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

Let the equation of the tangent at a point \( P \) on the parabola \( x^2 - 4x - 4y + 16 = 0 \) be \( 2x - y - 5 = 0 \). If the equation of the normal drawn at \( P \) to this parabola is \( ax + y + c = 0 \), then find \( ac \):

If \( \theta \) is the acute angle between the tangents drawn from the point \( (2, 3) \) to the hyperbola \( 5x^2 - 6y^2 - 30 = 0 \), then \( \tan \theta \) is:

The difference between the focal distances of any point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text{ is } 6. \text{ If } (\sqrt{13}, k) \text{ is an end point of a latus rectum of this hyperbola, then } k = \]

If \( f(x) = \begin{cases} 1 + 6x - 3x^2, & x \leq 1 \\ x + \log_2(b^2 + 7), & x > 1 \end{cases} \) is continuous at all real \( x \), then \( b \) is:

Let \( f(x) \) be a real valued function. If \( f'(x) \) is a constant for all \( x \in \mathbb{R}, f(0) = 1 \) and \( f'(0) = 2 \), then

If \( f : \mathbb{R} \to \mathbb{R} \) is defined by: \[ f(x) = \begin{cases} \sin x - \sin \left( \frac{x}{2} \right), & x<0 \\ \frac{x}{\sqrt{x^2 + \sqrt{x^2}}}, & x>0 \end{cases} \] If \( f \) is continuous on \( \mathbb{R} \), then \( f(0) = \)

If \( f(x) \) is a differentiable function, \( f'(x) \geq 5 \) for \( x \in [2,6] \), \( f(2) = 4 \) and \( f(3) = 15 \), then a possible value of \( f(6) \) is:

If \( y = \log x \), then the value of \( x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y \) at the point \( \left( \sqrt{e}, \sqrt{e} \right) \) is:

Let \( f(x) \) be a differentiable function, \( A(0, \alpha) \) and \( B(8, \beta) \) be two points on the curve \( y = f(x) \). Given \( f(0) = 2 \) and \( f'(4) = -\frac{3}{4} \). If the chord \( AB \) of the curve is parallel to the tangent drawn at the point \( (4, f(4)) \), then \( \beta \) is:

If the tangent drawn to the curve \( y = x^3 - ax^2 + x + 1 \) at each point \( x \in \mathbb{R} \), is inclined at an acute angle with the positive direction of \( X \)-axis, then the set of all possible values of \( a \) is: