1
The value of $${\sin ^2}5^\circ + {\sin ^2}10^\circ + {\sin ^2}15^\circ + {\sin ^2}20^\circ + .... + {\sin ^2}90^\circ $$ is
2
On simplifying $${{{{\sin }^3}A + \sin 3A} \over {\sin A}} + {{{{\cos }^3}A - \cos 3A} \over {\cos A}}$$ we get
3
If $$\sin x + \sin y = a$$ and $$\cos x + \cos y = b$$, then $${\tan ^2}\left( {{{x + y} \over 2}} \right) + {\tan ^2}\left( {{{x - y} \over 2}} \right)$$ is equal to
4
If $$p = \tan \left( { - {{11\pi } \over 6}} \right)$$, $$q = \tan \left( {{{21\pi } \over 4}} \right)$$ and $$r = \cot \left( {{{283\pi } \over 6}} \right)$$, then which of the following is/are correct?
1. The value of p $$\times$$ r is 2.
2. p, q and r are in GP.
Select the correct answer using the code given below.
5
$$\sin A + 2\sin 2A + \sin 3A$$ is equal to which of the following?
I. $$4\sin 2A{\cos ^2}\left( {{A \over 2}} \right)$$
II. $$2\sin 2A{\left( {\sin {A \over 2} + \cos {A \over 2}} \right)^2}$$
III. $$8\sin A\cos A{\cos ^2}\left( {{A \over 2}} \right)$$
Select the correct answer using the code given below.
6
If $$x = \sin 70^\circ \sin 50^\circ $$ and $$y = \cos 60^\circ \cos 80^\circ $$, then what is xy equal to?
7
If $$\sin {\theta _1} + \sin {\theta _2} + \sin {\theta _3} + \sin {\theta _4} = 4$$, then what is the value of $$\cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3} + \cos {\theta _4}$$ ?
8
What is the value of $$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)$$ ?
9
If $$x\cos \theta + y\sin \theta = z$$, then what is the value of $${(x\sin \theta - y\cos \theta )^2}$$ ?
10
If $$\sin 18^\circ = {{\sqrt 5 - 1} \over 4}$$, then what is the value of sin81$$^\circ$$ ?
11
What is $${{1 - \tan 2^\circ \cot 62^\circ } \over {\tan 152^\circ - \cot 88^\circ }}$$ equal to?
12
What is the maximum value of the function $$f(\theta )$$ ?
13
What is the minimum value of the function f($$\theta$$) ?
14
Consider the following statements
1. $$f(\theta ) = 2$$ has no solution.
2. $$f(\theta ) = {7 \over 2}$$ has a solution.
Which of the above statement(s) is/are correct?
15
What is the value of $$\sin {A \over 2}\sin {B \over 2}\sin {C \over 2}$$ ?
16
What is the value of $$\cos \left( {{{A + B} \over 2}} \right)\cos \left( {{{B + C} \over 2}} \right)\cos \left( {{{C + A} \over 2}} \right)$$ ?
17
What is tan ($$\alpha$$ + $$\beta$$) equal to?
18
What is $$\sin (\alpha + \beta )sec\alpha sec\beta $$ equal to?
19
If the equation possesses solution, then what is the minimum value of k?
20
If the equation possesses solution, then what is the maximum value of k?
21
Consider the following statements
1. If ABC is an equilateral triangle, then $$3\tan (A + B)\tan C = 1$$.
2. If ABC is a triangle in which A = 78$$^\circ$$, B = 66$$^\circ$$, then $$\tan \left( {{A \over 2} + C} \right) < \tan A$$
3. If ABC is any triangle, then $$\tan \left( {{{A + B} \over 2}} \right)\sin \left( {{C \over 2}} \right) < \cos \left( {{C \over 2}} \right)$$
Which of the above statements is/are correct?
22
If $$A = (\cos 12^\circ - \cos 36^\circ )(\sin 96^\circ + \sin 24^\circ )$$ and $$B = (\sin 60^\circ - \sin 12^\circ )(\cos 48^\circ - \cos 72^\circ )$$, then what is $${A \over B}$$ equal to?
23
The value of $$\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ $$ is equal to
24
The value of $$\sqrt 3 \cos ec20^\circ - \sec 20^\circ $$ is equal to
25
Angle $$\alpha$$ is divided into two parts A and B such that A $$-$$ B = x and tan A : tan B = p : q. The value of sin x is equal to
26
$$\sqrt {1 + \sin A} = - \left( {\sin {A \over 2} + \cos {A \over 2}} \right)$$ is true, if
27
In triangle ABC, if $${{{{\sin }^2}A + {{\sin }^2}B + {{\sin }^2}C} \over {{{\cos }^2}A + {{\cos }^2}B + {{\cos }^2}C}} = 2$$, then the triangle is
28
Match List-I with List-II and select the correct answer using the code given below the lists
Image
29
If $$\sin A = {3 \over 5}$$, where $$450^\circ < A < 540^\circ $$, then $$\cos {A \over 2}$$ is equal to
30
What is $${1 \over {\sin 10^\circ }} - {{\sqrt 3 } \over {\cos 10^\circ }}$$ equal to?
31
The maximum value of $$\sin \left( {x + {\pi \over 6}} \right) + \cos \left( {x + {\pi \over 6}} \right)$$ in the interval $$\left( {0,{\pi \over 2}} \right)$$ is attained at
32
If $$K = \sin \left( {{\pi \over {18}}} \right)\sin \left( {{{5\pi } \over {18}}} \right)\sin \left( {{{7\pi } \over {18}}} \right)$$, then what is the value of K?
33
The expression $${{\sin \alpha + \sin \beta } \over {\cos \alpha + \cos \beta }}$$ is equal to
34
If $$\sin \theta = 3\sin (\theta + 2\alpha )$$, then the value of tan ($$\theta$$ + $$\alpha$$) + 2 tan$$\alpha$$ is equal to
35
What is the value of tan18$$^\circ$$ ?
36
If $$\tan (\alpha + \beta ) = 2$$ and $$\tan (\alpha - \beta ) = 1$$, then tan(2$$\alpha$$) is equal to
37
Consider the following for triangle ABC
1. $$\sin \left( {{{B + C} \over 2}} \right) = \cos \left( {{A \over 2}} \right)$$
2. $$\tan \left( {{{B + C} \over 2}} \right) = \cot \left( {{A \over 2}} \right)$$
3. $$\sin (B + C) = \cos A$$
4. $$\tan (B + C) = - \cot A$$
Which of the above are correct?
38
If $$\sec \theta - \cos ec\theta = {4 \over 3}$$, then what is (sin$$\theta$$ $$-$$ cos$$\theta$$) equal to?
39
What is the maximum value of the function f(x) = 4sin2 x + 1 ?
40
What is $${{2\tan \theta } \over {1 + {{\tan }^2}\theta }}$$ equal to?
41
If $$\sec (\theta - \alpha )$$, $$\sec \theta $$ and $$\sec (\theta + \alpha )$$ are in AP, where cos $$\alpha$$ $$\ne$$ 1, then what is the value of sin2$$\theta$$ + cos$$\alpha$$ ?
42
If A + B + C = 180$$^\circ$$, then what is $$\sin 2A - \sin 2B - \sin 2C$$ equal to?
43
A is an angle in the fourth quadrant it satisfies the trigonometric equation $$3{(3 - {\tan ^2}A - \cot A)^2} = 1$$. Which one of the following is a value of A?
44
What is/are the solution(s) of the trigonometric equation $$\cos ec\,x + \cos x = \sqrt 3 $$ where 0 < x < 2$$\pi$$ ?
45
If $$\theta = {\pi \over 8}$$, then what is the value of $${(2\cos \theta + 1)^{10}}{(2\cos 2\theta - 1)^{10}}{(2\cos \theta - 1)^{10}}{(2\cos 4\theta - 1)^{10}}$$ ?
46
If cos $$\alpha$$ and cos $$\beta$$ (0 < $$\alpha$$ < $$\beta$$ < $$\pi$$) are the roots of the quadratic equation 4x2 $$-$$ 3 = 0, then what is the value of sec $$\alpha$$ $$\times$$ sec $$\beta$$ ?
47
If $$A = {\sin ^2}\theta + {\cos ^4}\theta $$, then for all real $$\theta$$, which one of the following is correct?
48
What is $${{\sin 5x - \sin 3x} \over {\cos 5x + \cos 3x}}$$ equal to?
49
What is $$\sin 105^\circ + \cos 105^\circ $$ equal to?
50
If x, x $$-$$ y and x + y are the angles of a triangle (not an equilateral triangle) such that tan (x $$-$$ y), tan x and tan (x + y) are in GP, then what is x equal to?
51
ABC is a triangle inscribed in a circle with centre O. Let $$\alpha$$ = $$\angle$$BAC, where 45$$^\circ$$ < $$\alpha$$ < 90$$^\circ$$. Let $$\beta$$ = $$\angle$$BOC. Which one of the following is correct?
52
If $${{\sin (x + y)} \over {\sin (x - y)}} = {{a + b} \over {a - b}}$$, then what is $${{\tan x} \over {\tan y}}$$ equal to?
53
If sin$$\alpha$$ + sin$$\beta$$ = 0 = cos$$\alpha$$ + cos$$\beta$$, where 0 < $$\beta$$ < $$\alpha$$ < 2$$\pi$$, then which one of the following is correct?
54
Suppose cos A is given. If only one value of $$\cos \left( {{A \over 2}} \right)$$ is possible, then A must be
55
If $$\cos \alpha + \cos \beta + \cos \gamma = 0$$, where $$0 < \alpha \le {\pi \over 2}$$, $$0 < \beta \le {\pi \over 2}$$, $$0 < \gamma \le {\pi \over 2}$$, then what is the value of sin$$\alpha$$ + sin$$\beta$$ + sin$$\gamma$$ ?
56
The maximum value of $$\sin \left( {x + {\pi \over 5}} \right) + \cos \left( {x + {\pi \over 5}} \right)$$, where $$x \in \left( {0,{\pi \over 2}} \right)$$, is attained at
57
What is the maximum value of $$16\sin \theta - 12{\sin ^2}\theta $$ ?
58
What is tan25$$^\circ$$ tan15$$^\circ$$ + tan15$$^\circ$$ tan50$$^\circ$$ + tan25$$^\circ$$ tan50$$^\circ$$ equal to?
59
What is $$\cot A + \cos ecA$$ equal to?
60
What is $$\cot \left( {{A \over 2}} \right)$$ $$-$$ $$\tan \left( {{A \over 2}} \right)$$ equal to?
61
If angle C of a triangle ABC is a right angle, then what is tan A + tan B equal to?
62
What is $$\cos 80^\circ + \cos 40^\circ - \cos 20^\circ $$ equal to?
63
If $$2\tan A = 3\tan B = 1$$, then what is $$\tan (A - B)$$ equal to?
64
If $$\tan A - \tan B = x$$ and $$\cot B - \cot A = y$$, then what is the value of $$\cot (A - B)$$ ?
65
What is the minimum value of $${{{a^2}} \over {{{\cos }^2}x}} + {{{b^2}} \over {{{\sin }^2}x}}$$ where a > 0 and b > 0 ?
66
What is $$\sin (\alpha + \beta ) - 2sin\alpha cos\beta + sin(\alpha - \beta )$$ equal to?
67
If $$\cos ec\theta = {{29} \over {21}}$$, where 0 < $$\theta$$ < 90$$^\circ$$, then what is the value of 4 sec$$\theta$$ + 4 tan$$\theta$$ ?
68
Consider the following statements
1. cos $$\theta$$ + sec $$\theta$$ can never be equal to 1.5 .
2. tan $$\theta$$ + cot $$\theta$$ can never be less than 2 .
Which of the above statements is/are correct?
69
What is the length of the chord of a unit circle which subtends an angle $$\theta$$ at the centre?
70
What is the least value of $$25\cos e{c^2}x + 36{\sec ^2}x$$ ?
71
What is the value of $${{\sin 34^\circ \cos 236^\circ - \sin 56^\circ \sin 124^\circ } \over {\cos 28^\circ \cos 88^\circ + \cos 178^\circ \sin 208^\circ }}$$ ?
72
tan54$$^\circ$$ can be expressed as
73
What is the value of $$\theta$$ ?
76
What is $$\cos (\alpha - \beta )$$ equal to?
77
What is $${\sin ^2}(\alpha - \beta ) + 2ab\cos (\alpha - \beta )$$ equal to?
78
If $$\sin \alpha + \cos \alpha = p$$, then what is $${\cos ^2}(2\alpha )$$ equal to?
79
If $$\tan \theta = {1 \over 2}$$ and $$\tan \phi = {1 \over 3}$$, then what is the value of $$(\theta + \phi )$$ ?
80
If $$\cos A = {3 \over 4}$$, then what is the value of $$\sin \left( {{A \over 2}} \right)\sin \left( {{{3A} \over 2}} \right)$$ ?
81
What is the value of $$\tan 75^\circ + \cot 75^\circ $$ ?
82
What is the value of $$\cos 46^\circ \cos 47^\circ \cos 48^\circ \cos 49^\circ \cos 50^\circ .......\cos 135^\circ $$ ?
83
If $$\sin 2\theta = \cos 3\theta $$, where $$0 < \theta < {\pi \over 2}$$, then what is sin$$\theta$$ equal to?
84
If the roots of the equation $${x^2} + px + q = 0$$ are tan 19$$^\circ$$ and tan 26$$^\circ$$, then which one of the following is correct?
85
What is $${(1 + \tan \alpha \tan \beta )^2} + {(\tan \alpha - \tan \beta )^2} - {\sec ^2}\beta {\sec ^2}\beta $$ equal to?
86
If $$\sin \theta + \cos \theta = \sqrt 2 \cos \theta $$, then what is $$(\cos \theta - \sin \theta )$$ equal to?
87
If $$\sin \theta = - {1 \over 2}$$ and $$\tan \theta = {1 \over {\sqrt 3 }}$$, then in which quadrant does $$\theta$$ lie?
88
If $$p = \cos ec\theta - \cot \theta $$ and $$q = {(\cos ec\theta + \cot \theta )^{ - 1}}$$, then which one of the following is correct?
89
$$
\text { What is } \frac{\sqrt{3} \cos 10^{\circ}-\sin 10^{\circ}}{\sin 25^{\circ} \cos 25^{\circ}} \text { equal to ? }
$$
NDA Mathematics 21 April 2024
96
If tan (π cos θ) = cot (π sin θ), $0<\theta<\frac{\pi}{2}$; then what is the value of $8 \sin ^2\left(\theta+\frac{\pi}{4}\right)$ ?
NDA Mathematics 3 September 2023
97
If $\tan \alpha=\frac{1}{7}$, $\sin \beta=\frac{1}{\sqrt{10}}$; $0<\alpha, \beta<\frac{\pi}{2}$, then what is the value of cos (α + 2β) ?
NDA Mathematics 3 September 2023
98
What is the period of the function?
NDA Mathematics 3 September 2023
99
What is the value of p + q?
NDA Mathematics 3 September 2023
102
For how many values of x does $\frac{1}{p}$ become zero?
NDA Mathematics 3 September 2023
103
What is a value of sin 3x + sin 3y?
NDA Mathematics 3 September 2023
104
What is a value of cos3 x + cos3 y?
NDA Mathematics 3 September 2023
105
What is cos 2β equal to ?
NDA Mathematics 16 April 2023
106
What is the value of sec2γ?
NDA Mathematics 16 April 2023
107
What is the minimum value of x ?
NDA Mathematics 16 April 2023
108
At what value of A does x attain the minimum value ?
NDA Mathematics 16 April 2023
109
The given equation can be reduced to
NDA Mathematics 16 April 2023
110
If sin2x = a - b√c, where a and b are natural numbers and c is prime number, then what is the value of a - b + 2c ?
NDA Mathematics 16 April 2023
111
What is the least value of m(θ) ?
NDA Mathematics 16 April 2023
112
Under what condition does m attain the least value ?
NDA Mathematics 16 April 2023
113
If f(x) = x(4x2 - 3), then what is f(sinθ) equal to ?
NDA Mathematics 16 April 2023
114
What is the value of cosec$\left(−\frac{73 \pi}{3}\right)$ ?
NDA Mathematics 4 September 2022
115
What is the value of $\cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)$ ?
NDA Mathematics 4 September 2022
116
What is the value of tan$\left(\frac{3\pi}{8}\right)$ ?
NDA Mathematics 4 September 2022
117
What is cos 36° − cos 72° equal to ?
NDA Mathematics 4 September 2022
118
If sec x = $\frac{25}{24}$ and x lies in the fourth quadrant, then what is the value of tan x + sin x ?
NDA Mathematics 4 September 2022
119
What is the value of tan2 165° + cot2 165° ?
NDA Mathematics 4 September 2022
120
What is the value of $\sin\left(2\text{n}\pi+\frac{5\pi}{6}\right)\sin\left(2\text{n}\pi−\frac{5\pi}{6}\right)$ where n ∈ Z ?
NDA Mathematics 4 September 2022
121
If 1 + 2(sin x + cos x)(sin x − cos x) = 0 where 0 < x < 360°, then how many values does x take ?
NDA Mathematics 4 September 2022
122
What is the maximum value of 3(sin x − cos x) + 4(cos3 x − sin3 x) ?
NDA Mathematics 4 September 2022
123
If f(α) = $\sqrt{\sec^2\alpha−1}$, then what is $\frac{f(\alpha)+f(\beta)}{1−f(\alpha) f(\beta)}$ equal to ?
NDA Mathematics 4 September 2022
124
What is value of cot2 15° + tan2 15° ?
NDA Mathematics 10 April 2022
125
In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to?
NDA Mathematics 10 April 2022
126
If $α + β = \frac{\pi}{4}$ and 2tan α = 1, then what is tan 2β equal to?
NDA Mathematics 10 April 2022
127
If tan (45° + θ) = 1 + sin 2θ, where $-\frac{\pi}{4}< θ <\frac{\pi}{4},$ then what is the value of cos 2θ?
NDA Mathematics 10 April 2022
128
Let sin 2θ = cos 3θ, where θ is acute angle. What is the value of 1 + 4 sin θ? (given that $\sin 18^\circ =\frac{\sqrt 5 - 1}{4}$)
NDA Mathematics 10 April 2022
129
If $\tan θ = - \frac{5}{12},$ then what can be the value of sin θ?
NDA Mathematics 10 April 2022
130
What is the value of $\cos^4\frac{7\pi}{8}+\cos^4\frac{5\pi}{8}?$
NDA Mathematics 10 April 2022
131
What is $\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right)$ equal to?
NDA Mathematics 10 April 2022
132
If tan α + tan β = 1 - tan α.tan β, where tan α tan β ≠ 1, then which of the following is one of the values of (α + β) ?
NDA Mathematics 10 April 2022
133
If (1 + tan θ) (1 + tan 9θ) = 2, then what is the value of tan(10θ)?
NDA Mathematics 10 April 2022
134
What is the value of sin 0° + sin 10° + sin 20° + sin 30° + ⋯ + sin 360°?
NDA Mathematics 10 April 2022
135
What is the maximum value of the functions $f(x) = \frac{1}{\tan x+\cot x},$ where $0 < x < \frac{\pi}{2}?$
NDA Mathematics 10 April 2022
136
If $\tan A=\dfrac{1}{7}$, then what is cos 2A equal to?
NDA Mathematics 18 April 2021
137
If $\rm tan\:x=-\dfrac{3}{4}$ and x is in the second quadrant, then what is the value of sin x ⋅ cos x?
NDA Mathematics 18 April 2021
143
If 7 sinθ + 24 cosθ = 25, then what is the value of (sin θ + cos θ)?
NDA Mathematics 18 April 2021
144
If 3cosθ = 4sinθ, then what is the value of tan (45° + θ)?
NDA Mathematics 18 April 2021
145
The number of integer values of k, for which the equation 2sinx = 2k + 1 has a solution, is
NDA Mathematics 18 April 2021