval [4,10]
9. If
𝑥
𝑥−𝑦
=log(
𝑎
𝑑𝑦
𝑥
) then prove that 𝑑𝑥 = 2 − 𝑦
𝑥−𝑦
10. If y=eax cos bx, then prove that y2-2ay1+(a2+b2)y=0
11. If x=a sin2t(1+cos 2t) and y=b cos2t(1-cos 2t) then find
12. Find the derivative of the function f(x)= cos-1[sin √
1+𝑥
2
𝑑𝑦
𝑑𝑥
𝜋
at x=
4
]+ xxat x=1
13. Find whether the following function is differentiate at x=1 and x=2 or not𝑓(𝑥) =
𝑥
𝑥<1
{2 − 𝑥
1≤𝑥≤2
2
−2 + 3𝑥 − 𝑥
𝑥>2
14. If y=xx, prove that y2 –
𝑦1 2 𝑦
𝑦
-
𝑥
=0
15. Verify Mean value theorem for the function f(x)= 2sin x+ sin 2x on [0,]
16. If x=ecos 2t& y=esin 2t, prove that
𝑑𝑦
=
𝑑𝑥
−𝑦𝑙𝑜𝑔𝑥
𝑥𝑙𝑜𝑔𝑦
17. Differentiate (sin2x)x +sin-1√3𝑥 with respect to x
𝜋(𝑥+1)
𝑘 sin (
2
18. Find k, if𝑓(𝑥) = { tan 𝑥−sin 𝑥
𝑥3
), 𝑥 ≤ 0
is continuous at x=0
𝑥>0
19. If x=sin t, y=sinpt, prove that (1-x2)y2– xy1+p2y=0
20. Find the derivative of 𝑓(𝑒 𝑡𝑎𝑛𝑥 )w.r.t.xatx = 0, it is given that f ꞌ(1) = 5.
63
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
ANSWERS OF PROFICIENCY EXERCISE
2.
(Sec32t)/ 2at
3.
not differentiable at x = 0
5. no point of discontinuity
2
6. y𝑒 −𝑥 [
1
𝑥
- 2x log x ]
8. c =8
11. b / a
12.
¾
17. ( sin2x)x [ 2x cot2x + log sin2x ] +
18.
3
2√3𝑥−9𝑥 2
½
20. 5
64
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Chapter 6: Application of Derivatives
POINTS TO REMEMBER






65
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT








If f be a Continuos function on [a,b] and differentiable on (a,b) hence
(a) f is increasing in [a,b] if f ꞌ(x) ≥ 0 for each x ∈ (𝒂, 𝒃)
(b) f is decreasing in [a,b] if f ꞌ(x) ≤ 0 for each x ∈ (𝒂, 𝒃)
(c) f is constant in [a,b] if f ꞌ(x) = 0 for each x ∈ (𝒂, 𝒃)
66
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT







67
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT






68
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT

69
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
70
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
71
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
72
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
73
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
74
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
75
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
85.
The money to be spent for the welfare of the employees of a firm is proportion to the rate of
change of its total revenue (marginal revenue ).if the total revenue (in Rs.) received from
the sale of x units of a product is given by R(x) = 3𝑥 2 + 36𝑥 + 5, Find the marginal
revenue ,when x = 5 and write which value does the question indicate ?
86. 1. In competition , a brave child tries to inflate a huge spherical balloon bearing slogans
against child labour at the rate of 900 cubic centimeter of gas per second .find the rate at
which the radius of the balloons is increasing, when its radius is 15cm.why is child Labour
not good for society ?
87. 2. The side of an equilateral triangle are increasing at the rate of 2cm/s .find the rate at which
the area increases ,when the side is 20cm.
88. 3. Find the interval in which the function f given by f(x)=4𝑥 3 − 6𝑥 2 − 72𝑥 + 30
is , (i)strictly increaseing (ii)strictly discreasing .
89.
Find the intervals in which the function f ( x ) = x4 – 8x3 + 22 x2 - 24x + 21 is increasing
or decreasing
90. 4. Using differential ,find the approximate value of √0.082.
91.5. Find the points of local maxima and minima of the function f(x) = x 3 − 6x 2 + 9x − 8.
92.6. Find all the point of local maxima and minima of f(x)= sin x- cos x on
7. [0 , 2𝜋]. A𝑙𝑠𝑜𝑓𝑖𝑛𝑑𝑙𝑜𝑐𝑎𝑙𝑚𝑎𝑥𝑖𝑚𝑎𝑎𝑛𝑑𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑣𝑎𝑙𝑢𝑒.
93.8. Show that the altitude of the right circular cone of maximum volume that can be inscribed in
a sphere of radius r is
4r
3
,also show that the maximum volume of the cone is
8
27
of the
volume of the sphere .
94.9. A jet of a enemy country is flying along the curved x 2 =4y .a soldier placed at point
10. ( -1 ,2)wants to shoot down the jet of enemy .when it is nearest to him .find the nearest
point to the soldier .how does this problem help soldier in the battle field ? justify your
answer .
95.11. The sum of the perimeter of a circle and square is k, where k is some constant .prove that
the sum of the area is least ,when the side of square is double the radius of circle.
96.12. A man ,2m tall walk at the rate of 5/3 m/s towards a street light which is 16/3 m above the
ground. At what rate is the tip of his shadow moving and at what rate is the length of the
shadow changing when he is 10/3 m from the base of the light .
97.13. Of all closed right circular cylindrical cans of volume 128πcm3 ,find the dimensions of the
can which has minimum surface area.
76
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
98.
If the sum of the lengths of the hypotenuse and a side of a right triangle is given ,then show
that the area of the triangle is maximum when the angle between them is 600
99.14. If an open box with square base is to be made of a given quantity of card board of areac 2
,then show that maximum volume of the box is
c3
6√3
cubic unit .
100. Prove that the curve y 2 = 4ax and xy = c 2 cut at the right angles ,if c 4 = 32α4
101.
15. Find the equation of the tangent to the curve x = 1- cos𝜃 and
16. y = 𝜃 − sin𝜃 at the 𝜃 =
π
4
102.
17. Find the points of local maxima , local minima and the point of inflection of the
function f( x )= x 5 − 5x 4 + 5x 3 - 1 also find the corresponding local maximum and local
minimum values.
103.
18. The volume of a spherical balloon is increasing at the rate of 25cm3 / sec. find the rate of
change of its surface area at the instant when the radius in 5cm.
104. Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius
5√3 cm in 500π cm3 .
105. A figure consists of a semi –circle with a rectangle on its diameter . Given the perimeter of
the figure , find its dimensions in order that the area may be maximum .
PROFICIENCY EXCERCISE
1. Show that the equation of normal at any point t on the curve x = 3 cost-cos3t and y=3
Sint–Sin3t is 4 ( yCos3t – x Sin3t) = 3 Sin 4t.
2. Find the internals in which f(x) = Sin3x-Cos3x, 0<x<π, is strictly increasing or strictly
decreasing.
3. The equation of tangent at (2, 3) on the curve y2=ax3+b is y=4x-5. Find the values of a
and b.
4. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can
be inscribed is6√3r.
5. Find the internals in which the function f(x) =
4 𝑆𝑖𝑛 𝑥
2+𝐶𝑜𝑠 𝑥
− 𝑥 , 0 ≤ x ≤ 2π is strictly
increasing or strictly decreasing.
6. Find the equation of tangents to the curve y = x3 + 2x – 4, which are perpendicular to
the line x+14y+3 =0.
77
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
7. Show that semi-vertical angle of a cone of maximum volume and given slant height is
cos −1
1
√3
8. Find the maximum and minimum values of f(x)=Secx+logCos2x, 0<x<2π
9. Find the angle of intersection of the curves y2=4ax and x2=4by
10. Tangent to the circle x2+y2=4 at any point on it in the first quadrant makes intercepts
OA and OB on x and y axis respectively, O being centre of the circle. Find the minimum
value of OA + OB.
11. Find the value of p for which the curves x2=9p (9 – y) and x2=p(y+1) cut each other at
right angles.
12. Find the maximum area of an isosceles triangle inscribed in the ellipse
𝑥2
16
+
𝑦2
9
=1,
with its vertex at one end of the major axis.
13. Find the point on the curve =
x
1+x2
, where the tangent to the curve has the greatest
slope.
14. Find the absolute maximum and absolute minimum values of the function f given by
f(x)=Cos2x+Sinx, x∈ [o,π]
15. If the function f(x)=2x3-9mx2+12m2x+1, where m>0 attains its maximum and minimum
at p and q respectively such that p2=q, then find the value of m.
16. Find the minimum value of (ax+by), where xy=C2.
17. Find the Co-ordinates of a point of the parabola y=x2+7x+2 which is closest to the
straight line y=3x-3.
78
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
8. [ 0, π ]
9. ( 0, e ]
10. x ≥ 1
11. (– ∞, 0 ) U (0, ∞)
12. (0 ,
13. Maximum value = 4, Minimum value = 0
14. a > 1
15. R
16. 7
17. (1/2, ¼)
18. (2, –3)
19. ¼
20. (1, 7)
21. (0, 0), (2, 4)
22. ½
23. – ¼
24. 8πr
25. 2πr cm2/cm, 6π cm2/cm
26. 72
27. – a/2b
28. Rs 80.
𝜋
6
)
29. a > 0
30. (4, 11) and (–4, – 31/3)
31. – 8/3 cm/sec
32. 4/45π cm/sec
33. (a) 0 cm/sec (b) 14 cm2/sec
34. 4/48π cm/sec
35. 7.11 cm/sec
36. ( 7/2, ¼ )
37. Y = ½
38. 2x + 3my = am2 ( 2 + 3m2 )
40. 48x – 24y = 23
41. 2x + 2y = a2
42. (8/3, 128/27) , (–8/3, – 128/27)
44. Increasing in (– ∞, 2) U (6, ∞), Decreasing in (2, 6)
46. (– ∞, –1) and (1, ∞)
47. 25/3
48. Increasing in (π/4, π/2) Decreasing in (0, π/4)
49. Strictly decreasing in (1, ∞)
53. 0.2083
54. 2.9907
55. 0.06083
56. 0.1925
57. 5.002
58. 45.46
60. 25, 10
65. Strictly increasing in [0, π/4) U (5π/4, 2π]
Strictly decreasing in ( π/4, 5π/4)
66. Strictly increasing in (1, 3) U (3, ∞)
Strictly decreasing in (–∞,–1) U (–1, 1)
79
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
π
67. Local Maxima at x = π/6
Local max. value =
√3
6
Local minima at x = − 6
−
π
Local minima value =
6
68. Strictly increasing in (–∞, 2) U (3, ∞)
70. x cos 𝜃 + y sin 𝜃= a
Strictly decreasing in ( 2, 3 )
71. (0,0) , (1,2) , (-1,-2)
72. x + y – 3 = 0
76. (4, -4)
77. Side = 4 cm
79.
85.
86.
144
𝜋+4
,
36𝜋
4𝜋
80. 3√3 𝑅 3
𝜋+4
66
1
𝜋
𝑐𝑚/𝑠𝑒𝑐
87.
20√3 cm2/sec
88.
st.inc (-∞,-2) U (3,∞) , st. dec (-2,3)
89.
inc – [1,2]U(3, ∞) ,
90.
0.2867
91.
Local maximum at x =1 , Local minimum at x =3
92.
local max .at x=
3𝜋
4
Local min. at x =
dec- (-∞, 1) U (2,3)
, Local max value = √2
7𝜋
4
, Local min. value = -√2
94.
(2, 1)
96.
8/3 m/sec , length of shadow decreasing at the rate 1 m/sec.
97.
r = 4 cm, h = 16 cm
101.
equation of tangent: (√2 − 1)𝑥 − 𝑦 = 2(√2 − 1) − 4
102.
local max .at x=1 ,
𝜋
Local max. value = 0
Local min. at x = 3 , Local min. value = -28
103.
10 cm2/sec
105.
2𝑃
Length = 𝜋+4 ,
80
𝑃
Breadth =𝜋+4
−√3
2
+
π
6
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
ANSWERS OF PROFICIENCY EXERCISE
2.
st. Inc.(0,π/4) U (7π/12, 11π/12), st.dec.( π/4, 7π/12) U (11π/12, π)
3.
a=2, b=-7
5. st.Inc.(0,π/2) U (3π/2, 2π), st.dec.( π/2, 3π/2)
6.
14x – y=20, 14x – y=-12
9.
Right angle or Ѳ
8.
=𝑡𝑎𝑛−1 [
Max.=2(1-log2) Min. = -1
3𝑎1/3 61/3
2
2
]
10.4√2
2(𝑎3 +𝑏3 )
11.
14.
0,4
Max.-5/4 , Min. -1
12. 9√3 sq.units
15.
m =2
16.
17.
(-2,-8)
2√𝑎𝑏 c
81
13. (0,0)
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Chapter 7: Integrals
POINTS TO REMEMBER




82
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT







83
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT



84
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT




85
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT



86
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT



87
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT



88
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT



89
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT

90
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
91
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
92
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
93
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
94
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
51.
95
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
52.
96
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
53.
97
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
54.
55.
98
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
56.
57.
58.
99
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
59.
100
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
60.
61.
62.
63.
64.
101
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
65.
66.
67.
68.
69.
𝜋
Evaluate: ∫0 𝑙𝑜𝑔(1 + cos 𝑥) 𝑑𝑥
If sinx , f( x ) and cosx are in A.P , then find the value of ∫ f(x)dx .
Evaluate ∫
e6logx−
e4logx−
e5logx
dx
e3logx
.
70.
Evaluate ∫ cos −1 ( sinx)dx .
71.
If ∫0 ( 3x 2 + 2x + k)dx = 0 , then find the value of k .
72.
73.
74.
75.
76.
77.
78.
79.
1
e2
Evaluate ∫e
1
x logx
1
Evaluate ∫0
Evaluate ∫
Evaluate
dx .
1
ex
+ e−x
dx .
e2x − e−2x
e2x + e −2x
dx
2
∫ 1−cos2xdx .
If ∫ ex ( tanx + 1 ) secx dx = ex f (x ) + C ,then write the value of f(x) .
Evaluate∫
2𝑥
√1+4 𝑥
Evaluate ∫
dx
ex
√5−4ex −e2x
dx .
π
If f(x) = cosx - cos2x + cos3 x + ………….. is a G.P , then find the value of ∫02 f (x) dx
.
80.
81.
82.
83.
84.
Evaluate ∫
Evaluate ∫
Evaluate
1−3sinx
dx
√tanx
dx
sinx.cosx
∫
Evaluate ∫
Evaluate
dx
x2
1−
x4
.
dx .
1
dx .
cos(x − α) cos( x−β )
1 𝑥 3 +|𝑥|+1
∫−1 𝑥 2 +2|𝑥|+1dx .
102
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
85.
86.
87.
88.
1
Evaluate ∫
sin4 x
2π
Evaluate ∫0
1
π
cos2 x+
93.
94.
4sin2 x
dx .
1
x2
1
92.
dx .
cos 2 x
Evaluate ∫02
3 dx
1
Evaluate ∫
x(x5 + 3)
π
4
π
−4
dx.
log(cosx + sinx)dx .
Evaluate ∫
π
Evaluate ∫0
.
+ x4
1
5+
π/4
4 cos x
dx .
log(1 + tan x ) dx .
Evaluate ∫0
∞
1
∫0
Evaluate
( x2 + a2 )
π/2
(x2 + b2 )
95.
Prove that
96.
Evaluate ∫0 log ( − 1) dx .
x
97.
98.
99.
100.
101.
102.
103.
dx.
dx .
√( 1+sin2x
Evaluate ∫
91.
x cos2 x + cos4 x
( sinx +cosx )2
Evaluate ∫02
89.
90.
+esinx
1
π
+sin2
∫0
1
log|tanx + cotx |dx .= π log 2 .
1
1
∫ (logx −
Evaluate
dx .
Evaluate ∫
√(
1
(logx)2
5x+3
x2 +
4x+
)dx .
dx .
10)
4
Evaluate ∫ Cos x dx
Evaluate ∫
𝑥2
𝑥 4 +𝑥 2 −2
5x
dx
x
5
5
x
Evaluate ∫ 5 .5 .5 dx
Evaluate ∫
Evaluate ∫
sin 2 x
𝑑𝑥
sin 5 x. sin 3x
sin x
sin 4x
dx
103
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
104.

Evaluate

105.
Evaluate
cos 2 x  cos 2
dx
cos x  cos 
2
dx
 1  tan x
0
106.
Evaluate ∫ e−3x cos 3 x dx .
107.
Evaluate ∫1 (e2−3x +
108.
Evaluate ∫
x2 + 4
x4 + 16
109.
Evaluate ∫
tanx+ tan3 x
1+ tan3 x
3
110.
3
Evaluate
2

x 2 + 1)dx , using limit sum.
dx
dx
x cos x dx
0
PROFICIENCY EXERCISE
𝑥 2 −3𝑥+1
1.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫
2.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶
∫−𝜋(𝐶𝑜𝑠𝑎𝑥 − 𝑆𝑖𝑛𝑏𝑥)2 𝑑𝑥
3.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶
∫
4.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶
5.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫ (𝑥−1)(𝑥 2 𝑑𝑥
+1)
6.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫0
7.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫ (𝑥+1)2 𝑑𝑥
8.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫
𝑑𝑥
1+𝑥 𝑡𝑎𝑛𝑥
9.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫ (𝑥−1)(𝑥 2 𝑑𝑥
+1)
10.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫0
11.
Find ∫
𝑑𝑥
𝐶𝑜𝑠𝑥(1+𝐶𝑜𝑠𝑥)
√1−𝑥 2
𝑑𝑥
𝜋
𝑆𝑖𝑛𝑥−𝑥 𝐶𝑜𝑠𝑥
𝑥(𝑥+𝑆𝑖𝑛𝑥)
𝐶𝑜𝑠 2 𝑥
𝜋/2
∫0
1+3𝑆𝑖𝑛2 𝑥
𝑑𝑥
𝑑𝑥
𝑥3
𝜋/4
1
𝐶𝑜𝑠 2 𝑥√2𝑆𝑖𝑛2𝑥
𝑑𝑥
𝑙𝑜𝑔𝑥
𝑥
𝑥4
𝜋/2 5𝑆𝑖𝑛𝑥+3𝐶𝑜𝑠𝑥
𝑆𝑖𝑛𝑥+𝐶𝑜𝑠𝑥
𝑑𝑥
1−𝐶𝑜𝑠𝑥
104
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
1
12.
Find ∫ 3(𝑥 5 3/5 𝑑𝑥
𝑥
+1)
13.
Evaluate: ∫0
14.
Evaluate: ∫𝜋/4 𝑒 2𝑥 (
15.
Find ∫
16.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫0
17.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫−𝜋/2
𝑑𝑥
1+𝑒 𝑥
18.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫−1 |𝑥 3 -x| dx
19.
𝐹𝑖𝑛𝑑 ∶ ∫ (𝑥 2
𝑑𝑥
+1)(𝑥 2 +4)
20.
Evaluate: ∫0
21.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫−2
𝑑𝑥
1+5𝑥
22.
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫0
𝜋/4
𝑆𝑒𝑐𝑥
1+2𝑆𝑖𝑛2 𝑥
𝜋/2
𝑑𝑥
1−𝑆𝑖𝑛2𝑥
1−𝐶𝑜𝑠2𝑥
(𝑥+3)𝑒 𝑥
(𝑥+5)3
) 𝑑𝑥
𝑑𝑥
2𝑆𝑖𝑛𝑥
𝜋/2
2𝑆𝑖𝑛𝑥 +2𝐶𝑜𝑠𝑥
𝑑𝑥
𝜋/2 𝐶𝑜𝑠𝑥
2
2𝑥+1
𝜋
𝑥𝑆𝑖𝑛𝑥
1+3𝐶𝑜𝑠 2 𝑥
2
𝜋
𝑑𝑥
𝑥2
𝑥
1+ 𝑆𝑖𝑛𝛼.𝑆𝑖𝑛𝑥
𝑑𝑥
105
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
106
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
107
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
108
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
109
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
110
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
111
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
112
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
113
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
57.
58.
59.
114
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
-𝜋 log 2
1
( sin 𝑥 − cos 𝑥) + C
2
𝑥3
3
+ C
115
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
𝜋
70.
𝑥-
2
𝑥2
2
71.
k= -2
72.
log 2
+C
𝜋
tan−1 𝑒 -
73.
4
1
log|𝑒 2𝑥 + 𝑒 −2𝑥 | +C
74.
2
75. -cotx + C
76.
f(x) = secx
1
77.
𝑥
+ √1 + 4𝑥 )] +C
[log(2
𝑙𝑜𝑔2
78.
sin−1 (
79.
80.
𝜋
2
𝑒 𝑥 +2
3
) +C
- 1
1
𝑥
log|
√2
tan −3−2√2
2
𝑥
2
| +C
tan − 3+2√2
81. 2√tan 𝑥+ c
82. - ¼ log|
83.
1
𝑥−1
| - ½ tan−1 𝑥 +C
𝑥+1
log|
sin(𝛼−𝛽)
cos(𝑥−𝛼)
cos(𝑥−𝛽)
|+C
84. 2log2
85.
1
√3
tan−1 [
tan 𝑥− cot 𝑥
√3
] +C
86. π
87. 2
88.
89.
𝜋
6
4
{ (x3/4+1) – log(x3/4+1)} +C
3
116
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
1
90.
15
91.
𝜋
-
4
𝑥5
𝑥 5 +3
|+C
log2
𝜋
92.
3
𝜋
93.
94.
log|
8
log2
𝜋
2𝑎𝑏(𝑎+𝑏)
96. 0
97.
𝑋
+C
log 𝑋
98 . 5 √𝑥 2 + 4𝑥 + 10 - 7 log |x+2 + √𝑥 2 + 4𝑥 + 10 | +C
99.
100.
101.
102.
1
sin 4𝑥
8
4
[ 3x + 2sin 2𝑥 +
1
6
log |
55
𝑥−1
𝑥+1
|+
√2
3
]+ C
tan−1
𝑥
+C
√2
5𝑥
(log 5)3
+c
1
1
3
5
logsin 3𝑥 - logsin 5𝑥 + C
1
1+𝑠𝑖𝑛𝑥
8
1−𝑠𝑖𝑛𝑥
103. - log |
|+
1
log |
4√2
1+√2𝑠𝑖𝑛𝑥
1−√2𝑠𝑖𝑛𝑥
| +C
104. 2sin 𝑥 + 2x cos 𝛼 + C
105.
106.
107.
𝜋
4
𝑒 −3𝑥
24
(sin 3𝑥- cos 3𝑥 ) +
1
32
3
3
- ( e-7 – e-1) +
1
108 .
2√2
tan−1(
𝑥 2 −4
2√2𝑥
3𝑒 −3𝑥
40
(sin 𝑥- 3cos 𝑥 ) + C
)+C
1
1
1
3
6
√3
109. - log | 1+ tanx| + log |tan2x – tanx +1| +
117
tan−1 (
2𝑡𝑎𝑛𝑥−1
√3
)+C
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
5
110.
-
2𝜋
1
𝜋2
ANSWERS OF PROFICIENCY EXERCISE
1.
3
2
𝑥
𝑆𝑖𝑛−1 x - √1 − 𝑥 2 +3√1 − 𝑥 2 +c
2
3. log|x|-log|x+Sinx|+c
1
1
1
2
4
2
9.
−𝑙𝑜𝑔𝑥
+ log|
𝑥+1
𝑥2
2
𝑥
|+C
1
1
+x+2log|x-1|-4log|𝑥 2 +1|- 𝑡𝑎𝑛−1 x +C
𝑥
13.
15.
19 .
20.
𝑒𝑥
(𝑥+5)
1
3
+
16.
6
6.
6
5
1
√2𝜋
4
2 +𝐶
14.
𝜋
17. 1
4
18.
1
1
1
𝑥
3
3
6
2
𝜋
𝑒 ⁄2
2
11
4
log|𝑥 2 +1|+ 𝑡𝑎𝑛−1 x - log|𝑥 2 +4|- 𝑡𝑎𝑛−1 - +C
√3𝜋2
9
21.
2𝑏
12. -2 [1 + 1/𝑥 5 ]2/5+C
11. log|Secx + tanx| - 2tan2 + C
√2+1
log(
)
3
2
𝑆𝑖𝑛2𝑏𝜋
10. 2𝜋
2
1
2𝑎
-
8. log|Cosx + xSinx| + C
𝑥+1
1
𝑆𝑖𝑛2𝑎𝜋
𝜋
4.
5. x + log|x-1|+ log|𝑥 2 +1|- 𝑡𝑎𝑛−1 x +C
7.
2𝜋 +
2.
8
22.
3
118
𝜋(𝜋−2𝛼)
2𝐶𝑜𝑠𝛼
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
CHAPTER 8 : APPLICATIONS OF INTEGRALS
POINTS TO REMEMBER

 Definite Integral ∫𝑏 𝑓(𝑦)𝑑𝑦 of the function f(y) from limit a to b represents the area enclosed by the graph of
𝑎
the function f(y), y-axis and the horizontal lines y=a and y=b
 Area bounded by the curve y=f(x) , the x-axis and the ordinates x=a and x=b using elementary
strip method is computed as follows
119
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Area of elementary strip = y. dx
𝑏
𝑏
Total area =∫𝑎 𝑦𝑑𝑥 =∫𝑎 𝑓(𝑥)𝑑𝑥
 Area bounded by the curve x=f(y) , the y-axis and the abscissa y=c and y=d using elementary
strip method is computed as follows
Area of elementary strip = x. dy
𝑑
𝑑
Total area =∫𝑐 𝑥𝑑𝑦 =∫𝑐 𝑓(𝑦)𝑑𝑦

120
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
121
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT


122
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT


Some Standard Integrals Required
123
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
124
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
11.
12.
13.
14.
15.
16.
17. Find the area of the region bounded by the curve y2 = 2y – x and the y-axis .
18. A farmer has a piece of land . He wishes to divide it equally in his two sons to maintain peace
and harmony in the family . If his land is denoted by the area bounded by the curve y2 = 4x and
the line x = 4 ,and to divide the area equally ,he draw a line x = a , then what is the value of a
.What is the equality among the people ? .
19. A farmer has a triangular piece of land whose area is bounded by the lines 2x +y = 4 ,3x –
2y = 6 and x – 3y + 5 = 0.Find the area of this field . Suppose a farmer has two children ,one
is a son and other is a daughter . farmerhas decided to divide this field into equal parts, which
value is shown by the farmer ?
20. Find the area of the region bounded by two parabolas y2 = 6x and x2 = 6 y .
21. Find the area enclosed by the curves x = 3 cos t and y = 2 sin t .
22. Find the area of the region bounded by the parabola y2 = x and the line y + x = 2 .
23.
Make a rough sketch of the region given below and find its area using integration.
{ (x, y) ∶ 0 ≤ y ≤ 2x + 3, 0 ≤ y ≤ x 2 + 3 , 0 ≤ x ≤ 3}.
125
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
24. Find the area of the region bounded by the curve y = √1 − x 2 , the line y = x and positive x − axis.
25. If the area enclosed between the curves y = ax 2 and x = ay 2 (a > 0) is 1 sq. unit, then find the
value of ‘a’
26. Find the area of the region between the curves x2 + y2 = 9 and (x -3 )2 + y2 = 9
PROFICIENCY EXERCISE
1. Using Integration find the area of the region
{ (x,y) : x2 + y2 ≤ 2ax , y2 ≥ ax ,
x ,y ≥ 0 }
2. Prove that the curves y2 =4x and x2=4y divide the area of square bounded by
x=0 , x=4 ,y=4 and y=0 into three equal parts.
3. Using Integration find the area of the triangle formed by negative x-axis and tangent
and normal to the circle x2+y 2=9 at (-1 , 2√2 )
4. Using the method of integration , find the area of the triangular region whose vertices
are (2 , -2) ; (4 ,3 ) and (1 ,2)
5. Using Integration find the area of the region{ (x,y) : y2≤ 6ax and x2+ y2≤ 16a2 }
6. Using Integration find the area of the region bounded by the curves y=√4 − 𝑥 2 ,
x2+y 2 - 4x = 0 and the x-axis.
7. Using Integration find the area of the triangle formedby positive x-axis and tangent
and normal to the circle x2+y 2=4 at ( 1,√3 )
8. If the area bounded by the parabola y2=16ax and the line y=4mx is
𝑎2
12
sq units , then
using integration, find the value of m.
9. Find the area of the region{ (x ,y ) : x2+ y2 ≤ 4 , x + y ≥ 2} using the method of
integration.
10. Using Integration find the area bounded by the curvesY= |x-1| and y=3 - |x|
11. Using Integration find the area of the region bounded by the line y-1=x ,
the x – axisand the ordinates x=-2 and x=3
126
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
12. Sketch the region bounded by the curves y= √5 − 𝑥 2 and y=|x-1| and
find its area using integration.
13. Using Integration find the area of the region bounded by the lines y =2+x , y= 2-x & x=2
14. Using Integration find the area of the region bounded by the line x – y+2=0
the curve x =√𝑦and y-axis.
ANSWERS
1.
2.
3.
4.
5.
6.
127
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17. 4sq. units
3
18. a = 423
19. 7sq. units
2
20. 12 sq. units
128
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
21. 6𝜋 sq. units
22. 7
sq. units
6
23. 50
sq. units
3
24. 𝜋sq. units
8
25.
a=
26.
1
√3
6𝜋 −
9√ 3
2
sq. units
ANSWERS OF PROFICIENCY EXERCISE
1.
𝜋
2
( 4 − 3) 𝑎2 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠
13
sq. units 5.
2
4.
6.
4𝜋
3
(
4√3
3
+
− √3sq. units
8. m = 2√2
14.
5𝜋
1
( 4 − 2) 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠
10
3
16𝜋
3
9√2sq. units
) 𝑎2 sq. units
7. 2√3sq. units
9. 𝜋 − 2 sq. units
10. 4 sq. units
12.
3.
11.
17
sq. units
2
13. 4 sq. units
sq. units
129
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
CHAPTER 9: DIFFERNTIAL EQUATIONS
POINTS TO REMEMBER









130
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT







131
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT





132
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
133
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
134
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
135
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
136
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
137
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
𝑑𝑦
( 1 + x2 ) 𝑑𝑥 + y = tan-1 x .
16. Write the integrating factor of the differential equation
5x ( y1) 2 – y2 - 6y = log x .
17. Write the degree of the differential equation
18. Find the particular solution of the differential equation
dy
dx
=
x ( 2 logx+1 )
siny+y cosy
, given that
y = π /2 when x = 1 .
𝑑𝑦
19. Solve the differential equation ( x - 1 )
𝑑𝑥
= 2x3y .
20. Solve the differential equation xy log ( y/x ) dx + [ y2 – x2 log ( y/x ) ]dy = 0
21. Find the particular solution of the differential equation
𝑑𝑦
𝑑𝑥
= 1 + x + y + xy
,
given that y = 0 when x = 1 .
22. Find the particular solution of the differential equation
log {
𝑑𝑦
𝑑𝑥
} =3x + 4y ,
given that y = 0 when x = 0.
23. Solve the differential equation ( x + 2 y2 )
𝑑𝑦
𝑑𝑥
= y , given when x = 2 ,y = 1 . If x denotes the
percentage of people who are polite and y denotes the percentage of the people who are
intelligent . Find x , when y = 2% . A polite person is always liked by all in society .Do you
agree ?Justify .
24. Find the equation of a curve passing through origin , if the slope of the tangent to the curve
at any point ( x , y ) is equal to the square of the difference of the abscissa and ordinate of
the point .
25. Solve the initial value problem
edy/dx = x + 1 , y = 5
when x = 0 .
26. Solve the differential equation ( 1 + y2 ) ( 1 + log x) dx + x dy = 0, given that when x = 1, y = 1
138
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
27. Solve the differential equation x log x
𝑑𝑦
𝑑𝑥
( 1 + x2 )
28. Solve the differential equation
2
+ y = ( ) log x , x > 0 .
𝑥
𝑑𝑦
𝑑𝑥
−1
+ y =𝑒 𝑚 tan 𝑥 .
x
29. Show that the differential equation 2y ex/y dx + ( y – 2x ey ) dy = 0 is homogenous and find
its particular solution , given that x = 0 when y = 1 .
30. Show that the general solution of the differential equation
𝑑𝑦
𝑑𝑥
+
y2 +
y+
x2 +
x
1
+1
= 0 is given by
( x + y +1 ) = A ( 1 – x – y – 2xy ) , where A is a parameter .
31. Find the general solution of the differential equation ( 1 + y2 ) + ( x - etan
32. Solve the differential equation ( x3 + x2 + x + 1 )
𝑑𝑦
𝑑𝑥
−1 y
)
𝑑𝑦
𝑑𝑥
= 0.
= 2x2 + x , given that y = 1 when x = 0 .
PROFICIENCY EXERCISE
1. Find the particular solution of the differential equation
𝑑𝑦
𝑑𝑥
=
𝑥𝑦
𝑥 2 +𝑦 2
,given that y=1 and x=0
2.Find the differential equation of the family of curves ( x-h)2+ (y – k)2= r 2, where h and k
are arbitrary constants.
3. Show that the differential equation
𝑑𝑦
=
𝑦2
𝑑𝑥 𝑥𝑦− 𝑥 2
is homogeneous and also solve it.
4. Find the particular solution of the differential equation
𝑑𝑦
𝑑𝑥
+ y tanx = 3x 2 + x3tanx , x≠
𝜋
2
given that y=0 when x =
𝜋
3
5. Solve the following differential equation , given that y = 0 when x =
Sin 2x
𝑑𝑦
𝑑𝑥
𝜋
4
- y = tanx
6. Find the differential equation for all straight lines , which are at a unit distance from the
origin.
139
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
7. Solve the differential equation :
x
𝑑𝑦
𝑑𝑥
+ y - x + xycot x
=0
( x≠ 0)
8. Solve the differential equation:
( 𝑥 2 + 3xy + y2) dx
- 𝑥 2 dy =0
given that y=0 when x= 1
9. Solve the differential equation:
( cot −1 𝑦 + x )dy = (1 + y2 ) dx
10. Solve the differential equation:
(x + 1)
𝑑𝑦
𝑑𝑥
- y =𝑒 3𝑥 ( x + 1 ) 3
11. Find the particular solution of the differential equation
𝑑𝑦
𝑑𝑥
=
− ( 𝑥+𝑦 cos 𝑥 )
1+ sin 𝑥
, given that y = 1 , when x = 0
ANSWERS
140
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
141
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
142
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
143
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
−1 𝑥
16.
𝑒 tan
18.
y sin y = x2logx +
20.
22.
𝑥2
2𝑦 2
17.
𝑦
1
𝑥
2
𝜋
2
[log | | + ] + log 𝑦 = 𝑐
4𝑒 3𝑥 + 3𝑒 −4𝑦 = 7
1
19. log 𝑦 = 2 [
3
+
𝑥2
2
21. log(1 + 𝑦) = 𝑥 +
23. x = 2y2 , 8%
24. (1 + 𝑥 − 𝑦) = ±(1 − 𝑥 + 𝑦)𝑒 2𝑥
25.
𝑥3
𝑦 = (𝑥 + 1) log|𝑥 + 1| − 𝑥 + 5
144
+ 𝑥 + log|𝑥 − 1| + 𝑐]
𝑥2
2
−
3
2
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
𝜋
26.tan−1 𝑦 =
4
−2
27. 𝑦𝑙𝑜𝑔𝑥 =
28. 𝑦𝑒
tan−1 𝑥
1
1
2
2
+ − (1 + 𝑙𝑜𝑔𝑥)2
(1 + 𝑙𝑜𝑔𝑥) + 𝑐
𝑥
𝑒 (𝑚+1) tan
=
−1 𝑥
+𝑐
𝑚+1
𝑥
29. 2𝑒 𝑦 + 𝑙𝑜𝑔𝑦 = 2
31. 𝑥𝑒
tan−1 𝑦
𝑒 2 tan
=
−1 𝑦
2
+𝑐
1
3
1
2
4
2
32. 𝑦 = log|𝑥 + 1| + log|𝑥 2 + 1| − tan−1 𝑥 + 1
ANSWERS OF PROFICIENCY EXERCISE
1.
−𝑥 2
+ log|𝑦|=0
2𝑦 2
2. ry2 + ( y12 + 1 )3/2= 0
3. y = xlog 𝑦 - cx
4. y = x3 -
2𝜋3
27
cos 𝑥
5. y = tan 𝑥 - √tan 𝑥
6. y = xy1±√𝑦1 2 + 1
7. y =
8. y=
1
𝑐
- cot 𝑥 +
𝑥
𝑥 sin 𝑥
𝑥 log |𝑥|
1− log| 𝑥|
−1 𝑦
9. x = (1 - cot −1 𝑦 ) + c 𝑒 − cot
10.
𝑦
= (x+1 )
𝑥+1
11. y =
𝑒 3𝑥
3
-
𝑒 3𝑥
9
+c
2− 𝑥 2
2 ( 1+ sin 𝑥 )
145
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
CHAPTER 10 : VECTOR ALGEBRA
POINTS TO REMEMBER






146
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT

l = cos , m = cos 𝛽 , n = cos 𝛾 are called the direction cosines of 𝑟⃗.







147
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT







148
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT



149
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT








150
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT






151
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT





⃗⃗ and ⃗𝒃⃗ are two non-zero vectors then ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ ⊥ ⃗𝒃⃗ .
(iii) If 𝒂
𝒂 . ⃗𝒃⃗ = 𝟎 ⇔ 𝒂

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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
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PRACTICE QUESTIONS
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21. Find the position vector of a point R which divides the line segment joining
points P(î + 2ĵ + k̂ ) andQ (−î + ĵ + k̂ ) in the ratio 2:1
(i) Internally (ii) externally
⃗⃗ and c⃗ are three vectors such that |a⃗⃗ | = 5 , | b
⃗⃗ |= 12 , |c⃗ | = 13
If a⃗⃗ , b
25.
and ⃗⃗⃗⃗⃗a
+ ⃗⃗
b + c⃗ = ⃗0⃗ , then find the value of ⃗⃗⃗
a. ⃗⃗
b + ⃗⃗
b.c⃗ +c⃗. a⃗⃗ .
26. Find the value of p for which the vectors 3i + 2j +9k and i -2pj + 3k are parallel .
27. If
a⃗⃗ and ⃗⃗
b are perpendicular vectors , | a⃗⃗ + ⃗⃗
b |= 13 and | a⃗⃗ |= 5 ,
then find the value of | ⃗⃗
b| .
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158
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159
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
160
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
58. Find a vector of magnitude 5units parallel to the resultant of the vectors a⃗⃗ =
̂
2i
⃗⃗ = î -2 ĵ +k̂.
−k̂andb
̂
+3j
59. A vector makes an angle π/4 with each of x-axis and y-axis. Find the angle made by it with zaxis.
π
60. If a unit vector a⃗⃗ makes an angle 3 withî,π/4 withĵ and an acute angle θ with k̂, then find θ and
hence find the components of a⃗⃗.
61. Let a⃗⃗ = 4î + 5ĵ − k̂ , ⃗⃗
b = î − 4ĵ + 5k̂ and c⃗ = 3î + ĵ − k̂ , find a vector ⃗⃗
d which is perpendicular
to both a⃗⃗⃗⃗and ⃗⃗
b and satisfying ⃗⃗
d.c⃗=21.
⃗⃗| = 5 and |c⃗| = 7 , find the angle between a⃗⃗ and ⃗⃗
⃗⃗,|a⃗⃗| = 3 , |b
62. Ifa⃗⃗ + ⃗⃗
b + c⃗=0
b.
63. Ifa⃗⃗ = 2𝑖̂ - 3𝑗̂ + 𝑘̂ ,
⃗⃗
b= - ̂𝑖 + 𝑘̂ ,
c⃗ = 2𝑗̂ – 𝑘̂ are three vectors , find the area of the
and
parallelogram having diagonals ( a⃗⃗ + ⃗⃗
b ) and ( ⃗⃗
b + c⃗ ) .
64. If
a⃗⃗
and
⃗⃗
b are two unit vectors such that ,a⃗⃗ + ⃗⃗
b is also a unit vector, then find the
angle between
a⃗⃗
and ⃗⃗
b .
65. Find the area of a triangle having the points A( 1 , 1 , 1) , B ( 1 , 2 , 3 ) and C ( 2 , 3 , 1 ) as
its vertices .
66. Show that the vectors a⃗⃗
, ⃗⃗
b and c⃗ are coplanar, if a⃗⃗ + ⃗⃗
b , ⃗⃗
b + c⃗ and ⃗c⃗ + a⃗⃗
are
coplanar.
67. Prove that in any triangle ABC , cos C =
a2
+ b2
2ab
−c2
,where a , b and c are the
magnitude of the sides opposite to the vertices A , B and C respectively .
68. If the vectors 𝑖̂-2x𝑗̂ -3y𝑘̂ and ̂𝑖 +3x𝑗̂ + 2y𝑘̂ are orthogonal to each other , then prove that
the locus of the point represents a circle .
⃗⃗, c⃗ be three vectors of magnitudes 3,4 and 5 respectively .if each one is the
69. Let a⃗⃗, b
perpendicular to sum of the other two vector ,prove that |𝑎 ⃗ + 𝑏 ⃗ + 𝑐 ⃗ | =5√2.
70. Find the angles of a triangle whose vertices are A(0,-1,-2), B(3,1,4) And C(5,7,1)
71. Let a⃗⃗ =x 2 î +2ĵ-2k̂ , ⃗⃗
b = î − ĵ + k̂ , and c⃗=x 2 î +5ĵ-4k̂
be three vectors . find the values of x for which the angle between a⃗⃗andb⃗⃗ is acute and
the angle betweena
⃗⃗ andc⃗ is obtuse.
⃗⃗ + c⃗ on a⃗⃗, where a⃗⃗ = 2î − 2ĵ + k̂, ⃗⃗
̂
72. Find the projection ofb
b = î + 2ĵ − 2k̂, and , c⃗=2î − ĵ + 4k,
⃗⃗ or ⃗⃗
73. If either a⃗⃗ =0
b = ⃗⃗
0 , then a⃗⃗. ⃗⃗
b = 0. but the converse need not be true justify your
answer with an example.
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PROFICIENCY EXERCISE
1. If 𝑟⃗ = x𝑖̂ + y𝑗̂ + z𝑘̂ , find ( 𝑟⃗ x 𝑖̂) . (𝑟⃗x 𝑗̂) + xy.
2. If 𝑎⃗ = 𝑖̂+2𝑗̂+𝑘̂, 𝑏⃗⃗ = 2𝑖̂ + 𝑗̂and 𝑐⃗⃗⃗= 3𝑖̂ – 4𝑗̂– 5𝑘̂ , then find a vector perpendicular to
both of the vector perpendicular to both of the vectors ( 𝑎⃗ – 𝑏⃗⃗) and ( 𝑐⃗ – 𝑏⃗⃗)
3. Find a unit vector perpendicular to the plane of triangle ABC, where the coordinates of its vertices are A(3, -1,2), B(1, -1, -3) and C(4, -3, 1)
4. The vectors 𝑎
⃗⃗⃗⃗= 3𝑖̂+ x𝑗̂and ⃗⃗⃗⃗
𝑏 = 2𝑖̂+𝑗̂+y𝑘̂ are mutually perpendicular.
⃗⃗⃗⃗ , then find the value of y.
If |𝑎⃗ | = |𝑏|
̂ , 2𝑖̂+4𝑗̂+6𝑘̂, 3𝑖̂+5𝑗̂+4𝑘̂
5. Show that the four points with position vectors 4𝑖̂ + 8̂+12𝑘
𝑗
and 5𝑖̂+8𝑗̂+5𝑘̂ are coplanar.
6. Find the value of a+b, if the points (2,a,3) , (3,-5,b) and (-1,11,9) are collinear.
7. Find x such that four points A(4,1,2) , B(5,x,6), C(5,1,-1) and D(7,4,0) are coplanar.
8. Find a unit vector in the direction of the sum of the vectors 2𝑖̂ + 3𝑗̂-𝑘̂ and 4𝑖̂-3𝑗̂+2𝑘̂.
⃗⃗⃗⃗⃗⃗
9. The two vectorŝ+𝑘
𝑗 ̂ and 3𝑖̂-𝑗̂+4𝑘̂ represent the two side vectors ⃗⃗⃗⃗⃗⃗
𝐴𝐵 and 𝐴𝐶
respectively of
∆ABC. Find the length of the median through A.
10. Write the number of vectors of unit length perpendicular to both the vectors
⃗⃗⃗⃗=
𝑎 2𝑖̂+𝑗̂+2𝑘̂ and 𝑏⃗⃗ = 𝑗̂+𝑘̂
11. The two adjacent sides of a parallelogram are 2𝑖̂ – 4𝑗̂ – 5𝑘̂ and 2𝑖̂+2𝑗̂+3𝑘̂. Find the
two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of
the parallelogram.
12. Find the angle between the vectors⃗⃗⃗⃗+𝑏
𝑎 ⃗⃗ and 𝑎⃗– 𝑏⃗⃗, if 𝑎⃗ = 2𝑖̂-𝑗̂+3𝑘̂ and 𝑏⃗⃗ = 3𝑖̂+𝑗̂-2𝑘̂
and hence find a vector perpendicular to both 𝑎⃗+𝑏⃗⃗ and 𝑎⃗-𝑏⃗⃗.
13. Find λ and μ if (𝑖̂+3𝑗̂+9𝑘̂) x (3𝑖̂ -𝜆𝑗̂+ μ 𝑘̂) = ⃗⃗⃗⃗
0
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
14. If⃗⃗⃗⃗
𝑎 and 𝑏⃗⃗ are unit vectors, then what is the angle between 𝑎⃗ and 𝑏⃗⃗ for 𝑎⃗ – √2𝑏⃗⃗
to be a unit vector?
15. Given that vectors⃗⃗⃗⃗,
𝑎 𝑏⃗⃗, 𝑐⃗⃗⃗form a triangle such that 𝑎⃗ = 𝑏⃗⃗+𝑐⃗, find p,q,r,s such that
area of triangle is 5√6 where ⃗⃗⃗⃗
𝑎 = p𝑖̂+q𝑗̂+r𝑘̂, 𝑏⃗⃗ = s𝑖̂+3𝑗̂+4𝑘̂ and 𝑐⃗ = 3𝑖̂+𝑗̂-2𝑘̂.
16. If | 𝑎
⃗⃗⃗⃗x 𝑏⃗⃗ |2 + | 𝑎⃗. 𝑏⃗⃗|2 = 400 and | 𝑎⃗ | = 5 , then write the value of | 𝑏⃗⃗|.
17. Write the position vector of the point which divides the join of points with position
vectors 3𝑎⃗ – 2𝑏⃗⃗ and 2𝑎⃗ + 3𝑏⃗⃗ externally in the ration 2:1.
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20.
√𝟑
21.
164
−𝟏
𝟑
𝟒
̂ , −𝟑𝒊̂ + 𝒌
̂
𝒊̂ + 𝒋̂ + 𝒌
𝟑
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
24.
√𝟑
𝒊̂
𝟐
𝟏
+ 𝒋̂
25. -169
𝟐
26. -1/3
58.
60.
15
√10
𝝅
𝟑
,
62. 𝝅/𝟑
27. 12
𝑖̂ +
𝟏
√𝟐
5
√10
𝒊̂ ,
𝑗̂
𝟏
𝟐
59. 𝝅/𝟐
𝒋̂,
𝟏
𝟐
𝒌̂
̂
61. 𝟕𝒊̂ − 𝟕𝒋̂ − 𝟕𝒌
63.
√𝟐𝟏
𝟐
𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
64. 2𝝅/𝟑
65.
165
√𝟐𝟏
𝟐
𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
70. ∠𝑨 =
𝝅
, ∠𝑩 =
𝟒
𝝅
, ∠𝑪 =
𝟐
𝝅
𝟒
71. (-3 , -2)∪(2 , 3)
72. 2
ANSWERS OF PROFICIENCY EXERCISE
1.0
3.
2. -
−10
√165
6.
8.
𝑖̂ −
−7
√165
𝑗̂ +
4
√165
𝑘̂
√37
𝑖̂ +
1
√37
𝑘̂
π
2
14.
15.
16. 4
̂
𝑘
√2
7.
4
9.
√34
2
11. 2√101 sq. units
10. 2
12.
√2
+
4. ±2√10
0
6
𝑗̂
, 2𝑖̂– 26 𝑗̂ – 10 𝑘̂
13.
λ = -9, μ = 27
π
4
p = -8,8
q=4
r=2
s= -11,5
17. 𝑎⃗ + 8𝑏⃗⃗
166
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
CHAPTER 11 : THREE DIMENSIONAL GEOMETRY
POINTS OF REMEMBER
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168
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169
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170
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171
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172
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173
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174
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PRACTICE QUESTIONS
175
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
176
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
177
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
178
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
55.
Find the direction cosines of a line which makes equal angles with the coordinate axes .
56.
Find the angle between the lines x/2 = y /2 = z/1 and ( x – 5 ) /4 = ( y – 2 ) / 1 = ( z -3 ) /
8.
57.
If the cartesian equation of a line is ( 3 –x ) / 5 = ( y + 4) / 7 = ( 2z- 6 ) /4 , then write
the vector equation of the line .
58.
Determine the direction cosines of the normal to the planex + y + z = 1 and the distance
from the origin .
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
59.
Cartesian equation of line AB is
2𝑥−1
2
4−𝑦
=
𝑧 +1
=
7
2
. Write the direction ratios of a line
parallel to AB .
60.
If a line in the space makes angles
find the value of
61.
α ,β
and γ with the coordinate axes
, then
cos2α + cos 2β + cos2γ + sin2α + sin2β + sin2γ .
Find the value of λ such that the line
𝑥−2
6
=
𝑦−1
𝜆
𝑧+5
=
is perpendicular to the
−4
plane 3x – y – 2z = 7.
62.
Find the distance between the lines l1 and l2 given by
𝑟⃗ = I + 2j – 4k + 𝜆 ( 2i +3j + 6k)
and 𝑟⃗ = 3i + 3j -5k + μ ( 4i + 6j +12k ) .
63.
Find the points on the line ( x + 2 ) / 3 = ( y + 1 ) /2 = ( z – 3 ) /2 at a distance of 5 units
from the point P ( 1 , 3 , 3 ).
64.
A plane meets the coordinates axes at A , B and C such that the centroid of the triangle
ABC is the point
𝑥
( α ,β ,γ ) . Show that the equation of the plane is
𝛼
+
𝑦
𝛽
+
𝑧
𝛾
=3
.
65.
Find the length and the foot of the perpendicular from the point ( 7 ,14 , 5) in the plane
2x + 4y – z = 2 .
66.
Find the coordinates of the point where the line
𝑥−2
3
=
𝑦+1
4
=
𝑧−2
2
intersect the plane x – y
+ z -5 = 0.Also find the angle between the line and the plane .
67.
The points ( 1, 1, p ) and ( -3, 0,1 ) are equidistant from the plane 𝑟⃗ . ( 3i + 4j -12 k) + 13
= 0, then find the value of p .
68.
Find the direction cosines of the two lines which are connected by the relations l - 5m +
3n = 0 and
69.
7 l2 + 5 m2 – 3n2= 0 .
Find the vector and Cartesian equation of the line passing through the point ( 2 , 1 , 3 )
and perpendicular to the lines
𝑥−1
1
=
180
𝑦−2
2
=
𝑧−3
3
𝑥
𝑎𝑛𝑑 −3 =
𝑦
2
=
𝑧
5
.
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
70.
If straight lines having direction cosines given by al + bm+cn = 0 and fmn + gnl + hlm = 0
are perpendicular , then show that f/a + g/b + h/c = 0.
71.
Prove that the lines ( x -2 ) / 1 = ( y -4 ) /4 = ( z -6 ) /7 and
( x + 1 ) /3 = ( y + 3 ) /5 = ( z + 5 ) /7 are coplanar . Also find the plane containing these
lines .
72.
Find the vector equation of the plane through the points ( 2 , 1 , -1) and ( -1 ,3 ,4 ) and
perpendicular to the plane x – 2y + 4z = 10 .
73.
If a variable line in two adjacent positions has direction cosines l ,m ,n and l+ δl , m+ δm
,n + δn, show that
the
small angle
δ θ between two positions is given by
( δθ)2 = ( δl)2 + (δm ) 2 + ( δn )2 .
74.
Find the image of the point having position vector 𝑖̂ + 3𝑗̂ + 4𝑘̂ in the plane 𝑟⃗.(2𝑖̂ –𝑗̂+ 𝑘̂)+3
=0
75.
If the edges of a rectangular parallelepiped are a , b , c, prove that the angles between
the four diagonals are given by
𝑎2 ±
𝑏2 ±
𝑐2
𝑎2 +
𝑏2 +
𝑐2
𝑐𝑜𝑠 −1 (
181
).
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
PROFICIENCY EXERCISE
1. Find the vector and Cartesian equation of the line through the point (1, 2, -4) and
perpendicular to the two line𝑟⃗ = (8 𝑖̂ – 19 𝑗̂ + 10 𝑘̂ ) + λ(3 𝑖̂ – 16 𝑗̂ + 7 𝑘̂ ) and𝑟⃗ = (15 𝑖̂ + 29 𝑗̂ + 5
𝑘̂ ) + μ ( 3 ̂𝑖 + 8 𝑗̂– 5 𝑘̂ ).
2. Find the co-ordinate of the point P where the line through A ( 3, -4, -5 ) and B ( 2, -3, 1 )
crosses the plane passing through three points L ( 2, 2, 1 ), M (3, 0, 1 ) and N ( 4, -1, 0 ). Also
find the ratio in which P divides the line segment AB.
3. Find the co-ordinates of the point where the line through the points A (3, 4, 1 ) and B ( 5, 1, 6 )
crosses the XZ – plane. Also find the angle which this line makes with the XZ – plane.
4. Find the position vector of the foot of perpendicular and the perpendicular distance from the
point P with position vector 2 𝑖̂ + 3 𝑗̂ + 4 𝑘̂ to the plane ⃗⃗⃗𝑟 . ( 2𝑖̂ + 𝑗̂ + 3 𝑘̂ ) = 26. Also find the
image of P in the plane.
5. Show that the lines
𝑥−1
3
=
y−1
−1
, 𝑧 + 1 = 0and
𝑥−4
2
=
z+1
3
, 𝑦 = 0, intersect. Find their point of
intersection.
6. Write the sum of intercepts cut off by the plane⃗⃗⃗.
𝑟 ( 2𝑖̂ + 𝑗̂– 𝑘̂ ) – 5 = 0 on the three axes.
7. Find the equation of the plane which contains the line of intersection of the planes 𝑟⃗⃗⃗. ( 𝑖̂ –
2 ̂𝑗 + 3 𝑘̂ ) = 4 and 𝑟⃗⃗⃗. ( -2 𝑖̂ + ̂𝑗 + 𝑘̂ ) + 5 = 0 and whose intercept on x – axis is equal to that of
an y – axis. Hence write the vector equation of a plane passing through the point ( 2, 3, -1 )
and parallel to the plane obtained above.
8. Find the co-ordinates of the foot of perpendicular drawn from the point A ( -1, 8, 4 ) to the
line joining the points B ( 0, -1, 3 ) and C ( 2, -3, -1 ). Hence find the image of the point A in the
line BC.
9. Find the co-ordinates of the point where the line⃗⃗⃗𝑟 = ( -𝑖̂ – 2̂𝑗 – 3 𝑘̂ ) + ( 3𝑖̂ + 4 ̂𝑗 + 3 𝑘̂ ) meets
the plane which is perpendicular to the vector 𝑛
⃗⃗⃗⃗= 𝑖̂ + 𝑗̂+ 3𝑘̂ and at a distance of 4/√11 from
origin.
10. Find the equation of the plane containing the two parallel lines
z+1
𝑥−2
6
3
. Also, find if the plane thus obtained contains the line
182
=
𝑥−1
2
y−1
1
=
=
y+1
−1
z−2
5
z
𝑥
3
4
= and
or not.
=
y−2
−2
=
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
11. If lines
𝑥−1
2
y+1
=
3
=
z−1
4
and
𝑥−3
1
=
y−k
2
=
z
1
interest then find the value of k and hence find
the equation of the plane containing these lines.
12. Find the equation of a plane passing through the points P ( 6, 5, 9 ) and parallel to the plane
determined by the points A ( 3, -1, 2 ), B ( 5, 2, 4) and C ( -1, -1, 6). Also find the distance of
this plane from the points A.
13. Find the equation of the plane passing through the points ( -1, 2, 0 ), ( 2, 2, -1 ) and parallel to
the line
𝑥−1
1
=
2y+1
2
=
z+1
−1
.
14. Find the shortest distance between the lines x + 1 = 2y = - 12 z and x = y + 2 = 6z – 6.
15. From the points P (a, b, c) perpendiculars PL and PM are drawn to YZ and ZX planes
respectively. Find the equation of the plane OLM.
16. Let P(3, 2, 6) be a point in the space and Q be a point on the line 𝑟⃗⃗⃗= (𝑖̂ – 𝑗̂ + 2𝑘̂ ) + μ ( - 3𝑖̂ + 𝑗̂ +
5 𝑘̂ ), then find the value of u for which the vector ⃗⃗⃗⃗⃗⃗
𝑃𝑄 is parallel to the plane x – 4y + 3z = 1.
17. Find the vector and Cartesian equations of the plane which bisects the line joining the points
(3, -2, 1) and (1, 4, -3) at right angles.
18. Show that the following two lines are coplanar.
𝑥−𝑎+𝑑
𝛼−𝛿
=
y−a
𝛼
=
z−a−d
α+δ
𝑥−𝑏+𝑐 y−b
z−b−c
=
=
𝛽−𝛾
𝛽
β+γ
19. Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0)
and (0,1,0) and makes angle π/4with the plane x+y=3. Also find the equation of the plane.
20. Find the equation of a plane which passes through the points (3,2,0) and contains the line
𝑥−3
1
=
y−6
5
=
z−4
4
.
21. Find the distance of the point 3𝑖̂ – 2̂𝑗 + 𝑘̂ from the plane 3x + y – z + 2 = 0 measured parallel
to the line
𝑥−1
2
=
y+2
−3
=
z−1
1
. Also find the foot of the perpendicular from the given point
upon the given plane.
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1
1
√3
√3
55 .± , ±
,±
1
2
56.cos-1( )
3
√3
57 .𝑟⃗ = (+ 3𝑖̂ – 4𝑗̂+ 3 ̂𝑘 ) + λ (-5 𝑖̂ + 7̂𝑗 + 2 𝑘̂)
58. direction cosine
1
,
1
√3 √3
,
1
√3
distance =
1
√3
59. 1, -7, 2
60. 1
61. λ = -2
62.
√293
7
𝑢𝑛𝑖𝑡𝑠
63. (4, 3, 7) , ( -2, -1, 3)
65. foot of plane (1,2,8) ,length = 3√21
66 . (22 , -1,2) sin-1(
69.
𝑥−2
2
=
𝑦−1
−7
=
1
)
67. 1,
√87
𝑧−3
7
3
71. x-2y+z = 0
4
74 (-3,5,2) or -3 𝑖̂+5𝑗̂+2𝑘̂
72. 18x + 17y+4z = 49
ANSWERS OF PROFICIENCY EXERCISE
1.
𝑥−1
24
=
y−2
36
=
z+4 𝑥−1
72
&
2
=
y−2
3
=
z+4
6
2. P ( 1, −2, 7 ) externalIy 2: 1
17
23
3. ( 3 , 0 , 3 ) , 𝜃 = sin-1(3/√38)
4. P.V of F = 3 𝑖̂ + 7/2𝑗̂+ 11/2𝑘̂ Image of P (4, 4, 7) ⊥ distance = √14/2
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5. ( 4, 0, -1 )
6.
5/
2
7. x + y – 4z – 1 = 0
,
𝑟⃗⃗⃗. ( ̂𝑖 + 𝑗̂ – 4𝑘̂) – 9 = 0
8. F ( -2, 1, 7 ), A ( -3, -6, 10 )
9. (2, 2, 0 )
10. 8x + y – 5z = 7 ,
given line is contained by plane.
11. k = 9/2, 5x – 2y – z = 6
12.
6
,3x – 4y + 3z = 19.
√34
13. x + 2y + 3z = 3
14. 2
15. bcx + acy – abz = 0
16. u = ¼
17. x – 3y + 2z + 3 = 0
19.
1, 1, ±√2 ,x + y ±√2 z = 1
20. x – y + z = 1
9
−30
11
11
21. 4√14 , ( ,
19
, )
11
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CHAPTER 12 : LINEAR PROGRAMMING
POINTS TO REMEMBER
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PRACTICE QUESTIONS
1.
2.
3.
4.
5.
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6.
7.
8.
9.
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10. On completing the construction of his house , a person discovers that 100 sq feet of plywood
scrap and 80 sq feet of white pine scrap are in usable form for the construction of tables and
book cases .It takes 16 sq feet of plywood and 8 sq feet of white pine to make a table and 12 sq
feet of plywood and 16 sq feet of white pine to construct a book case . On selling the finished
product to a local furniture house , he can earn a profit of Rs 25 on each table and Rs 20 on
each book case . How should he most profitably use the left over wood ? Use graphical
method to solve the linear programming problem .
11. A merchant plan to sell two types of personal computers a desktop model and a portable model
whose cost are obtained on multiplying
α and β (α < β ) by 1000 respectively , where α
and β are the roots of the quadratic equation
p 2 - 65 p + 1000 = 0 . He estimates that the
total monthly demand of computers will not exceed 250 units . Determine the number of units of
each type of computers which the merchant should stock to get maximum profit , if he does not
want to invest more than Rs 70 lakh and his profit on the desktop model is Rs 4500 and on the
portable model is Rs 5000. Make an LPP and solve it graphically .
12. A House wife wishes to mix together two kinds of food ,X and Y, in the such a way that the
mixture contains at least 10 units of vitamin A,12 unit of vitamin B and 8 units of vitamin C.
The vitamin contents of one kg of food is given the below :
Vitamin A
Vitamin B
Vitamin C
FOOD X
1
2
3
FOOD Y
2
2
1
One kg of the food X costs Rs6 and one kg of food of Y costs Rs10.find the least cost of the
mixture which will produce the diet .
13. There is a factory located at each of the two place P and Q. from these location ,a certain
commodity is delivered to each of these depots situated at A,B and C . The weekly requirements
of the depots are respectively 5,5 and 4 units of the commodity while the production capacity of
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
the factories at P and Q are respectively 8 and 6 units .the cost of transportation per unit is given
below
COST (IN RS)
FROM
A
B
C
P
16
10
15
Q
10
12
10
TO
How many units should be transported from each factory to each depot in order that the
transportation cost is minimum ,formulate the above LPP mathematically and then solve it .
14. A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profit
from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To
control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10
litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect
fish and wild life using a pond which collects drainage from this land. How much land should be
allocated to each crop so as to maximize the total profit of the society? Write the advantage of
cooperative society in a village?
15.
A toy company manufactures two types of dolls, A and B. Market tests and available
resources have indicated that the combined production level should not exceed 1200 dolls per
week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the
production level of dolls of type A can exceed three times the production of dolls of other type
by at most 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on
dolls A and B, how many of each
should be produced weekly in order to maximize the profit?
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16. A small firm manufactures gold rings and chains. The total number of rings and
chains
manufactured per day is at most 24.It takes 1 hour to make a ring and 30 minutes to make a
chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs.300 and
that on a chain is Rs.190, find the number of rings and chains that should be manufactured per
day, so as to earn the maximum profit. Make it as L.P.P.and solve it graphically.
17. David wants to invest at most Rs12,000 in Bonds A and B. According to the rule, he has to invest
at least Rs 2,000 in Bond A and at least Rs4,000 in Bond B. If the rate of interest in bonds A and
B respectively are 8% and 10% per annum, formulate the problem as L.P.P. and solve it
graphically for maximum interest. Also determine the maximum interest received in a year.
PROFICIENCY EXERCISE
1. A manufacturer produces two products A and B. Both the products are processed on
two different machines. The available capacity of first machine is 12 hours and that of second
machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and
each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each
unit of product A is sold at Rs 7 profit and that of B at a profit of Rs 4. Find the production
level per day for maximum profit graphically.
2. A company manufactures two types of cardigans: type A and type B. It costs Rs360 to
make a type A cardigan and Rs 120 to make a type B cardigan .The company can make at
most 300 cardigans and spend at most Rs 72000 a day . The number of cardigans of type B
cannot exceed the number of cardigans of type A by more than 200. The company makes a
profit of Rs 100 for each cardigan of type A and Rs 50 for every cardigan of type B. formulate
this problem as a linear programming problem to maximise the profit of the company. Solve
it graphically and find maximum profit.
3. A retired person wants to invest an amount of Rs 50,000 this broker recommends investing
in two type of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested
amount .He decides to invest at least Rs 20,000 in bond ‘A’ and at least Rs 10,000 in bond ‘B’.
He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear
programming problem graphically to maximise his returns .
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4. In order to supplement daily diet, a person wishes to take X and Y tablets . The contents
(in milligrams per tablet) of iron , calcium and vitamins in X and Y are given as below.
Tablets
Iron
Calcium
X
6
3
Y
2
3
Vitamin
2
4
The person needs to supplement at least 18 milligram of iron, 21 miligram of calcium and 16
miligram of vitamins . The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How
many tablets of each type should the person take in order to satisfy the above requirement
at the minimum cost ? Make an LPP and solve graphically.
5. A company manufactures three kinds of calculators A, B and C in its two factories first
and second. The company has got an order for manufacturing at least 6400 calculators of
kind A, 4000 of kind B and 4800 of kind C. The daily output of factory first is of 50 calculators
of kind A, 50 calculators of kind B and 30 calculators of kind C . The daily output of factory
second is of 40 calculator of kind A ,20 of kind B and40 of kind C. The cost per day to run
factory first is Rs 12000 and factory second Rs 15000. How many days do the two factories
have to be in operation to produce the order with the minimum cost? Formulate this
problem as an L P P and solve it graphically.
6. A dealer in a rural area wishes to purchase some sewing machines. He has only Rs57,600
to invest and has space for at most 20 items. An electronic machine costs him Rs 3600 and a
manually operated machine cost Rs 2400. He can sell an electronic machine at a. profit of RS
220 and a manually operated machine at a profit of Rs 180.Assuming that he can sell all the
macinnes that he buys, how should he invest his money in order to maximize his profit ?
Make it as an LPP and solve it graphically.
7.A dietician wants to develop a special diet using two foods X and Y. Each packet (contain
30g) of food X contains 12 units of calcium ,4 units of iron , 6 units of cholesterol and 6 units
of vitamin A. Each packet of same quantity of food Y contains 3 units of calcium, 20 units of
iron, 4 units of cholesterol and 3 units of vitamin A .The diet requires at least 240 units of
calcium, at least 460 units of iron and at most 300 units of cholesterol. Make an LPP, to find
how many packets of each food should be used to minimise the amount of vitamin A in the
diet, and solve it graphically.
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
8. The standard weight of a special purpose brick is 5 kg and it must contain two basic
ingredients B1 and B2. B1 cost Rs 5 per kg and B2 cost Rs 8 per kg. Strength considerations
dictate that the brick should contain not more than 4 kg of B1 and minimum 2 kg of B2 since
demand for the product is likely to be related to the price of the brick, find the minimum cost
of brick satisfying the above conditions formulate this situation as an LPP and solve it
graphically.
9. Minimize and maximize Z=5x+2y subject to the following constraints:
x-2y≤ 2
3x+2y ≤ 12
-3x+2y ≤ 3
x≥ 0 , y≥ 0
10. The postmaster of a local post office wishes to hire extra helpers during Deepawali
season, because of a large increase in the volume of mail handling and delivery. Because of
the limited office space and budgetary conditions, the number of temporary helpers must
not exceed 10. According to past experience, a man can handle 300 letters and 80 packages
per day. On the average, and a woman can handle 400 letters and 50 packets per day The
postmaster believes that the daily volume of extra mail and packages will be no less than
3400 and 680 respectively. A man receives Rs 225 a day and a woman recivesRs 200 a day.
How many men and women helpers should be hired to keep the pay roll at a
minimum?formulate an LPP and solve it graphically.
ANSWERS
2.
3.
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
4.
5.
6.
7.
8.
9.
10. Table -4, Book case -3,max.profit = Rs 160
11. Max profit – Rs 11,50,000 , Desktop model - 200, Portable model -50
12. Mini. Cost – Rs 52, when 2 kg of food X & 4 kg of food Y are mixed
13. The optimal transportation strategy will be to deliver 0, 5, 3 units from the
factory at P and 5,0 & 1 units from the factory at Q to the depots A, B & C resp.
Minimum cost Rs. 155
14. Max. profit-Rs 4,95,000, Crop A -30 hectares ,Crop B – 20 hectares
15. 800 dolls of type A, 400 dolls of type B, Max. Profit Rs16000
16. Gold rings -8, Chains – 16, Max. profit = Rs 5440
17. Investment in bond A – Rs 2000, Investment in Bond B = Rs 10000
Maximum Intrest received – Rs 1160.
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
ANSWERS OF PROFICIENCY EXERCISE
1. Max. Profit = Rs. 26. Product A = 2 units, Product B = 3 units.
2 .Max.Profit = Rs. 22500, Cardigens of type A = 150 ,Cardigens of type B = 150
3 .Max returns = Rs. 4900, Investment in bond A = Rs. 40000
Investment in bond A = Rs. 10000
4 . Minimum cost = Rs. 8, Tablet X = 1, Tablet Y = 6
5 . Minimum cost = Rs. 1860000, Factory I = 80 days, Factory II = 60 days.
6 . Max. Profit = Rs. 3920, Electronic sewing machines = 8,
Manually operated sewing machines = 12.
7. Mini. Amount of Vitamin A = 150 units, Food X = 15 packets, Food Y = 20 packets.
8. Mini cost of brick= Rs. 31, when 3 kg of B1 and 2 kg of B2 are taken.
9 . Z is minimum at x = 0, y = 0 and mini value = 0
7
3
2
4
Z in maximum at x = , y =
, and max. value = 19
10 . Men =6, Women = 4.
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DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
Chapter 13 : PROBABILITY
Points to Remember
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If three events A,B,C are independent then P(A  B  C) = 1 - P(A’) P(B’) P(C’)
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PRACTICE QUESTIONS
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41. Events E and F are such that
P ( not E or not F ) = 0.25 . State whether E and F are
mutually exclusive .
42. If A and B are two events such that P ( A ) = ½ , P ( B ) = 1/3 and P ( A ∩ B ) = ¼ ,then
find P (
𝐴̅
𝐵
).
43. A die is tossed thrice . find the probability of getting an even number at least once .
44. Given two independent events A and B such that P ( A ) = 0.3 , P ( B ) = 0.6 . Find
and not B ) .
45. For the following probability distribution
Find
X
1
2
3
4
P(X)
1/10
1/5
3/10
2/5
E ( X2) .
46. Find the variance of the Binomial distribution B ( 5 ,1/4 ).
212
P(A
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
47. If E and F are events with P( E ) = 3/5, P( F ) =3 /10 , and P( E and F ) = 1/5 .Then find
whether the events E & F are independent or not .
48. There are three coins , one is a two headed coin ( having head on both faces ) , another is a
biased coin that comes up heads 75 % of the time and third is an unbiased coin . One of the
three coins is chosen at random and tossed ,it shows head . What is the probability that it
was the two headed coin ? .
49. A couple has two children . Find the probability that both the children ( i) males if it is known
that atleast one of the child is male .( ii )females , if it is known that the elder child is a female
50. The probabilities of A , B , C solving a problem are 1/3 , 2/7 and 3/8 respectively .If all the
three solve the problem simultaneously , find the probability that exactly one of them will
solve it.
51. If each element of a second order determinant is either zero or one , what is the probability
that the value of the determinant is positive ? ( assuming that the individual entries of the
determinant are chosen independently , each value being assumed with probability ½ )
52. The probabilities of two students A and B coming to the school in time are 3/7 and 5 /7
respectively. Assuming that the events , ‘A coming in time ‘ and ‘ B coming in time ‘ are
independent , find the probability of only one of them coming to school in time . Write at
least one advantage of coming to the school in time .
53. A speaks truth in 60 % of the cases , while B in 90 % of the cases . In what percent of the
cases are they likely to contradict each other in stating the same fact ? . In cases of
contradiction ,do you think the statement of B will carry more weight as he speaks truth in
more number of cases than A ? .
54. Suppose you have two coins which appear identical in your pocket . You know that one is fair
and other is two headed . If you take one coin out ,toss it and get a head , what is the
probability that it was a fair coin ?.
55. A company has two plants to manufacture motorcycles. 70% motorcycles are manufactured
at the first plant ,while 30 % are manufactured at second plant. At the first plant ,80 %
motorcycles are rated of standard quality ,while at the second plant 90% are rated of
standard quality . A motorcycle is randomly picked up ,is found to be of standard quality . Find
the probability that it has come out from the second plant .
56. An experiment succeeds thrice as often
as it fails . Find the probability that in the next 5
trials there will be at least 3 successes .
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57. From a lot of 15 bulbs which include 5 defectives , a sample of 4 bulbs is drawn one by one
with replacement. Find the probability distribution of number of defective bulbs . Hence ,find
the mean of the distribution .
58. In a hostel , 60 % of the students read Hindi newspaper , 40 % read English newspaper and
20% read both Hindi and English newspapers. A student is selected at random ( a ) find the
probability that he /she reads neither Hindi nor English newspaper . ( b) if he /she reads
Hindi newspaper , find the probability that he/ she reads English newspaper ( c) if he /she
reads English newspaper , find the probability he /she reads Hindi newspaper .
59. Two cards are drawn simultaneouslywith out replacement from a well shuffled deck of 52
cards . Find the mean and variance of the number of red cards .
60. In a college , 70 % students pass in Physics , 75 % pass in Mathematics and 10 %
students fail in both .One student is chosen at random . What is the probability that
( a ) he passes in Physics & Mathematics ?
( b) he passes Mathematics given that he passes in Physics ?
( c) he passes in Physics given that he passes in Mathematics ?.
61. In answering a question on a multiple choice questions test with four choices per questions
.A student knows the answer , guesses or copies the answer .If ½ is the probability that he
knows the answer ,1/4 is the probability that he guesses it and ¼ that he copies it .Assuming
that a student
who copies the answer will be correct with probability ¾ . What is the
probability that the student knows the answer given that he answered it correctly ? Anil does
not know the answer to one of the question in the test .The evaluation process has negative
marking . Which value would Anil violate ,if he restores to unfair .How would his personality
would be hampered ?
62. A shopkeeper sells three types of flower seeds A , B and C. They are sold as a mixture ,
where the proportion are 4:4:2 respectively . The germination rate of the three types of seeds
are 45% ,60 %, 35 % .Calculate the probability ( a ) of a randomly chosen seed to germinate
( b) that it will not germinate given that the seed is type C . ( c ) that it is of the type B ,given
that a randomly chosen seed does not germinate .
63. X is taking up subjects Mathematics , Physics and Chemistry
in the examination . His
probabilities of getting grade A in these subjects are 0.2 ,0.3 and 0.5 respectively . Find the
probabilities that he gets
( a ) grade A in all subjects
( b)
( c ) grade A in two subjects .
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grade A in no subjects
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PROFICIENCY EXERCISE
1. Three persons A,B, and C apply for a job of manager in a private company. Chances of
their selection (A,B and C) are in ratio 1:2:4. The probabilities that A,B and C can
introduce changes to improve profits of the company are 0.8,0.5 and 0.3 respectively.
If the change does not take place, find the probability that it is due to the appointment
of C.
2. An urn contains 3 white and 6 red balls. Four balls are drawn one by one with
replacement from the urn. Find the probability distribution of the number of red balls
drawn. Also find mean and variance of the distribution.
3. In a game, a man wins Rs. 5 for getting a number greater than 4 and loses Rs 1
otherwise, when a fair die is thrown. The man decided to throw a die thrice but to quit
as and when he gets a number greater than 4. Find the expected value of the amount
he wins/loses.
4. A bag contains 4 balls. Two balls are drawn at random (without replacement) and are
found to be white. What is the probability that all balls in the bag are white?
5. Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn
one by one successively with replacement, then find the probability distribution of
number of bad oranges drawn. Hence find the mean and variance of the distribution.
6. A committee of 4 students is selected at random from a group consisting of 7 boys and
4 girls. Find the probability that there are exactly 2 boys in the committee, then that at
least one girl must be there in the committee.
7. A random variable X has the following probability distribution.
X
0
1
2
3
4
5
6
2
2
P(x)
C
2C
2C
3C
𝐶
2𝐶
7𝐶 2 +C
Find the value of C and also calculate mean of the distribution.
8. A,B and C throw a pair of dice in that order alternately till one of them gets a total of 9
and wins the games. Find their respective probabilities of winning, if A starts first.
9. A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white
balls and 3 black balls. Two balls are drawn (without replacement) at random from one
of the bags and were found to be one white and one black. Find the probability that
the balls were drawn from bag Y.
10. Three numbers are selected at random ( without replacement) from first six positive
integers. Let X denote the largest of the three numbers obtained. Find the probability
distribution of X. Also find, the mean and variance of the distribution.
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11. A box has 20 pens of which 2 are defective. Calculate the probability that out of 5
pens drawn one by one with replacement, at most 2 are defective.
12. Let X, denote the number of colleges where you will apply after your results and
P(X=x) denotes your probability of getting admission in x number of colleges. It is given
that
Kx if x=0 or 1
P(X=x) =
2Kx if x=2
K(5-x) if x=3 or 4
0
if x>4
Where K is a positive constant. Find the value of K. Also ,find the probability that you will
get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2college.
13. Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two
balls are transferred at random from bag A to B and then a ball is drawn from bag B at
random. If the balls drawn from bag B is found to be red, find the probability that two
red balls were transferred from A to B.
14. An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads
obtained.
15. If A and B are independent events such that P (𝐴̅ ∩ B) =2/15 and P(A ∩ 𝐵̅)=1/6 then
find P(A) and P (B).
16. For 6 trials of an experiment ,let X be a binomial variate which satisfies the relation.
9P (x=4) = P(x=2) Find the probability of success.
17. An urn contains 3 red and 5black balls. A ball is drawn at random its colour is noted and
returned to the urn. Moreover, 2 additional balls of the colour noted down, are put in
the urn and then two balls are drawn at random (without replacement) from the urn.
Find the probability that both the balls drawn are of red colour.
216
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
18. From a set of 100 cards numbered 1 to 100 one card is drawn at random. Find the
probability that the number on the card is divisible by 6or 8, but not by 24.
19.Assume that the chances of a patient having a heart attack is 40% It is also assumed that a
meditation and yoga course reduces the risk of heart attack by 30% and the prescription
of a certain drug reduces its chance by 25%.At a time a patient can choose any one of the
two options with equal probabilities.It is given that after going through. One of the two
options the patient selected at random suffers a heart attack. Find the probability that the
patient followed a course of meditation and yoga.
20. An unbiased coin is tossed ‘n’ times. Let the random variable X denote the number of
times the head occurs. If P(x=1), P (x=2) and P(x=3) are in A.P, Find the value of n.
21. How many times must a fair coin be tossed so that the probability of getting at least one
head is more than 80%?
22. A die is thrown three times. Events A and B are defined as below.
A:
5 on the first and 6 on the second throw.
B: 3 or 4 on the third throw.
Find the probability of B, given that A has already occurred.
23. Find the probability distribution of the number of doublets in four throws of a a pair of
dice. Also find the mean and variance of this distribution.
217
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
218
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
219
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
41. No
43.
42.
7
44.
8
45. 10
46.
47. Not
48.
49.
1
3
,
1
50.
2
220
1
4
0.12
15
16
4
9
25
56
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
51.
3
52.
16
53. 42%
55.
54.
27
56.
83
57. Mean =
26
49
1
3
459
512
4
3
𝑥𝑖
0
1
2
3
4
𝑝𝑖
16
81
32
81
24
81
8
81
1
81
58. a) 1/5
b) 1/3
c) ½
59. Mean = -1 , Variance = 0.49
60. a)
61.
11
20
b)
11
c)
14
11
15
2
, Honesty
3
16
62. a) 0.49
b) 0.65
c)
63. a) 0.03
b) 0.28
c) 0.22
221
51
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
ANSWERS OF PROFICIENCY EXERCISE
1.
7
10
8
8
3
9
2. Mean = , Variance =
3.
𝑥𝑖
0
1
2
3
4
𝑝𝑖
1
81
8
81
24
81
32
81
16
81
19
4.
9
4
16
5
25
5. Mean = , Variance =
6.
5
𝑥𝑖
0
1
2
3
4
𝑝𝑖
256
625
256
625
96
625
16
625
1
625
126
295
7. C =
1
10
, Mean = 2.66
8. A wins =
9.
3
81
,
217
B wins =
72
217
, 𝐶 wins =
9
17
10. Mean = 5.25, Variance = 0.79
222
64
217
DAV PUBLIC SCHOOL THERMAL COLONY PANIPAT
11.
𝑥𝑖
3
4
5
6
𝑝𝑖
1
20
3
20
6
20
10
20
99144
100000
12. K =
13.
1
i)
8
1
8
18
19.
5
iii)
8
7
8
14. Mean = 2 , Variance = 1
133
1
1
5
6
15. P(A) = , P(B) =
17.
ii)
1
16.
1
18.
8
4
1
5
14
20. n = 7
29
21. n≥ 3
22.
2
5
3
9
23. Mean = , Variance =
1
3
𝑥𝑖
0
1
2
3
4
𝑝𝑖
625
1296
500
1296
150
1296
20
1296
1
1296
223
Design of Question Paper
224
SAMPLE QUESTION PAPER
MATHEMATICS (041)
CLASS XII – 2016-17
Time allowed : 3 hours
Maximum Marks : 100
General Instructions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
All questions are compulsory.
This question paper contains 29 questions.
Question 1- 4 in Section A are very short-answer type questions carrying 1 mark each.
Question 5-12 in Section B are short-answer type questions carrying 2 marks each.
Question 13-23 in Section C are long-answer-I type questions carrying 4 marks each.
Question 24-29 in Section D are long-answer-II type questions carrying 6 marks each.
Section-A
Questions 1 to 4 carry 1 mark each.
1.
State the reason why the Relation R 
 a, b  : a  b  on
2
the set R of real numbers is not
reflexive.
2.
If A is a square matrix of order 3 and 2 A  k A , then find the value of k.
3.
If a and b are two nonzero vectors such that a  b  a  b , then find the angle between a and b .
4.
If  is a binary operation on the set R of real numbers defined by a  b  a  b  2, then find the
identity element for the binary operation  .
Section-B
Questions 5 to 12 carry 2 marks each.
1
5.
Simplify cot 1
6.
Prove that the diagonal elements of a skew symmetric matrix are all zeros.
7.
If y  tan 1
8.
If x changes from 4 to 4.01, then find the approximate change in log e x.
9.
 1 x  x
Find  
e dx.
2 
 1 x 
x2 1
for x  1.
dy
2
3
5x
1
1


.
,
x
, then prove that
2
2
dx 1  4 x 1  9 x 2
1  6x
6
6
2
10. Obtain the differential equation of the family of circles passing through the points
 a,0 .
 a,0 and
11. If a  b  60, a  b  40 and a  22, then find b .
12. If P ( A) 
2
1
1
, P ( B )  , P ( A  B )  , then find P( A / B ).
5
3
5
Section-C
Questions 13 to 23 carry 4 marks each.
 1 2 
, then using A1 , solve the following system of equations: x  2 y  1, 2 x  y  2.

2 1 
13. If A  
1

2
x

1,
x


2 at x  1 .
14. Discuss the differentiability of the function f ( x)  
2
3  6 x, x  1

2
OR
For what value of k is the following function continuous at x  

6
?
 3 sin x  cos x

,x  


6

x
f ( x)  
6



k, x  
6

15.


If x  a sin pt , y  b cos pt , then show that a 2  x 2 y
d2y 2
 b  0.
dx 2
16. Find the equation of the normal to the curve 2 y  x 2 , which passes through the point (2, 1).
OR
 


Separate the interval 0,  into subintervals in which the function f ( x)  sin 4 x  cos 4 x is
2
strictly increasing or strictly decreasing.
17. A magazine seller has 500 subscribers and collects annual subscription charges of Rs.300 per
subscriber. She proposes to increase the annual subscription charges and it is believed that for
every increase of Re 1, one subscriber will discontinue. What increase will bring maximum
income to her? Make appropriate assumptions in order to apply derivatives to reach the
solution. Write one important role of magazines in our lives.
18. Find
  cos
2
sin x
dx.
x  1 cos2 x  4 
19. Find the general solution of the differential equation 1  tan y  dx  dy   2xdy  0.
OR



x



x

Solve the following differential equation:  1  e y  dx  e y  1 
20. Prove that a {(b  c )  (a  2b  3c )}   a

x
 dy  0.
y
c  .
b
21. Find the values of a so that the following lines are skew:
x 1 y  2 z  a x  4 y 1


,

 z.
2
3
4
5
2
22. A bag contains 4 green and 6 white balls. Two balls are drawn one by one without replacement.
If the second ball drawn is white, what is the probability that the first ball drawn is also white?
23. Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find
the probability distribution of the number of diamond cards drawn. Also, find the mean and the
variance of the distribution.
Section-D
Questions 24 to 29 carry 6 marks each.
24. Let f : 0,    R be a function defined by f ( x)  9 x 2  6 x  5. Prove that f is not invertible.
Modify, only the codomain of f to make f invertible and then find its inverse.
OR
Let  be a binary operation defined on Q  Q by  a, b    c, d    ac, b  ad  , where Q is the set
of rational numbers. Determine, whether  is commutative and associative. Find the identity
element for  and the invertible elements of Q  Q .
 a  b
c
25.
Using properties of determinants, prove that
a
b
OR
2
c
b  c 
a
b
c
2
 2a  b  c .
3
a
c  a
b
2
p
q
If p  0, q  0 and
q
r
p  q q  r
p  q
q  r  0, then, using properties of determinants, prove
0
that at least one of the following statements is true: (a) p, q, r are in G. P., (b)  is a root of the
2
equation px  2qx  r  0.
26.
Using integration, find the area of the region bounded by the curves y  5  x2 and y  x  1 .

2
27.
Evaluate the following:
x sin x cos x
dx.
4
x  cos 4 x
 sin
0
OR
4
Evaluate
 (x  e
2x
)dx as the limit of a sum.
0
28.
Find the equation of the plane through the point (4, -3, 2) and perpendicular to the line of
intersection of the planes x – y + 2z – 3 = 0 and 2x – y -3z = 0. Find the point of intersection of
the line r  iˆ  2 ˆj  kˆ   (iˆ  3 ˆj  9kˆ) and the plane obtained above.
29.
In a mid-day meal programme, an NGO wants to provide vitamin rich diet to the students of an
MCD school .The dietician of the NGO wishes to mix two types of food in such a way that
vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C.
Food 1 contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C. Food 2 contains 1
unit per Kg of vitamin A and 2 units per kg of vitamin C. It costs Rs 50 per kg to purchase Food
1 and Rs 70 per kg to purchase Food 2. Formulate the problem as LPP and solve it graphically
for the minimum cost of such a mixture?
--0--0--0--
MATHEMATICS (041)
CLASS XII – 2016-17
Marking Scheme
Section A
3
1.
1 1
1 1
     ,   R. Hence, R is not reflexive.
2 2
2 2
[1]
2.
k  23  8
[1]
3.
sin   cos    45
[1]
4.
e  R is the identity element for  if a  e  e  a  aa  R  e  2
[1]
Section B
5.
Let sec1 x   . Then x  sec  and for x < -1,

2
 
Given expression = cot 1 ( cot  )
= cot 1 (cot(   ))    sec1 x as 0     
6.
[1/2]
[1/2]

2
[1]
Let A be a skew-symmetric matrix. Then by definition A   A
[1/2]
 the (i, j)th element of A  the (i, j)th element of ( A)
[1/2]
 the ( j, i)th element of A  the ( i, j) th element of A
[1/2]
For the diagonal elements i  j  the (i, i)th element of A  the (i, i)th element of A
 the (i, i)th element of A  0 Hence, the diagonal elements are all zero.
7.
y  tan 1
3x  2 x
 tan 1 3 x  tan 1 2 x
1  3x2 x
dy
3
2


2
dx 1  9 x 1  4 x 2
8.
[1/2]
[1]
[1]
Let y  log e x, x  4,  x  .01
[1/2]
dy 1

dx x
[1/2]
1
 dy 
dy      x 
 .0025
400
 dx  x  4

1
2x  x

  1  x2 1  x2 2  e dx
 

[1]

9. Given integral  

1
d
1
2 x
e x  c as
(
)
2
2
2
1 x
dx 1  x
1  x2 
[1]
[1]
10.
x2   y  b   a 2  b2or , x 2  y 2  2by  a 2 .........(1)
2
dy
dy
2 x  2 y  2b  0  2b 
dx
dx
2x  2 y
dy
dx
Substituting in (1), ( x 2  y 2  a 2 )
11.
2
2

a b  a b  2 a  b
2
2
[1]
dy
dx .........(2)
[1/2]
dy
 2 xy  0
dx
[1/2]

[1]
2
12.
 b  2116
[1/2]
 b  46
[1/2]
P( A / B ) 
P( A  B )
P( B )
[1/2]

1  P( A  B)
1  P( B)
[1/2]

1   P( A)  P( B)  P( A  B) 
1  P( B)
[1/2]

7
10
[1/2]
Section C
13.
A 5
[1/2]
 1 2
adjA  

 2 1 
[1+1/2]
A1 
adjA 1  1 2
 
A
5 2 1 
[1/2]
 x
 1
Given system of equations is AX  B, where X    , B   
 y
2
[1/2]
3
5
X  A1B   
4
 5 
[1/2]
3
4
x ,y
5
5
[1/2]
14.
1
Lf ( )  lim
2 h 0
1
1
1
f (  h)  f ( )
2(  h)  1  0
2
2  lim 2
2
h  0
h
h
1
Rf ( )  lim
2 h 0
1
1
1
f (  h)  f ( )
3  6(  h)  0
2
2  lim
2
 6
h 0
h
h
1
1
1
Lf ( )  Rf ( ) , f is not differentiable at x 
2
2
2
[1+1/2]
[1+1/2]
[1]
OR

2sin( x  )
6
lim f ( x)  lim



x 
x 
x
6
6
6
[2]
2
[1]

f ( )  k
6
[1/2]
For the continuity of f ( x) at x  


, f ( )  lim f ( x)  k  2
6
6 x
[1/2]
6
15.
dx
dy
 ap cos pt ,
 bp sin pt
dt
dt
[1]
dy bp sin pt
b

  tan pt
dx ap cos pt
a
[1/2]
d 2 y bp sec2 pt dt

dx 2
a
dx
[1]

16.
bp sec2 pt
1
b 2
d2y
2
2

 2

(
a

x
)
y
 b2  0
a
pa cos pt (a  x 2 ) y
dx 2
Let the normal be at ( x1 , y1 ) to the curve 2 y  x 2 .
( x1 , y1 ) 
[1+1/2]
dy
 x The slope of the normal at
dx
1
1

x1
 dy 
 
 dx ( x1 , y1 )
[1]
The equation of the normal is y  y1 
The point (2, 1) satisfies it 1  y1 
1
( x  x1 )
x1
[1/2]
1
(2  x1 )  x1 y1  2.....(1)
x1
2
Also, 2 y1  x1 ......(2)
[1/2]
[1/2]
2
1
Solving (1) and (2), we get x1  2 3 , y1  2 3
[1/2]
2
3
The required equation of the normal is x  2 y  2  2
2
3
[1]
OR
f ( x)  4sin 3 x cos x  4cos3 x sin x   sin 4 x
f ( x)  0  x 

[1]
4
In the interval

Sign of f’(x)
-ve as
(0, )
4
0  4x  
 
+ve as
( , )
4 2
17.
[1]
  4 x  2
Conclusion
Marks

[0, ]
f is strictly decreasing in
4
 
[ , ]
f is strictly increasing in 4 2
[1]
[1]
Increase in subscription charges = Rs x, Decrease in the number of subscriber = x. Obviously, x
is a whole number.
[1/2]
Income is given by y = (500 – x)(300 + x). Let us assume for the time being
0  x  500, x  R
[1]
dy
dy
 200  2 x,
 0  x  100
dx
dx
[1/2]
 d2y 
d2y
 2,  2 
 2  0
dx 2
 dx  x 100
[1/2]
y is maximum when x = 100, which is a whole number. Therefore, she must increase the
subscription charges by Rs 100 to have maximum income.
[1/2]
Magazines contribute, a great deal, to the development of our knowledge. Through valuable and
subtle critical and commentary articles on culture, social civilization, new life style we learn
a lot of interesting things. Through reading magazines, our mind and point of view are
consolidated and enriched.
18.
cos x  t   sin xdx  dt The given integral  
Put t 2  y,
[1]
dt
(t  1)(t 2  4)
2
1
A
B


( y  1)( y  4) y  1 y  4
1  ( y  4) A  B( y  1), 0  A  B, 1  4 A  B  A 
The given integral  
[1/2]
1
1
,B 
3
3
1
dt
1
dt
1
1
t
  2
  tan 1 t  tan 1  c
2

3 (t  1) 3 (t  4)
3
6
2
1
1
cos x
  tan 1 (cos x)  tan 1
c
3
6
2
[1]
[1]
[1]
[1/2]
19.
1  tan y  dx  (1  tan y  2 x)dy 
2 dy
I .F .  e
 1 tan y
e

dx
2

x 1
dy 1  tan y
(  sin y  cos y )  (cos y sin y )
dy
cos y sin y
[1]
 eloge (cos y sin y ) y  (cos y  sin y)e y
[2]
x(cos y  sin y)e y   (cos y  sin y)e y dy  x(cos y  sin y)e y  e y sin y  c
[1]
OR
x  y
x
x
  1 e
x  y
dx  y 
x
y

 f ( ), hence homogeneous [1/2]
We have (1  e ) dx    1 e dy 
x
dy
y
y 
(1  e y )
x
x  vy,
dx
dv
v y
dy
dy
[1/2]
1  ev
dy
 ev  v dv   y
[1]
loge ev  v   loge y  loge c
[1]
 loge (ev  v) y  log e c
[1/2]
x
x
 (ev  v) y  c  A  (e y  ) y  A , the general solution
y
20.
LHS  a  (b  a  2b  b  3b  c  c  a  2c  b  3c  c )
[1]
 a  (b  a )  3a  (b  c )  a  (c  a )  2a  (c  b ) as b  b  c  c  0
[1]
 3  a b
[1]
 3  a b
21.
[1/2]
c   2  a c b 
c   2  a b
c    a b
c 
[1]
As 2 : 3: 4  5: 2 :1, the lines are not parallel
[1/2]
Any point on the first line is (2  1,3  2, 4  a)
[1]
Any point on the second line is (5  4, 2  1,  )
[1]
Lines will be skew, if, apart from being non parallel, they do not intersect. There must not exist
a pair of values of  ,  , which satisfy the three equations simultaneously:
2  1  5  4,3  2  2  1, 4  a   Solving the first two equations, we get   1,   1
[1]
These values will not satisfy the third equation if a  3
22.
[1/2]
Let E1  First ball drawn is white, E2  First ball drawn is green, A  Second ball drawn is
white
[1]
The required probability, by Bayes’ Theorem, =
P( E1 / A) 
23.
P( E1 )  P( A / E1 )
P( E1 )  P( A / E1 )  P( E2 )  P( A / E2 )
[1]
6 5

5
10 9


6 5 4 6 9
  
10 9 10 9
[2]
Let X denote the random variable. X=0, 1, 2 n = 2, p = ¼, q = ¾
[1/2]
xi
0
pi
1
2
2
9
3
C 0  4   16
2
2
13 6
C 1 4  4   16
Total
[1+1/2]
2
2
1
1
C 2  4   16
xipi
0
6/16
2/16
1/2
xi 2 pi
0
6/16
4/16
5/8
Mean =  xi pi 
1
2
[1/2]
[1/2]
Variance =  xi 2 pi    xi pi 

Marks
2
[1/2]
5 1 3
 
8 4 8
[1/2]
Section D
24.
x 0,  , y  9x2  6x  5  (3x  1)2  6  5..........(1) Range f =  5,   codomain f, hence, f
is not onto and hence, not invertible
[2]
Let us take the modified codomain f = 5, 
[1/2]
Let us now check whether f is one-one. Let x1 , x2 0,   such that
f ( x1 )  f ( x2 )   3x1  1  6   3x2  1  6  3x1  1  3x2  1  x1  x2 Hence, f is one-one.
2
2
Since, with the modified codomain = the Range f, f is both one-one and onto, hence invertible.
[1+1/2]
From (1) above, for any y   5,   , (3x  1) 2  y  6  x 
f 1 :  5,    0,   , f 1 ( y) 
y  6 1
3
y  6 1
3
[1]
[1]
OR
Let  a, b  ,  c, d   Q  Q . Then b + ad may not be equal to d + cb. We find that
1, 2   2,3   2,5 ,  2,3  1, 2   2,7   2,5 Hence,  is not commutative.
[1]
Let
 a, b ,  c, d  , e, f  Q  Q,( a, b  c, d )  e, f   ace, b  ad  acf   a, b   (c, d )  e, f )
Hence,  is associative.
[1]
 x, y   Q  Q is the identity element for  if
 x, y    a, b   a, b   x, y    a, b   a, b  Q  Q i.e.,
 xa, y  xb   ax, b  ay    a, b i.e., xa  a, y  xb  b  ay  b , (x, y) = (1, 0) satisfies these
equations. Hence, (1, 0) is the identity element for 
[2]
 c, d   Q  Q is the inverse of  a, b  Q  Q if
 c, d    a, b    a, b    c, d   1, 0  , i.e., (ac, b  ad )  (ca, d  cb)  (1, 0)  c 
1
b
,d 
. The
a
a
1 b
)
a a
inverse of  a, b   Q  Q, a  0 is ( ,
 a  b
25.
LHS =



1
abc
1
abc
2
c2
b2
b2
 a  b  c  a  b  c 

2
[1/2]
a2
c  a
2
0
0
 b  c  a  (b  c  a)
c2
a2
(b  c  a )(b  c  a )
(b  c  a )(b  c  a )
c  a 
a  b  c
2
0
(b  c  a )
0
abc
a  b  c
a  b  c
(b  c  a ) (b  c  a )
2
a  b  c
0
(2a )
abc
a  b  c
2
 ac  bc  c 
0
(2ac)
abcca
a  b  c
abcca
2
 ac  bc 
a2
0
(C1  C1  C3 , C2  C2  C3 )
c2
a2
c  a 
0
[1]
2
[1/2]
2
0
c2
(b  c  a ) a 2 ( R3  R3  ( R1  R2 ))
(2c)
2ca
2

c2
b  c 
a2
[2]
[1]
c2
(ba  ca  a 2 ) a 2
(2ca)
2ca
c2
c2
(ba  ca ) a 2 (C1  C1  C3 , C2  C2  C3 )
0
2ca
[1/2]
[1]
a  b  c

2
a  b  c

2
2c 2 a 2
abcca
b
2
a  b
c
c
(b  c) a (C1  C1  C3 , C2  C2  C3 )
0
1
a
0
 ab  ac  b
2
[1]
 bc  ac   2(a  b  c )3
[1/2]
OR
pq
q2
1
pq
pr
Given equation 
pq
p  q q  r
0
q 2  pr
1

pq
pr
pq
p  q q  r
q 2  pr

pq
q 2  pr

q

[1]
q 2  pr
pq  pr  0 ( R1  R1  R2 )
0
0
1
pq
pr
p  q q  r
q 2  pr

p
pq
pq  q 2
pq  pr  0
0
[1]
1
pq  pr  0
0
[1]
0
1
1
q
r
q  r  0
p  q q  r
0
[1]
0
0
1
q
q q  r  0(C2  C2  C3 )
p  q q  r
0
[1]
q 2  pr 2
(q   rq  pq 2  q 2 )  0  (q 2  pr )(2q  r  p 2 )  0  q 2  pr  0 (i.e., p, q, r
q
are in GP) or 2q  r  p 2 =0(i.e.,  is a root of the equation 2qx  r  px 2 =0
[1]
26.
Figure [1 Marks]
2
Solving y  5  x , y  x  1 we get ( x  1)2  5  x 2  x 2  x  2  0  x  2, 1
[1]
The required area = the shaded area =
1
2
1
1
2
2
 ( 5  x  (1  x))dx   ( 5  x  ( x  1))dx
1
2
[2]
2

 x2

1
x 
x2 
  ( 5  x dx   1  x dx   (( x  1))dx   x 5  x 2  5sin 1

x


 x




2
2  1  2
5  1 
1
1
1
1
2
1
2
2
[1+ ½ ]
2
2 
 1 5
    (sin 1
 sin 1
)  sq units
5
5 
 2 2
27.


2
2

[1/2]




(  x)sin(  x) cos(  x)
 x) cos x sin x
2 (
x sin x cos x
2
2
2
2
dx
I  4
dx  
dx , I  
4
4
sin x  cos 4 x
cos
x

sin
x
4 
4 
0
0
0
sin (  x)  cos (  x)
2
2
[1]


( ) cos x sin x
2I   2 4
dx
cos x  sin 4 x
0
2

[1/2],



2

cos x sin x
cos x sin x

tan x sec x
cos ec 2 x cot x
2 I  ( )[ 
dx

dx
]

(
)[
dx

4
 4
 cot 4 x  1 dx] [2]
2 0 cos 4 x  sin 4 x
2 0 1  tan 4 x
 cos x  sin x

4
2
4
2
4

1
4
0
1
1
2I  ( )[ 
dt  
dp] substituting
2
4 0 1 t
1  p2
1
tan 2 x  t , cot 2 x  p  2 tan x sec2 xdx  dt , 2cot x cos ec 2 xdx  dp


2
2 I  ( )[tan 1 t ]10  ( )[tan 1 p]10 
4
4
8
I
[1]
[1]
2
[1/2]
16
OR
4
n
f ( x)  x  e ,  f ( x)dx  lim h f (rh), nh  4
2x
n , h0
0
n
n
n
r 1
1
1
f (rh)  rh  e 2 rh ,  f (rh)  h r   e 2 rh
h
n(n  1) 2 h e2 nh  1
e
2
e2 h  1
4

f ( x)dx  lim [nh
n  , h 0
0
 lim[4
h 0
[1]
r 1
nh  h 2 h e8  1 1
 e 2h
 ]
e 1 2
2
2h
4  h 2 h e8  1 1
e8  1
 e 2h
 ]  8
e 1 2
2
2
2h
[1]
[2]
[1]
[1]
28.
iˆ ˆj kˆ
n  b1  b2  1 1 2  5iˆ  7 ˆj  kˆ
2 1 3
[2]
The equation of the plane is r  n  (5iˆ  7 ˆj  kˆ)  (4iˆ  3 ˆj  2kˆ), i.e., r  (5iˆ  7 ˆj  kˆ)  1
The position vector of any point on the given line is (1   )iˆ  (2  3 ) ˆj  (1  9 ) kˆ
We have (1   )5  (2  3 )7  (1  9)1  1
29.
[1]
[1]
[1]
  1
[1/2]
The position vector of the required point is  ˆj  8kˆ
[1/2]
Let x kg of Food 1 be mixed with y kg of Food 2. Then to minimize the cost, C = 50x + 70y
subject to the following constraints: 2 x  y  8, x  2 y  10, x  0, y  0
[2]
Graph [2]
At
C
(0, 8)
Rs 560
(2,4)
Rs 380
(10.0)
Rs 500
Marks
[1]
In the half plane 50x + 70y < 380, there is no point common with the feasible region. Hence, the
minimum cost is Rs 380.
[1]
--0-0-0--