RELATIONS AND FUNCTIONS
One mark questions
Q.1 If f:R R is defined by f(x)= 3x+2, find f(f(x)).
Q.2 If the binary operation * define on A , is define as a*b = 2a + b – ab for all a,b
3*4.
Q.3 If f(x) = x2 +1, g(x) =
.Find the value of
find gof(5).
Q.4 State the reason for the relation R in the set {1,2,3} given by R = {(1,2),(2,1)} not to be transitive.
Q.5 Let * be a binary operation on N given by a*b =HCF (a,b) for all a,bεN. Write the value of 22*4.
Q.6 Let set A = {a,b,c} and R is the relation in A given by R = {(a,a),(a,b),a,c),(b,a),(c,c)}. Is R symmetric?
Give reasons.
Q.7 Let f:RR be defined by f(x)= x2. Is f one-one ?
Q.8 If a binary operation *‘ on the set Z , is defined by a * b = a + b2 Then find the value of 2 * 4
Four marks questions:
Q.9 Check whether the relation R in R defined by R = { (a,b) : a
b3 } is an equivalence relation.
Q.10) Show that the relation R defined in the set A of all triangles as R = {(T1, 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?
Q.11 If f(x) =
4x  3
2
2
, x  , show that f o f(x) = x, for all x  . What is the inverse of f ?
6x  4
3
3
Q.12 Check whether the binary operation * define on R by a*b= ab + 1 is (i) commutative (ii)
associative.
Q.13 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 4 }. Prove that R is an equivalence relation.
Q.14 Consider f:R+  [-5,∞), given by f(x)= 9x2 +6x -5 .Show that f is invetible and find f -1 .
Q.15 Let f:N N be defined by f(n) ={
.
Find whether the function f is bijective.
Q.16 Let N be the set of all natural numbers & R be the relation on N × N defined by R =
{ (a , b) R (c , d) iff a + d = b + c}. Show that R is an equivalence relation.
One Mark Questions
1.
2.
3.
4.
5.
6.
7.
8.
9.
If f(x) = 27 and g(x) = find g.f(x)
( )
If f(x) = x+7 and g(x) = x-7 , x
Let * be a binary operation on N given by a*b = L.C.M (a,b) for all a, b
The binary operation * : R*R
is defined as a*b = 2a+ b Find (2*3)*4
Find f.g if f(x) = and g(x)= logx
Show that the binary operation * : R*R
is defined as a*b = a+ 2b is not commutative.
Give an example of a relation which is symmetric but neither reflexive nor trSolutionitive.
Prove that thee function f:R
given by f(x)= 2x is one-one.
Consider f:R
given by f(x)= 4x+3 and f is invertible . Write the inverse of f .
10. Let * be the binary operation on N given by a*b =L.C.M. of a and b .Find 20 * 16 .
Four Marks Questions
1. Consider
( )
(
) given by ( )
. Prove that f is
invertible with
√
2. A binary operation * is defined on the set X=R-{-1} BY x*y=x+y+xy,
. Check whether *
is commutative and associative. Find its identity element and also find the inverse of each
element of X.
3. Show that the relation
4.
*(
) |
in the set
*
+ Let be defined on
i) is a binary operation on
6. Let f:N
+ given by
+is an equivalence relation. Write all the equivalence classes
|
of .
5. Let
*
as
Prove that
ii) the given operation is commutative as well as associative.
be defined by f(x) ={
, for all n
N. Examine whether the function is
onto, one to one or bijective?
7. f : N→R be a function defined as ( )
is the range of f. Hence find the inverse of f.
.Show that f:N→S is invertible .where S
Six Mark Questions
1. Consider f : R→[-5,∞) given by f(x) = 9x2+6x-5. Show that f is Invertible. Find the inverse of f.
2. Define equivalence relation. Let T be the set of all triangle in a plane with R a relation in T given
by R = {(T1,T2):T1 is similar to T2}. Show that R is an equivalence relation.
3. Let
and let * be binary operation on A defined by (a, b)*(c, d) = (a+c, b+d). Show
that
(i)
(A,*) is commutative.
(ii)
(A, *) is associative
(iii) (A, *) has an identity and find it.
4. Show that operation * on Q – {1},defined by a * b = a + b – ab for all a, b  Q – {1} satisfies
(i)
the closure property,
(ii)
the associative property
(iii) the commutative property
(iv)
What is the identity element?
(v)
For each a  Q – {1}, find the inverse of a.
5. Let f and g be real valued functions, such that (fog)(x) = cosx3 and (gof)(x) = cos3x, find the
functions f and g.
6. Consider f: {1,2,3} → {a, b ,c} and g : {a ,b ,c} → {apple ,ball, cat} defined by f(1) = a, f(2) = b ,
f(3) = c , g(a) = apple , g(b) = ball , g(c) = cat. Show that f, g and gof are invertible. Find out f -1 ,
g-1 and (gof)-1 and show that (gof)-1 = f-1o g-1.
7. Define a relation with example. Show that the relation R in the set Z of integers given by
R = {(a, b): 2 divide a – b} is an equivalence relation.
8. Define binary operation with example. Determine which of the following binary operations on
the set N are associative and which are commutative.
(i)
(ii)
VALUE BASED
1. Show that the relation R defined in the set A of all books in the library of a college given by R =
{(a,b) ; a and b have the same number of pages} is an equivalent relation . Write how the habit
of book reading is useful for us?
2. The table gives a relation between a person’s age and the person’s maximum recommended
heart rate.
a) What is the domain?
b) What is the range?
c) The range element 200 corresponds to what element in the domain?
d) Complete the ordered pair: (50,_)
e) Complete the ordered pair: (_, 190)
f)
Age (years) x
20
30
40
50
60
Maximum Recommended Heart Rate (Beats per
minute) y
200
190
180
170
160
What are the ways to remain healthy and lead a longer life? How will you inculcate such habits
in your life?
INVERSE TRIGONOMETRIC FUNCTIONS
One mark questions:
Q.1) Find the principal value of cosec-1( √ )
Q.2) Find the principal value of cot-1(√ )
Q.3) Find the principal value of
Q.4) Find the value of sin0
(
)
( )1.
Q.5) If sin(sin-1 + cos-1x ) = 1. Find the value of x.
Q.6) Find the Principal value of cos -1 ( cos
13
)
6
 1 
Q.7) Find the principal value of cos -1  
 2 
Q.8) Find the principal value of cosec -1 (  2 )
1
1
Q.9) Find the value of cos -1   + 2 sin -1  
2
2
Q.10) Write the principal value of
.
Four marks Question
Q.1) Show that
=
Q.2) Show that
=
Q.3) Prove that
Q.4) 2
√ =
(
)=
,x ,
(
)
Q.5) Prove that cot.
/=7
Q.6) Prove that
+
Q.7) Solve tan-1(
-
=
) = tan-1x. x > 0
 1  sin x  1  sin x  x
Q.8) Cot -1 
  , x ε (0, π/4)
 1  sin x  1  sin x  2
Q.9) Prove that : tan-10
√
√
√
√
1=
cos-1x , -1/√
–
HOTS
Q.1) Prove the following: tan 0
Q.2) Prove the following: cos,
( )1+tan 0
*
( )1=
)+- = √
(
.
.
Questions for practice
1. Write
/ in simplest form.
.√
2. Solve for x :
.
3. Solve the equation
tan-1 
4. Solve for
(
/
.
/
. /.
 2x
2
1 x
1 x2

 + cot-1 

 2x
)
.
 
 = , x >0
 3
/
5. Solve for x:
6. Prove that :
. /
. /
. /
7. Prove that:
( )
. /
. /
8. Prove that:
. /
.
HOTS QUESTIONS
1. If
2.
+
Prove the following:-
= θ, then prove that 9 -12xycos =36
√
+
= .
.
3. Write in the simplest form:4.
Solve for x :-
√
√
√
√
+
= 2
Matrices
One marks problem
Q.1) If A =[aij] is a 2x2 matrix such that aij = i2 – j2
Q.2) Let A =0
1 and C = 0
Q.3) If [
] [ ] = [ ] , find x,y,z.
Q.4) If AT = 0
Q.5) If A = 0
1 and B = 0
1 find 3A – C.
1, find ( A + 2B )T
1 , find A + AT
Q.6) If 0
Q.7) If A = 0
Q.8) If A = 0
1= 0
1 , find the value of y .
1, find k such that A2 = 8A + KI
1, find f(A), where f(x) = x2 -5x + 7
 x  2 y 3 y  0  3

Q.9) Find the value of x and y from the following 
2  8 2 
 4x
FOUR MARKS QUESTION
1 2 3
Q.1) if A   3  2 1; Then show that A3-23A-40I = 0


4 2 1

 0
Q.2) IF A  

 tan
2

 tan

2  and I is the identity matrix of order 2 then show that

0 

cos x  sin x 0
F ( x)   sin x cos x 0  show that F(x).F(y)=F(x+y)
 0
0
1 
Q.3) If
 6 2 2 
Q.4) Express the matrix A   2 3  1 as the sum of symmetric matrix and skew-symmetric


 2  1 3 
matrix.
 cos 
Q.5) A  
 sin 
sin  
 cos n sin n 
; then prove that An  

; n  N .
cos  
 sin n cos n 
1 2 
Q.6) By using elementary operations find the inverse of the matrix A  

2  1
Six marks questions
Q.1) By using elementary operations find the inverse of the matrix
0 1 2 
1 2 3


3 1 1
 1 3 2 
Q.2) By using elementary operations find the inverse of the matrix  3 0 1
 2 1 0 
Questions for practice
Q.1
If A = [
] then find the value of A2 – 3A + 2I
Q.2
Express the matrix A = [
] as the sum of a symmetric and a skew symmetric matrix.
Q.3
Find the matrix X such that [
]X =[
Q.4
Find the matrix X so that X0
1=0
Q.5
A= [
], then prove that An= [
]
1
], n
Determinants
ONE MARKS QUESTION
Q.1
If |
| =|
| , then write the value of x .
Q.2
Evaluate : |
Q.3
.If A = 0
Q.4
Evaluate the determinant |
|
1 ,then show that |
| = 4| |
|
Q.5
Solve for x : |
Q.6
Evaluate : |
| = 11
|
Evaluate : |
|
Evaluate :|
Q.7
|
Without expanding Evaluate |
|
SECTION B (4 marks)
Prove the followings by using properties of determinants:
Q.1) |
| = (a-b)(b-c)(c-a)
Q.2) |
Q.3)
Q.4)
Q.5
Q.6)
Q7)
Q.8)
|
x
x2
x3
y
y2
y3

2
 
z
z2
z3
= (5x+4) (4-x)2
= xyz(x – y)(y – z)(z – x)


2

2
    
a
ab
a  2b
a
a  b a  2b
= (    )(    )(   ) (      ) )
a  2b
ab
a.
= 9 (a + b) b2
abc
2a
2a
2b
bca
2b
2c
2c
cab
= (a + b + c) 3
a  b  2c
a
b
c
b  c  2a
b
= 2(a + b + c) 3
c
a
c  a  2b
1 a
1
1
 1 1 1
1 1  b 1  abc 1    
 a b c
1
1 1 c
Q.9) |
| = 1 +a2+b2+c2
Q.10) Using properties of determinants, solve the following for x:
ax ax ax
ax ax ax 0
ax ax ax
Q.11) 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 others (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 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 for honesty and supervision is twice the number of awardees for
helping others. Represent the above situation algebraically and convert in matrix form. Apart from
these values, namely, honesty, cooperation and supervision, suggest one more value which the
management
Questions for Practice
1. Given A = [
] and B = [
], find AB and use this result to solve the following
system of equations:
x – y = 3, 2x + 3y + 4z = 17 & y + 2z = 7.
2. If A = [
], find A– 1. Using A– 1, solve the following system of equations: 2x + y + 3z = 9, x
+ 3y – z = 2 & –2x + y + z = 7.
3. Using the properties of determinants prove that
a
b
c
a  b b  c c  a  a 3  b 3  c 3  3abc
bc ca ab


1 a 2  bc a 3
4. 1 b 2  ca b 3   (a  b)(b  c)(c  a)a 2  b 2  c 2 
1 c 2  ab c 3
a2
bc
ac  c 2
b2
ac  4a 2 b 2 c 2
5. a 2  ab
ab
b 2  bc
c2
6. |
| = 2|
7. |
| = (x-y) (y-z) (z-x) (xy +yz+zx)
Value based
|
8. A school wants to award its students for the values of Honesty, Regularity and Hard work with a
total cash award of Rs 6,000. Three times the award money for Hardwork added to that given for
honesty amounts to Rs 11,000. The award money given for Honesty and Hardwork 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, namely, Honesty, Regularity
and Hardwork, suggest one more value which the school must include for awards.
9. 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 ` x each, ` y each and ` z each for the three respective
values to 3, 2 and 1 students respectively with a total award money of ` 1,600. School B wants to
spend ` 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award
money to the three values as before). If the total amount of award for one prize on each value is `
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.
10. 10 students were selected from a school on the basis of values for giving awards and were divided
into three groups. The first group comprises hard workers, the second group has honest and law
abiding students and the third group contains vigilant and obedient students. Double the number of
students of the first group added to the number in the second group gives 13, while the combined
strength of first and second group is four times that of the third group. Convert this situation in
matrix form and solve. Apart from the values, hard work, honesty and respect for law, vigilance and
obedience, suggest one more value, which in your opinion, the school should consider for awards.
HOTS
x
1. If xyz, and y
z
x2 1  x3
y2 1 y3  0
z2 1 z3
, then prove that xyz  -1
x 1 x  2 x  a
2. If a,b,c are in AP then show that x  2 x  3 x  b  0
x3 x4 xc
3. If a, b, c all are positive and pth, q th, and r th terms of a G P respectively. then prove that
log a
log b
log c
p 1
q 10
r 1
4. Using the properties of Determinants, show that
(
|
)
(
| = 2abc(a+ b
)
(
+ c)3
)
5. Show that x = 2 is a root of equation |
| = 0 and solve it Completely.
1 1 2   2 0 1 
6. Use the product 0 2 3  9 2 3 To solve the following system of equations



 3 2 4   6 1 2 
x-y+2z =1,2y-3z=1,3x-2y+4z=2.
2 3 10
4 6 5
6 9 20
   4;    1and    2
x y z
x y z
x y z
7. Solve by matrix method.
CONTINUITY AND DIFFERNTIABILITY
4 marks Questions Continuity
Find the unknown constant if the functions are continuous
k x 2 ,
Q.1) f  x   
3,
x2
x2
if x  2
2 x  1,

if x  2
Q.2) f ( x)  a,
 x  1, if x  2

2 x  1, if x  2

if x  2
Q.3) f ( x)  k ,
3x  1, if x  2

3ax  b, if x  1

x 1
Q.4) f ( x)  11, if
5ax  2b, if x  1

5, x  2

Q.5) f  x   ax  b, 2  x  10
21, x  10

 kx  1, x  
Q.6) f ( x)  
cos x , if x  
Q.7)
Q.8)
 k ( x 2  2),
f ( x)  
 3x  1,
  ( x 2  2 x),
f ( x)  
 4 x  1,
if x  0
if x  0
if x  0
if x  0
ax  1,
Q.9) Find the relation between a and b so that f  x   
bx  3,
x3
is continues at x = 3.
x3
 x 2  ax  b, 0  x  2

Q.10) The function f (x) is defined as follows: f ( x)  
3 x  2, 2  x  4
 2ax  5b, 4  x  8

4 marks Questions Diffferentiation
Q.1) If y =xcos x + (cos x)sinx, find
Q.2) If y= x x  (cos x)sin x find
dy
.
dx
dy
dx
Solution
Q.3) If x=a sin pt and y=b cospt, find the value of
dy
Q.4) If x= a(cost+log tant/2) and y= asint find
.
dx
d2y

at t 
Q.5) If x = 4 sin3t & y = 4 cos3t, find
2
3
dx
Q.6) Differentiate
( x  3)( x 2  4)
with respect to
3x 2  4 x  5
 cos x  sin
Q.7) If y= (sin x) x  tan 1 
 cos x  sin
Q.8) If xy = ex-y , then prove that
Q.9) Find
at t=0?
x.
x
dy
 find
.
dx
x
dy
log x

dx (log ex) 2
dy
If xy + yx = ab
dx
Q.10) If ( cos x) y = ( cos y )x
find
dy
dx

Q.11) The value of c in Rolle ’s Theorem for the function f(x) x3-3x in the interval 0, 3

Q.12) It is given that the function f ( x)  x 3  bx 2  ax  5, x  1,3 Rolles theorem holds with c  2 
Find the values of a and b?
Q.13) Verify Mean Value theorem f(x) = sin x – sin 2x , x  0,  
PRACTICE QUESTIONS
 kx 2
if x  2
1. Find the value of k for which f(x)= 
is continuous at x=2
x

3
if
x

2

2. Find the relationship between a and b so that the function f defined by
 ax  1 if x  3
f(x) = 
is continuous at x = 3.
bx  3 if x  3
3. Find the value of k so that the function f is continuous at x = /2,
1
3
  cos x 

k
; x 


  2x 
2
f ( x)   
5; x  

2

m( x 2  x)
4. Find the value of m, for which the function f ( x)  
;x  0
cos x; x  0
is continuous at x =0?
5. Find the derivative of √
6.
Find the derivative of
√ .
(
).
√
4 Marks Questions
1. Find all the points of discontinuity of the functions defined by
 x  2, x  1

f ( x )   x  2,1  x  2
0, x  2

 sin x
,x  0

2. Find all the point of discontinuity of f ( x)   x
 x  1, x  0
3. Show that the function defined by f(x) = cos (x2) is a continuous function.
4. Discuss the continuity of Sinx + Cos x.
5.
Determine the values of a,b,c for which the function f(x) may be continuous at x =0

 sin(a  1) x  sin x

x

f ( x)  
c

2
 x  bx  x

b x3
6. Let f(x)=
7.
x0
x0
x0

3
 1  sin x
 3cos 2 x

f ( x)  
a
 b(1  sin x)

2
 (  2 x)

x
x
x

2

2
if f(x) is continuous at x= .
Find a and b.

2
2 x  1; x  0
The path of a moving bike is given by f ( x)  
2 x  1; x  0
Find the critical point (at which function is discontinuous)on the path? Whether the rider should pass
that point or not? Justify your answer?
8.
 cos 3 x  cos x

, x  0 Show that the functions is continuous at
If f ( x)  
x2
 4, x  0
x= 0.
Mention any
two factors that will affect the continuity of thought while writing the exam.
9.
 x 2  ax  b,0  x  2

If the function f(x ) is defined as f ( x)  3 x  2,2  x  4
2ax  5b,4  x  8

If f (x ) is continuous on 0, 8 , find the values of a and b ?
 x  3, x  3

Find all the points of continuity of f where f is defined as f ( x)   2 x,3  x  3
6 x  2, x  3

10.
11. Show that the function defined by g ( x)  x  x is discontinuous at all integral values, where x
denotes greatest integer less than or equal to x.
 1  sin x  1  sin x
12. Differentiate w. r. t. x , y = 
, 0<x<π/2.
1

sin
x

1

sin
x

13. Find the derivative of y= (sinx)cosx
14. If x  3 sin   sin 3 , y  3 cos   cos 3 , Find dy/dx at  =π/3 ?
15. Prove that the derivative of
16. Find
if ( x 2  y 2 ) 2  xy .
17. If √
√
(
2 cot x 2 is
 2 2x
sin x 2 sin 2 x 2
), then find
.
18. Find the derivative of with respect to if
19. If
(
)
(
then prove that
20. If y = (sin-1x)2 then prove that (1-x2) y2 –x y1 = 2.
21. If y
22. Differentiate
prove that (x2+1)
+xy+1=0
x
with respect to sinx .
sin x
23. If x 1  y  y 1  x  0 , -1<x<1, Prove that
dy
1

dx
1 x2
24. Verify Mean Value theorem f(x) = f ( x)  x( x  3)e  x / 2 ; x   3,0 ,
)