List of top Questions asked in MHT CET

Let \( x \) be the length of each of the equal sides of an isosceles triangle and \( \theta \) be the angle between these sides. If \( x \) is increasing at the rate \( \frac{1}{2} \) m/hour and \( \theta \) is increasing at the rate \( \frac{\pi}{3} \) rad/hour, then the rate at which area of the triangle is increasing when \( x = 12 \) m and \( \theta = \frac{\pi}{4} \) is
The abscissae of the points of the curve \[ y = x^3 \quad \text{are in the interval} \quad [-2, 2], \] where the slope of the tangents can be obtained by the mean value theorem for the interval \( [-2, 2] \), are
The solution set for minimizing the function \( z = x + y \) with constraints \[ x + y \geq 2, \quad x + 2y \leq 8, \quad x \leq 3, \quad x, y \geq 0 \] contains
The distance of the point \( P(3, 8, 2) \) from the line \[ \frac{x}{2} = \frac{y - 3}{4} = \frac{z - 3}{3}, \quad \text{measured parallel to the plane} \quad 3x + 2y - 2z + 15 = 0, \] is
The angle between the lines \[ \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-3}{-1}, \quad \frac{x-1}{2} = \frac{y-3}{3} = \frac{z-1}{4}, \] where \( l, m, n \) are roots of the equation \[ x^2 + x^2 - 4x - 4 = 0, \] is
If \( I(\theta) = \cos \theta_1 \cos \theta_2 \cos \theta_3 \dots \cos \theta_n \), then \[ \tan \theta_1 + \tan \theta_2 + \tan \theta_3 + \cdots + \tan \theta_n = ? \]
Evaluate the limit: \[ \lim_{n \to \infty} \left[ \frac{1}{1 - \pi} + \frac{8}{1 + \pi} + \cdots + \frac{x^2}{n\pi} \right]. \]
The correct simplified circuit diagram for the logical statement \[ \left[ (q \land ( \sim q \lor r)) \land (\sim p \lor (p \land r)) \right] \] is
If \( n^+ C_{n+1} - n^3 C_n = 15(n + 2) \), then \( n = \)
In a triangle ABC, with usual notations, \[ \tan \left( \frac{A}{4} \right) = \frac{5}{6}, \quad \tan \left( \frac{C}{2} \right) = \frac{2}{5}, \] then
If \( \{ (p \land \sim q) \land (p \land r) \} \rightarrow (p \lor q) \) has truth value false, then truth values of the statements \( p, q, r \) are respectively
Evaluate the integral \[ \int_0^{\pi/2} \frac{\cos^2 x \sin^2 x}{\cos^2 x + \sin^2 x} \, dx. \]
The differential equation of all circles having their centres on the line \( y = 5 \) and touching the X-axis is

If \( \sin\left(\frac{\pi}{4}\cot\theta\right) = \cos\left(\frac{\pi}{4}\tan\theta\right) \), then the general solution of \( \theta \) is: 

The angle between the lines whose direction cosines are \[ \frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2} \quad \text{and} \quad \frac{-\sqrt{3}}{4}, \frac{1}{2}, \frac{1}{\sqrt{3}} \] is
With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If \( a = 1 \), then \( \angle A = \)
The value of \( m \in \mathbb{R} \), when the angle between the vectors \[ \mathbf{p} = m\hat{i} - 6\hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{q} = \hat{i} + 2\hat{j} + 2m\hat{k} \] is obtuse, is
The equation of the curve passing through the origin and satisfying the equation
\[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2, \] is
The value of \[ \tan 20^\circ \cdot \tan 80^\circ \cdot \cot 50^\circ = ? \]
The c.d.f. of a discrete random variable \( X \) is Then \( P(X = -3) / P(X \leq 0) \) is:

A random variable \( X \) has p.m.f. \( P(X = x) = \frac{{}^{4}C_x}{2^4}, \quad x = 0, 1, 2, 3, 4 \), and \( \mu \) and \( \sigma^2 \) are the mean and variance respectively of the random variable \( X \), then: 

The locus of point of intersection of the tangents to the circle \( x^2 + y^2 = 16 \), such that the angle between them is 60°, is
The locus of the points represented by \( |z + 3| - |z - 3| = 6 \), where \( z \) is a complex number, is ....

If \( f(x) = \frac{k \sin x + 2 \cos x}{\sin x + \cos x} \) is strictly increasing for all real values of \( x \), then: 

Evaluate the integral: \[ \int \frac{\sin x}{5\sin^2 x + 6\cos^2 x} \, dx \]