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` KUKATPALLY CENTRE

IMPORTANT QUESTIONS

FOR

INTERMEDIATE PUBLIC EXAMINATIONS

IN

MATHS-IA 2017-18



INTERMEDIATE PUBLIC EXAMINATION, MARCH 2017

Total No. of Questions - 24 Reg.

Total No. of Printed Pages - 2 No.

Part - III MATHEMATICS, Paper-I (A)

(English Version) Time : 3 Hours] [Max. Marks : 75

SECTION - A 10 2 = 20 M

I. Very Short Answer Type questions:

1. If 0, , , ,6 4 3 2

A and :f A B is a surjection defined by cosf x x then find B .

2. If 2f x , 2g x x , 2h x x for all x R , then find fo goh x .

3. If

3 2 1

2 2 0

1 3 1

A ,

3 1 0

2 1 3

4 1 2

B and X A B then find X .

4. If 1 2

0 1A then find 'AA .

5. 2 5a i j k and 4b i mj nk are collinear vectors then find m and n .

6. Find the vector equation of the line passing through the point 2 3i j k and parallel to the

vector 4 2 3i j k .

7. Find the angle between the vectors 2 3i j k and 3 2i j k .

8. If 4

sin5

and is not in the first quadrant, find the value of cos .

9. Prove that 3 5

cos48 cos128

.

10. If 5

cosh2

x , find the value of (i) cosh 2x and (ii) sinh 2x .

SECTION – B 5 4 = 20 M

II. Short Answer Type questions:

(i) Attempt any five questions

(ii) Each question carries four marks

11. Show that

2

2

2

1

1

1

a a

b b a b b c c a

c c

.

12. If , ,a b c are non-coplanar find the point of intersection of the line passing through the points

2 3a b c , 3 4 2a b c with the line joining the points 2 3a b c , 6 6a b c .

13. If 2a i j k , 2 4b i j k and c i j k then find a b b c .

14. (i) Find the range of 13cos 3 3sin 4x x .

(ii) Evaluate o o

2 21 1sin 82 sin 22

2 2.



15. Solve 21 sin 3sin cos .

16. Show that 1 113 2cot sin sin tan

17 3.

17. In ABC , if 1 1 3

a c b c a b c, show that 60C .

SECTION – C 5 7 = 35 M III. Long Answer Type questions:

(i) Attempt any five questions

(ii) Each question carries seven marks

18. (i) If :f Q Q is defined by 5 4f x x , x Q , show that f is a bijection and find 1f .

(ii) If 4,5 , 5,6 , 6, 4f and 4, 4 , 6,5 , 8,5g then find f g and fg .

19. Using mathematical induction, prove 1 2 3 2 3 4 3 4 5 ......upto n terms

1 2 3

4

n n n n, n N .

20. Solve the following system of equations by using Cramer’s rule:

2 3 9; 6; 2x y z x y z x y z

21. (i) Show that

1 2 1

3 2 3

1 1 2

A is non-singular and find 1A .

(ii) If

1 2 2

2 1 2

2 2 1

A then show that 2 4 5 0A A I .

22. Find the shortest distance between the skew lines:

6 2 2 2 2r i j k t i j k and 4 3 2 2r i k s i j k .

23. If 2A B C S , then prove that

cos cos cos 1 4cos cos cos2 2 2

S A S B CS A S B C .

24. Show that in a ABC , 2 2 2

2 2 2 21 2 3

1 1 1 1

2

a b c

r r r r.

BLUE PRINT (MATHS-IA)

S.No. Chapter Name Weightage Marks

ALGEBRA

1. Functions 11 (7 + 2 + 2)

2. Mathematical induction 7 (7)

3. Matrices & Determinants 22 (7 + 7 + 4 + 2 + 2)

VECTOR ALGEBRA

4. Addition of vectors 8 (4 + 2 + 2)

5. Product of vectors 13 (7 + 4 + 2)

TRIGONOMETRY

6. Trigonometry upto transformations 15 (7 + 4 + 2 + 2)

7. Trigonometric equations 4 (4)

8. Inverse trigonometric functions 4 (4)

9. Hyperbolic functions 2 (2)

10. Properties of triangles 11 (7 + 4)



VERY SHORT ANSWER QUESTIONS

1.

A. If 0, , , ,6 4 3 2

A and :f A B is a surjection defined by cosf x x then find B .

B. If : 1f R R is defined by 1

log1

xf x

x, then show that

2

22

1

xf f x

x.

C. If : \ 0f R R is defined by 1

f x xx

then prove that 2 2 1f x f x f .

D. Find the domains of the following real valued functions:

a) 2 1f x x x b) 10

3log

xf x

x

c)

10

12

log 1f x x

x d)

1f x

x x

e) 2

1

2f x

x x f) logf x x x

g) 2

1

6 5f x

x x h) 2

2

11

3 2f x x

x x

i) 3 3x x

f xx

j) 1

log 2f x

x

E. Find the ranges of the following real valued functions:

a) x x b) 2

sin

1

x

x c)

2 4

2

x

x d) 29 x e) 2log 4 x

F. If : 0f R R is defined by 33

1,f x x

x then show that

10f x f

x

G. If :f R R is defined by 2

2

1

1

xf x

x, then show that tan cos2f

H. If 2, 1,0,1,2A and :f A B is a surjection defined by 2 1f x x x , then find B .

2.

A. If 1

11

xf x x

x then find fofof x and fofofof x .

B. If : 4,5 , 5,6 , 6, 4f and : 4, 4 , 6,5 , 8,5g then find

a) 4f b) fg c) f d) 2f

C. If :f R R , :g R R are defined by 3 1,f x x 2 1g x x , then find

i) 2 1fof x ii) 2fog iii) 2 3gof a



D. Prove that the real valued function 121x

x xf x

e is an even function on \ 0R .

E. Determine whether the following functions are even or odd.

a) 2log 1f x x x b) 1

1

x

x

ef x x

e

F. If :f R R and :g R R are defined by 22 3f x x and 3 2g x x , then find

i) fog x , ii) gof x , iii) fof o , iv) 3go fof .

G. Find the inverse of the following functions:

a) If ,a b R , :f R R defined by f x ax b 0a

b) : 0,f R defined by 5xf x c) : 0,f R defined by 2

logf x x .

H. If 1

1

xf x

x, 1x , show that 1fof x x

I. If the function :f R R defined by 3 3

2

x x

f x , then show that

2f x y f x y f x f y .

J. If the function :f R R defined by 4

4 2

x

xf x , then show that 1 1f x f x .

3.

A. Construct 3 2 matrix whose elements are defined by 1

32

ija i j

B. Define Trace of matrix and give an example.

C. If 2 4

1A

k and 2A O , then find the value of k.

D. Find the trace of

1 3 5

2 1 5

2 0 1

E. If 1 2

3 4A ,

3 8

7 2B and 2X A B then find X.

F. Define symmetric and skew-symetric matrix and give an example.

G. If 2 4

5 3A then find 'A A and 'AA .

H. If 0

0

iA

i, find 2A .

I. Find the products of i)

5

1 4 2 1

3

ii)

12 1 4

26 2 3

1



J. If 3 2 8 5 2

2 6 2 4

x y

z a then find the values of , ,x y z and a .

K. 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.

4.

A. Define singular matrix and give an example.

B. Define rank of a matrix.

C. If

1 0 0

2 3 4

5 6

A

x

and det A = 45 find x .

D. If

1 5 3

2 4 0

3 1 5

A and

2 1 0

0 2 5

1 2 0

B then find 3 4 'A B .

E. Find the Rank of the Matrix

1 4 1

2 3 0

0 1 2

F. If

7 2

1 2

5 3

A and

2 1

4 2

1 0

B then find AB’ and BA’.

G. If

3 0 0

0 3 0

0 0 3

A , then find 4A .

H. If cos sin

sin cosA , show that AA’ = A’A = I.

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 2 ,i j k 4 2 2i j k and 6 3 13i j k

respectively and AB AC , then find the value of .

C. If , and be the angles made by the vectors 3 6 2i j k with the positive directions of the

coordinates axes, then find cos , cos and cos .

D. If OA i j k , 3 2AB i j k , 2 2BC i j k and 2 3CD i j k , then find vector OD.

E. 2 5a i j k and 4b i mj nk are collinear vectors, then find m and n .

F. Find the angles made by the straight line passing through the points 1, 3,2 and 3, 5,1 with

the coordinate axes.

G. Let 2 4 5 ,a i j k b i j k and 2c j k . Find the unit vector in the opposite direction

of a b c .



6.

A. Find the vector equation of plane passing through the points 2 5 ,i j k 5 j k , and

3 5i j .

B. If the vectors 3 4i j k and 8 6i j k are collinear vectors, then find and .

C. If OA i j k , 3 2AB i j k , 2 2BC i j k & 2 3CD i j k , then find the vector OD .

D. Let a i j k and 2 3b i j k find

(i) The projection vector of b and a and its magnitude.

(ii) The vector components of b in the direction of a and perpendicular to a .

E. For any vector a , show that 22 2

2a i a j a k a .

F. If 2, 3p q and ,6

p q , then find 2

p q .

G. Find the distance of a points 2,5, 3 from the plane . 6 3 2 4r i j k .

H. Find the vector equation of the line passing through the point 2 3i j k and parallel to the

vector 4 2 3i j k .

I. OABC is a parallelogram. If OA= a and OC c , find the vector equation of the side BC.

J. 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.

K. Find the vector equation of the line joining the points 2 3i j k and 4 3i j k .

L. Find the vector equation of the plane passing through the points 0,0,0 , 0,5,0 , and 2,0,1 .

M. Find the vector equation of the plane passing through the points. 2 5 , 5i j k j k .

N. If the vectors 3 5i j k and 2 i j k are perpendicular to each other, find .

O. Show that 2i a i j a j k a k a for any vector a .

P. Compute i j j k k i .

7.

A. If 11,a 23b and 30,a b then find the angle between the vectors ,a b and a b .

B. If a i j k and 2 3b i j k , then find the projection vector of b on a and its magnitude.

C. If , ,P Q R and S are points whose position vectors are ,i k 2 ,i j 2 3i k and 3 2i j k

respectively, then find the component of RS and PQ .

D. Find the angle between the planes . 2 2r i j k and . 3 6 4.r i j k

E. Let 1e and 2e be unit vectors containing angle . If 1 2

1sin

2e e , then find .

F. Show that 2i a i j a j k k a a for any vector .a

G. Prove that for any three vectors , ,a b c 2b c c a a b a b c .



H. If 2 3a i j k and 3 2b i j k , then show that a b and a b are perpendicular to each

other.

I. Let a and b be non-zero, non-collinear vectors. If a b a b , then find the angle between

a and b .

J. If 2 2 3 ,a i j k 3 2b i j k , then find the angle between 2a b and 2a b .

K. Find the area of the parallelogram for which the vectors 2 3a i j and 3b i k are adjacent

sides.

L. If 2 3a i j k and 3 5b i j k are two sides of a triangle, then find its area.

M. Let 2a i j k and 3 4b i j k . If is the angle between a and b , then find sin .

N. Find the area of the triangle whose vertices are 1,2,3A , 2,3,1B and 3,1,2C .

O. For any three vectors, , ,a b c , prove that 2

b c c a a b a b c .

8.

A. If 3sin 4cos 5, then find the value of 4sin 3cos .

B. If cos sin 2 cos , prove that cos sin 2 sin

C. If tan 20o , then show that 2tan160 tan110 1

21 tan160 .tan110

o o

o o

D. Prove that 4 2 6 63 sin cos 6 sin cos 4 sin cos 13

E. Prove that 2 3 7

cot .cot .cot ....cot 116 16 16 16

F. Prove that 2 2 2 2 2tan cot sec cos sec .cosec ec .

G. Prove that tan sec 1 1 sin

tan sec 1 cos.

H. Prove that 1 cot cos 1 tan sec 2ec

I. Prove that 2 2 2 2sin cos cos sec tan cot 7ec

J. Prove that 3 5 7 9

cot .cot .cot .cot .cot 120 20 20 20 20

9.

A. Draw the graph of 2cosy x in 0, .

B. Find the period of the function defined by 2tan 4 9 ...f x x x x n x

C. Find the extreme values of 5cos 3cos 83

x x over R.



D. Find the periods of the functions:

a) tan5f x x b) 4 9

cos5

xf x c) sinf x x

E. Find the range of (i) 7cos 24sin 5x x (ii) 13cos 3 3sin 4x x

F. Prove that 1 3

4sin10 cos10o o

G. Prove that tan70 tan 20 2 tan 50o o o

H. Prove that 2 21 1 3 1sin 52 sin 22

2 2 4 2

o o

I. Draw the graph of sin 2y x in , .

J. If 45oA B , then prove that (i) 1 tan 1 tan 2A B (ii) cot 1 cot 1 2A B

K. Prove that cos9 sin9

cot 36cos9 sin9

o oo

o o

L. Draw the graph of the tanx between 0 and 4

.

M. Draw the graph of the cos2x in the interval 0, .

N. Find the minimum and maximum values of (i) 3cos 4sinx x (ii) sin2 cos2x x

O. If is not an odd multiple of 2

and if tan 1 , then show that 1 sin 2 cos2

tan1 sin 2 cos2

.

P. Prove that 4 cos66 sin84 3 15o o

Q. Prove that cos cos 120 cos 240 0o o

10.

A. Prove that, for any x R , 3sinh 3 3sinh 4sinhx x x

B. If 5

cosh2

x , find the values of (i) cosh 2x and (ii) sinh 2x

C. If sinh 5,x show that log 5 26e

x

D. Show that 1 1 1tanh log 3

2 2 e

E. If 3

sinh ,4

x find cosh 2x and sinh 2x

F. Prove that (i) cosh sinh cosh sinh ,n

x x nx nx for any n R

(ii) cosh sinh cosh sinh ,n

x x nx nx for any n R

G. If sinh 3x then show that log 3 10e

x



SHORT ANSWER QUESTIONS

11.

A. If

1 2 2

2 1 2

2 2 1

A then show that 2 4 5 0A A I

B. If

1 2 2

3 2 1 2

2 2 1

A then show that 1 'A A .

C. If

3 3 4

2 3 4

0 1 1

A then show that 1 3A A .

D. If 2

, then show that 2 2

2 2

cos cos sin cos cos sin0

cos sin sin cos sin sin

E. If 1 0

0 1I and

0 1

0 0E then show that

3 3 23aI bE a I a bE .

H. i) If

2 3

2 3

2 3

1

1 0

1

a a a

b b b

c c c

and

2

2

2

1

1 0

1

a a

b b

c c

then show that 1abc

ii) without expanding the determinant, prove that

a)

2

2

2

11

1 1

1 1

a abc b c

ca c a b b

ab a b c c

b) Show that 3 3 3 3

b c c a a b

a b b c c a a b c abc

a b c

c)

2 2 3

2 2 3

2 2 3

1

1

1

a a bc a a

b b ca b b

c c ab c c

d) Show that

1

1

1

bc b c

ca c a a b b c c a

ab a b

e) 2 2 2

1 1 1

ax by cz a b c

x y z x y z

yz zx xy

f) Show that 4

y z x x

y z x y xyz

z z x y

I. If cos sin

sin cosA then show that for all the positive integers n,

cos sin

sin cos n n n

An n

J. If 3 4

1 1A then for any integer 1n show that

1 2 4

1 2n n n

An n

K. If A and A are invertible then show that AB is also invertible and 1 1 1AB B A .

L. For any nxn matrix A prove that A can be uniquely expressed as sum of a symmetric matrix and

skew symmetric matrix.

M. Show that the determinant of skew-symmetric matrix of order 3 is always zero.



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. , ,a b c are non-coplanar vectors. Prove the following four points are coplanar.

a) 4 3 ,a b c 3 2 5 ,a b c 3 8 5 ,a b c 3 2a b c .

b) 6 2 ,a b c 2 3 ,a b c 2 4 ,a b c 12 3a b c

D. If ,a b c d b c d a and , ,a b c are non-coplanar vectors, then show that

0a b c d .

E. Find the equation of the line parallel to the vector 2 2i j 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 15AP , find

the position vector of P .

F. Let ,a b be non-collinear vectors. If 4 2 1x y a x y b and

2 2 2 3 1y x a x y b are such that 3 2 then find x and y .

G. Is the triangle formed by the vectors 3 5 2 ,i j k 2 3 5i j k and 5 2 3i j k equilateral?

H. If the points whose position vectors are ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ3 2 ,2 3 4 , 2i j k i j k i j k and ˆˆ ˆ4 5i j k are

coplanar, then show that 146

17.

I. In the two dimensional plane, prove by using vector method, the equation of the line whose

intercept on the axes are ' 'a and ' 'b is 1yx

a b.

J. Show that the line joining the pair of points 6 4 , 4a b c c and the line joining the pair of

points 2 3 , 2 5a b c a b c intersect at the point 4c when , ,a b c are non-coplanar

vectors.

K. If , ,a b c , are non coplanar find the point of intersection of the line passing through the points

2 3 ,3 4 2a b c a b c with the line joining the points 2 3 , 6 6a b c a b c .

L. Find the vector equation of the plane passing through points ˆ ˆˆ ˆ ˆ4 3 ,3 7 10i j k i j k and

ˆˆ ˆ2 5 7i j k and show that the point ˆˆ ˆ2 3i j k lies in the palne.

13.

A. If 0a b c , 3,a 5b and 7c , then find the angle between a and b .

B. If 2,a 3b and 4c 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. Let a and b be vectors, satisfying 5a b and , 45oa b . Find the area of the triangle

having 2a b and 3 2a b as two of its sides.

D. For any two vectors a and b , show that 2 2 2 2

1 1 1 .a b a b a b a b .



E. If 2 3 ,a i j k 2b i j k and 3 2c i j k . Find the a b c

F. ,a b , c are non-zero vectors and a is perpendicular to both b and c . If 2,a 3,b 4c

and 2

,3

b c , then find a b c

G. G is the centroid of ABC and ,a ,b c are the lengths of sides BC , CA and AB respectively.

Prove that 22 2 2 2 2 23 9a b c OA OB OC OG where ' 'O is any point.

H. For any two vectors a and b , 2 2 22 2. . . .a b a a b b a b a b a b

I. Find the vector having magnitude 6 units and perpendicular to both 2i k and 3 j i k .

J. Find unit vector perpendicular to the plane passing through the points 1,2,3 , 2, 1,1 and

1,2, 4 .

K. 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 .

L. 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 .

M. 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

N. 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 .

O. Find the volume of the tetrahedron whose vertices are 1,2,1 , 3,2,5 , 2, 1,0 and 1,0,1 .

P. Find , in order that the four points 3,2,1 , 4, ,5 , 4,2, 2A B C and 6,5, 1D be coplanar.

Q. For any three vectors , ,a b c prove that 2

b c c a a b a b c .

R. Let ,a b and c be unit vectors such that b is not parallel to c and 1

2a b c b . Find the

angle made by a with each of b and c .

S. ,a b and c are non-zero and non-collinear vectors and 0, is the angle between b and c .

If 1

3a b c b c a , then find sin .

T. If , ,a b c are unit vectors such that a is perpendicular to the plane of ,b c and the angle between

b and c is 3

, then find a b c .

14.

A. Prove that 2 4 8 1

cos .cos .cos7 7 7 8

B. Find the value of 6 6 4 42 sin cos 3 sin cos

C. If cos 0 , tan sin m and tan sin n , then show that 2 2 4m n mn



D. Eliminate from the following:

i) 3 3cos , sinx a y b ii) 4 4cos , sinx a y b

iii) sec tan , sec tanx a y b iv) cot tanx and sec cosy

E. Find the maximum and minimum values of

i) 3sin 4cosx x ii) cos 2 2 sin 33 3

x x

F. If sec sec 2sec and cos 1, then show that cos 2 cos2

.

G. If , ,x y z are non zero real numbers and if 2 4

cos cos cos3 3

x y z for some R ,

then show that 0xy yz zx

H. Prove that 3 7 9 1

1 cos 1 cos 1 cos 1 cos10 10 10 10 16

I. For A R , prove that

a) 3

sin 20 sin 40 sin60 sin8016

o o o o

b) 1

sin .sin 60 sin 60 sin 34

A A A A

c) 1

cos .cos 60 cos 60 cos34

A A A A and hence deduce that

d) 2 3 4 1

cos cos cos cos9 9 9 9 16

J. Let ABC be a triangle such that cot cot cot 3A B C , then prove that ABC is an equilateral

triangle.

K. If A is not an integral multiple of 2

, prove that

a) tan cot 2cos 2A A ec A b) cot tan 2cot2A A A

L. If is not an integral multiple o f2

, prove that tan 2tan2 4tan4 8cot8 cot

M. If 3A is not an odd multiple of 2

, prove that tan .tan 60 .tan 60 tan3A A A A and hence

find the value of tan6 tan 42 tan66 tan78o o o o .

N. If , are the solutions of the equation cos sina b c ( , ,a b c are non-zero numbers) then

show that

i) 2 2

2sin sin

bc

a b ii)

2 2

2 2sin .sin

c a

a b

O. If A is not an integral multiple of , prove that sin16

cos .cos2 .cos 4 .cos816sin

AA A A A

A and hence

deduce that 2 4 8 16 1

cos .cos .cos .cos15 15 15 15 16



P. If 1

sin sin4

x y and 1

cos cos3

x y , then show that

a) 3

tan2 4

x y b)

7cot

24x y

Q. Prove that 4cos12 cos 48 cos72 cos36o o o o

R. If 4

cos cos5

x y and 2

cos cos7

x y , find the value of 14tan 5cot2 2

x y x y

S. If cos cos ,a b cos 0 , then prove that tan cota b a b .

T. Prove that 4 4 4 43 5 7 3sin sin sin sin

8 8 8 8 2

U. Prove that 2 2 3cos 76 cos 16 cos76 cos16

4

V. If 04

A B , 24

sin25

A B , 4

cos5

A B , find the value of tan2A .

15.

A. If x is acute and sin 10 cos 3 68o ox x find x .

B. Solve 2 27 sin 3cos 4

C. Solve 22cos 3sin 1 0

D. Solve 4sin sin2 sin4 sin3x x x x

E. If 0 , solve 1

cos .cos2 .cos34

F. Find all values of x in , satisfying the equation 21 cos cos .... 38 4x

G. Solve the following equations and write general solution.

i) 24cos 3 2 3 1 cos ii) 2 26tan 2 cos cos2x x x

iii) 2

1 sin 2 sin 3 cos3x x x iv) 2 22sin sin 2 2x x

H. If tan cot,p q , and p q then show that the solutions are in A.P. with C.D p q

. I. Solve

tan 3cot 5sec

J. Solve 21 sin 3sin cos

K. Solve 2 sin cos 3x x

L. Solve sin2 cos2 sin cosx x x x

M. Find the general solution of 1 2

tan ,sec3 3

x x

N. If tan cos cot sin , then prove that 1

cos4 2 2

O. Find the common roots of the equations cos2 sin2 cotx x x and 2 22 cos cos 2 1x x .

P. Solve the equation 26 cos 7 sin cos 0x x x .



Q. If 1

tan tancos

x xx

and 0,2x , find the values of x .

R. If 2

3x y and

3sin sin

2x y then find x and y .

S. Solve sin3 4sin sin sinx x where ,n n z .

16.

A. Prove that 1 14 1sin 2 tan

5 3 2

B. If 1 1 1sin sin sin ,x y z then prove that 4 4 4 2 2 2 2 2 2 2 2 24 2x y z x y z x y y z z x

C. Solve 5 12

arc sin arc sin2x x

. 0x

D. Prove that 2

1 12

1cos tan sin cot

2

xx

x.

E. If 1 1 1cos cos cos ,p q r then prove that 2 2 2 2 1p q r pqr

F. If 1 1 1sin sin sin ,x y z then prove that 2 2 21 1 1 2x x y y z z xyz

G. Solve 1 1sin sin 23

x x

H. If 2 2

1

2 2

1 1tan ,

1 1

x x

x x then prove that 2 sin 2x

I. Prove that 1 1 14 7 117sin sin sin

5 25 125

J. Prove that 1 1 14 5 16sin sin sin

5 13 65 2

K. Prove that 1 1 41cot 9 cos

4 4ec

L. Prove that 1 11 1cos 2tan sin 4tan

7 3

M. If 1 1cos cosp q

a b, then prove that

2 22

2 2

2.cos sin

p pq q

aba b.

N. If , ,a b c are distinct non-zero real numbers having the same sign, prove that

1 1 11 1 1cot cot cot or 2

ab bc ca

a b b c c a

O. If 2

1 1 12 2 2

2 1 2sin cos tan ,

1 1 1

p q x

p q x then prove that

1

p qx

pq.

P. i) If 1 1 1tan tan tan ,x y z then prove that x y z xyz

ii) If 1 1 1tan tan tan2

x y x , then prove that 1xy yz zx



Q. Solve the following equation for x : 1 1cos sin2 6

xx

R. Prove that (i) 1 1 11 1 1

2 5 8 4Tan Tan Tan (ii) 1 1 13 3 8

4 5 19 4Tan Tan Tan

S. Solve 1 1 12

1 1 2tan tan tan

2 1 4 1x x x

17.

(I)

A. If cot ,2

A b c

a find angle B .

B. In ABC , prove that 1 2 3

1 1 1 1

r r r r

C. Show that 21 2 3 r r r r

D. In ABC , show that cos 2b c A s

E. In ABC , if 3 ,a b c b c a bc find A .

F. If 4,a 5,b 7c , find cos2

B.

G. In ABC , find 2 2cos cos2 2

C Bb c .

H. Prove that 2 2cos cosa b C c B b c

I. In ,ABC if 1 1 3

a c b c a b c, show that 60C

J. In ,ABC show that 2 2

2

sin

sin

B Cb c

B Ca

K. Show that 2 2 2cot cot cotabc

a A b B c CR

(II)

A. If 1: : 2 : 5 : 12r R r , then prove that the triangle is right angled at A.

B. In an equilateral triangle, find the value of r

R.

C. If 90oA , show that 2 r R b c

D. If cot : cot : cot 3 : 5 : 72 2 2

A B C, show that : : 6 : 5 : 4a b c

E. If 3 ,b c a then find the value of cot cot2 2

B C.

F. If seca b c , prove that 2

tan sin2

bc A

b c.



G. Prove that 1 2 3

1 2 2 3 3 1

r r ra

r r r r r r

H. Prove

2

2 2 2

cot cot cot2 2 2

cot cot cot

A B Ca b c

A B C a b c

I. 2 2 2

cot cot cot4

a b cA B C

J. Prove that 2

cot cot cot2 2 2

A B C s.

K. If 60 ,C then show that (i) 1a b

b c c a (ii)

2 2 2 20

b a

c a c b

L. Show that in ABC , cos cosa b C c B

M. If 1 2 3, ,p p p are the altitudes of the vertices , ,A B C of a triangle respectively, show that

2 2 21 2 3

1 1 1 cot cot cotA B C

p p p.

N. If : : 7 : 8 : 9a b c , find cos : cos : cosA B C

O. If cot ,cot ,cot2 2 2

A B C are in A.P., then prove that , ,a b c are in A.P.

P. If 2 1 3 1 2 32r r r r r r . Show that 90A .

Q. If 2 2

2 2

sin

sin

a b C

A Ba b, prove that ABC is a right angled.

R. Show that 2 2sin 2 sin 2 2 sinb C c B bc A

S. If 2 2 2sin ,sin ,sin2 2 2

A B C are in H.P., then show that , ,a b c are in H.P.

LONG ANSWER QUESTIONS

18.

A. Let : , :f A B g B C be bijections. Then 1 1 1gof f og .

B. Let :f A B , AI and BI be identity functions on A and B respectively. Then A BfoI f I of .

C. 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 Bfog I and Agof I and, in this case, 1g f .

D. Let 1, , 2, , 4, , 3,f a c d b and 1 2, , 4, , 1, , 3,g a b c d , then show that

1 1 1gof f og .

E. If the function f is defined by 2

3 2, 3

2, 2 2

2 1, 3

x x

f x x x

x x

. Then find the values of

4 , 2.5 , 2 , 4 , 0 , 7f f f f f f



F. Let :f A B be a bijection. Then 1Bfof I and 1

Af of I .

G. Let :f A B , :g B C and :h C D . Then ,ho gof hog of that is, composition of functions

is associative.

H. If : , :f R R g R R are defined by 4 1f x x and 2 2g x x then find

i) gof x (ii) 1

4

agof (iii) fof x (iv) 0go fof

I. Let 1,2,3 , , , , , ,A B a b c C p q r . If : , :f A B g B C are defined by

1, , 2, , 3,f a c b , , , , , ,g a q b r c p then show that 11 1f g g f .

K. If :f Q Q is defined by 5 4f x x x Q then show that f is a bijection and find 1f .

L. If the function f is defined by

2, 1

2, 1 1

1, 3 1

x x

f x x

x x

, then find the values of

a) 3f b) 0f c) 1,5f d) 2 2f f e) 5f

19.

A. Show that 49 16 1n n is divisible by 64 for all positive integers n .

B. 1 1 1 1

...1.3 3.5 5.7 2 1 2 1 2 1

n

n n n

C. 2 1 3 13.5 2n n is divisible by 17.

D. 1 2 3

1.2.3 2.3.4 3.4.5 ...upto terms4

n n n nn

E. 3 3 3 3 3 3

21 1 2 1 2 3....upto terms 2 9 13

1 1 3 1 3 5 24

nn n n

F.

22 2 2 2 2 2 1 2

1 1 2 1 2 3 ....upto terms12

n n nn

G. Use Mathematical Induction to prove the formula 22 3.2 4.2 ... upto terms .2 ,nn n n N .

H. Show that, n N , 1 1 1

....upto terms1.4 4.7 7.10 3 1

nn

n

I. If x and y are natural numbers and x y , using mathematical induction, show that n nx y is

divisible by x y , for all n N

K. Using mathematical induction, show that m mx y is divisible by x y , if m is an odd natural

number and ,x y are natural numbers.

L. Use mathematical induction to prove that 2 1 3 1

2.4 3n n

is divisible by 11, n N .

M.

2 6 112.3 3.4 4.5 ...upto terms

3

n n nn



N. Prove by Mathematical induction, for all n N 2 .....a ar ar upto n terms 1

, 11

na rr

r

O. Show that 4 3 1n n divisible by 9 for all positive integers n .

P. Use mathematical induction to prove the statement,

2

2

3 5 7 2 11 1 1 ...... 1 1

1 4 9

nn

n

20.

A. Show that 2

b c c a a b a b c

c a a b b c b c a

a b b c c a c a b

.

B. Show that 3

2 2

2 2

2 2

a b c a a

b b c a b a b c

c c c a b

C. Find the value of x if

2 2 3 3 4

4 2 9 3 16 0

8 2 27 3 64

x x x

x x x

x x x

D. Show that

2 2 22

22 2 2 3 3 3

2 2 2

2

2 3

2

bc a c ba b c

b c a c ac b a a b c abc

c a b b a ab c

.

E. If

1 2 2

2 1 2

2 2 1

A then show that the adjoint of A is 3 'A . Find 1A .

F. Show that

2

3

2 2 1 1

2 1 2 1 1

3 3 1

a a a

a a a .

G. Show that 2 2 2

3 3 3

a b c

a b c abc a b b c c a

a b c

H. If

1 2 1

0 1 1

3 1 1

A then find 3 23 3A A A I .

I. Show that

2 3

2 3

2 3

1

1

1

a a

b b a b b c c a ab bc ca

c c

J. Show that 3

2

2 2

2

a b c a b

c b c a b a b c

c a c a b

.



K. Show that

2

2 4

2

a a b c a

a b b b c a b b c c a

c a c b c

.

L. If

1 2 3

0 1 4

2 2 1

A then find 1

'A

M. If 1 1 1

2 2 2

3 3 3

a b c

A a b c

a b c

is a non-singular matrix, then show that A is invertible and 1

det

adj AA

A.

21.

A. Solve the following equations by Gauss-Jordan method

3 4 5 18x y z , 2 8 13x y z , 5 2 7 20x y z

B. Solve 3 4 5 18x y z , 2 8 13x y z , 5 2 7 20x y z by using matrix inversion method.

C. Solve 3 4 5 18x y z , 2 8 13x y z , 5 2 7 20x y z by using Cramer’s Rule.

D. Apply the test of rank to examine whether the following equations are consistent.

2 3 8x y z , 2 4x y z , 3 4 0x y z and if consistent find the complete solution.

E. Find the nontrivial solutions if any, for the following system of equations:

2 5 6 0x y z , 3 8 0x y z , 3 4 0x y z

F. Solve the following equations by Gauss-Jordan method

5 6 4 15x y z , 7 4 3 19x y z , 2 6 46x y z

G. Solve the following system of equations by Gauss – Jordan method.

3x y z , 2 2 3x y z , 1x y z .

H. By using Gauss-Jordan method, show that the following system has no solution.

2 4 0x y z , 2 2 5x y z , 3 6 7 2x y z .

I. Examine whether the following systems of equations are consistent or inconsistent and if

consistent find the complete solution. 4x y z , 2 5 2 3x y z , 7 7 5x y z .

J. Show that the following system of equations is consistent and solve it completely:

3x y z , 2 2 3x y z , 1x y z

22.

A. Find the shortest distance between the skew lines 6 2 2 2 2r i j k t i j k and

4 3 2 2r i k s i j k

B. If 1, 2, 1A , 4,0, 3B , 1,2, 1C and 2, 4, 5D , find the distance between AB

and CD.

C. Let , ,a b c be three vectors. Then prove that

i) . .a b c a c b b c a ii) . .a b c a c b a b c



D. Let a i j , b j k , c k i . Find unit vector d such that . 0a d b c d .

E. A line makes angles 1 2 3, , and 4 with the diagonals of a cube. Show that

2 2 2 21 2 3 4

4cos cos cos cos

3.

F. Let 2b i j k , 3c i k . If a is a unit vector then find the maximum value of a b c .

G. For any four vectors , ,a b c and d . .

.. .

a c a da b c d

b c b d and in particular

2 22 2 .a b a b a b .

H. Find the volume of the parallelopiped whose coterminus edges are represented by the vectors

2 3 ,i j k 2i j k and 2i j k .

I. 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

J. Find the equation of the plane passing to the points 2,3, 1 , 4,5,2A B and 3,6,5C .

K. Show that in any triangle the altitudes are concurrent.

23.

A. If A, B, C are angles in a triangle, then prove that sin2 sin2 sin2 4cos sin cosA B C A B C

B. If A, B, C are angles in a triangle, then prove that sin sin sin 4sin sin cos2 2 2

A B CA B C

C. If A B C , then prove that 2 2 2cos cos cos 2 1 sin sin sin2 2 2 2 2 2

A B C A B C

D. In triangle ABC, prove that cos cos cos 4cos cos cos2 2 2 4 4 4

A B C A B C

E. If 2A B C S , then prove that sin sin sin 4cos cos sin2 2 2

S A S B CS A S B C

F. If 0A B C , then prove that 2 2 2cos cos cos 1 2 cos cos cosA B C A B C

G. If 2A B C S , then prove that

cos( ) cos( ) cos( ) cos 4cos cos cos2 2 2

A B CS A S B S C S

H. If A, B, C are angles in a triangle, then prove that 2 2 2sin sin sin 2sin sin cosA B C A B C

I. If 2

A B C , then prove that cos2 cos2 cos2 1 4sin sin sinA B C A B C .

J. If 3

2A B C , then prove that 2 2 2cos cos cos 2 cos cos sinA B C A B C

K. If 0A B C , then prove that sin2 sin2 sin2 4sin sin sinA B C A B C

L. If 2A B C D , then prove that

sin sin sin sin 4cos sin cos2 2 2

A B A C A DA B C D



24.

A. Show that 2 2 2cos cos cos2 2 2

A B Ca b c s

R

B. Prove that 3 3 3cos cos cos 3a B C b C A c A B abc

C. If 2 2 2 28a b c R , then prove that the triangle is right angled.

D. Show that cos cos cos 1r

A B CR

E. In ABC , prove that 1 2 3 4r r r r R .

F. Show that 2 2 2cos cos cos 22 2 2 2

A B C r

R.

G. If 13,a 14,b 15c , show that 65

,8

R 4,r1

21

2r ,

212r and

314r .

H. If 1 2 32, 3, 6r r r and 1,r prove that 3, 4a b and 5c .

I. Show that 2 22 2 2cos sin

2 2

A Ab c b c a

J. Prove that cos cos cosa A b B c C

bc a ca b ab c

K. If 3

cos cos cos ,2

A B C then show that the triangle is equilateral.

L. Show that 2 2 2

2 2 2 2 21 2 3

1 1 1 1 a b c

r r r r

M. Show that 31 2 1 1

2

rr r

bc ca ab r R

N. Show that 3 1 2 4 cosr r r r R B .

O. Show that 2 3 3 11 2

3 1 2

bc r r ca r rab r r

r r r

P. Prove that 2 2 2 2 2 2 2 21 2 3 16r r r r R a b c

Q. If 1 2 3

, ,p p p are altitudes drawn from vertices , ,A B C to the opposite sides of a triangle

respectively, then show that

i)

1 2 3

1 1 1 1

p p p r ii)

1 2 3 3

1 1 1 1

p p p r iii)

2 3

31 2 3

8

8

abcp p p

abcR

R. Show that i) 12 3

2 3

rra r r

r r ii) 1 2

1 21 2

4R r rr r

r r

wish you all the best

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