List of top Mathematics Questions asked in MHT CET

Evaluate the following product: \[ \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \cdots \times \tan 89^\circ \]

If \( \int_{0}^{1} (5x^2 - 3x + k) \, dx = 0 \), then \( k = \)

The eccentricity of the ellipse given by the equation \[ 9x^2 + 16y^2 = 144 \] is

The L.P.P. to maximize \( z = x + y \), subject to \[ x + y \le 30,\; x \le 15,\; y \le 20,\; x + y \ge 15,\; x \ge 0,\; y \ge 0 \] has

\(y = mx + \dfrac{2}{m}\) is the general solution of

The bacteria increases at the rate proportional to the number of bacteria present. If the original number $N$ doubles in $4$ hours, then the number of bacteria in $12$ hours will be

The differential equation of all lines perpendicular to the line \[ 5x + 2y + 7 = 0 \text{ is} \]

Evaluate the integral \[ \int \frac{1 + 2e^{-x}}{1 - 2e^{-x}} \, dx \]

The shortest distance between the lines \( 1 + x = 2y = -12z \) and \( x = y + 2 = 6z - 6 \) is

If \( A + B + C = 180^\circ \), then the value of \( \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right) + \tan \left( \frac{B}{2} \right) \tan \left( \frac{C}{2} \right) + \tan \left( \frac{C}{2} \right) \tan \left( \frac{A}{2} \right) \) is

The acute angle between the line \[ \vec{r} = (i + 2j + k) + \lambda (i + j + k) \] and the plane \[ \vec{r} \cdot (2i - j + k) = 5 \] is

If \( y = e^{4x}\cos 5x \), then find \( \dfrac{d^2y}{dx^2} \) at \( x = 0 \).

The domain of the function \( f(y) = \frac{\cos^{-1(y - 5)}{\sqrt{25 - y^2}} \) is}

Which of the following matrix is invertible?
\[ A_1 = \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix} \] \[ A_2 = \begin{pmatrix} -1 & -2 & 3 \\ 4 & 5 & 7 \\ 2 & 4 & -6 \end{pmatrix} \] \[ A_3 = \begin{pmatrix} 1 & 0 & 0 \\ 5 & 2 & 1 \\ 7 & 2 & 1 \end{pmatrix} \] \[ A_4 = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{pmatrix} \]

If \( CP \) and \( CD \) is a pair of semi-conjugate diameters of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), then \( CP^2 + CD^2 = \)

If \( \tan \theta + \sin \theta = a \) and \( \tan \theta - \sin \theta = b \), then the values of \( \cot \theta \) and \( \csc \theta \) are respectively:

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get at least one correct answer is

In \( \triangle ABC \) with usual notations, \( a = 4 \), \( b = 3 \), \( \angle A = 60^\circ \), then \( c \) is a root of the equation

If \( x = a(1 - \cos\theta) \), \( y = a(\theta - \sin\theta) \), then \( \frac{d^2y}{dx^2} = \)

If \( \sin x + \sin^2 x = 1 \), then \( \cos^8 x + 2 \cos^6 x + \cos^4 x \) is

If \( P(\theta) \) lies on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and \( S \) and \( S' \) are foci of the hyperbola, then \( SP \cdot SP' = \)

The function \[ f(x) = \frac{x + 1}{9x + x^3} \] is

If \( f(x) = \log(\sin x) \), \( x \in \left[ \frac{\pi}{6}, \frac{5\pi}{6} \right] \), then the value of \( c \) by applying L.M.V.T. is

If \[ f(x)=\frac{(e^{2x}-1)\sin x^\circ}{x^2}, \quad x\neq0 \] is continuous at \(x=0\), then \(f(0)=\)

The integral \[ \int_0^{\frac{\pi}{2}} \log \left( \frac{\sqrt{1 - \cos 2x}}{\sqrt{1 + \cos 2x}} \right) dx \] is