MHT CET 2026 15th April Evening Shift

The approximate value of $(0.007)^{\frac{1}{3}}$ is

The surface area of a spherical ball is increasing at the rate of $4\pi\ \text{cm}^2$/second. The rate at which the radius is increasing when the surface area is $16\pi\ \text{cm}^2$ is

The maximum value of $\left(\dfrac{1}{x}\right)^x$, $x > 0$ is

The point on the curve $9y^2 = x^3$ where the normal to the curve makes equal intercepts with the co-ordinate axes is

The area of the region bounded by the curves $y = |x - 4|$, $x = 3$ and $x = 5$, and the X-axis is

The angle between the tangents drawn from the origin to the circle $(x-7)^2 + (y+1)^2 = 25$ is

If $\omega$ is a complex cube root of unity, then the value of $\sin\left[\pi(\omega^{10} + \omega^{23}) - \dfrac{\pi}{4}\right] =$

$\int\limits_{0}^{\pi} |\sin 2x|\, \text{d}x =$

If $f(x)$ is an even function, then $\int\limits_{-2}^{2} (|x| + f(x)\sin x)\, \text{d}x$ is

A spherical mothball has initial radius 3 cm. Due to evaporation, the radius of the ball reduces to 1 cm in 4 months. In how many months would the mothball evaporate completely if the volume is lost at a rate proportional to the surface are...

The solution of $\dfrac{\text{d}y}{\text{d}x} = \sin(x + y) + \cos(x + y)$ is

The order and degree of the differential equation $3 - \left(\dfrac{\text{d}^3 y}{\text{d}x^3}\right)^{\frac{7}{3}} = \left(\dfrac{\text{d}y}{\text{d}x}\right)^5$ are respectively

The solution of the differential equation $e^{-x}(y+1)\text{d}y + (\cos^2 x - \sin 2x)y\, \text{d}x = 0$, given that $y = 1$ when $x = 0$ is

If $y = \tan^{-1}\sqrt{\dfrac{1 + \sin 2x}{1 - \sin 2x}}$, then $\dfrac{\text{d}y}{\text{d}x}$ at $x = \dfrac{\pi}{6}$ is

The derivative of $\sin\left(\log\left(\dfrac{x+3}{x}\right)\right)$ is

If $e^y + xy = e$, then the ordered pair $\left(\dfrac{\text{d}y}{\text{d}x}, \dfrac{\text{d}^2 y}{\text{d}x^2}\right)$ at $x = 0$ is equal to

$\int \dfrac{x^2\, \text{d}x}{(x^2 + 2)(x^2 + 5)} =$

The value of $\int \dfrac{\sin^6 x + \cos^6 x}{\sin^2 x \cdot \cos^2 x}\, \text{d}x$ is

$\int \text{cosec}^{-1}\left(\sqrt{\dfrac{a+x}{x}}\right)\, \text{d}x =$

The value of $\tan^{-1}\left(\dfrac{\cos\left(\frac{19\pi}{4}\right) - 1}{\sin\left(\frac{\pi}{4}\right)}\right)$ is equal to

If $\lim\limits_{x \to \infty} \dfrac{(2x-1)^{19} \cdot (3x+2)^{11}}{(6x-5)^{30}} = 2^a \cdot 3^b$, then $a + b =$

The value of f(0) so that the function $f(x) = \dfrac{(256 - 8x)^{\frac{1}{4}} - 4}{16 - 4(64 + 3x)^{\frac{1}{3}}}$, $x \neq 0$ is continuous at $x = 0$, is

The difference between the maximum and minimum values of the objective function $Z = 3x + 5y$, subject to the constraints $x + 3y \leq 60$, $x + y \geq 10$, $x - y \leq 0$, $x, y \geq 0$ is

If $x$, y, z are the sides of a right angled triangle, where z is the largest side, then $\dfrac{1}{\log_{x+z} y} + \dfrac{1}{\log_{z-x} y} =$

The dual of the statement pattern $(p \wedge \sim q) \longrightarrow (q \wedge \sim p)$ is equivalent to

Negation of the statement "If an integer is greater than 4 and less than 5, then it is a multiple of 3", is

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, $C = \begin{bmatrix} 7 & 3 \\ 0 & 6 \end{bmatrix}$ and $AB = C$, then the inverse of matrix B is

The element in the third row and the second column in the inverse matrix of a matrix $\begin{bmatrix} 1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 4 \end{bmatrix}$ is

The equation of the common tangent touching the circle $(x-3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the X-axis is

A bag contains 23 balls, of which 7 are identical. The number of ways of selecting 12 balls from the bag is....

If 6 boys and 3 girls are to be seated in a row for a photograph, then the probability that the end seats are occupied by the girls and no two girls are side by side is

For the following probability distribution, the standard deviation of the random variable X isX234p(X=$x$)0.20.50.3

A random variable X has the following probability distributionX12345678P(X=$x$)0.150.230.120.100.200.080.070.05For the event E = { X is a prime number } and F = { X < 4 }, P(E$\cup$F) =

The probability that event A happens in a trial is 0.4. If three independent trials are made, then the probability that A happens at least once is.......

In a triangle ABC, with the usual notations, $\angle B = \dfrac{\pi}{3}$, $\angle C = \dfrac{\pi}{4}$. If D divides BC internally in the ratio 1:3, then $\dfrac{\sin \angle BAD}{\sin \angle CAD} =$

Let $P_1$, $P_2$, $P_3$ be the altitudes of a triangle ABC from the vertices A, B, C respectively. If $\triangle$ denotes the area of the triangle and s is the semi-perimeter of the triangle, then $\dfrac{\cos A}{P_1} + \dfrac{\cos B}{P_2} ...

The equation of the line passing through the point of intersection of the lines $x + 2y + 6 = 0$ and $2x - y = 2$ and making an intercept 5 on the y-axis is

The line $x + y = 3$ intersects the pair of straight lines $x^2 - 3xy + y^2 = 0$ at points A and B. Then the co-ordinates of the mid-point of AB are

The co-ordinates of the point where the line joining the points $(3,5,-7)$ and $(-2, 1, 8)$ is intersected by the YOZ plane are

If the lines $\vec{r} = (\hat{i} + m\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (4\hat{i} + \hat{j}) + \mu(5\hat{i} + m\hat{j} + \hat{k})$ intersect each other, then m =

If the lines $x = -1 + s$, $y = 3 - \lambda s$, $z = 1 + \lambda s$ and $x = \dfrac{t}{2}$, $y = 1 + t$, $z = 2 - t$ with parameters s and t, are coplanar, then $\lambda =$

The equation of the plane passing through the point $(1,2,1)$ and perpendicular to the planes $x + 2y + 2z - 7 = 0$ and $3x + 3y + 2z - 5 = 0$ is

The distance of the point $(1,0,-3)$ from the plane $x-y-z=9$ measured parallel to the line $\dfrac{x-2}{2} = \dfrac{y+2}{3} = \dfrac{z-6}{-6}$ is

If $\sec 4\theta - \sec 2\theta = 2$, then $\theta =$

If $33\theta = \pi$, then the value of $\cos\theta \cos2\theta \cos4\theta \cos8\theta \cos16\theta$ is

A unit vector coplanar with $\hat{i} + \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} + \hat{k}$ and perpendicular to $\hat{i} + \hat{j} + \hat{k}$ is

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = |\vec{b}| = |\vec{c}| = 3$ and $\theta$ is the angle between $\vec{b}$ and $\vec{c}$ then $\tan^2\theta + \cot^2\theta =$

The altitude of the parallelopiped, whose coterminous edges are the vectors $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - \hat{k}$, $\vec{c} = \hat{i} + \hat{j} + 3\hat{k}$, where $\vec{a}$, $\vec{b}$ are the sid...

Let ABCD be a quadrilateral with $\overline{AB} = \vec{a}$, $\overline{AD} = \vec{b}$ and $\overline{AC} = 3\vec{a} + 2\vec{b}$. If its area is $\alpha$ times the area of the parallelogram with AB, AD as adjacent sides, then the value of $\...

Let $\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - \vec{a}| = 2\sqrt{2}$ and the angle between $\vec{a} \times \vec{...