MHT CET 2025 20th April Morning Shift

20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are

If the curves $y^2=6 x$ and $9 x^2+b y^2=16$ intersect each other at right angles, then the value of $b$ is

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is

The area of the region bounded by $\frac{x^2}{9}+\frac{y^2}{4}=1$ and the line $\frac{x}{3}+\frac{y}{2}=1$ is

The equation of the circle passing through the point $(1,1)$ and having two diameters along the pair of lines $x^2-y^2-2 x+4 y-3=0$ is

The equation $|z+1-i|=|z-1+i|$ represents a (where z is a complex number)

The value of $\int_{-1}^1\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) \mathrm{d} x$ is

\(\int_{-2}^2\left|x^2-x-2\right| \mathrm{d} x=\)

The differential equation whose solution represents the family $x^2 y=4 \mathrm{e}^x+\mathrm{c}$, where c is an arbitrary constant, is

The money invested in a company is compounded continuously. ₹ 400 invested today becomes ₹ 800 in 6 years, then at the end of 33 years, it will become .. $(\sqrt{2}=1.4142)$

The general solution of

$x(x-1) \frac{\mathrm{d} y}{\mathrm{~d} x}=x^3(2 x-1)+(x-2) y$ is

The sum of the degree and order of the differential equation $\sqrt{\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}}=\sqrt[5]{\frac{\mathrm{d} y}{\mathrm{~d} x}-5}$ is

If $y=\log _{\mathrm{e}} x^3+3 \sin ^{-1} x+\mathrm{kx}^2$ and $y^{\prime}\left(\frac{1}{2}\right)=2 \sqrt{3}$, then $k=$

If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$ is

If $\mathrm{y}=x^x+x^{\frac{1}{x}}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

The function defined by $\mathrm{f}(x)=\frac{2 x+3}{3 x+4}, x \neq-\frac{4}{3}$ is

\(\int \frac{d x}{2+\cos x}=\)

If $\int \frac{2 x+3}{(x-1)\left(x^2+1\right)} d x$
\(=\log _e\left\{(x-1)^{\frac{5}{2}}\left(x^2+1\right)^2\right\}-\frac{1}{2} \tan ^{-1} x+\mathrm{A}\)
where A is an arbitrary constant, then the value of $a$ is

\(\int \sin ^5 x \mathrm{~d} x=\)

If $\tan ^{-1}\left(\frac{x}{2}\right)+\tan ^{-1}\left(\frac{y}{2}\right)+\tan ^{-1}\left(\frac{z}{2}\right)=\frac{\pi}{2} \quad$ then $x y+y z+z x=$

\(\mathop {\lim }\limits_{x \to 0} \frac{\mathrm{e}^{\tan x}-\mathrm{e}^x}{\tan x-x}=\)

If $\mathrm{f}(x)=\left\{\begin{array}{ll}\operatorname{m} x+1, & x \leqslant \frac{\pi}{2} \\ \sin x+\mathrm{n}, & x>\frac{\pi}{2}\end{array}\right.$, is continuous at $x=\frac{\pi}{2},(\mathrm{~m}, \mathrm{n} \in \mathbb{Z})$ then

The shaded region in the following figure represents a solution set of

If a statement $q$ has truth value False and $(\mathrm{p} \wedge \mathrm{q}) \leftrightarrow \mathrm{r}$ has truth value True then which of the following has truth value true?

The logically equivalent statement of $(\sim \mathrm{p} \wedge \mathrm{q}) \vee(\sim \mathrm{p} \wedge \sim \mathrm{q}) \vee(\mathrm{p} \wedge \sim \mathrm{q})$ is

If $A=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]$, then $A^T A^{-1}=$

The area of the triangle formed by the lines joining the vertex of the parabola $x^2=20 y$ to the end of its latus rectum is

A regular polygon has 20 sides. The number of triangles that can be drawn by using the vertices but not using the sides are

If two numbers $p$ and $q$ are chosen randomly from the set $\{1,2,3,4\}$, one by one, with replacement, then the probability of getting $\mathrm{p}^2 \geq 4 \mathrm{q}$ is

If X is a binomial variable with range $\{0,1,2,3,4\}$ and $\mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=4)$ then the parameter ' $p$ ' of the binomial distribution is

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. If X is the random variable of getting queens, then the value of $2 E(X)+3 E\left(X^2\right)$ for the number of queens is

A random variable $X$ has the following probability distribution
$$ \begin{array}{|l|c|c|c|c|c|} \hline \mathrm{X}: & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(\mathrm{X}): & \mathrm{k} & 2 \mathrm{k} & 4 \mathrm{k} & 2 \mathrm{k} & \mathrm{k}...

In a triangle ABC , with usual notations if $\frac{2 \cos \mathrm{~A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{2 \cos \mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ca}}$ then $\angle \mathrm{A...

In a triangle ABC with usual notations, if $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in arithmetic progression, then, $\tan \frac{\mathrm{A}}{2} \cdot \tan \frac{\mathrm{C}}{2}=$

With usual notations, in a triangle $A B C$, if $\theta$ is any real number, then $a \cos (B-\theta)+b \cos (A+\theta)$ is

The acute angle between the diagonals of a parallelogram whose vertices are $\mathrm{A}(2,-1)$, $B(0,2), C(2,3)$ and $D(4,0)$ is

The distance between the lines represented by the equation $4 x^2+4 x y+y^2-6 x-3 y-4=0$ is

If the plane $\frac{x}{3}+\frac{y}{2}-\frac{z}{4}=1$ cuts the co-ordinate axes at points $\mathrm{A}, \mathrm{B}$ and C , then the area of the triangle ABC is

If the shortest distance between the lines $\frac{x-\mathrm{k}}{2}=\frac{y-4}{3}=\frac{\mathrm{z}-3}{4}$ and $\frac{x-2}{4}=\frac{y-4}{6}=\frac{\mathrm{z}-7}{8}$ is $\frac{13}{\sqrt{29}}$, then $\mathrm{k}=$

The direction cosines of the line $x-y+2 z=5$ and $3 x+y+z=6$ are

If the angle between the planes $x-2 y+3 z-5=0$ and $x+\alpha y+2 z+7=0$ is $\cos ^{-1}\left(\frac{1}{14}\right)$ then the difference between the values of $\alpha$ is

The acute angle between the lines $x=-2+2 \mathrm{t}, y=3-4 \mathrm{t}, \mathrm{z}=-4+\mathrm{t}$ and $x=-2-\mathrm{t}, y=3+2 \mathrm{t}, \mathrm{z}=-4+3 \mathrm{t}$ is

If $\tan 3 \theta=\cot \theta$, then $\theta=$

The value of $\sqrt{3} \cot 20^{\circ}-4 \cos 20^{\circ}$ is equal to

If $A+B=\frac{\pi}{2}$ then the maximum value of $\cos \mathrm{A} \cdot \cos \mathrm{B}$ is

If $\bar{a}=4 \hat{i}+3 \hat{j}+\hat{k}, \bar{b}=\hat{i}-2 \hat{j}+2 \hat{k}$ then $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{b}})))=$

If $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ are unit vectors such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\frac{1}{2}, \overline{\mathrm{c}} \cdot \overline{\mathrm{d}}=\frac{1}{2}$ and the angle between $\overline{\mathrm{a}} \...

The magnitude of a vector which is orthogonal to the vector $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and is coplanar with the vectors $\hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}+2 \hat{j}+\hat{k}$ is

Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=4, \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=2$. If $\overline{\mathrm{c}}=(2 \overline{\mathrm{a}} \times \overline{\mathrm{b}...

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors and $\theta$ is the angle between them, then $\overline{\mathrm{a}}+\overline{\mathrm{b}}$ is a unit vector when $\theta$ is