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JUNIOR MATHS IA CHAPTER WISE IMPARTANT QUESTIONS for IPE

By MNRAO, Maths Sr.Faculty

FUNCTIONS VERY SHORT ANSWER QUESTIONS

1. Which of the following are injections or surjections or bijections ? Justify your answers

i) :f R R→ defined by ( )2 1

3

xf x

+=

ii) ( ): 0,f R → ∞ defined by ( ) 2xf x =

iii) ( ): 0,f R∞ → defined by ( ) logef x x=

iv) [ ) [ ): 0, 0,f ∞ → ∞ defined by ( ) 2f x x=

2. Is ( ) ( ) ( ) ( ){ }1,1 , 2,3 , 3,5 , 4,7g = is a function from { }1,2,3,4A = to { }1,3,5,7B = ? If this is

given by the formula ( )g x ax b= + , then find a and b.

3. 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+ + − =

4. If ( ) ( ) ( )22, , 2f x g x x h x x= = = for all x R∈ , then find ( ( )( )( )fo goh x

5. Find the inverse of the following functions

i. If , , :a b R f R R∈ → defined by ( ) ( )0f x ax b a= + ≠

ii. ( ): 0,f R → ∞ defined by ( ) 5xf x =

iii. ( ): 0,f R∞ → defined by ( ) 2logf x x=

6. Define surjection

7. Find the domains of the following real valued functions.

i. ( )( )( )2

1

1 3f x

x x=

− +

ii. ( )( )1

log 2f x

x=

−

iii. ( )2

1

1f x

x=

−

iv. ( ) 2 25f x x= −

8. Find the ranges of the following real valued functions

i. 2 4

2

x

x

−

−

9. If ( ) ( ) ( ){ }1, 2 , 2, 3 , 3, 1f = − − then find

i. 2 f ii. 2 f+ iii. 2f iv. f

10. Find the domains of the following real valued functions

i. ( )2 2x x

f xx

+ + −=


SHORT ANSWER QUESTIONS

1. If the function f is defined by ( )

2, 1

2, 1 1, then find the valuesof

1, 3 1

x x

f x x

x x

+ >

= − ≤ ≤ − − < < −

(i) ( )3f , (ii) (0)f , (iii) ( )1,5f − , (iv) ( ) ( )2 2f f+ − , (v) ( )5f −

2. If { }: 1f R R− ± → is defined by ( )1

log1

xf x

x

+=

−, then show that ( )

2

22

1

xf f x

x

=

+

3. If : , :f R R g R R→ → defined by ( ) ( ) 23 2, 1f x x g x x= − = + then find

(i) ( )( )1 2gof− , (ii) ( )( )1gof x −

4. 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

− − −=

5. If ( ) ( )1

11

xf x x

x

+= ≠ ±

− then find ( )( )fofof x and ( )( )fofofof x

6. Let : , : :f A B g B C and h C D→ → → . Then ( ) ( )ho gof hog of= , that is, composition of

functions is associative.

7. If : , :f R R g R R→ → are defined by ( ) ( ) 24 1 2f x x and g x x= − = + then find

(i) ( )( )gof x (ii) ( )1

4

agof

+

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

8. If :f Q Q→ is defined by ( ) 5 4f x x= + for all x Q∈ , show that f is a bijection and find 1f −

LONG ANSWER QUESTIONS

1. Let : , :f A B g B C→ → be bijections. Then :gof A C→ is a bijection.

2. Let : , A Bf A B I and I→ be identity functions on A and B respectively. Then A BfoI f I of= =

3. Let : , :f A B g B C→ → be bijections. Then ( )1 1 1gof f og

− − −= .

4. Let :f A B→ be a bijection. Then 1 1

B Afof I and f of I− −= = .

5. 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 B Afog I and gof I= = and in this case, 1g f

−=

2. MATHEMATICAL INDUCTION


Using mathematical induction, prove the following for all n N∈

1. 2.3 3.4 4.5 ...+ + + upto n terms ( )2 6 11

3

n n n+ +=


2. ( )( )

1 1 1 1...

1.3 3.5 5.7 2 1 2 1 2 1

n

n n n+ + + + =

− + +

3. 4 3 1nn− − is divisible by 9

4. 1.2.3 2.3.4 3.4.5 ...+ + + upto n terms ( )( )( )1 2 3

4

n n n n+ + +=

5. 3 3 3 3 3 3

1 1 2 1 2 3....

1 1 3 1 3 5

+ + ++ +

+ + + upto n terms 2

2 9 1324

nn n = + +

6. ( ) ( )2 2 2 2 2 21 1 2 1 2 3 ...+ + + + + + upto n terms

( ) ( )2

1 2

12

n n n+ +=

7. Show that 49 16 1nn+ − is divisible by 64 for all positive integers n .

8. Show that 1 1 1

, ...1.4 4.7 7.10

n N∀ ∈ + + + upto n terms 3 1

n

n=

+.

3. ADDITION AND SCALAR MULTIPLICATION OF VECTORS

VERY SHORT ANSWER QUESTIONS

1. Let 2 3a i j k= + + and 3b i j= + . Find the unit vector in the direction of a b+ .

2. If , 3 2 , 2 2OA i j k AB i j k BC i j k= + + = − + = + − and 2 3CD i j k= + + , then find the vector

OD .

3. 2 5a i j k= + + and 4b i mj nk= + + are collinear vectors, then find m and n .

4. 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− + .

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

6. Find the vector equation of the plane passing through the points

2 5 , 5 ,i j k j k− + − − and 3 5i j− +


1. If , ,a b c are linearly independent vectors, then show that

i. 2 3 , 2 3 4 , 2a b c a b c b c− + − + − − + are linearly dependent

ii. 3 2 ,2 4 ,3 2a b c a b c a b c− + − − + − are linearly independent.

2. If , ,a b c are non-coplanar vectors, then test for the collinearity of the following points whose

position vectors are given

i. 3 4 3 , 4 5 6 , 4 7 6a b c a b c a b c− + − + − − +

3. If , ,a b c are noncoplanar 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− + − +

4. Find the vector equation of the plane which passes through the points 2 4 2i j k+ + , 2 3 5i j k+ +

and parallel to the vector 3 2 .i j k− + Also find the point where this plane meets the line joining

the points 2 3i j k+ + and 4 2 3i j k− +


5. Find the vector equation of the plane passing through points 4 3 , 3 7 10i j k i j k− − + − and

2 5 7i j k+ − and show that the point 2 3i j k+ − lies in the plane

6. If the points whose position vectors are 3 2 , 2 3 4 , 2i j k i j k i j k− − + − − + + and 4 5i j kλ+ +

are coplanar, then show that 146

7λ

−=

7. Let A B C D E F be a regular hexagon with centre ‘O’. Show that

AB + AC + AD + AE + AF=3 AD=6 AO

8. In the two dimensional plane, prove by using vector methods, the equation of the line whose

intercepts on the axes are ‘a’ and ‘b’ is 1x y

a b+ = .


1. If i,j,k are unit vectors along the positive directions of the coordinate axes, then show that the four

points 4 5 , ,3 9 4i j k j k i j k+ + − − + + and 4 4 4i j k− + + are coplanar.

2. In ABC∆ , if ‘O’ is the circumcentre and H is the orthocentre, then show that

(i) OA OB OC OH+ + = (ii) 2HA HB HC HO+ + =

4. MULTIPLICATION OF VECTORS


1. Find the angle between the vectors 2 3i j k+ + and 3 2i j k− +

2. Find the angle between the planes ( )2 2 3r i j k− + = and ( )3 6 4r i j k+ + =

3. Find the radius of the sphere whose equation is ( )2 2 4 2 2r r i j k= − +

4. If ( )( )2 4 2 2 2 0r i j k r i j k− + − + − + = is the equation of the sphere, then find its centre.

5. 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 .

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

7. If 2, 3 4a b and c= = = each of , ,a b c is perpendicular to the sum of the other two vectors,

then find the magnitude of a b c+ +

8. If ( )2, 3 ,6

p q and p qπ

= = = , then find 2

p q×

9. If 2

43

pi j pk+ + is parallel to the vector 2 3i j k+ + , find p .

10. Compute ( ) ( )2 3 4 2j i k i j k× − + + × .

11. Find unit vector perpendicular to both 2 3i j k and i j k+ + + +

12. Find the area of the parallelogram having 2a j k and b i k= − = − + and adjacent sides

13. Find the area of the parallelogram whose diagonals are 3 2 3 4i j k and i j k+ − − +

14. Find the area of the triangle having 3 4 5 7i j and i j+ − + as two of its sides.

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

16. Compute i j j k k i − − −


17. Find the volume of the parallelopiped having coterminus edges , 2i j k i j and i j k+ + − + − .

18. Find t for which the vectors 2 3 , 2 3i j k i j k and j tk− + + − − are coplanar

19. Determine λ , for which the volume of the parallelopiped having conterminus edges

, 3 3i j i j and j kλ+ − + is 16 cubic units.

20. Find the volume of the tetrahedron having the edges , 2i j k i j and i j k+ + − + + .

21. If a,b,c are mutually perpendicular unit vectors, then find the value of [ ]2

a b c

22. 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

23. Let a and b be non-zero, non collinear vectors. If a b a b+ = − , then find the angle between

a and b

24. Find the angle between the planes 2 3 6 5 6 2 9 4x y z and x y z− − = + − =

25. If , ,P Q R and S are points whose position vectors are

, 2 , 2 3 3 2i k i j i k and i j k− − + − − − respectively, then find the component of RS on PQ.

26. Find the equation of the sphere with the line segment joining the points

( ) ( )1, 3, 1 2, 4,1A and B− − as diameter.

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

28. Let 2a i j k= − + and 3 4b i j k= + − . If θ is the angle between a and b , then find sin θ .

29. For any four vectors , ,a b c and d , prove that

( )( ) ( )( ) ( )( ) 0b c a d c a b d a d c d× × + × × + × × =


1. If 0a b c+ + = , then prove that a b b c c a× = × = × .

2. If 2 , 2 4a i j k b i j k and c i j k= + − = − + − = + + , then find ( ) ( ).a b b c× ×

3. Find the vector area and the area of the parallelogram having 2 2 2a i j k and b i j k= + − = − +

as adjacent sides

4. If 13, 5 . 60a b and a b= = = then find a b×

5. Find unit vector perpendicular to the plane passing through the points

( ) ( ) ( )1, 2,3 , 2, 1,1 1, 2, 4and− − .

6. If 2 3 3 ,a i j k b i j k and c i j k= + + = + − = − + , then compute ( )a b c× × and verify that it is

perpendicular to a.

7. If 7 2 3 , 2 8a i j k b i k and c i j k= − + = + = + + , then compute

( ),a b a c and a b c× × × + . Verify whether the cross product is distributive over vector addition.

8. 3 2 , 3 2 , 4 5 2 3 5a i j k b i j k c i j k and d i j k= − + = − + + = + − = + + ,

then compute the following.

i. ( ) ( )a b c d× × × and ii. ( ) ( ).a b c a d b× − ×

9. , ,a b c are non-zero vectors and a is perpendicular to both b and c. If

( ) 22, 3, 4 ,

3a b c and b c

π= = = = , then find a b c

10. If ,a b and c are non-coplanar vectors, then prove that the four points with position vectors

2 3 , 2 3 , 3 4 2a b c a b c a b c+ − − + + − and 6 6a b c− + are coplanar.

11. Find the volume of the tetrahedron whose vertices are ( ) ( ) ( )1, 2,1 , 3, 2,5 , 2, 1,0− and ( )1,0,1− .


12. Show that the equation of the plane passing through the points with position vectors

3 5 , 5 7i j k i j k− − − + + and parallel to the vector 3 7i j k− + is 3 2 0x y z+ − =

13. If ( ) ( ) ( )2 2 21, , , 1, , 1, ,A a a B b b and C c c= = = are non-coplanar vectors and

2 3

2 3

2 3

1

1 0

1

a a a

b b b

c c c

+

+ =

+

,

then show that 1 0a b c + =

14. If 2 3 , 2 , 2a i j k b i j k c i j k= − + = + + = + + then find

( ) ( )a b c and a b c× × × ×

15. Show that the points 2 , 3 5 3 4 4i j k i j k and i j k− + − − − − are the vertices of a right angled

triangle. Also find the other angles.

16. Prove that the smaller angle θ between any two diagonals of a cube is given by 1

cos3

θ = .

17. With the usual notation the following is true in any ABC∆ .

sin sin sin

a b c

A B C= =

18. If 0 ,A B π≤ ≤ , then ( )sin sin cos cos sinA B A B A B− = −

19. Show that ( ) ( ) ( ) 2i a i j a j k k a a× × + × × + × × = for any vector a.

20. Prove that for any three vectors , , , 2a b c b c c a a b a b c + + + =

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

b c c a a b a b c × × × =

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

2a b c b× × = . Find the angles

made by a with each of b and c.


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

2 2 2 2

1 2 3 4

4cos cos cos cos

3θ θ θ θ+ + + =

2. 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.

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

CD.

4. If 2 , 2 , 2a i j k b i j k c i j k= − + = + + = + − find ( ) ( )a b c and a b c× × × ×

5. If 2 3 , 2 3 2a i j k v i j k and c i j k= − − = + − = + − , verify that ( ) ( )a b c a b c× × ≠ × ×

6. If 2 3 , 2 , 4a i j k b i j k c i j k= + − = − + = − + − and d i j k= + + , then compute ( ) ( )a b c d× × ×

7. Let 1 2andθ θ be non-negative real numbers such that 1 2θ θ π+ ≤ . Then

(i) ( )1 2 1 2 1 2cos cos cos sin sinθ θ θ θ θ θ− = +

(ii) ( )1 2 1 2 1 2cos cos cos sin sinθ θ θ θ θ θ+ = −

8. In any triangle prove that the altitudes are concurrent.

9. 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= − − + − −


10. Find the equation of the plane passing through the point ( )3, 2, 1A = − − and parallel to the vectors

2 4 3 2 5b i j k and c i j k= − + = + −

11. 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 × × × = −

12. Let , ,a b c be three vectors. Then

(i) ( ) ( ) ( ). .a b c a c b b c a× × = − (ii) ( ) ( ) ( ). .a b c a c b a b c× × = −

MATRICES

VERY SHORT ANSWER TYPE QUESTIONS

1) If 3 2 8 5 2

2 6 2 4

x y

z a

− − = + − −

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

02. Find the trace of

1 3 5

2 1 5

2 0 1

− −

03. Find the following products wherever possible

2 2 1 2 3 4

1 0 2 2 2 3

2 1 2 1 2 2

− − − −

04.

3 4 913 2 0

0 1 50 4 1

2 6 12

− −

05. If A =

1 2 3

2 5 6

3 7x

−

is a symmetric matrix, then find x.

06. If 2 2 2 2, 1a ib c id

A a b c dc id a b

+ + = + + + = − + −

then find the inverse of A

07. If

1 2 5 1 2

0 2 2 0 2

1 1 1 1 1 1

x y x y

z

a

− − − − = − + −

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

08. If 2 3 1 1 2 1

6 1 5 0 1 3A and B

− = = − −

the find the matrix X such that A + B – X = 0 What is

the order of the matrix X ?

09. If 1 2 3 8

,3 4 7 2

A B

= =

and 2X + A = B then find X.

SHORT ANSWER TYPE QUESTIONS

01. If 1 0 0 1

and E0 1 0 0

I

=

then show that ( )3 3 23 .aI bE a I a bE+ = +


02. If

0 2 1

2 0 2

1 0

A

x

= − − −

is a skew symmetric matrix, then find x.

03. FI

1 5 3

2 4 0

3 1 5

A

= − −

and

2 1 0

0 2 5

1 2 0

B

− = −

then find '3 4A B−

04. If

1 2 2

2 1 2

2 2 1

A

− − − = − −

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

05. If

3 3 4

2 3 4

0 1 1

A

− = − −

then show that 1 3A A− =

06.

1 2 3

2 3 4

0 1 2

07. If cos sin

sin cosA

θ θ

θ θ

= −

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

sin cos

nn n

An n

θ θ

θ θ

= −

08. If

1 22 1 2

3 01 3 4

5 4

A and B

− − = = − −

then verify that (AB)’ = B’A’

LONG ANSWER TYPE QUESTIONS

01. 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 abc = -1

02. 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

+ + + + = + + + +

03. Show that

2 2 2 2

2 2 2

2 2 2

2

2

2

a b c bc a c b

b c a c ac b a

c a b b a ab c

−

= − −

= ( )2

3 3 3 3a b c abc+ + −

04. 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

− + + + − + = + + + + + −

05. ( )( )( )( )

2 3

2 3

2 3

1

1

1

a a

b b a b b c c a ab bc ca

c c

= − − − + =


06. 9x y z+ + =

2 5 7 52x y z+ + =

2 0x y z+ − =

07. 2 8 13x y z− + =

3 4 5 18x y z+ + =

5 2 7 20x y z− + =

08. 9x y z+ + =

2 5 7 52x y z+ + =

2 0x y z+ − =

09. 0x y z+ − =

2 0x y z− + =

3 6 5 0x y z+ − =

10. 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

+ + + + + + = + + +

11. 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

− − − − = + + − −

12. 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

− − − − − − = − − −

13. 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 A is invertible and 1

det

AdjAA

A

− =

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

3

2 2 3

1

x y z

x y z

x y z

+ + =

+ − =

+ − =

15. Solve the following simultaneous linear equations by using ‘Cramer’s rule.

3 4 5 18x y z+ + =

2 8 13x y z− + =

5 2 7 20x y z− + =

16. 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− + =

17. Let A and B are invertible matrices. Then 1, 'A A− and AB are invertible further.

i) ( )1

1A A−− = ii) ( ) ( )

1 11 1A A

− −= iii) ( )1 1 1AB B A

− − −=


5. TRIGONOMETRIC RATIOS AND TRANSFORMATIONS


1. If ( )cos 0 1t tθ = < < and θ does not lie in the first quadrant, find the values of

(i) sin θ (ii) tan θ

2. If A,B,C are angles of a triangle, then prove that 2 3

cos cos 02 2

A B C A C+ + − + =

3. If A,B,C,D are angles of a cyclic quadrilateral, then prove that

(i) sin sin sin sinA C D B− = −

4. Draw the graph of ( )sin 2 ,y x in π π= −

5. Find the periods of the functions

(i) ( ) sinf x x=

6. Find the value of (i) 2 21 1sin 82 sin 22

2 2

o o−

(ii) 2 21 1cos 112 sin 52

2 2

o o−

7. Find the period of the function defined by ( ) 4 4sin cosf x x x= + for all x R∈ .

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

x xπ

+ + +

over R.

9. Find the maximum and minimum values of the following functions over R.

(i) ( ) 5sin 12cos 13f x x x= + −

10. Prove that (i) 2 2 5 1sin 24 sin 6

8

o o −− =

11. If 4

tan3

A = , find the values of

(i) sin 2A (ii) cos 2A

12. If 12

tan5

A−

= and 630 720o oA< < find the values of

(i) sin2

A (ii) cos

2

A

13. If θ is not an odd multiple of 2

π and if tan 1θ ≠ − , then show that

1 sin 2 cos 2tan

1 sin 2 cos 2

θ θθ

θ θ

+ −=

+ +

14. If tanb

aθ = , then prove that cos 2 sin 2a b aθ θ+ = .

15. Prove that 1 3

4sin10 cos10o o

− = .

16. Find the extreme values of the following functions over R.

(i) 2 2

cos . cos .cos3 3

x x xπ π

+ −

17. If cos 0, tan sin mθ θ θ> + = and tan sin nθ θ− = , then show that 2 2 4m n mn− =

18. Prove that 2 1 1 3 1sin 52 sin 22

2 2 4 2

o o+

− =


19. Show that cos 42 cos78 cos162 0o o o+ + =

20. Prove that 1

sin 21 cos9 cos84 cos64

o o o o− =

21. Find the values of 2 2cos 45 sin 15o o−


1. If 4

tan3

θ−

= and θ does not lie in 4th quadrant, prove that

5sin 10cos 9sec 16cos 4cot 0ecθ θ θ θ θ+ + + + =

2. Prove that ( ) ( ) ( )4 2 6 63 sin cos 6 sin cos 4 sin cos 13θ θ θ θ θ θ− + + + + =

3. If 2sin

1 cos sinx

θ

θ θ=

+ +, find the value of

1 cos sin

1 sin

θ θ

θ

− +

+

4. If 3cos sinec aθ θ− = and 3sec cos bθ θ− = , then prove that ( )2 2 2 2 1a b a b+ =

5. If 225oA B+ = , then prove that cot cot 1

.1 cot 1 cot 2

A B

A B=

+ +

6. If 04

A Bπ

< < < , ( ) ( )7 12

sin , cos25 13

A B A B+ = − = , find the value of tan 2A

7. If ( ) ( )cos ,cos , cosθ α θ θ α− + are in H.P., Then prove that 2cos 1 cosθ α= +

8. Prove the following

(i) 2 4 8 1

cos cos .cos7 7 7 8

π π π=

9. If 3

cos5

α = and 5

cos13

β = and ,α β are acute angles, then prove that

(i) 2 16cos

2 65

α β+ =

10. If A is not an integral multiple of π , prove that

sin16

cos .cos 2 .cos 4 .cos816sin

AA A A A

A= and hence deduce that

2 4 8 16 1cos .cos .cos .cos

15 15 15 15 16

π π π π=

11. Prove that ( )4 cos66 sin 84 3 15o o+ = +

12. If 1

sin sin4

x y+ = and 1

cos cos3

x y+ = , then show that

(i) 3

tan2 4

x y+=

13. If neither ( )15oA − nor ( )75o

A − is an integral multiple of 180o , prove that

( ) ( ) 4cos 2cot 15 tan 15

1 2sin 2

o o AA A

A− + + =

−

14. Prove that (if none of the denominators is zero)

2.cot ,cos cos sin sin

2sin sin cos cos

0 ,

n n n A Bif n is evenA B A B

A B A Bif n is odd

− + + + = − −


15. If ( ) ( )sec sec 2secθ α θ α θ+ + − = and cos 1α ≠ , then show that cos 2 cos2

αθ = ±

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

cos cos cos3 3

x y zπ π

θ θ θ

= + = +

for some

Rθ ∈ , then show that 0xy yz zx+ + =

17. If 270oA B C+ + = , then prove that

(i) sin 2 sin 2 sin 2 4 sin sin cosA B C A B C+ − = −

18. If 2

sec tan3

θ θ+ = , find the value of sinθ and determine the quadrant in which θ lies.

19. If cos sin 2 cosθ θ θ+ = , prove that cos sin 2 sinθ θ θ− = .

20. If 45oA B+ = then prove that

(i) ( )( )1 tan 1 tan 2A B+ + =

21. If 12 3

sin , cos13 5

A B= = and neither A nor B is in 1st quadrant, then find the quadrant in which

A+B lies.

22. Let ABC be a triangle such that cot cot cot 3A B C+ + =

23. Prove that (i) 5 1

sin184

o −=

24. For A R∈ , prove that

(i) ( ) ( )1

cos .cos 60 cos 60 cos34

A A A A+ − = and hence deduce that

(ii) 2 3 4 1

cos cos cos cos9 9 9 9 16

π π π π=

25. If ,α β are the solutions of the equation cos sina b cθ θ+ = (a,b,c are non-zero real numbers) then

show that

(i) 2 2

2sin sin

bc

a bα β+ =

+

(ii) 2 2

2 2sin .sin

c a

a bα β

−=

+

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

8 8 8 8 2

π π π π+ + + =

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

4

o o o o+ − =

28. If sin sinx y a+ = and cos cosx y b+ = , find the value of

(i) tan2

x y+

29. For any Rα ∈ , prove that

( ) ( ) ( )2 2 2 1cos 45 cos 15 cos 15

2

o o oα α α− + + − − =

30. Suppose ( )α β− is not an odd multiple of ,2

mπ

is a non zero real number such that

( )( )

sin 11

cos 1

mm and

m

α β

α β

− −≠ − =

− +. Then prove that tan .tan

4 4m B

π πα

− = +

31. If A,B,C are angles of a triangle, prove that

(i) cos 2 cos 2 cos 2 4cos cos cos 1A B C A B C+ + = − −


32. If A, B, C are angles in a triangle, then prove that

(i) sin sin sin 4cos cos cos2 2 2

A B CA B C+ + =

33. If 90oA B C+ + = , then show that

(i) 2 2 2sin sin sin 1 2sin sin sinA B C A B C+ + = −

34. If A, B, C are angles of a triangle, then prove that

2 2 2sin sin sin 1 2cos cos sin2 2 2 2 2 2

A B C A B C+ − = −

35. If ( ) ( )cos cos ,cos 0, sin 0a bθ α θ α θ α+ = − ≠ ≠ , then prove that

( ) ( )tan cota b a bθ α+ = −

36. Prove that tan .tan tan tan 34 3

A A A Aπ π

+ − =

and deduce that

tan 6 tan 42 tan 66 tan 78 1o o o o =

37. If 9

πα = , then prove that 16sin sin 2 sin 3 sin 4 3α α α α = and

16cos cos 2 cos3 cos 4 1α α α α =



(i) 2 2 2cos cos cos 2 1 sin sin sin2 2 2 2 2 2

A B C A B C + + = +

2. In triangle ABC, prove that

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

A B C A B Cπ π π− − −+ + =

(ii) sin sin sin 1 4cos cos sin2 2 2 4 4 4

A B C A B Cπ π π− − −+ − = +


( ) ( ) ( )cos 2 cos 2 cos 2 cos 2 4cos cos cosA B C D A B A C A D+ + + = + + +

4. If 2A B C S+ + = , then prove that

(i) ( ) ( )sin cos cos 1 4cos cos cos2 2 2

S A S B CS A S B C

− −− + − + = − +

5. If A, B, C are angles in a triangle, then prove that

sin sin sin 1 4sin , sin , sin2 2 2 4 4 4

A B C A B Cπ π π− − −+ + = +

6. 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− + − + − + =


6. TRIGONOMETRIC EQUATIONS


1. Solve the following equations

(i) [ ]5 1

cos 2 , 0, 24

θ θ π+

= ∈

(ii) [ ]2 3cos , 0,

4θ θ π= ∈

2. Find the general solution of the following equation

(i) 2 25cos 7sin 6θ θ+ =

3. Find the general solution of the following equation

(i) 3 1

sin ,cos2 2

θ θ−

= =


1. Solve the following equation and write general solution

(i) ( )24cos 3 2 3 1 cosθ θ+ = +

2. If 2 3

sin ,3 2

x y and x sin yπ

+ = + = find x and y

3. Solve the following and write the general solution (i) sin 7 sin 4 sin 0θ θ θ+ + =

(ii) sin 3 cos 2x x+ =

4. Solve the following equation

(i) tan sec 3,0 2θ θ θ π+ = ≤ ≤

5. Solve sin sin 2 sin 3 cos cos 2 cos3x x x x x x+ + = + +

6. If 2sin 3 sin 2cos sin 2 2cosx x x x x+ + = + , find the general solution

7. Solve the equation 26 cos 7 cos 0x sin x x− + + =

8. If 1

tan tancos

x xx

= + and [ ]0, 2x π∈ find the values of x .

9. If x is acute and ( ) ( )sin 10 cos 3 68o ox x+ = − find x .

10. Solve 21 sin 3sin cosθ θ θ+ =

11. If 1 2,θ θ are solutions of the equation cos 2 sin 2a b cθ θ+ = ,

1 2tan tan 0and a cθ θ≠ + ≠ then find the values of

(i) 1 2tan tanθ θ+ (ii) 1 2tan .tanθ θ

12. If 0 θ π< < , solve 1

cos .cos 2 .cos34

θ θ θ =

13. If [ ]1

cos cos .cos , 0,63 3 4

x x x xπ π

π

+ − = ∈

, then find the sum of all the solutions of the

equations.

14. Solve the equation sin 7 sin 4 sin 0 02

πθ θ θ θ

+ + = ≤ ≤


15. If cos 2 sin 2a b cθ θ+ = has solutions 1 2andθ θ such that 1 2tan tanθ θ≠ and if 0a c+ ≠ , then

prove that ( )1 2tanb

aθ θ+ =

16. If tan sin cot cos2 2

π πθ θ

=

, then prove that

1sin

4 2

πθ

+ = ±

17. Solve the equation tan tan 2 tan 3 0x x x+ + =

7. INVERSE TRIGONOMETRIC FUNCTIONS


1. Evaluate the following

(i) 1 3

2Sin−

−

(iv) ( )1 3Cot− −

2. Evaluate the following

(i) 1 3sin

2 2Sin

π − −

−

3. Find the values of

(i) 1 4sin 2

3Sin

−

4. Find the values of the following

(i) 1 1

2Sin

− −

(ii) 1 3

2Cos−

−


(i) 1 4sin

3Sin

π−

(ii) 1 4tan

3Tan

π−


(i) 1 12 2cos

3 3Cos Sin− −

− −

(ii) ( ) ( )2 1 2 1sec 3 cos 2Cot ec Tan− −+


1. Prove that

(i) 1 1 13 12 33

5 13 65Sin Cos Cos

− − −+ =

2. Find the values of

(i) 1 13 12sin

5 13Cos Cos− −

+

3. Show that ( ) ( )2 1 2 1sec 2 cos 2 10Tan ec Cot− −+ =


4. Prove that

(i) 1 1 11 1

2 5 8 4Tan Tan Tan

π− − −1+ + =

(ii) 1 1 1 11 201(18)

7 8 43Tan Tan Cot Cot

− − − −1+ = +

6. Prove that 1 11 1 2tan tan

4 2 4 2

a a bCos Cos

b b a

π π− − + + − =

7. Solve for x :

(i) ( )1 1 11Sin x Sin x Cos x− − −− + =

(ii) 1 11sin os 1

5Sin C x

− − + =

8. Prove that

(i) 1 1 13 5 3232

5 13 325Sin Cos Cos

− − − − =

(ii) 1 14 12

5 3 2Sin Tan

π− −+ =

9. If 1 1 1Cos p Cos q Cos r π− − −+ + = , then prove that 2 2 2 2 1p q r pqr+ + + =

10. If 2

1 1 1

2 2 2

2 1 2

1 1 1

p q xSin Cos Tan

p q x

− − − −− =

+ + − , then prove that

1

p qx

pq

−=

+

11. If 1 1 1Sin x Sin y Sin z π− − −+ + = , then prove that

2 2 21 1 1 2x x y y z z xyz− + − + − =

12. (i) If 1 1 1Tan x Tan y Tan z π− − −+ + = , then prove that x y z xyz+ + =

(ii) If 1 1 1

2Tan x Tan y Tan z

π− − −+ + = , then prove that 1xy yz zx+ + =

13. Solve the following equation for x .

(i) 2

1 1 1

2 2 2

2 1 23 4 2

1 1 1 3

x x xSin Cos Tan

x x x

π− − −−− + =

+ + −

(ii) 1 1

tan cos sin cot , 02

arc arc xx

= ≠

14. If 0, 0 1x y and xy< < < , then 1 1 1

1

x yTan x Tan y Tan

xy

− − − ++ =

−

15. Prove that 1 1 14 7 117

5 25 125Sin Sin Sin

− − −+ =

16. Prove that 1 1 14 5 16

5 13 25 2Sin Sin Sin

π− − − + + =

17. Prove that 1 14 12

5 3 2Sin Tan

π− −+ =

18. If 1 1 1Sin x Sin y Sin z π− − −+ + = , 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+ + + = + +

19. If 1 1P qCos Cos

a bα− −+ = , then prove that

2 22

2 2

2.cos sin

p pq q

a ab bα α− + =

20. Prove that ( ){ }2

1 1

2

1cos sin

2

xTan Cot x

x

− − + = +


21. Prove that arc 34

sec arc cos 175 4

arc ecπ

+ =

22. If [ )0,1x ∈ , then prove that

2

1 1

2 2 2

1 2 1 1 2tan

2 1 2 1 1

x x xSin Cos

x x x

− − −+ =

+ + −

23. Find the number of ordered pairs ( ),x y satisfying the equations 1 1 2

3Sin x Sin y

π− −+ = and

1 1

3Cos x Cos y

π− −− = .

24. Solve : ( ) ( )1 1 1 81 1

31Tan x Tan x Tan

− − −+ + − =

8. HYPERBOLIC FUNCTIONS


1. If 3

sinh4

x = , find ( )cosh 2x and ( )sinh 2x

2. Prove that (i) ( ) ( ) ( )cosh sinh cosh sinhn

x x nx nx− = − , for any n R∈

3. Prove that tanh tanh

2cossec 1 sec 1

x xechx

hx hx+ = −

− + for 0x ≠

4. For any x R∈ , Prove that ( )4 4cosh sinh cosh 2x x x− =

5. For ,x y R∈ prove that

(i) ( )sinh sinh cosh cosh sinhx y x y x y+ = +

(ii) ( )cosh cosh cosh sinh sinhx y x y x y+ = +

6. For any x R∈ prove that ( )1 2sinh log 1ex x x− = + +

7. If 5

cosh2

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

8. If ,4 4

π πθ

− ∈

and log cot4

exπ

θ

= +

, then prove that

(i) cosh sec 2x θ= and (ii) sinh tan 2x θ= −

9. If sinh 5x = , show that ( )log 5 26ex = +

10. Show that 1 1 1tanh log 3

2 2e

− =

11. If 1

tanh4

x = , then prove that 1 5

log2 3

ex

= +

9. PROPERTIES OF TRIANGLES


1. If 2 ., 3 ., 4 ,a cms b cms c cms= = = then find cos A.

2. If the angles are in the ration 1 : 5 : 6, then find the ratio of its sides


3. Prove that ( )cos sin cos sinb a C A a A C− =

4. Show that 2 2cos cos2 2

C Bb c s+ =

5. In ABC∆ , prove that 1 2 3 4r r r r R+ + − =

6. In ABC∆ , if ( )( ) 3a b c b c a bc+ + + − = find A

7. In ABC∆ , show that ( )cos 2b c A s+ =∑

8. In ABC∆ , show that ( )( )

2 2

2

sin

sin

B Cb c

a B C

−−=

+

9. Show that 2 2sin 2 sin 2 2 sinb C c B bc A+ =


1. Prove that ( )2 sin

0sin sin

B Ca

B C

−=

+∑

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

bc a ca b ab c+ = + = +

3. If 60oC = , then show that

(i) 1a b

b c c a+ =

+ +

(ii) 2 2 2 2

0b a

c a c b+ =

− −

4. In ABC∆ , prove that 1 2 3 4 cosr r r r R C+ + − =

5. Show that 2

cos cos cosa A b B c CR

∆+ + =

6. Prove that ( )( )( ) 2

1 2 2 3 3 1 4r r r r r r Rs+ + + =


1. Prove that 2

cot cot cot2 2 2

A B C s+ + =

∆

2. For any angle θ , show that ( ) ( )cos cos cosa b C c Bθ θ θ= + + −

3. If cos cos cos 3/ 2A B+ + = , then show that the triangle is equilateral

4. If cot ,cot , cot2 2 2

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

5. Prove that 3 2 2

1 2 3

1 1 1 1 1 1 4abc R

r r r r r r r s

− − − = =

∆

6. Show that cos cos cos 1r

A B CR

+ + = +

7. 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)

( )2 3

1 2 3 3

8

8

abcPP P

R abc

∆= =

8. If 13, 14, 15a b c= = = , show that 1 2

65 21, 4, , 12

8 2R r r r= = = = and 3 14r =


9. If 1 2 32, 3, 6r r r= = = and 1r = , Prove that 3, 3 5a b and c= = =

10. In , tan cot2 2

B C b c AABC prove that

b c

− − ∆ =

+

11. Show that 2 2 2cot cot cotabc

a A b B c CR

+ + =

12. In ABC∆ , if 1 1 3

a c b c a b c+ =

+ + + +, show that 60oC =

13. Prove that 2 2 2

cot cot cot4

a b cA B C

+ ++ + =

∆

14. Prove that ( ) ( ) ( )3 3 3cos cos cos 3a B C b C A c A B abc− + − + − =

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

2 2 2

1 2 3

1 1 1 cot cot cotA B C

p p p

+ ++ + =

∆

16. Show that 31 2 1 1

2

rr r

bc ca ab r R+ + = −

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