Let P be a point on the parabola y² = 4ax, where a > 0. The normal to the parabola at P meets the x -axis at a point Q. The area of the triangle PFQ where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair (a, m) is

If \[ \cos \alpha = k \cos \beta \] then \[ \cot \left( \frac{\alpha + \beta}{2} \right) \] is equal to

If \( A = \left\{ \frac{x}{9x^2 + 20} \right\} \) and \( f: A \to \mathbb{R} \) is defined by \( f(x) = 2x^3 - 15x^2 + 36x - 48 \), then the maximum value of \( f(x) \) is

If one root of the quadratic equation \(a(b-c)x^2+b(c-a)x+c(a-b)=0\) is 1, then the other root is

Let \( \alpha, \beta, \) and \( \gamma \) be real numbers. Consider the following system of linear equations:
\( x + 2y + z = 7 \)
\( x + \alpha z = 11 \)
\( 2x - 3y + \beta z = \gamma \)
Match each entry in List I to the correct entries in List II

List I		List II
(P)	If \( \beta = \frac{1}{2}(7\alpha - 3) \) and \( \gamma = 28 \), then the system has	(1)	a unique solution
(Q)	If \( \beta = \frac{1}{2}(7\alpha - 3) \) and \( \gamma \neq 28 \), then the system has	(2)	no solution
(R)	If \( \beta \neq \frac{1}{2}(7\alpha - 3) \) where \( \alpha = 1 \) and \( \gamma \neq 28 \), then the system has	(3)	infinitely many solutions
(S)	If \( \beta \neq \frac{1}{2}(7\alpha - 3) \) where \( \alpha = 1 \) and \( \gamma = 28 \), then the system has	(4)	\( x = 11, y = -2 \) and \( z = 0 \) as a solution
		(5)	\( x = -15, y = 4 \) and \( z = 0 \) as a solution

The distance ‘s’ in meters travelled by a particle in ‘t’ seconds is given by s = \(\frac{2t^3}{3}-18t+\frac{5}{3}.\) The acceleration when the particle comes to rest is

The function \( f(x) = \frac{\sqrt{3x^2 - 5x - 2}}{2x^2 - 7x + 5} \) has discontinuous points at \( x = \dots \)

If y = a sin x + b cos x, then y² + \((\frac{dy}{dx})^2\) is a

If the points of contact of the tangents drawn from \( (0,0) \) to the curve \( y = x^2 + 3x + 4 \) are \( (\alpha, \beta) \) and \( (\gamma, \delta) \), then \( \beta + \delta = \):

Evaluate the integral \( \int x^3 (\log x)^2 dx \):

Let A=\(\begin{pmatrix} 0&0 &1 \\ 1&0 &0 \\ 0&0 &0 \end{pmatrix}\), B=\(\begin{pmatrix} 0&1 &0 \\ 0&0 &1 \\ 0&0 &0 \end{pmatrix}\) and P=\(\begin{pmatrix} 0&1 &0 \\ x&0 &0 \\ 0&0 &y \end{pmatrix}\) be an orthogonal matrix such that B=PAP^-1 holds. Then