MHT CET 2025 21st April Morning Shift

If the function $\mathrm{f}(x)=x(x+3) \mathrm{e}^{-\frac{x}{2}}$ satisfies all the conditions of Rolle's theorem in $[-3,0]$, then c is

The radius of the base of a cone is increasing at the rate $3 \mathrm{~cm} /$ minute and the altitude is decreasing at the rate $4 \mathrm{~cm} /$ minute . The rate at which the lateral surface area is changing, when the radius is 7 cm and ...

The function f defined by $\mathrm{f}(x)=(x+2) \mathrm{e}^{-x}$ is

The function $x^5-5 x^4+5 x^3-10$ has a maximum, when $x$ is equal to

The area inside the parabola $y^2=4 \mathrm{a} x$, between the lines $x=\mathrm{a}$ and $x=4 \mathrm{a}$ is equal to

The minimum distance and maximum distance of the point $\mathrm{P}(2,-7)$ from the circle $x^2+y^2-14 x-10 y-151=0$ are respectively _______units

The complex numbers $\sin x+i \cos 2 x$ and $\cos x$ - $\mathrm{i} \sin 2 x,(\mathrm{i}=\sqrt{-1})$ are conjugate to each other for,

The value of $\int_0^2\left[x^2\right] \mathrm{d} x$ is (where $[x]$ denotes the greatest integer function not greater than $x$ )

$\int_2^4 \frac{\log x^2}{\log x^2+\log \left(36-12 x+x^2\right)} \mathrm{d} x$ is equal to

Which of the following is not a homogeneous function?

The general solution of $\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x y \mathrm{e}^{x^2}$ is

A particular solution of $\frac{\mathrm{d} y}{\mathrm{~d} x}=(x+9 y)^2$, when $x=0, y=\frac{1}{27}$ is

The assets of a person reduced in his business such that the rate of reduction is proportional to the square root of the existing assets. If the assets were initially ₹ 10 lakhs and due to loss they reduce to ₹ 10000 after 3 years, then the...

If $x=\log \mathrm{t}, \mathrm{t}>0$ and $y=\frac{1}{\mathrm{t}}$ then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=$

If $(\mathrm{a}+\mathrm{b} x) \mathrm{e}^{\frac{y}{x}}=x$, then $x^3 \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}$ is equal to

If $y=\frac{\mathrm{K}^{\cos ^{-1} x}}{1+\mathrm{K}^{\cos ^{-1} x}}$ and $\mathrm{t}=\mathrm{K}^{\cos ^{-1} x}$, then $\frac{\mathrm{d} y}{\mathrm{dt}}=$

If $\quad \int \frac{3 \sin x \cos x}{4 \sin x+7} \mathrm{~d} x=\mathrm{A} \sin x-\mathrm{Blog}(4 \sin x+7)+\mathrm{c}$ where c is the constant of integration, then the value of $\mathrm{A}+\mathrm{B}$ is equal to

$\int \sqrt{x^2-6 x-16} \mathrm{~d} x$ equals

\(\int \frac{\mathrm{d} x}{x\left(x^2+1\right)}=\)

The value of $2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{3}{8}$ is

If $\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{3}{5}+\sin ^{-1} x=\frac{\pi}{2}$ then $x=$

\(\mathop {\lim }\limits_{x \to \infty } \frac{\mathrm{e}^{x^4}-1}{\mathrm{e}^{x^4}+1}=\)

If $f(x)= \begin{cases}\frac{8^x-4^x-2^x+1}{x^2}, & \text { if } x>0 \\ e^x \sin x+x+\lambda \log 4, & \text { if } x \leqslant 0\end{cases}$
is continuous at $x=0$ then the value of $1000 \mathrm{e}^\lambda=$

The L.P.P. , minimize $z=30 x+20 y, x+y \leq 8$, $x+2 y \geq 4,6 x+4 y \geq 12, x \geqslant 0, y \geqslant 0$ has

The logical statement
\([\sim(\sim p \vee q) \vee(p \wedge r) \wedge(\sim q \wedge r)]\)
is equivalent to

If the statement pattern $(p \wedge q) \rightarrow(r \vee \sim s)$ is false, then the truth values of $p, q, r$ and $s$ are respectively

The third element in the second row of adjoint of a matrix $A=\left[a_{i j}\right]_{3 \times 3}$ (where $a_{i j}=2 i+j$ ) is

The Y-intercept of the common tangent to the parabolas $y^2=32 x$ and $x^2=108 y$ is

The greatest possible number of points of intersection of 8 distinct straight lines and 4 distinct circles is

The following is the probability distribution of X
$$ \begin{array}{|c|c|c|c|c|} \hline \mathrm{X} & 0 & 1 & 2 & 3 \\ \hline \mathrm{P}(\mathrm{X}=x) & \frac{1+\mathrm{p}}{5} & \frac{2-2 \mathrm{p}}{5} & \frac{2-\mathrm{p}}{5} & \frac{2 \ma...

A random variable X takes values $0,1,2,3$, ........ with probabilities. $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{2}\right)^x, \mathrm{k}$ is a constant, then $P(X=1)=$

If a random variable X follows the Binomial distribution $B(10, \quad p)$ such that $5 \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)$, then the value of $\frac{\mathrm{P}(\mathrm{X}=5)}{\mathrm{P}(\mathrm{X}=6)}$ is equal to

The probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two 'I's do not come together is

With usual notation, in triangle ABC , $\mathrm{m} \angle \mathrm{A}=30^{\circ}$ then the value of $\left(1+\frac{\mathrm{a}}{\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{c}}\right)\left(1+\frac{\mathrm{c}}{\mathrm{b}}-\frac{\mathrm{a}}{\mathrm{b}...

In a triangle ABC , with usual notations if $\mathrm{a}=4, \mathrm{~b}=8, \angle \mathrm{C}=60^{\circ}$, then the value of $\angle \mathrm{B}$ and the ratio $\cos \mathrm{A}: \cos \mathrm{C}$ respectively are,

In $\triangle A B C$, with usual notations, $a \cos B=b \cos A, a \cos C \neq c \cos A$ then $\mathrm{A}(\triangle \mathrm{ABC})$ $\qquad$ sq. units.

If the line $2 x+y=\mathrm{k}$ passes though the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ internally in the ratio $3: 2$ then, $(k+1):(k-1)=$

The joint equation of the bisectors of the angles between the lines $x=5$ and $y=3$ is

The equation of the plane containing the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{-2}$ and the point $(0,5,0)$ is

The shortest distance between the lines $\bar{r}=(4 \hat{i}-\hat{j})+\lambda(\hat{i}+2 \hat{j}-3 \hat{k})$ and $\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+4 \hat{j}-5 \hat{k})$ is

If the distance of the point $\mathrm{P}(1,-2,1)$ from the plane $x+2 y-2 z=\alpha$, where $\alpha>0$ is 5 units, then the foot of the perpendicular from P to the plane is

\(3 \tan ^6 10^{\circ}-27 \tan ^4 10^{\circ}+33 \tan ^2 10^{\circ}=\)

If $\mathrm{f}(x)=\cos (\log x)$ then $\mathrm{f}\left(x^2\right) \cdot \mathrm{f}\left(y^2\right)-\frac{1}{2}\left[\mathrm{f}\left(\frac{x^2}{y^2}\right)+\mathrm{f}\left(x^2 y^2\right)\right]$ has the value

Three vectors $\hat{\mathrm{i}}-\hat{\mathrm{k}}, \lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+(1-\lambda) \hat{\mathrm{k}}$ and $\mu \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+(1+\lambda-\mu) \hat{\mathrm{k}}$ represents coterminous edges of a...

The maximum value and minimum value of the volume of the parallelopiped having coterminous edges $\hat{\mathrm{i}}+x \hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{j}}+x \hat{\mathrm{k}}$ and $x \hat{\mathrm{i}}+\hat{\mathrm{k}}$ are respe...

Let $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ be unit vectors at an angle $\frac{\pi}{3}$ with each other. If $(\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})) \cdot(\overline{\mathrm{a}} \times \...

The lines $\overline{\mathrm{r}}=\overline{\mathrm{a}}+\lambda(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$ and $\overline{\mathrm{r}}=\overline{\mathrm{c}}+\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})$ will intersect...

The line of intersection of the planes $\bar{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=1 \quad$ and $\quad \bar{r} \cdot(\hat{i}+4 \hat{j}-2 \hat{k})=2 \quad$ is parallel to the vector

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors such that $|\overline{\mathrm{a}}+\overline{\mathrm{b}}|=\sqrt{3}$, then the angle between $\bar{a}$ and $\bar{b}$ is

Let $\bar{a}, \bar{b}$, and $\bar{c}$ be unit vectors. Suppose that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=0$ and if the angle between $\overline{\mathrm{b}}$ and $\overline{\mat...