List of top Questions asked in MHT CET

If \[ \mathbf{a} = \hat{i} + \hat{j} + \hat{k}, \quad \mathbf{b} = \hat{i} - \hat{j} + 2\hat{k}, \quad \mathbf{c} = x\hat{i} + \hat{j} + (x - 1)\hat{k} \] If the vector \( \mathbf{c} \) lies in the plane of \( \mathbf{a} \) and \( \mathbf{b} \), then \( x = \)

The approximate value of \( \cot^{-1} (1 \cdot 001) \) is

If the line \( 6x - y - 4 = 0 \) touches the curve \( y^2 = ax^3 + b \) at the point (1, 2), then \( a + b = \)

If \[ \tan^{-1} x + \tan^{-1} y = c \text{ is the general solution of the differential equation} \]

With usual notations, if in \( \triangle ABC \), \( s \) is the semi-perimeter and \( (s - a)(s - b) = (s - c) \), then \( \triangle ABC \) is

Evaluate the integral \[ \int x^3 e^{x^2} \, dx \]

Evaluate the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}}{\sqrt[3]{\sec x} + \sqrt[3]{\csc x}} \, dx \]

If \[ \cos x + \cos y = -\cos \alpha, \quad \sin x + \sin y = -\sin \alpha, \quad \text{then} \quad \cot \left( \frac{x + y}{2} \right) = \]

The statement pattern \( \sim (p \vee q) \vee (\sim p \wedge q) \) is equivalent to

The direction cosines of a line which makes equal acute angles with the co-ordinate axes are

The adjoint of the matrix \[ A = \begin{pmatrix} 2 & 3
-3 & 5 \end{pmatrix} \] is

If \[ y = \left( \frac{x^2 + 1}{x} \right)^x \text{ and } \frac{dy}{dx} = y \left[ g(x) + \log \left( \frac{x^2}{x+1} \right) \right], \text{ then } g(x) = \]

The integral \[ \int_0^{\frac{\pi}{2}} \frac{1 - \cot x}{\csc x + \cos x} \, dx = \]

If the equation \[ ax^2 + hxy + by^2 = 0, \quad c \neq 0 \text{ represents a pair of coincident lines, then} \]

For \( \theta \in \left( 0, \frac{\pi}{2} \right) \), if \( \tan 3\theta \cdot \tan 2\theta \cdot \tan \theta + \tan 2\theta + \tan \theta = 1 \), then \( \theta = \)

If \( f : \mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R} \) are defined by \[ f(x) = x^2 - 3x + 4 \quad \text{and} \quad g(x) = 2x + 1, \quad \text{then the value of} \quad x \text{ for which} \quad f(x) = f \circ g(x) \text{ is} \]

For the following shaded region, the linear constraints are:

A metal has half-life period of 10 days. A sample originally has a mass of 1000 mg, then the mass remaining after 50 days is

If \[ A = \{ x / x \text{ is a prime number}, 0 \leq x \leq 9 \}, \text{ then the number of elements of the power set of } A \text{ is} \]

If the equation \( ax^2 + hy^2 + cx + cy = 0, c \neq 0 \) represents a pair of lines, then

If the p.m.f. of a random variable \( X \) is \[ \begin{array}{|c|c|c|c|c|} X & 1 & 2 & 3 & 4 & 5
P(X = x) & k & \frac{k}{3} & \frac{k}{4} & \frac{k}{2} & \end{array} \] then \( k = \)

If \( \mu \) and \( \sigma^2 \) are mean and variance of a random variable \( X \) whose p.m.f. is given by \[ P(X = x) = \binom{6}{x} \left( \frac{1}{4} \right)^x \left( \frac{3}{4} \right)^{6 - x}, \quad x = 0, 1, 2, 3, 4, 5, 6, \text{ then the value of } 2\mu + 12\sigma^2 = \]

The solution of \( rdx + (x - r^2) dr = 0 \) is

If \[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{pmatrix} \] then \( A^4 A^{-1} = \)

If the lines \[ \frac{x - 1}{5} = \frac{y + 1}{3} = \frac{3 - z}{\lambda} \quad \text{and} \quad \frac{x + 1}{4} = \frac{1 - 3y}{15} = \frac{z + 1}{1} \] are perpendicular to each other, then \( \lambda = \)