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Let the set of all $a \in R$ such that the equation $\cos 2x + a \sin x = 2a - 7$ has a solution be $[p, q]$ and $r = \tan 9^\circ - \tan 27^\circ - \frac{1}{\cot 63^\circ} + \tan 81^\circ$, then $pqr$ is equal to ___________.

For \( \alpha,\beta \in \left(0, \dfrac{\pi}{2}\right) \), let \( 3\sin(\alpha+\beta)=2\sin(\alpha-\beta) \) and a real number \( k \) be such that \( \tan \alpha = k \tan \beta \). Then, the value of \( k \) is equal to:

If the value of \(\displaystyle \frac{3\cos 36^\circ + 5\sin 18^\circ}{5\cos 36^\circ - 3\sin 18^\circ} \) is \(\displaystyle \frac{a\sqrt{5} - b}{c} \), where \( a \), \( b \), \( c \) are natural numbers and \( \gcd(a,c)=1 \), then \( a+b+c \) is equal to :

If \( 2 \sin^{3} x + \sin 2x \cos x + 4 \sin x - 4 = 0 \) has exactly \( 3 \) solutions in the interval \( \left[ 0, \dfrac{n\pi}{2} \right], \; n \in \mathbb{N} \), then the roots of the equation \( x^{2} + nx + (n - 3) = 0 \) belong to:

Suppose \( \theta\in\left[0,\dfrac{\pi}{4}\right] \) is a solution of \( 4\cos\theta-3\sin\theta=1 \). Then \( \cos\theta \) is equal to:

If \( y(\theta) = \frac{2 \cos \theta + \cos 2\theta}{\cos 3\theta + 4 \cos 2\theta + 5 \cos \theta + 2} \), then at \( \theta = \frac{\pi}{2} \), \( y'' + y' + y \) is equal to:

If \( \sin x = -\frac{3}{5} \), where \( \pi < x < \frac{3\pi}{2} \), then \( 80(\tan^{2} x - \cos x) \) is equal to:

The sum of the solutions \( x \in \mathbb{R} \) of the equation \( \dfrac{3\cos 2x + \cos^{3} 2x}{\cos^{6} x - \sin^{6} x} = x^{3} - x^{2} + 6 \) is:

For $0 < c < b < a$, let $(a+b-2c)x^2 + (b+c-2a)x + (c+a-2b) = 0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements:

(I)$\quad$ If $\alpha \in (-1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$

(II)$\quad$ If $\alpha \in (0, 1)$, then $b$ may be the geometric mean of $a$ and $c$

$\text{If } \tan A = \frac{1}{\sqrt{x(x^2 + x + 1)}}, \; \tan B = \frac{\sqrt{x}}{\sqrt{x^2 + x + 1}} \quad \text{and} \quad \tan C = (x^{-3} + x^{-2} + x^{-1})^{1/2}, \; 0 < A, B, C < \frac{\pi}{2}, \text{ then}$ $A + B \text{ is equal to:}$

The number of solutions of \( \sin^{2}x + (2 + 2x - x^{2})\sin x - 3(x-1)^{2} = 0 \), where \( -\pi \le x \le \pi \), is \( \_\_\_\_\_\_\_ \).

Let \( |\cos \theta\cos(60-\theta)\cos(60+\theta)\le \frac{1}{8},\ \theta\in[0,2\pi] \). Then, the sum of all \( \theta\in[0,2\pi] \), where \( \cos 3\theta \) attains its maximum value, is:

If \( \alpha \), \( -\frac{\pi}{2} < \alpha < \frac{\pi}{2} \) is the solution of \( 4\cos\theta + 5\sin\theta = 1 \), then the value of \( \tan \alpha \) is:

Let a rectangle \(ABCD\) of sides \(2\) and \(4\) be inscribed in another rectangle \(PQRS\) such that the vertices of the rectangle \(ABCD\) lie on the sides of the rectangle \(PQRS\). Let \(a\) and \(b\) be the sides of the rectangle \(PQRS\) when its area is maximum. Then \( (a+b)^2 \) is equal to:

The number of solutions of the equation \( 4\sin^{2}x - 4\cos^{3}x + 9 - 4\cos x = 0 \); \( x \in [-2\pi, 2\pi] \) is:

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JEE Main 2024 5th April Evening Shift

The number of solutions of \( \sin^{2}x + (2 + 2x - x^{2})\sin x - 3(x-1)^{2} = 0 \), where \( -\pi \le x \le \pi \), is \( \_\_\_\_\_\_\_ \).