Trigonometry Questions - JEE

Match List-I (number of solutions in given intervals) with List-II (counts): P) $\cos3x\cos6x=\cos4x\cos7x$, $x\in[0,\pi/2)$ Q) $\sin2x\sin6x=\cos x\cos3x$, $x\in[0,\pi/2]$ R) $\cos3x\cos7x=\cos2x\cos8x$, $x\in[0,\pi/2]$ S) $\sin5x\cos3x=\sin6x\sin2x$, $x\in[0,\pi/2)$ List-II: 1)1, 2)2, 3)3, 4)4, 5)5

$\displaystyle\sum_{n=1}^\infty\cot^{-1}\!\left(\frac{(2n^2+2n+1)(n^2+n+1)}{n^4+2n^3+2n^2+2n+2}\right)=\sec^{-1}\!\left(\frac{5}{\lambda}\right)$. Then $\lambda$ equals:

The value of $x \in \left(0,\dfrac{\pi}{2}\right)$ satisfying $\dfrac{\sqrt{5}-1}{\sin x} + \dfrac{\sqrt{10+2\sqrt{5}}}{\cos x} = 8$ is

If the solution of the equation $\log_{\cos x}\cot x + 4\log_{\sin x}\tan x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, is $\sin^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha$, $\beta$ are integers, then $\alpha + \beta$ is equal to:

The number of solutions of the equation: $x^2 + (x+1)\sin\frac{\pi x}{6} = \frac{3+x}{2}$; $-2 \leq x \leq 0$

The period of the function $f(x) = e^{\sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cos\left(x + \frac{\pi}{3}\right)}$ is:

If $u=\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}+\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}$, then the difference between maximum and minimum values of $u^2$ is given by:

A quadrilateral $ABCD$ in which $AB = a$, $BC = b$, $CD = c$ and $DA = d$ is such that one circle can be inscribed in it and another circle can be circumscribed about it. $\cos A =$

The number of solutions of $\tan^{-1}4x+\tan^{-1}6x=\dfrac{\pi}{6}$, where $-\dfrac{1}{2\sqrt{6}}<x<\dfrac{1}{2\sqrt{6}}$, is equal to

The value of $\displaystyle\prod_{r=1}^{7} \cos\frac{r\pi}{15}$ is

If the angles of elevation of the top of a tower from three collinear points $A$, $B$ and $C$ on a line leading to the foot of the tower are $30°$, $45°$ and $60°$ respectively, then the ratio $AB : BC$ is

Range of $f(x)=\sin^{-1}x+\tan^{-1}x+\sec^{-1}x$ is:

If $[\sin^{-1}\cos x - \sin^{-1}\tan x - 1] = 1$ whose $[.]$ denotes the greatest integer function, then $x$ belongs to:

Roots $r,s,t$ of $x(x-2)(3x-7)=2$ are real and positive. $\tan^{-1}r+\tan^{-1}s+\tan^{-1}t=$

If $\dfrac{\tan(A-B)}{\tan A}+\dfrac{\sin^2C}{\sin^2A}=1$, $A,B,C\in\left(0,\dfrac{\pi}{2}\right)$, then

Two rays are drawn through a point $A$ at an angle of $30°$. A point $B$ is taken on one of them at a distance $a$ from the point $A$. A perpendicular is drawn from the point $B$ to the other ray and another perpendicular is drawn from its foot to $AB$ to meet $AB$ at another point from where the similar process is repeated indefinitely. The length of the resulting infinite polygon line is:

Let $\dfrac{\pi}{2}<\theta<\pi$ and $\cot\theta=-\dfrac{1}{2\sqrt{2}}$. Then the value of $\sin\!\left(\dfrac{15\theta}{2}\right)(\cos8\theta+\sin8\theta)+\cos\!\left(\dfrac{15\theta}{2}\right)(\cos8\theta-\sin8\theta)$ is equal to

The least value of $\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}$ is: (Where $A, B, C$ are interior angles of a triangle)

If the equation $a_1 + a_2 \cos 2x + a_3 \sin^2 x = 1$ is satisfied by every real value of $x$, then the number of possible values of the triplet $(a_1, a_2, a_3)$ is:

For a triangle ABC, the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

If $\pi < \theta < \frac{3\pi}{2}$ and $\cos \theta = -\frac{3}{5}$, then $\tan \left(\frac{\theta}{2}\right)$ is equal to

A tower of height 50 m is located on top of a hill opposite to a tower $T_2$ of height 80 m on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_1$, then the width (in m) of the road between the feet of the towers $T_1$ and $T_2$ is

Let S = \left\{x \in \mathbb{R} : 0 < x < 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}. If n(S) denotes the number of elements in S then:

Let S be the set of all solutions of the equation \cos^{-1}(2x) - 2\cos^{-1}(\sqrt{1-x^2}) = \pi, \quad x \in \left[-\frac{1}{2}, \frac{1}{2}\right]. Then \sum_{x \in S} 2\sin^{-1}(x^2 - 1) \text{ is equal to}

Let $\theta \in \left(0, \frac{\pi}{4}\right)$ and $t_1 = (\tan \theta)^{\tan \theta}$, $t_2 = (\tan \theta)^{\cot \theta}$, $t_3 = (\cot \theta)^{\tan \theta}$ and $t_4 = (\cot \theta)^{\cot \theta}$, then:

In a $\triangle ABC$, $\angle B=\frac{\pi}{3}$ and $\angle C=\frac{\pi}{4}$, also $D$ divides $BC$ internally in the ratio $1:3$, then $\frac{\sin\angle BAD}{\sin\angle CAD}$ is equal to:

$\text{cosec}^{-1}(\cos x)$ exists if:

If $\cot x=\dfrac{5}{12}$ for some $x\in\left(\pi,\dfrac{3\pi}{2}\right)$, then $\sin7x\left(\cos\dfrac{13x}{2}+\sin\dfrac{13x}{2}\right)+\cos7x\left(\cos\dfrac{13x}{2}-\sin\dfrac{13x}{2}\right)$ is equal to

If $\alpha,\beta$ are roots of $x^2-3x+2=0$, then $\tan^{-1}\alpha+\tan^{-1}\beta=$

Values of $x$ satisfying $\sin^{-1}(x^2-5x+7)=2\tan^{-1}1$:

If $(x-a)\cos\theta + y\sin\theta = (x-a)\cos\phi + y\sin\phi = a$, $\tan\frac{\theta}{2} - \tan\frac{\phi}{2} = 2e$ and $\theta, \phi$ are unequal angles less than $360°$, then $y^2$ is equal to:

$f(x)=\cot^{-1}\!\sqrt{x(x+3)}+\cos^{-1}\!\sqrt{x^2+3x+1}$ is defined on set $S$. $S$ equals:

If $E = \cos^2 71° + \cos^2 49° + \cos 71° \cos 49°$, then the value of $10E$ is equal to

All $x$ satisfying $(\sin^{-1}x)^2-(\cos^{-1}x)^2>0$:

\tan^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right) \text{ is equal to}

The area bounded by the curve $y = |\cos^{-1}(\sin x)| + |\frac{\pi}{2} - \cos^{-1}(\cos x)|$ and the $x$-axis, where $\frac{\pi}{2} \leq x \leq \pi$, is equal to

When the elevation of the sun changes from $45°$ to $30°$, the shadow of a tower increases by 60 units, then the height of the tower is

If $\cos \alpha + \cos \beta = a$, $\sin \alpha + \sin \beta = b$ and $\alpha - \beta = 2\theta$, then $\tan \frac{\alpha}{\tan \frac{\alpha}{2}} = \frac{a^2 + b^2 - 3}{\text{}}$

If $P$ be any interior point of the equilateral $\triangle ABC$ of side length $2$ units and also $x_a, x_b, x_c$ be the distances of $P$ from the sides $BC, CA, AB$ respectively, then $x_a + x_b + x_c =$

The value of $\cot^{-1}\left(2^2-\frac{1}{2}\right)+\cot^{-1}\left(2^3+\frac{1}{2^2}\right)+\cot^{-1}\left(2^4+\frac{1}{2^3}\right)+\ldots \infty$ is:

If $\cos^{-1}(2x^2-1)=2\pi-2\cos^{-1}x$, then:

A chimney of 20 m height standing on the top of a building subtends an angle whose tangent is $\frac{1}{4}$ at a distance of 70 m from the foot of the building, then the height of building is

The domain of $f(x)=\sin^{-1}(\log_2(x/3))$ is:

All $x$ satisfying $(\sin^{-1}x)^2-(\cos^{-1}x)^2>0$:

If $\alpha,\beta$ are roots of $x^2-3x+2=0$, then $\tan^{-1}\alpha+\tan^{-1}\beta=$

$\tan^{-1}\!\left(1-x^2-\dfrac{1}{x^2}\right)+\sin^{-1}\!\left(x^2+\dfrac{1}{x^2}-1\right)$, $x\ne 0$, equals:

If $\displaystyle\sum_{n=1}^{\infty} \cot^{-1}\!\left(2 + \frac{n(n+1)}{2}\right) = \tan^{-1} a$, then $a$ is equal to

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45°. It flies off horizontally straight away from O. After 1 second, the elevation of the bird from O is reduced to 30°. The speed (in m/s) of the bird is

A tower is observed from three collinear points A, B, C on ground such that angles of elevation are $\alpha$, $2\alpha$, $3\alpha$. If $AB : BC = ?$

If $p = \cos 55°$, $q = \cos 65°$ and $r = \cos 175°$, then the value of $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{pq}$ is equal to:

Trigonometry Questions (1127)