MHT CET 2025 25th April Morning Shift

$\mathrm{f}(x)=\frac{x}{2}+\frac{2}{x}, x \neq 0$ is strictly decreasing in

The rate of change of the volume of a sphere with respect to its surface area, when the radius is 5 m is

The area bounded by the parabolas $y=9 x^2, y=\frac{x^2}{16}$ and the line $y=1$ is

If ${ }^n \mathrm{C}_0+\frac{1}{2}{ }^n \mathrm{C}_1+\frac{1}{3}{ }^n \mathrm{C}_2\($+\ldots \frac{1}{n}^n C_{n-1}+\frac{1}{n+1}{ }^n C_n=\frac{1023}{10} \,\,\, then \,\,\,\,\mathrm{n}=\)

A pair of tangents are drawn to the circle $x^2+y^2+6 x-4 y-12=0$ from a point $\mathrm{P}(-4,-5)$, then the area enclosed between these tangents and the area of the circle is

Let $z$ be the complex number with $\operatorname{Im}(z)=10$ and satisfying $\frac{2 \mathrm{z}-\mathrm{n}}{2 \mathrm{z}+\mathrm{n}}=2 \mathrm{i}-1$, where $\mathrm{i}=\sqrt{-1}$, for some natural number ' $n$ ' then

\(\int_1^3 \frac{\log x^2}{\log \left(16 x^2-8 x^3+x^4\right)} d x=\)

\(\int\limits_0^1 \frac{1}{2+\sqrt{x}} d x=\)

The rate at which the population of a city increases varies as the population. In a period of 20 years, the population increased from 4 lakhs to 6 lakhs. In another 20 years the population will be

The differential equation $x \frac{\mathrm{~d} y}{\mathrm{~d} x}=2 y$ represents ________

$y=\mathrm{e}^x(\mathrm{~A} \cos x+\mathrm{B} \sin x)$ is the solution of the differential equation

The solution of the equation $x^2 y-x^3 \frac{\mathrm{~d} y}{\mathrm{~d} x}=y^4 \cos x$, where $y(0)=1$, is

If $\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$ is

The eccentricity of the curve represented by $x=3(\cos t+\sin t), y=4(\cos t-\sin t)$ is

If $\mathrm{f}(x)=\log \left(\frac{1+x}{1-x}\right)$ and $\mathrm{g}(x)=\frac{3 x+x^3}{1+3 x^2}$, then $(\mathrm{fog})(x)=$

$\int \frac{\mathrm{d} x}{3 \cos 2 x+5}$ equals

\(\int \frac{1}{\mathrm{e}^x+1} \mathrm{~d} x=\)

$\int \mathrm{e}^x\left(\frac{x+5}{(x+6)^2}\right) \mathrm{d} x$ is

If $y=\tan ^{-1}\left(\frac{4 x}{1+5 x^2}\right)+\cot ^{-1}\left(\frac{3-2 x}{2+3 x}\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

The principal value of $\cos ^{-1}\left[\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right]$ is

\(\lim\limits_{x \rightarrow 5} \frac{\sqrt{2-2 \cos \left(x^2-12 x+35\right)}}{(x-5)}=\ldots \ldots\)

If Rolle's theorem holds for the function $x^3+\mathrm{a} x^2+\mathrm{b} x, 1 \leq x \leq 2$ at the point $\frac{4}{3}$, then the values of $a$ and $b$ are respectively

If $\mathrm{f}(x)=\frac{\sin \left(\pi \cos ^2 x\right)}{3 x^2}, x \neq 0$ is continuous at $x=0$ then $\mathrm{f}(0)=$

The solution set of the constraints $|x-y| \leq 1, x, y \geq 0$ is

If $y=\log _3\left(\log _3 x\right)$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=3$ is $\ldots \ldots$

The negation of the statement "The triangle is an equilateral or isosceles triangle and the triangle is not isosceles and it is right angled" is

If the statements $p, q$ and $r$ are true, false and true statements respectively, then the truth value of the statement pattern $[\sim \mathrm{q} \wedge(\mathrm{p} \vee \sim \mathrm{q}) \wedge \sim \mathrm{r}] \vee \mathrm{p}$ and the trut...

If $A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]_{3 \times 3}$, then $A^{-1}=$

If a random variable X has the following probability distribution of X
$$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \mathrm{X}=x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mathrm{P}(\mathrm{X}=x) & 0 & \mathrm{k} & 2 \mathrm{k} & 2 \mathrm{...

A pair of fair dice is thrown 4 times. If getting the same number on both dice is considered as a success, then the probability of two successes are

A family has 3 children. The probability that all the three children are girls, given that at least one of them is a girl is

Let X denote the number of hours you study on a Sunday. It is known that
$$ \mathrm{P}(\mathrm{X}=x)=\left\{\begin{array}{cc} 0.1 & , \text { if } x=0 \\ \mathrm{k} x & , \text { if } x=1 \text { or } 2 \\ \mathrm{k}(5-x) & , \text { if } x...

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

If in triangle ABC , with usual notations $\sin \frac{\mathrm{A}}{2} \cdot \sin \frac{\mathrm{C}}{2}=\sin \frac{\mathrm{B}}{2}$ and 2 s is the perimeter of the triangle, then the value of $s$ is

The joint equation of the bisector of the angle between the lines $2 x^2+11 x y+3 y^2=0$ is

A line passes through $\mathrm{P}(-4,1)$ and meets the co-ordinate axes at points A and B . If P divides the segment AB internally in the ratio $1: 2$, then the equation of the line is

The length of the foot of the perpendicular from the point $\left(1, \frac{3}{2}, 2\right)$ to the plane $2 x-2 y+4 z+17=0$ is

Let M and N be foots of the perpendiculars drawn from the point $\mathrm{P}(\mathrm{a}, \mathrm{a}, \mathrm{a})$ on the lines $x-y=0, \mathrm{z}=1$ and $x+y=0, \mathrm{z}=-1$ respectively and if $\angle \mathrm{MPN}=90^{\circ}$ then $\mathr...

If the lines $\frac{1-x}{2}=\frac{7 y+4}{2 \lambda}=\frac{2 z-5}{2}$ and $\frac{7-7 x}{3 \lambda}=\frac{y-1}{7}=\frac{6-\mathrm{z}}{5}$ are at right angle, then the value of $\lambda$ is

The lines $\frac{x-0}{1}=\frac{y-2}{2}=\frac{z+3}{\lambda}$ and $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{\lambda}$ are coplanar and $p$ is the plane containing these lines, then which of following point does not lie on the plane.

In 3-dimensional space, the equation $x^2-8 x+12=0$ represents ....

The number of values of $x$ in the interval $[0,3 \pi]$ satisfying the equation $2 \sin ^2 x+5 \sin x-3=0$ is

The principal solution of of $(5+3 \sin \theta)(2 \cos \theta+1)=0$ are

The maximum value of the function $a \sin x+b \cos x$ is

If $\sec x+\tan x=2,0 < x < \frac{\pi}{2}$ then $\sin \frac{x}{4}=$

If $\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{j}-\hat{k}$ then a vector $\bar{c}$ such that $\overline{\mathrm{a}} \times \overline{\mathrm{c}}=\overline{\mathrm{b}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=3$ is

A tetrahedron has vertices $\mathrm{O}(0,0,0), \mathrm{A}(1,2,1)$, $B(2,1,3), C(-1,1,2)$. Then the angle between the faces OAB and ABC will be

If $\bar{a}, \bar{b}, \bar{c}$ are non coplanar unit vectors such that $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\frac{\overline{\mathrm{b}}+\overline{\mathrm{c}}}{\sqrt{2}}$ then the angle between $\...

If the area of a parallelogram whose diagonals are represented by vectors $3 \hat{i}+\lambda \hat{j}+2 \hat{k}$ and $\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ is $\frac{\sqrt{117}}{2}$ sq. units, then $\lambda=$

The position vectors of the points $A, B, C$ are $\hat{i}+2 \hat{j}-\hat{k}, \hat{i}+\hat{j}+\hat{k}, 2 \hat{i}+3 \hat{j}+2 \hat{k}$ respectively. If $A$ is chosen as the origin, then the cross product of position vectors of $B$ and $C$ are