Qus : 1 2 The expression $\frac{tanA}{1-cotA}+\frac{cotA}{1-tanA}$ can be written as
1 sinA cosA + 1 2 secA cosecA + 1 3 tanA + cotA 4 secA + cosecA Go to Discussion Solution
Qus : 2 1 Angle of elevation of the top of the tower from 3
points (collinear) A, B and C on a road leading to the
foot of the tower are 30°, 45° and 60°, respectively.
The ratio of AB and BC is
1 $\sqrt(3):1$ 2 $\sqrt(3):2$ 3 $1:\sqrt(3)$ 4 $2:\sqrt(3)$ Go to Discussion Solution According to the given information, the figure should be as follows.
Let the height of tower = h
Qus : 3 1 If $3 sin x + 4 cos x = 5$, then $6tan\frac{x}{2}-9tan^2\frac{x}{2}$
1 1 2 3 3 4 4 6 Go to Discussion Qus : 4 3
Largest value of $cos^2\theta -6sin\theta cos\theta+3sin^2\theta+2 $ is
1 4 2 0 3 $$4+\sqrt{10}$$ 4 $$4-\sqrt{10}$$ Go to Discussion
Solution Qus : 5 1
Number of point of which f(x) is not differentiable $f(x)=|cosx|+3$ in $[-\pi, \pi]$
1 2 2 3 3 4 4 None of these Go to Discussion Solution
Points of Non-Differentiability of \( f(x) = |\cos x| + 3 \)
Step 1: \( \cos x \) is differentiable everywhere, but \( |\cos x| \) is not differentiable where \( \cos x = 0 \).
Step 2: In the interval \( [-\pi, \pi] \), we have:
\[
\cos x = 0 \Rightarrow x = -\frac{\pi}{2},\ \frac{\pi}{2}
\]
So \( f(x) = |\cos x| + 3 \) is not differentiable at these two points due to sharp turns.
✅ Final Answer:
\( \boxed{2 \text{ points}} \)
Qus : 6 2 If A > 0, B > 0 and A + B = $\frac{\pi}{6}$ , then the minimum value of $tanA + tanB$
1 $$\sqrt{3}-\sqrt{2}$$
2 $$\sqrt{3}-2\sqrt{3}$$ 3 $$\frac{2}{\sqrt{3}}$$ 4 $$\sqrt{2}-\sqrt{3}$$ Go to Discussion Solution On differentiating x= tanA + tan(π/6-A)
we get :
dx/dA = sec²A-sec²(π/6-A)
now putting
dx/dA=0
we get
cos²(A) = cos²(π/6-A) so 0≤A≤π/6
therefore
A=π/6-A from here we get A = π/12 = B
so minimum value of that function is
2tanπ/12 which is equal to 2(2-√3)
Qus : 8 3 If $cosec\theta-cot \theta=2$, then the value of $cosec\theta$ is
1 5/2 2 3/5 3 4/5 4 5/4 Go to Discussion
Solution Qus : 9 1 The solution of the equation ${4\cos }^2x+6{\sin }^2x=5$ are
1 $$x=n\pi\pm\frac{\pi}{4}$$ 2 $$x=n\pi\pm\frac{\pi}{3}$$ 3 $$x=n\pi\pm\frac{\pi}{2}$$ 4 $$x=n\pi\pm\frac{2\pi}{3}$$ Go to Discussion Qus : 10 4 The value of $\tan \Bigg{(}\frac{\pi}{4}+\theta\Bigg{)}\tan \Bigg{(}\frac{3\pi}{4}+\theta\Bigg{)}$ is
1 -2 2 2 3 1 4 -1 Go to Discussion Solution
We are given:
\[
\text{Evaluate } \tan\left(\frac{\pi}{4} + \theta\right) \cdot \tan\left(\frac{3\pi}{4} + \theta\right)
\]
✳ Step 1: Use identity
\[
\tan\left(A + B\right) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
\]
But we don’t need expansion — use known angle values:
\[
\tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 + \tan\theta}{1 - \tan\theta}
\]
\[
\tan\left(\frac{3\pi}{4} + \theta\right) = \frac{-1 + \tan\theta}{1 + \tan\theta}
\]
✳ Step 2: Multiply
\[
\left(\frac{1 + \tan\theta}{1 - \tan\theta}\right) \cdot \left(\frac{-1 + \tan\theta}{1 + \tan\theta}\right)
\]
Simplify:
\[
= \frac{(1 + \tan\theta)(-1 + \tan\theta)}{(1 - \tan\theta)(1 + \tan\theta)}
= \frac{(\tan^2\theta - 1)}{1 - \tan^2\theta} = \boxed{-1}
\]
✅ Final Answer:
\[
\boxed{-1}
\]
Qus : 11 4 If $\sin x=\sin y$ and $\cos x=\cos y$, then the value of x-y is
1 $\pi/4$ 2 $n \pi/2$ 3 $n \pi$ 4 $2n \pi$ Go to Discussion Solution
Given:
\[
\sin x = \sin y \quad \text{and} \quad \cos x = \cos y
\]
✳ Step 1: Use the identity for sine
\[
\sin x = \sin y \Rightarrow x = y + 2n\pi \quad \text{or} \quad x = \pi - y + 2n\pi
\]
✳ Step 2: Use the identity for cosine
\[
\cos x = \cos y \Rightarrow x = y + 2m\pi \quad \text{or} \quad x = -y + 2m\pi
\]
? Combine both conditions
For both \( \sin x = \sin y \) and \( \cos x = \cos y \) to be true, the only consistent solution is:
\[
x = y + 2n\pi \Rightarrow x - y = 2n\pi
\]
✅ Final Answer:
\[
\boxed{x - y = 2n\pi \quad \text{for } n \in \mathbb{Z}}
\]
Qus : 12 1 If $a_1, a_2, a_3,...a_n$, are in Arithmetic Progression
with common difference d, then the sum $(sind) (cosec a_1 . cosec a_2+cosec a_2.cosec a_2+...+cosec a_{n-1}.cosec a_n)$ is equal to
1 $$cot a_1 - cot a_n$$ 2 $$sin a_1 - sin a_n$$ 3 $$cosec a_1 - cosec a_n$$ 4 $$a_1-a_n$$ Go to Discussion
Solution Qus : 13 2 In a ΔABC, if $\tan ^2\frac{A}{2}+\tan ^2\frac{B}{2}+\tan ^2\frac{C}{2}=k$ , then k is always
1 $>1$ 2 $\geq 1$ 3 =2 4 =1 Go to Discussion
Solution Qus : 14 3 The general value of $\theta$, satisfying the equation $\sin \theta=\frac{-1}{2},\, \tan \theta=\frac{1}{\sqrt[]{3}}$
1 $$n\pi+\frac{\pi}{6},n\in I$$ 2 $$n\pi+{\lgroup{-1}\rgroup}^n(\frac{7\pi}{6}),n\in I$$ 3 $$2n\pi+\frac{7\pi}{6},n\in I$$ 4 $$2n\pi+\frac{11\pi}{6},n\in I$$ Go to Discussion Qus : 15 1 If
then the value of
is
1 38/3 2 38 3 114 4 None of these Go to Discussion
Solution Qus : 16 4 If tan x = - 3/4 and 3π/2 < x < 2π, then the value of sin2x is
1 7/25 2 -7/25 3 24/25 4 -24/25 Go to Discussion Solution
Qus : 17 3 The value of $\tan 9{^{\circ}}-\tan 27{^{\circ}}-\tan 63{^{\circ}}+\tan 81{^{\circ}}$ is equal to
1 5 2 3 3 4 4 6 Go to Discussion
Solution Qus : 18 3 If cosθ = 4/5 and cosϕ = 12/13, θ and ϕ both in the fourth quadrant, the value of cos( θ + ϕ )is
1 -16/65 2 -33/65 3 33/65 4 16/65 Go to Discussion Solution
Qus : 20 3 Express (cos 5x – cos7x) as a product of sines or cosines or sines and cosines,
1 2 cos4x cosx 2 2 sin 4x sin x 3 2 sin 6x sin x 4 2 cos 6x cos x Go to Discussion Solution
Qus : 21 2 If $32\, \tan ^8\theta=2\cos ^2\alpha-3\cos \alpha$ and $3\, \cos \, 2\theta=1$, then the general value of $\alpha$ =
1 $n\pi\pm\frac{\pi}{3}$ 2 $2n\pi\pm\frac{2\pi}{3}$ 3 $2n\pi\pm\frac{\pi}{3}$ 4 $n\pi\pm\frac{2\pi}{3}$ Go to Discussion Qus : 22 3 If |k|=5 and 0° ≤ θ ≤ 360°, then the number of distinct solutions of 3cosθ + 4sinθ = k is
NIMCET 2021
1 0 2 1 3 2 4 infinite Go to Discussion Qus : 24 3 If $a\, \cos \theta+b\, \sin \, \theta=2$ and $a\, \sin \, \theta-b\, \cos \, \theta=3$ , then ${a}^{2^{}}+{b}^2=$
1 6 2 5 3 13 4 10 Go to Discussion
Solution Qus : 25 3 The value of tan 1° tan 2° tan 3° ... tan 89° is:
1 $0$ 2 $\frac{1}{\sqrt{2}}$ 3 $1$ 4 $2$ Go to Discussion
Solution Qus : 26 2 If $P=sin^{20} \theta + cos^{48} \theta $ then the inequality that holds for all values of is
1 $$P\geq 1$$ 2 $$0<P\leq 1$$ 3 $$1 < P < 3$$ 4 $$0\leq P \leq 1$$ Go to Discussion
Solution Qus : 27 2 If $sin x + a cos x = b$, then $|a sin x - cos x|$ is:
1 $$\sqrt{a^{2}+b^{2}+1}$$ 2 $$\sqrt{a^{2}-b^{2}+1}$$ 3 $$\sqrt{a^{2}+b^{2}-1}$$ 4 None of above Go to Discussion
Solution Qus : 28 1 If $0 < x < \pi $ and $cos x + sin x = \frac{1}{2}$ , then the value of tan x is
1 $\frac{4-\sqrt{7}}{3}$ 2 $\frac{4+\sqrt{7}}{3}$ 3 $\frac{1+\sqrt{7}}{4}$ 4 $\frac{1-\sqrt{7}}{4}$ Go to Discussion
Solution Qus : 29 1 If tan A - tan B = x and cot B - cot A = y, then cot (A - B) is equal to
1 $\frac{1}{x}+\frac{1}{y}$ 2 $\frac{1}{x}-\frac{1}{y}$ 3 $-\frac{1}{x}+\frac{1}{y}$ 4 $-\frac{1}{x}-\frac{1}{y}$ Go to Discussion
Solution Qus : 30 3 The value of sin 20° sin 40° sin 80° is
1 $\frac{1}{2}$ 2 $\frac{\sqrt{3}}{2}$ 3 $\frac{\sqrt{3}}{8}$ 4 $\frac{1}{8}$ Go to Discussion
Solution Qus : 31 3 In a right angled triangle, the hypotenuse is four times the perpendicular drawn to it from the opposite vertex. The value of one of the acute angles is
1 $45^{o}$ 2 $30^{o}$ 3 $15^{o}$ 4 None of these Go to Discussion
Solution Qus : 32 2 If $\prod ^n_{i=1}\tan ({{\alpha}}_i)=1\, \forall{{\alpha}}_i\, \in\Bigg{[}0,\, \frac{\pi}{2}\Bigg{]}$ where i=1,2,3,...,n. Then maximum value of $\prod ^n_{i=1}\sin ({{\alpha}}_i)$.
1 $$\frac{1}{2^n}$$ 2 $$\frac{1}{2^{n/2}}$$ 3 1 4 None of these Go to Discussion
Solution Qus : 33 4 Solve the equation sin2 x - sinx - 2 = 0 for for x on
the interval 0 ≤ x < 2π
1 2 3 4 None of these Go to Discussion
Solution Qus : 34 1 If $\frac{tanx}{2}=\frac{tanx}{3}=\frac{tanx}{5}$ and x + y + z = π, then the
value of tan2 x + tan2 y + tan2 z is
1 38/3 2 38 3 114 4 None of these Go to Discussion
Solution Qus : 35 1 Find the value of sin 12°sin 48°sin 54°
1 1/8 2 1/6 3 1/2 4 1/4 Go to Discussion
Solution Qus : 37 1 The value of $tan(\frac{7\pi}{8})$ is
1 $1-\sqrt{2}$ 2 $1+\sqrt{2}$ 3 $\sqrt{2}+\sqrt{3}$ 4 $\sqrt{2}-\sqrt{3}$ Go to Discussion
Solution Qus : 38 2 The value of
is
1 tanθ - secθ 2 tanθ + secθ 3 cotθ - secθ 4 cotθ + secθ Go to Discussion Qus : 39 4 The value of sin 10°sin 50°sin 70° is
1 1/4 2 1/2 3 3/4 4 1/8 Go to Discussion Solution sin10° sin50° sin70°
= sin10° sin(60°−10°) sin(60°+10°)
= 1/4 sin3x10°
=1/4x1/2=1/8
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then the value of 
is
, then value of 
is