MATHEMATICS

Daily Practice Problems

Target IIT JEE 2025
CLASS : XI/XII/XIII DPP 1 : TRIGONOMETRY

PART - I : SUBJECTIVE QUESTIONS

1. What are the most general values of which satisfy the equations :
1
(a) sin = (b) tan (x – 1) = 3
2

2
(c) tan = – 1 (d) cosec = .
3
(e) 2cot 2 = cosec2

2. Solve sin9 = sin

3. Solve cot + tan = 2cosec

4. Solve sin2 = cos3

5. Solve cot = tan8

6. Solve cot – tan = 2.

7. Solve cosec = cot + 3.

8. Solve tan2 tan = 1

9. Solve tan + tan2 + 3 tan tan2 = 3.

10. Solve sin + sin3 + sin5 = 0.

11. Solve cos + sin = cos 2 + sin 2.

12. Find all values of  between 0° & 180° satisfying the equation cos 6  + cos 4 + cos 2 + 1 = 0 .

13. Solve cos2 x + cos2 2 x + cos2 3 x = 1 .

14. Solve sin2n – sin2(n – 1) = sin2, where n is constant and n  0, 1

15. Solve tan2 – (1 + 3 ) tan + 3 =0

16. Find all the angles between 0° and 90° which satisfy the equation sec 2 .cosec2 + 2 cosec2 = 8

17. Solve 4 cos – 3 sec = 2 tan

   2 
18. Solve: tan x . tan  x   . tan  x   = 3.
 3  3 

19. Solve 3 sin – cos = 2 20. Solve 5 sin + 2 cos = 5

PART - II : OBJECTIVE QUESTIONS

* Marked Questions may have more than one correct option.
1. The solution set of the equation 4sin.cos – 2cos – 2 3 sin + 3 = 0 in the interval (0, 2) is

 3 7    5  3  5    5 11 
(A)  ,  (B)  ,  (C)  , , ,  (D)  , , 
4 4 3 3   4 3 3  6 6 6 

RESONANCE T RIGONOMETRY # 1
2. All solutions of the equation 2 sin + tan = 0 are obtained by taking all integral values of m and n in:
2 2
(A) 2n + , n  (B) n or 2m  ± where n, m 
3 3
 
(C) n or m  ± where n, m  (D) n or 2m  ± where n, m 
3 3

   2 
3. The general solution of the equation tan x + tan  x   + tan  x   = 3 is
 3 3
n  n  n 
(A)  , n  (B)  , n  (C)  , n  (D) none
4 12 3 6 3 12

4. Total number of solutions of equation sinx . tan4x = cosx belonging to (0, ) are :
(A) 4 (B) 7 (C) 8 (D) 5

 
5. If x  0 ,  , the number of solutions of the equation sin 7x + sin 4x + sin x = 0 is:
 2
(A) 3 (B) 5 (C) 6 (D) None

6. The general solution of equation sinx + sin5x = sin2x + sin4x is:

n n n 2 n
(A) ; n  (B) ;n (C) ;n (D) ;n
2 5 3 3

7.* sinx, sin2x, sin3x are in A.P if

(A) x = n/2, n  (B) x = n, n  (C) x = 2n, n  (D) x = (2n +1), n 

8.* sin x + sin2x + sin 3x = 0 if

(A) sin x = 1/2 (B) sin 2x = 0 (C) sin 3x = 3 /2 (D) cos x =  1/2

9.* cos4x cos8x  cos5x cos9x = 0 if

(A) cos12x = cos 14 x (B) sin13 x = 0 (C) sinx = 0 (D) cosx = 0

10. The general solution of the equation 2cos2x = 3.2cos2x  4 is

(A) x = 2n, n  (B) x = n, n  (C) x = n, n  (D) x = n, n 

11. If 2 cos2 ( + x) + 3 sin ( + x) vanishes then the values of x lying in the interval from 0 to 2 are
(A) x = /6 or 5/6 (B) x = /3 or 5/3 (C) x = /4 or 5/4 (D) x = /2 or 5/2

cos 3 1
12. = if
2 cos 2  1 2
 
(A)  = n + , n  (B)  = 2n  , n 
3 3
 
(C)  = 2n ± , n  (D)  = n + , n 
6 6

13. If cos 2 + 3 cos  = 0, then

 17  3    17  3 
(A)  = 2n ±  where  = cos–1  
 (B)  = 2n ±  where  = cos–1  

 4   4 

  17  3 
(C)  = 2n ±  where  = cos–1  
 (D) none of these
 4 

RESONANCE T RIGONOMETRY # 2
14. If sin  + 7 cos  = 5, then tan (/2) is a root of the equation
(A) x2  6x + 1 = 0 (B) 6x 2  x  1 = 0 (C) 6x 2 + x + 1 = 0 (D) x 2  x + 6 = 0

1
15. The most general solution of tan = – 1 and cos = is :
2
7 7
(A) n + , n  (B) n + (– 1)n , n 
4 4
7
(C) 2n  + , n  (D) none of these
4

1
16. A triangle ABC is such that sin(2A + B) = . If A, B, C are in A.P. then the angle A, B, C are
2
respectively.
5     5   5  5 
(A) , , (B) , , (C) , , (D) , ,
12 4 3 4 3 12 3 4 12 3 12 4

17.* sinx  cos2x 1 assumes the least value for the set of values of x given by:
(A) x = n + (1)n+1 (/6) , n  (B) x = n + (1)n (/6) , n 
n
(C) x = n + (1) (/3), n  (D) x = n (1)n (/6) , n 

18.* The general solution of the equation cosx . cos6x = – 1, is :

(A) x = (2n + 1), n  (B) x = 2n, n 
(C) x = (2n – 1), n  (D) none of these

ANSWER KEY
  
1. (a) n + (– 1)n , n  (b) n + + 1, n (c) n – , n 
4 3 4
 
(d) n + (– 1)n , n  (e) n ± , n 
3 4
m ( 2m  1) 
2. , m  or , m  3. 2n ± , n 
4 10 3

 1    1   1 
4.  2n   , n  or 2n – , n  5.  n   , n  6. n   , n 
 2 5 2  2 9  4 2

2   1 
7. 2n + , n  8. (2n + 1) , n  9. n   , n 
3 6  3 3

n  1 2n 
10. , n  or  n   , n  11. 2 n , n or  + , n 
3  3  3 6

12. 30°, 45°, 90°, 135°, 150°

  
13. x = (2 n  1) , n  or x = (2 n  1) , n or x = n ± ,n
4 2 6

m  1 
14. m, m  or , m  or m   , m 
n 1  2 n

RESONANCE T RIGONOMETRY # 3
  
15. n + , n  or n + , n  16. 45° and 60°
3 4

 3 n 
17. n + (– 1)n , n  or n – (– 1)n , n  18. x=  , n
10 10 3 9

   3
19. n + + (– 1)n , n  20. 2n+ , n  or 2n + 2 where  = tan–1 ,n
6 4 2 7

PART - II
1. (D) 2. (B) 3. (C) 4. (D) 5. (B)

6. (C) 7.* (ABCD) 8.* (BD) 9.* (ABC) 10. (B)

11. (A) 12. (B) 13. (A) 14. (B) 15. (C)

16. (B) 17.* (AD) 18.* (AC)

RESONANCE T RIGONOMETRY # 4
 MATHEMATICS
Daily Practice Problems
Target IIT JEE 2025
CLASS : XI/XII/XIII SPECIAL DPP ON TRIGONOMETRY(-2) DPP. NO.- 2
SINGLE CORRECT TYPE
1 3
Q.1 If sin   and cos    , then the general value of  is (n  Z)
2 2
5  7 
(A) 2n  (B) 2n  (C) 2n  (D) 2n 
6 6 6 4

Q.2 If cos p + cosq = 0, then the different values of  are in A.P. where the common difference is
  2 3
(A) (B) (C) (D)
pq pq pq pq

Q.3 If cos  + cos 7 + cos 3 + cos 5 = 0, then  is equal to (n  Z)

(A) n (B) n/2 (C) n/4 (D) n/8

Q.4 If sin , 1, cos 2 are in G.P., then  is equal to (n  Z)

 n 1 
(A) n  ( 1)n (B) n   1 (C) 2n (D) none of these
2 2

    1
Q.5 The sum of all the solutions of the equation cos  cos     cos      ,   [ 0, 6]
3  3  4
100 
(A) 15 (B) 30 (C) (D) none of these
3

Q.6 The total number of solution of sin4x + cos4x = sin x cos x in [ 0, 2] is equal to
(A) 2 (B) 4 (C) 6 (D) none of these

Q.7 Number of solutions of sin 5x + sin 3x + sin x = 0 for 0  x   is

(A) 1 (B) 2 (C) 3 (D) none of these

Q.8 The sum of all the solution of cot  = sin 2, (   n, n integer), 0     is
(A) 3/2 (B)  (C) 3/4 (D) 2

MORE THAN CORRECT TYPE

Q.9 If 4 sin4x + cos4x = 1, then x is equal to (n  Z)

1 2 2n 
(A) n  (B) n   sin (C) (D) 2n 
5 3 4

3 3
Q.10 If sin   sin  cos   cos   1, then  is equal to (n  Z)
 
(A) 2n (B) 2n  (C) 2n  (D) n 
2 2

Dpps on Trigonometry ( - 2) [1]

Q.11 A general solution of tan2   cos 2  1 is (n  Z)
  
(A) n  (B) 2n  (C) n  (D) n 
4 4 4

1
Q.12 If sin x  cos x  y  for x  [ 0, ] , then
y
(A) x   / 4 (B) y = 0 (C) y = 1 (D) x  3  / 4

Q.13 sin   3 cos   6x  x 2  11, 0    4 , x  R, holds for

(A) no values of x and 
(B) one value of x and two values of 
(C) two values of x and two values of 
(D) two point of values of (x, )

Q.14 If sin2x - 2 sin x -1 = 0 has exactly four different solutions in x  [ 0, n], then value/values of n
is/are (n  N)
(A) 5 (B) 3 (C) 4 (D) 6

Q.15 For the smallest values of x and y, the equation 2(sin x + sin y) - 2 cos (x - y) = 3 has a solution
then which of the following is/are true ?
xy xy 1
(A) sin 1 (B) cos  
2  2  2
(C) number of ordered pairs (x, y) is 2
(D) number of ordered pairs (x, y) is 3

MATCH THE COLUMNS

Q.16 Column I (Equation) Column II (Solution)

   
(A) cos22x + cos2x = 1 (P) x  n    n  n  Z
 4  6
n
(B) cos x  3 sin x  3 (Q) x ,nZ
3

2

(C) 1  3 tan x  1  3 tan x  (R) x   2n  1 , n  Z
6
   
(D) tan 3x - tan 2x - tan x = 0 (S) x  2n    2n   , n  Z
 2  6

Dpps on Trigonometry ( - 1) [2]

 MATHEMATICS
Daily Practice Problems
Target IIT JEE 2025
CLASS : XI/XII/XIII SPECIAL DPP ON TRIGONOMETRY(-2) DPP. NO.- 3
SINGLE CORRECT TYPE
Q.1 The number of solutions of 12 cos3 x - 7 cos2 x + 4 cosx = 9 is
(A) 0 (B) 2 (C) infinite (D) none of these

Q.2 The equation cos x + sin x = 2 has

(A) only one solution (B) two solutions
(C) no solution (D) infinite number of solutions

Q.3 The number of solutions of 2 sin2 x + sin2 2x = 2, x  [ 0, 2] is

(A) 4 (B) 5 (C) 7 (D) 6

Q.4 General solution of tan  + tan 4 + tan 7 = tan  tan 4 tan 7 is

(A)  = n / 12, where n  Z (B)  = n / 9, where n  Z
(C)  = n + /12, where n  Z (D*) none of these

Q.5 The number of solutions of sec2 + cosec2 + 2 cosec2 = 8, 0     / 2 is

(A) 4 (B) 3 (C) 0 (D) 2

Q.6 The total number of solutions of tan x + cot x = 2 cosec x in [ 2, 2] is
(A) 2 (B) 4 (C) 6 (D) 8

3 1 3
Q.7 Number of roots of cos2 x  sin x   1  0 which lie in the interval [, ] is
2 4
(A) 2 (B) 4 (C) 6 (D) 8

Q.8 Let   [ 0, 4] satisfy the equation (sin  + 2)(sin  + 3)(sin  + 4) = 6. If the sum of all the
values of  is of the form k, then the value of k is
(A) 6 (B) 5 (C) 4 (D) 2

Q.9 If x, y  [ 0, 2] and sin x + sin y = 2, then the value of x + y is

(A)  (B) /2 (C) 3 (D) none of these

Q.10 For n  Z, the general solution of ( 3  1) sin   ( 3  1) cos   2 is (n  Z)

  n  
(A)   2n   (B)   n  ( 1) 
4 12 4 12
 n  
(C)   2n  (D)   n  ( 1) 
4 4 12

MORE THAN CORRECT TYPE

Q.11 For the equation 1 - 2x - x2 = tan2 (x + y) + cot2 (x + y)
(A) exactly one value of x exists (B) exactly two value of x exists
(C) y  1  n   / 4, n  Z (D) y  1  n   / 4, n  Z

Dpps on Trigonometry ( - 2) [3]

Q.12 If x  y   / 4 and tan x + tan y = 1, then (n  Z)
(A) sin x = 0 always (B) when x  n   / 4 then y  n
(C) when x  n then y  n  (  / 4) (D) when x  n + / 4 then y  n  (  / 4)
Q.13 If x  y  2 / 3 and sin x / sin y = 2, then
(A) the number of values of x  [0, 4] are 4
(B) number of values of x  [0, 4] are 2
(C) number of values of y  [0, 4] are 4
(D) number of values of y  [0, 4] are 8

Q.14 If cos(x   / 3)  cos x  a has real solutions, then

(A) number of integral values of a are 3
(B) sum of number of integral values of a is 0
(C) when a = 1, number of solutions for x  [0, 2] are 3
(D) when a = 1, number of solutions for x  [0, 2] are 2

MATCH THE COLUMNS

Q.15 Column I (Equation) Column II (Number of solutions)

(A) x3 + x2 + 4x + 2 sinx = 0 in 0  x  2 (P) 4
x x x- 2 -x-2
(B) sin e cos e = 2 +2 (Q) 1
(C) sin 2x + cos 4x = 2 (R) 2
(D) 30 |sin x| = x in 0  x  2 (S) 0

INTEGER TYPE
Q.16 Number of values of p for which equation sin3x + 1 + p3 - 3 p sin x = 0 (p > 0) has a root is

Q.17 Number of roots of the equation | sin x cos x |  2  tan2 x  cot 2 x  3, x  [0, 4] , are

Dpps on Trigonometry ( - 1) [4]

 MATHEMATICS
Daily Practice Problems
Target IIT JEE 2025
CLASS : XI/XII/XIII SPECIAL DPP ON TRIGONOMETRY(-2) DPP. NO.- 4
SINGLE CORRECT TYPE
Q.1 The number of solution of the equation tan x tan 4x = 1 for 0 < x <  is
(A) 1 (B) 2 (C) 5 (D) 8

1
Q.2 One root of the equation cos x  x   0 lies in the interval
2
        3 
(A)  0,  (B)   ,0  (C)  ,   (D)  , 
 2  2  2   2 

Q.3 The range of y such that the equation in x, y + cos x = sin x has a real solution is
(A) [-2, 2] (B) [  2, 2] (C) [-1, 1] (D) [-1/2, 1/2]

Q.4 The number of solution of sin x + sin 2x + sin 3x = cos x + cos 2x + cos 3x, 0  x  2, is
(A) 7 (B) 5 (C) 4 (D) 6

Q.5 The equation sin4x + cos4x + sin2x +  = 0 is solvable for

(A)  5 2    1
2 (B)  3    1 (C)  3 2    1 / 2 (D)  1    1

Q.6 Number of ordered pairs which satisfy the equation x2 + 2x sin (xy) + 1 = 0 are (where
y  [ 0, 2] )
(A) 1 (B) 2 (C) 3 (D) 0

2
Q.7 If (1  tan )(1  tan ) sec 2   2tan  0, then the number of values of  in the interval
(-/2, /2) are
(A) 1 (B) 2 (C) 3 (D) 4

   
Q.8 Number of solutions of tan  sin    cot  cos   ,   [ 0, 6] , is
2  2 
(A) 5 (B) 7 (C) 4 (D) 5

5
Q.9 The number of solutions of  cos r x  5 in the interval [ 0, 2] is
r 1

(A) 0 (B) 2 (C) 5 (D) 10

Q.10 The number of values of x for which sin 2x + cos 4x = 2 is

(A) 0 (B) 1 (C) 2 (D) infinite

COMPREHENSION TYPE
Paragraph for Ques. 11 - 13
Consider the cubic equation
x 3  (1  cos   sin ) x 2  (cos  sin   cos   sin )x  sin  cos   0 whose roots are x1, x2
and x3.

Dpps on Trigonometry ( - 2) [5]

Q.11 The value of x12  x 22  x 32 equals
(A) 1 (B) 2 (C) 2 cos  (D) sin (sin   cos )

Q.12 Number of values of  in [0, 2] for which at least two roots are equal
(A) 3 (B) 4 (C) 5 (D) 6

Q.13 Greatest possible difference between two of the roots if   [0, 2] is
(A) 2 (B) 1 (C) 2 (D) 2 2

Paragraph for Ques 14 - 16

Consider the system of equations sin x cos 2y = (a2 - 1)2 + 1, cos x sin 2y = a + 1

Q.14 Number of values of a for which the system has a solution is

(A) 1 (B) 2 (C) 3 (D) infinite

Q.15 Number of values of x  [ 0, 2], when the system has a solution for permissible values of a,
is/are
(A) 1 (B) 2 (C) 3 (D) 4

Q.16 Number of values of y  [ 0, 2], when the system has a solution for permissible values of a,
are
(A) 2 (B) 3 (C) 4 (D) 5

INTEGER TYPE
sin x sin3x sin 9x  
Q.17 Number of solution(s) of the equation    0 in the interval  0, 4 
cos 3x cos 9x cos 27x  

2x 2x
Q.18 Number of solutions of the equation  3 1    3 1  23x is

Dpps on Trigonometry ( - 1) [6]

 MATHEMATICS
Daily Practice Problems
Target IIT JEE 2025
CLASS : XI/XII/XIII SPECIAL DPP ON TRIGONOMETRY(-2) DPP. NO.- 5
SINGLE CORRECT TYPE
Q.1 The equation tan4x - 2 sec2x + a = 0 will have at least one solution if
(A) 1  a  4 (B) a  2 (C) a  3 (D) none of these

  1 
Q.2 The sum of all roots of sin   log3     0 in (0, 2) is
  x 
(A) 3/2 (B) 4 (C) 9/2 (D) 13/3

Q.3 Number of solutions the equation cos() . cos () = 1 has

(A) 0 (B) 2 (C) 1 (D) infinite

MORE THAN CORRECT TYPE

cosec 2
x1 2
Q.4 For 0  x  2, then 2 y  y 1  2
2
(A) is satisfied by eaxctly one value of y
(B) is satisfied by eaxctly two value of x
(C) is satisfied by x for which cos x = 0
(D) is satisfied by x for which sin x = 0

Q.5 If sin2x - a sin x + b = 0 has only one solution in (0, ) then which of the following statements
are correct ?
(A) a  ( ,1]  [ 2,  ) (B) b  ( ,0 ]  [1, )
(C) a = 1 + b (D) none of these

Q.6 If cos3  cos3, then the value of sin  can be given by

   2   2 
(A)  sin  (B) sin     (C) sin    (D) sin   
3   3   3 

Q.7 Which of the following set can be the subset of the general solutions of 1 + cos 3x = 2 cos 2x
(n  Z)
  
(A) n  (B) n  (C) n  (D) 2n
3 6 6

INTEGER TYPE
Q.8 The maximum integral value of a for which the equation a sin x + cos 2x = 2a - 7 has a solution
is

Q.9 Number of solution of the equation sin4x - cos2 x sin x + 2 sin2 x + sin x = 0 in 0  x  3  is

Q.10 Number of roots of the equation (3 + cos x)2 = 4 - 2 sin8x, x  [0, 5 ] are

Dpps on Trigonometry ( - 2) [7]

 ANSWER KEY
DPP-2
Q.1 A Q.2 C Q.3 D Q.4 B Q.5 B Q.6 A Q.7 C
Q.8 A Q.9 AB Q.10 AB Q.11 ACD Q.12 AC Q.13 BD

Q.14 AC Q.15 ABC Q.16 A - R, B - S, C - P, D - Q

DPP-3
Q.1 C Q.2 C Q.3 D Q.4 D Q.5 D Q.6 B Q.7 B

Q.8 B Q.9 A Q.10 A Q.11 AD Q.12 BC Q.13 AC Q.14 ABD

Q.15 A - Q, B - S, C - S, D - P Q.16 1 Q.17 0

DPP-4
Q.1 C Q.2 A Q.3 B Q.4 D Q.5 C Q.6 B Q.7 B

Q.8 B Q.9 B Q.10 A Q.11 B Q.12 C Q.13 A Q.14 A

Q.15 B Q.16 D Q.17 6 Q.18 1

DPP-5
Q.1 C Q.2 C Q.3 C Q.4 ABC Q.5 ABC Q.6 ACD Q.7 BD
Q.8 6 Q.9 4 Q.10 3

n
Dpps on Trigonometry ( - 1) [8]