List of top Questions asked in MHT CET

Evaluate the integral \[ \int \cos \left( \frac{\pi x}{6} \right) \cos \left( \frac{\pi x}{3} \right) \, dx. \]

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 correct simplified circuit diagram for the logical statement \[ \left[ (q \land ( \sim q \lor r)) \land (\sim p \lor (p \land r)) \right] \] 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 \( n^+ C_{n+1} - n^3 C_n = 15(n + 2) \), then \( n = \)

If \( I(\theta) = \cos \theta_1 \cos \theta_2 \cos \theta_3 \dots \cos \theta_n \), then \[ \tan \theta_1 + \tan \theta_2 + \tan \theta_3 + \cdots + \tan \theta_n = ? \]

The length of the altitude through the point \( D \) of the tetrahedron where the vertices of the tetrahedron are
\[ A(2, 3, 1), B(4, 1, -2), C(6, 3, 7), D(-5, -4, 8), \] is

Evaluate the limit: \[ \lim_{n \to \infty} \left[ \frac{1}{1 - \pi} + \frac{8}{1 + \pi} + \cdots + \frac{x^2}{n\pi} \right]. \]

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

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

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

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

The equation of the curve passing through the origin and satisfying the equation
\[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2, \] 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 volume of the tetrahedron whose coterminous edges are represented by
\[ \mathbf{a} = -12\hat{i} + p\hat{k}, \quad \mathbf{b} = -3\hat{j} - \hat{k}, \quad \mathbf{c} = -2\hat{i} + \hat{j} - 15\hat{k}, \] is 570 cu. units, then \( p = \)

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 value of \[ \tan 20^\circ \cdot \tan 80^\circ \cdot \cot 50^\circ = ? \]

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 distance between the lines represented by \[ 16x^2 + 9y^2 + 48x - 24xy - 36y + 35 = 0 \] is ......... units

The c.d.f. of a discrete random variable \( X \) is Then \( P(X = -3) / P(X \leq 0) \) is:

The locus of the points represented by \( |z + 3| - |z - 3| = 6 \), where \( z \) is a complex number, is ....

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

If \[ \sqrt{\log_3 x^{16}} + 9 \log_2 7 \sqrt[3]{x} = 5, \text{ then } x = \dots \]

The direction ratios of the line of intersection of the planes \( x - y + z - 5 = 0 \) and \( x - 3y - 6 = 0 \), are: