`KUKATPALL Y
CENTRE
Total No. of Questions - 24
Reg.
Total No. of Printed Pages - 16
No.
Part - III
MATHEMATICS, Paper-I (A)
(English Version)
Time : 3 Hours]
[Max. Marks : 75
SECTION - A
I.
1.
10  2 = 20 M
Very Short Answer Type questions:
(A) Find the domains of the following real valued functions:
a) f x 
1
b) f x  2  x  1  x
6x  x 2  5
c) f x  x 2  1 
1
d) f x 
x 2  3x  2
1
e) f x 
1
x x
f) f x  log x   x 
x2  x  2
 
 3  x 
g) f x  log 

10  x 
h) f x  x  2 
1
log
10
1  x 
3 x  3x
x
1
j) f x  
log 2  x 
(B) Find the ranges of the following real valued functions:
i) f x 
a) log 4  x 2

b)  x   x
c)

sin   x 
2
1  x 
d)
x2  4
x2
e)
9  x2
   
C) If A  0, , , ,
and f : A  B is a surjection defined by f x  cos x then find B .
6 4 3 2
If A  2, 1,0,1, 2 and f : A  B is a surjection defined by f x  x 2  x  1 , then find B .
1
2
D) If f : R \ 0  R is defined by f x  x  then prove that  f x  f x 2  f 1 .
x
1
1
If f : R  0  R is defined by f x  x 3  3 , then show that f x  f    0
x
x
 
If f : R  R is defined by f x 
1  x2
1  x2
, then show that f tan    cos 2 .
If f : R  1  R is defined by f x  log
 2 x 
1 x
  2 f x .
, then show that f 
 1  x 2 
1 x
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2.
A) If f : R  R and g : R  R are defined by f x  2 x 2  3 and g x  3x  2 , then find
i)  fogx , ii)  gof x , iii) fof o , iv) go  fof 3 .
If f : R  R , g : R  R are defined by f x  3x  1, g x  x 2  1 , then find


i) fof x 2  1
ii) fog 2
iii) gof 2 a  3
If f : 4, 5 ,5,6 ,6, 4 and g : 4, 4 ,6,5 ,8,5 then find
a) f  4
b) fg
c)
d) f 2
f
x1
x  1 then find  fofof x and  fofofof x .
x 1
B) Find the inverse of the following functions:
a) If a , b  R , f : R  R defined by f x  ax  b  a  0
If f x 
b) f : R  0,  defined by f x  5x
C) If f x 
x1
, x  1 , show that fof 1 x  x
x1
c) f : 0,   R defined by f x  log 2 x .
Determine whether the following functions are even or odd.
 e x  1 




a) f x  x 
b) f x  log x  x 2  1 


 e x  1 
Prove that the real valued function f x 
x
x
e 1

x
 1 is an even function on R \ 0 .
2
3x  3x
, then show that f x  y  f  x y  2 f x f  y .
2
4x
If the function f : R  R defined by f x 
, then show that f 1  x  1  f x .
4x  2
If the function f : R  R defined by f x 
3.
  3 2y  8  5
2 
a) If  zx 
  2 a  4 then find the values of x, y, z and a.
2
6

 

1
3
 2
0
b) Find the trace of  2 1
5
5 
1 




c) If A   13 24 , B  73 82 and 2X  A  B then find X.




 5
d) Find the products of i) 1 4 2   1
 3
 

 1
ii)  26 12 34  2

  1
 


e) If A  0i 0i  , find A 2 .




f) If A  21 k4  and A 2  O , then find the value of k.


g) Define Trace of matrix and give an example.
h) Define symmetric and skew-symetric matrix and give an example.
i) A certain bookshop has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics
books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 respectively. Find the total amount the
bookshop will receive by selling all the books, using matrix algebra.
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 2  4
 then find A  A' and AA' .
j) If A  
5 3 


k) Construct 3 2 matrix whose elements are defined by aij 
4.
1
i 3j
2
3 0 0
a) If A  0 3 0 , then find A 4 .
0 0 3

 sin  
b) If A  cos
 , show that AA’ = A’A=I.
 sin  cos  
1
 2 1 0 
5
3
4
0  and B   0 2 5 then find 3A-4B’.
 3 1 5
 1 2 0
c)If A   2
7
d) If A  1
 5
2 1
2 
2  and B   4
2  then find AB’ and BA’.
1 0 
3 
1
0 0
3 4 and det A = 45 find x.
 5 6 x
e) If A   2
f) Define singular matrix and give an example.
g) Define rank of a matrix.
 1 4 1
0 
 0 1 2 
h) Find the Rank of the Matrix  2 3
5.
A) ABCDE is a pentagon. If the sum of the vectors AB, AE, BC, DC, ED and AC is  AC , then find
the value of  .
B) If the position vectors of the point A, B and C are 2i  j  k , 4i  2 j  2 k and 6i  3 j  13k
respectively and AB  λAC , then find the value of  .
C) If OA  i  j  k , AB  3i  2 j  k , BC  i  2 j  2 k and CD  2i  j  3k , then find the vector
OD.
D) a  2 i  5 j  k and b  4i  mj  nk are collinear vectors, then find m and n .
E) Let a  2 i  4 j  5 k , b  i  j  k and c  j  2 k . Find the unit vector in the opposite direction
of a  b  c .
F) If  ,  and  be the angles made by the vectors 3i  6 j  2 k with the positive directions of the
coordinates axes, then find cos , cos and cos  .
G) Find the angles made by the straight line passing through the points 1, 3, 2 and 3, 5,1
with the coordinate axes.
6.
A) Find the vector equation of the line passing through the point 2i  3 j  k and parallel to the
vector 4i  2 j  3k .
B) OABC is a parallelogram. If OA= a and OC  c , find the vector equation of the side BC.
C) If a , b , c are the position vectors of the vertices A, B and C respectively of ABC , then find the
vector equation of the median through the vertex A.
D) Find the vector equation of the line joining the points 2i  j  3k and 4i  3 j  k .
E) Find the vector equation of plane passing through the points i  2 j  5 k , 5 j  k , and
3i  5 j .
F) Find the vector equation of the plane passing through the points 0,0,0 , 0,5,0 , and 2,0,1 .
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7.
A) If a  i  2 j  3 k and b  3i  j  2 k , then show that a  b and a  b are perpendicular to each
other.
B) Let a and b be non-zero, non-collinear vectors. If a  b  a  b , then find the angle between
a and b .
C) If a  11, b  23 and a  b  30, then find the angle between the vectors a , b and a  b .
D) If a  i  j  k and b  2i  3 j  k , then find the projection vector of b on a and its magnitude.
E) If P , Q , R and S are points whose position vectors are i  k , i  2 j , 2i  3 k and 3i  2 j  k
respectively, then find the component of RS and PQ .
F) Find the angle between the planes r.2 i  j  2 k  and r.3i  6 j  k   4.
1
e1  e2  sin  , then find  .
2
H) If a  2 i  2 j  3 k , b  3i  j  2 k , then find the angle between 2a  b and a  2b .
G) Let e1 and e2 be unit vectors containing angle  . If
I) Find the area of the parallelogram for which the vectors a  2i  3 j and b  3i  k are adjacent
sides.
J) If a  i  2 j  3 k and b  3i  5 j  k are two sides of a triangle, then find its area.
K) Let a  2i  j  k and b  3i  4 j  k . If  is the angle between a and b , then find sin  .
L) Find the area of the triangle whose vertices are A1,2, 3 , B2,3,1 and C 3,1,2 .
M) Show that i  a  i  j a  j  k k  a  2 a for any vector a.
N) Prove that for any three vectors a , b , c  b  c c  a a  b   2  abc  .
2
O) For any three vectors, a , b , c , prove that  b c c  a a  b    abc  .
8.
A) Prove that cot
B)

2
3
7
.cot
.cot ....cot
1
16
16
16
16
If 3sin   4 cos   5, then find the value of 4 sin   3 cos  .
C) If cos   sin   2 cos  , prove that cos   sin   2 sin 
2
D) Prove that tan   cot    sec 2   cos ec 2  sec2 .cos ec 2  .
E)
If tan 20 o   , then show that
F)
Prove that
tan 160 o  tan 110 o
1  tan 160 o.tan 110 o

1 2
.
2
tan   sec   1 1  sin 

.
tan   sec   1
cos 
G) Prove that 1  cot   cos ec 1  tan   sec    2


H) Prove that 3sin   cos  4  6 sin   cos  2  4 sin 6   cos6   13


I)
Prove that sin   cos ec 2  cos   sec  2  tan 2   cot 2   7
J)
Prove that cot

3
5
7
9
.cot .cot .cot .cot
1
20
20
20
20
20
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9.
A) Draw the graph of y  cos 2 x in 0,  .
B)
Draw the graph of y  sin 2 x in  ,  .

C) Find the period of the function defined by f x   tan x  4x  9x  ...  n 2 x

D) Find the periods of the functions:
 4x  9 
b) f x  cos 

 5 
a) f x  tan 5x
c) f x  sin x
E)
 1 o
 1 o
3 1
Prove that sin 2 52   sin 2 22  
 2
 2
4 2
F)
Prove that tan 70o  tan 20o  2 tan 50 o
G) If A  B  45o , then prove that
i) 1  tan A1  tan B  2
H) Prove that
cos 9o  sin 9 o
o
cos9  sin 9
o
ii) cot A  1cot B  1  2
 cot 36 o

.
4
I)
Draw the graph of the tan x between 0 and
J)
Draw the graph of the cos 2x in the interval 0,  .


K) Find the extreme values of 5 cos x  3 cos x    8 over R.

3
L)
Find the range of
ii) 13 cos x  3 3 sin x  4
i) 7 cos x  24 sin x  5
M) Find the minimum and maximum values of
i) 3 cos x  4 sin x
ii) sin 2 x  cos 2 x
N) If  is not an odd multiple of
O) Prove that
P)
1
sin 10

o

3
cos10o

1  sin 2  cos 2
and if tan   1 , then show that
 tan  .
2
1  sin 2  cos 2
4

Prove that 4 cos 66 o  sin 84o  3  15
o
o
Q) Prove that cos   cos 120     cos 240     0
10.
A) Prove that, for any x  R , sinh 3x  3sinh x  4 sinh 3 x
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B)
If cosh x 
5
, find the values of (i) cosh 2 x  and (ii) sinh 2x
2
C) If sinh x  5, show that x  log
e
D)
1 1
Show that tanh1    log 3
2 2
e
E)
If sinh x 
F)
Prove that
5 
26 
3
, find cosh 2x and sinh 2x
4
n
(i) cosh x  sinh x  cosh nx  sinh nx , for any n  R
n
(ii) cosh x  sinh x  cosh nx  sinh nx , for any n  R
G) If sinh x  3 then show that x  log
e
3 
10 
SECTION – B
II.
11.
5  4 = 20 M
Short Answer Type questions:
(i) Attempt any five questions
(ii) Each question carries four marks

 cos n sin n 
 sin  
n
a) If A  cos
 then show that for all the positive integers n, A   sin n cos n
 sin  cos  


b) A trust fund has to invest Rs. 30,000 in two different types of bonds. The first bond pays 5%
interest per year, and the second bond pays 7% interest per year. Using matrix multiplication,
determine how to divide Rs. 30000 among the two types of bonds if the trust fund must obtain
an annual total interest of
a) Rs. 1800
b) Rs. 2000
 1 2 2
c) If A   2 1 2 then show that A 2  4A  5I  0
 2 2 1

1  2n 4n 
4
n
d) If A   31 
 then for any integer n  1 show that A   n
1  2n 
 1

 1 2 2
e) If 3A   2 1 2 then show that A1  A ' .
2 2 1
 3 3 4
f) If A   2 3 4 then show that A1  A 3 .
 0 1 1
g) Two factories I and II produce three varieties of pens namely Gel, Ball and Ink pens. The sale in
rupees of these varieties of pens by both the factories in the month of September and October in a
year are given by the following matrices A and B.
September sales (in Rupees)
Gel
Ball
Ink
1000 2000 3000 Factory I

A 
 5000 3000 1000  Factory II


FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
October sales (in Rupees)
Gel
Ball
Ink
 500 1000 600 Factory I

A 
 2000 1000 1000 Factory II


i) Find the combined sales in September and October for each factory in each variety.
ii) Find the decrease in sales from September to October.
b c 1
bc
h) i) Show that ca c  a 1   a  bb  c c  a
ab a  b 1
b c c  a a b
ii) Show that a  b b  c c  a  a 3  b 3  c 3  3abc
a
c
yz
x
x
y
zx
y
z
z
xy
iii) Show that
a a2
b
1  a3
iv) If b b 2
c c2
 4xyz
a a2
1
1  b 3  0 and b b 2
1  0 then show that abc  1
1  c3
1
c c2
v) without expanding the determinant, prove that
a a2
i) b b
2
c c2
1 bc
bc
1 a2
a3
ax
by
cz
2
3
2
2
2
ca  1 b
ab
1 c2
bc
b
ii) x
1
c3
y
1
a
b
c
z  x y
z
1
yz zx xy
1 a a2
iii) 1 ca c  a  1 b b 2
1 ab a  b
1 c
c2
12.
A) Let ABCDEF be a regular hexagon with centre ‘O’. Show that
AB+AC+AD+AE+AF=3AD=6AO.
B) In ABC , if ‘O’ is the circumcentre and H is the orthocentre, then show that
i) OA + OB + OC = OH
ii) HA + HB + HC = 2 HO
C) Is the triangle formed by the vectors 3i  5 j  2 k , 2i  3 j  5k and 5i  2 j  3 k equilateral?
D) If a  b  c  d , b  c  d   a and a , b , c are non-coplanar vectors, then show that
abc  d 0 .
E) a , b , c are non-coplanar vectors. Prove the following four points are coplanar.
a) a  4b  3c , 3a  2b  5c , 3a  8b  5c , 3a  2b  c .
b) 6 a  2b  c , 2 a  b  3c , a  2b  4c , 12 a  b  3c
F) Find the equation of the line parallel to the vector 2i  j  2 k , and which passes through the
point A whose position vectors is 3i  j  k . If P is a point on this line such that AP  15 ,
find the position vector of P .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
13.
A) If a  b  c  0 , a  3, b  5 and c  7 , then find the angle between a and b .
If a  2, b  3 and c  4 and each of a , b , c is perpendicular to the sum of the other two
vectors, then find the magnitude of a  b  c .
C) G is the centroid of ABC and a , b , c are the lengths of sides BC , CA and AB respectively.
B)


Prove that a 2  b 2  c 2  3 OA2  OB2  OC 2  9 OG2 where 'O ' is any point.
2
2
2
D) For any two vectors a and b , a  b  a.ab.b  a.b  a 2 b 2   a.b
E)
Let a and b be vectors, satisfying a  b  5 and  a , b  45o . Find the area of the triangle
having a  2b and 3a  2b as two of its sides.
F)
Find the vector having magnitude
6 units and perpendicular to both 2i  k and 3 j  i  k .
G) Find unit vector perpendicular to the plane passing through the points 1,2,3 , 2, 1,1 and
1,2, 4 .
H) If a , b and c represent the vertices A, B and C respectively of ABC , then prove that
a  b  b c   c  a is twice the area of ABC .
I)
If A, B, C and D are four points, then show that ABCD  BC  AD  CA BD is four times
the area of ABC .
J)



For any two vectors a and b , show that 1  a 2 1  b 2  1  a.b 2  a  b  a  b 2 .
K) Let a , b , c be three vectors. Then
i)  a b c   a.c  b  b.c  a
L)
ii) a b c    a.c  b   a.bc
If a  i  2 j  3 k , b  2i  j  k and c  i  3 j  2 k . Find the a b  c 
M) a , b , c are non-zero vectors and a is perpendicular to both b and c . If a  2, b  3, c  4
2
and b , c 
, then find  a b c 
3
N) If  b c d c a d    a b d    a b c  , then show that the points with position vectors
a , b , c and d are coplanar.
O) If a , b , c are the position vectors of the points A, B and C respectively, then prove that the
vector a  b  b  c  c  a is perpendicular to the plane of ABC .
14.
A) If sec   tan  
B)
2
, find the value of sin  and determine the quadrant in which  lies.
3

 
Find the value of 2 sin 6   cos6   3 sin 4   cos 4 

C) If cos   0 , tan   sin   m and tan   sin   n , then show that m2  n2  4 mn
D) Eliminate  from the following:
E)
i) x  a cos 3  , y  b sin 3 
ii) x  a cos 4  , y  b sin 4 
iii) x  a sec   tan   , y  b sec   tan  
iv) x  cot   tan  and y  sec   cos 
Let ABC be a triangle such that cot A  cot B  cot C  3 , then prove that ABC is an
equilateral triangle.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
F)
Find the maximum and minimum values of




ii) cos x    2 2 sin x    3



3
3
i) 3sin x  4 cos x
G) If A is not an integral multiple of

, prove that
2
a) tan A  cot A  2 cos ec 2 A
b) cot A  tan A  2 cot 2 A
H) If  is not an integral multiple o f
I)

, prove that tan   2 tan 2  4 tan 4  8 cot 8  cot 
2
For A  R , prove that
1
a) sin A.sin 60  A sin 60  A  sin 3 A
4
1
b) cos A.cos 60  A cos 60  A  cos 3 A and hence deduce that
4
c) sin 20 o sin 40o sin 60o sin 80o 
J)

2
3
4
1
d) cos cos
cos cos

9
9
9
9
16
3
16

, prove that tan A.tan 60  A.tan 60  A  tan 3 A and
2
hence find the value of tan 6 o tan 42 o tan 66o tan 78 o .
If 3A is not an odd multiple of
K) If  ,  are the solutions of the equation a cos   b sin   c ( a , b , c are non-zero numbers) then
show that
i) sin   sin  
L)
2bc
ii) sin .sin  
a2  b2
c 2  a2
a2  b2
If A is not an integral multiple of  , prove that cos A.cos 2 A.cos 4 A.cos8 A 
hence deduce that cos
M) If sin x  sin y 
a) tan
xy 3

2
4
2
4
8
16
1
.cos .cos .cos

15
15
15
15
16
sin 16 A
and
16 sin A
1
1
and cos x  cos y  , then show that
4
3
ii) cot x  y  
7
24
N) Prove that 4 cos12 o cos 48o cos72 o  cos 36o
O) If cos x  cos y 
P)
xy
xy
4
2
and cos x  cos y  , find the value of 14 tan
 5 cot
5
7
2
2
If sec     sec      2 sec  and cos   1 , then show that cos    2 cos

.
2


2 
4 
Q) If x , y , z are non zero real numbers and if x cos   y cos     z cos    for some


3
3
  R , then show that xy  yz  zx  0
R) If a cos      b cos     , cos   0 , then prove that  a  b tan   a  b cot  .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
S)
Prove that sin 4
T)
Prove cos

3
5
7 3
 sin 4
 sin 4
 sin 4

8
8
8
8
2

2
3
4
5
1
.cos .cos .cos .cos

11
11
11
11
11 32

 
3 
7  
9  1
U) Prove that 1  cos 1  cos 1  cos 1  cos  

10 
10 
10 
10  16
15.




A) If x is acute and sin x  10 o  cos 3x  68o find x .
B)
Solve 7 sin 2   3 cos 2   4
C) Solve 2 cos 2   3 sin   1  0
2
D) Find all values of x in  ,  satisfying the equation 81coscos x....  4 3
E)
Solve tan   3 cot   5sec 
F)
Solve 1  sin 2   3sin  cos 
G) Solve
2 sin x  cos x  3
H) Solve 4 sin x sin 2 x sin 4 x  sin 3x
I)
If 0     , solve cos  .cos 2 .cos 3 
J)
Solve sin 2 x  cos 2 x  sin x  cos x
K) Find the general solution of tan x  
L)
1
4
1
2
,sec x 
3
3
Solve the following equations and write general solution.
ii) 4 cos 2   3  2  3  1 cos 
i) 6 tan 2 x  2 cos 2 x  cos 2 x
2
iii) 1  sin 2 x  sin 3x  cos 3x
iv) 2 sin 2 x  sin 2 2 x  2


1
M) If tan  cos    cot  sin   , then prove that cos     


4
2 2
N) Find the common roots of the equations cos 2 x  sin 2 x  cot x and 2 cos 2 x  cos 2 2 x  1 .
O) Solve the equation
P)
If tan x  tan x 
6  cos x  7 sin 2 x  cos x  0 .
1
and x  0,2  , find the values of x .
cos x
16.
A) Prove that sin1
4
7
117
 sin1
 sin1
5
25
125
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
B)
Prove that sin1
 16  
4
5
 sin1  sin1   
 25  2
5
13
41 

4
4
4
1

D) Prove that sin1  2 tan1 
5
3 2


1
1
E) Prove that cos 2 tan1   sin 4 tan1 


7
3
C) Prove that cot1 9  cos ec1
F)
If sin1 x  sin1 y  sin 1 z   , then prove that

x 4  y 4  z 4  4 x 2 y 2 z 2  2 x 2 y 2  y 2 z2  z 2 x 2

p
q
p2 2 pq
q2
 cos1   , then prove that

.cos  
 sin 2  .
a
b
ab
a2
b2
5
12 
H) Solve arc sin    arc sin  . x  0
x
x
2
G) If cos1

3
I)
Solve sin1 x  sin1 2 x 
J)
x2  1


Prove that cos  tan1 sin cot1 x  
.


x2  2
 

K) If cos1 p  cos1 q  cos1 r   , then prove that p 2  q 2  r 2  2 pqr  1
L)
If sin1
 1  q 2 

  tan1 2 x , then prove that x  p  q .
 cos1 
2
2
1  pq
 1  q 
1 p
1  x2
2p
M) If a , b , c are distinct non-zero real numbers having the same sign, prove that
 ab  1 
1  bc  1 
1  ca  1 
cot1 
  cot 
  cot 
   or 2
 ab 
 b c 
 c  a 
N) If sin1 x  sin1 y  sin 1 z   , then prove that x 1  x 2  y 1  y 2  z 1  z 2  2 xyz
O) i) If tan1 x  tan1 y  tan1 z   , then prove that x  y  z  xyz
ii) If tan1 x  tan1 y  tan1 x 
P)

, then prove that xy  yz  zx  1
2


 1  x 2  1  x 2 
 , then prove that x 2  sin 2
If   tan1 


2
2
 1  x  1  x 
Q) Solve the following equation for x :
x 
cos1 x  sin1   
2 6
17.
(I)
A) In ABC , show that
B)
b  c cos A  2s
In ABC , if  a  b  cb  c  a  3bc , find A .
C) If a  4, b  5, c  7 , find cos
B
.
2
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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D) In ABC , find b cos 2
C
B
 c cos 2 .
2
2
A bc

, find angle B .
2
a
E)
If cot
F)
Prove that a b cos C  c cos B  b 2  c 2
G) In ABC , if
1
1
3


, show that C  60
ac bc abc
H) In ABC , show that
b2  c 2
a
2

sin B  C 
sin B  C 
I)
abc
Show that a 2 cot A  b 2 cot B  c 2 cot C 
R
J)
In ABC , prove that
1
1
1 1
  
r1 r2 r3 r
K) Show that r r1 r2 r3  2
(II)
L)
Prove that
r1 r2  r3 
r1r2  r2 r3  r3r1
a
M) If r : R : r1  2 : 5 : 12 , then prove that the triangle is right angled at A.
N) In an equilateral triangle, find the value of
r
.
R
O) If A  90 o , show that 2 r  R  b  c
P)
In ABC , if r1  8, r2  12, r3  24, find a , b , c .
A
B
C
2
 cot  cot
2
2
2   a  b  c
Q) Prove
cot A  cot B  cot C
a2  b2  c 2
cot
SECTION – C
III.
5  7 = 35 M
Long Answer Type questions:
(i) Attempt any five questions
(ii) Each question carries seven marks
18.
1
A) Let f : A  B, g : B  C be bijections. Then  gof 
B)
 f 1 og1 .
Let f : A  B , I A and I B be identity functions on A and B respectively. Then
foI A  f  I Bof .
C) Let f : A  B be a bijection. Then fof 1  I B and f 1 of  I A .
D) Let f : A  B be a function. Then f is a bijection if and only if there exists a function
g : B  A such that fog  I B and gof  I A and, in this case, g  f 1 .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
E)
Let f : A  B , g : B  C and h : C  D . Then ho  gof   hog of , that is, composition of
functions is associative.
F)
Let
f  1, a ,2, c ,4, d ,3, b
g1 2, a , 4, b ,1, c , 3, d ,
and
then
show
that
 gof 1  f 1 og1 .
19.
A) Use Mathematical Induction to prove the formula
2  3.2  4.2 2  ...  upto n terms  n.2 n , n  N .
B)
Show that, n  N ,
1
1
1
n


 ....upto n terms 
1.4 4.7 7.10
3n  1
C) If x and y are natural numbers and x  y , using mathematical induction, show that xn  y n
is divisible by x  y , for all n  N
D) Using mathematical induction, show that xm  y m is divisible by x  y , if m is an odd
natural number and x , y are natural numbers.
E)
Show that 49n  16n  1 is divisible by 64 for all positive integers n .
F)
Use mathematical induction to prove that 2.42 n1  33 n1 is divisible by 11, n  N .
Using Mathematical Induction, prove the following, for all n  N


n n 2  6n  11
G)
2.3  3.4  4.5  ...upto n terms 
H)
1
1
1
1
n


 ... 

1.3 3.5 5.7
2 n  12 n  1 2n  1
I)
12  2 2  ....  n 2 
J)
3.52 n1  2 3n1 is divisible by 17.
K)
1.2.3  2.3.4  3.4.5  ...upto n terms 
3
n3
3
n n  1n  2n  3
4
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L)
13 13  2 3 13  2 3  33
n


 ....upto n terms   2n 2  9n  13

1
13
13 5
24 
2
n n  1 n  2
M) 12  12  2 2  12  2 2  32  ....upto n terms 
12

20.
 
bc

ca ab
a b
c
a) Show that c  a a  b b  c  2 b c a .
ab bc
1 a2
b) Show that 1 b2
1 c2
c) Show that
ca
c
a b
a3
b3  a  bb  cc  a ab  bc  ca
c3
abc
2a
2a
3
2b
bca
2b
 a  b  c
2c
2c
c a b
x2
2x  3 3x  4
2x  9 3x  16  0
x  8 2x  27 3x  64
d) Find the value of x if x  4
e) Show that
a  b  2c
a
b
3
c
b  c  2a
b
 2 a  b  c  .
c
a
c  a  2b
2
2bc  a2
a b c
f) Show that b c a 
c2
c a b
b2
2a
ab ca
ca
cb
c2
2ac  b2
a2
b2
2
a2
 a 3  b3  c 3  3abc .
2ab  c2


g) Show that a  b 2b b  c  4 a  bb  cc  a  .
21.
2c
 1 2 3 


1
h) If A   0 1 4 then find  A '
2 2 1


1 2 2 


i) If A   2
1 2  then show that the adjoint of A is 3A' . Find A1 .


 2 2 1 


a) Solve the following equations by Gauss-Jordan method
3x  4y  5z  18 ,
2x  y  8z  13 , 5x  2y  7z  20
b) Solve the following equations by Gauss-Jordan method
5x  6y  4z  15 ,
7x  4y  3z  19 , 2x  y  6z  46
c) Solve the following system of equations by Gauss – Jordan method.
x  y  z  3 , 2x  2y  z  3 , x  y  z  1 .
d) By using Gauss-Jordan method, show that the following system has no solution.
2x  4y  z  0 , x  2y  2z  5 , 3x  6y  7z  2 .
e) Solve 3x  4y  5z  18 ,
2x  y  8z  13
,
5x  2y  7z  20
by using matrix inversion
method.
f) Solve 3x  4y  5z  18 , 2x  y  8z  13 , 5x  2y  7z  20 by using Cramer’s Rule.
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g) Examine whether the following systems of equations are consistent or inconsistent and if
consistent find the complete solution. x  y  z  4 , 2x  5y  2z  3 , x  7y  7z  5 .
h) Apply the test of rank to examine whether the following equations are consistent.
2 x  y  3z  8 , x  2 y  z  4 , 3x  y  4 z  0 and if consistent find the complete solution.
i) Show that the following system of equations is consistent and solve it completely:
x  y  z  3 , 2x  2 y  z  3 , x  y  z  1
j) Find the nontrivial solutions if any, for the following system of equations:
2 x  5y  6 z  0 , x  3y  8z  0 , 3x  y  4 z  0
22.
A) Let a , b , c be three vectors. Then prove that
i)  a  b  c   a.c b  b.c  a
B)
For
any
four
vectors
ii) a b  c    a.c b  a.bc
a,b,c
and
 a  b .c  d 
d
a.c a.d
b.c b.d
and
in
particular
 a b 2  a2 b 2   a.b 2 .
C) Find the volume of the parallelopiped whose coterminus edges are represented by the vectors
2i  3 j  k , i  j  2 k and 2i  j  k .
D) For
any
four
vectors
a,b,c
and
d
 a  b c  d   a c d b  b c d  a
and
 a  b c  d   a b d c   a b c d
E)
Find the shortest distance between the skew lines r  6i  2 j  2 k   t i  2 j  2 k  and
r  4i  k   s 3i  2 j  2 k 
F)
Find
the
shortest
distance
between
the
lines
r  6 i  2 j  2 k   i  2 j  2 k 
and
r   4 i  k    3i  2 j  2 k 
G) If A  1, 2, 1 , B  4,0, 3 , C  1,2, 1 and D  2, 4, 5 , find the distance between
AB and CD.
H) Let b  2i  j  k , c  i  3k . If a is a unit vector then find the maximum value of  a b c  .
I)
23.
Let a  i  j , b  j  k , c  k  i . Find unit vector d such that a.d  0  b c d  .
All example problems page. 283 - 288
Exercise 6F 2(ii) , 3(i, ii), 4(i,ii), 5(i,ii,iii), 10(i,ii)
24.
 B  C  b  c
A
A) In ABC , tan 
cot
 
 2  bc
2
B)
2
Show that b  c cos 2
C) Show that
A
A
2
 b  c  sin 2  a 2
2
2
c  b cos A cos B

b  c cos A cos C
D) If a  b  c sec  , prove that tan  
E)
Show that a cos 2
2 bc
A
sin .
b c
2
A
B
C

 b cos 2  c cos 2  s 
2
2
2
R
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F)
Prove that a 3 cos B  C   b 3 cos C  A  c 3 cos  A  B  3abc
G) Prove that
a cos A b cos B c
cos C

 
 
bc
a
ca
b
ab
c
H) If b  c  3a , then find the value of cot
B
C
cot .
2
2
3
, then show that the triangle is equilateral.
2
I)
If cos A  cos B  cos C 
J)
If a 2  b 2  c 2  8 R 2 , then prove that the triangle is right angled.
K) Page. 389 (Examples 27,28)
L)
Page. 392 (Problems 13 to 18)
1
M) Show that
r2

1
r12

1
r2 2

1
r32

a2  b 2  c 2
2
r
r
r
1
1
N) Show that 1  2  3  
bc ca ab r 2 R
O) Show that r  r3  r1  r2  4 R cos B .
P)
ab  r1r2 bc  r2 r3 ca  r3r1


r3
r1
r2
Show that
Q) In ABC , prove that r1  r2  r3  r  4 R .
R) Show that cos A  cos B  cos C  1 
r
R

S)
Prove that r12  r2 2  r32  r 2  16 R 2  a 2  b 2  c 2
T)
If p , p , p
1
2
3

are altitudes drawn from vertices A, B, C to the opposite sides of a triangle
respectively, then show that
i)
1
1
1
1



p
p
p
r
1
2
3
ii)
1
1
1
1



p
p
p
r
1
2
U) If a  13, b  14, c  15 , show that R 
3
3
iii) p p p 
1 2
3
 abc 2
8R 3

83
abc
65
21
, r  4, r 
, r  12 and r  14 .
1
2
3
8
2
V) If r1  2, r2  3, r3  6 and r  1, prove that a  3, b  4 and c  5 .
wish you all the best
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