MHT CET 2025 22nd April Evening Shift

Let $f$ be a function which is continuous and differentiable for all $x$. If $\mathrm{f}(1)=1$ and $\mathrm{f}^{\prime}(x) \leq 5$ for all $x$ in $[1,5]$, then the maximum value of $\mathrm{f}(5)$ is

The function $\mathrm{f}(x)=\sin ^4 x+\cos ^4 x$ increases if

The normal to the curve $x=9(1+\cos \theta)$, $y=9 \sin \theta$ at $\theta$ always passes through the fixed point

An open tank with a square bottom is to contain 4000 cubic cm . of liquid. The dimensions of the tank so that the surface area of the tank is minimum, is

If a curve $y=a \sqrt{x}+b x$ passes through the point $(1,2)$ and the area bounded by this curve, line $x=4$ and the X -axis is 8 sq . units, then the value of $a-b$ is

The equations of the tangents to the circle $x^2+y^2=36$ which are perpendicular to the line $5 x+y=2$, are

$\mathrm{z}=\frac{3+2 \mathrm{i} \sin \theta}{1-2 \mathrm{i} \sin \theta},(\mathrm{i}=\sqrt{-1})$ will be purely imaginary if $\theta=$

\(\int_0^{\frac{\pi}{4}}(\sqrt{\tan x}+\sqrt{\cot x}) d x=\)

The value of the integral $\int_1^2 \frac{x \mathrm{~d} x}{(x+2)(x+3)}$ is

A wet substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet, hung in the open air, loses half its moisture during the first hour, then $90 \%$ of the moisture will be lost in ________ hours.

The general solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\sin \left(\frac{x+y}{2}\right)=\sin \left(\frac{x-y}{2}\right)$ is

The equation of the curve passing through $\left(2, \frac{9}{2}\right)$ and having the slope $\left(1-\frac{1}{x^2}\right)$ at $(x, y)$ is

If $\mathrm{f}(1)=3, \mathrm{f}^{\prime}(1)=2$, then $\frac{\mathrm{d}}{\mathrm{dx}}\left\{\log \left[\mathrm{f}\left(\mathrm{e}^x+2 x\right)\right]\right\}$ at $x=0$ is

Derivative of $x^{\left(x^x\right)}$ is

The values of $b$ and $c$ for which the identity $\mathrm{f}(x+1)-\mathrm{f}(x)=8 x+3$ is satisfied, where $\mathrm{f}(x)=\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}$, are

The foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$. The equation of the hyperbola with eccentricity 2 is

The foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$. The equation of the hyperbola with eccentricity 2 is

\(\int \frac{x^3}{x^4+5 x^2+4} d x=\)

\(\int \sec ^{\frac{2}{3}} x \cdot \operatorname{cosec}^{\frac{4}{3}} x d x=\)

\(\int \mathrm{e}^{2 x} \frac{(\sin 2 x \cos 2 x-1)}{\sin ^2 2 x} \mathrm{~d} x=\)

Considering only the principal values of the inverse trigonometric function, the value of $\tan \left(\cos ^{-1} \frac{1}{5 \sqrt{2}}-\sin ^{-1} \frac{4}{\sqrt{17}}\right)$ is

The number of solutions of $\tan ^{-1}\left(x+\frac{2}{x}\right)-\tan ^{-1}\left(\frac{4}{x}\right)-\tan ^{-1}\left(x-\frac{2}{x}\right)=0$ are

The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ w.r.t. $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^2}}{1-2 x^2}\right)$ at $x=0$ is

\(\mathop {\lim }\limits_{x \to 1}\left(\log _3 3 x\right)^{\log _x 8}=\ldots\)

If $f(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2} & , \text { if } x<0 \\ \frac{a}{\sqrt{x}} & , \text { if } x=0 \\ \frac{(16+\sqrt{x})^{\frac{1}{2}}-4}{16} & , \text { if } x>0\end{array}\right.$
is continuous at $x=0$, then $\mathr...

The correct constraints for the given feasible region are ….

$p:$ If 7 is an odd number then 7 is divisible by 2 .
q : If 7 is prime number then 7 is an odd number. If $V_1$ and $V_2$ are respective truth values of contrapositive of p and q then $\left(\mathrm{V}_1, \mathrm{~V}_2\right) \equiv$

If $p$ : switch $S_1$ is closed, $q$ : switch $S_2$ is closed, $r$ : switch $S_3$ closed, then the symbolic form of the following switching circuit is equivalent to
Switching Circuit:

If matrix $\quad A=\frac{1}{11}\left[\begin{array}{rrr}-1 & 7 & -24 \\ 2 & a & 4 \\ 2 & -3 & 15\end{array}\right] \quad$ and $A^{-1}=\left[\begin{array}{rrr}3 & 3 & 4 \\ 2 & -3 & 4 \\ b & -1 & c\end{array}\right]$, then the values of $a, b,...

If four digit numbers are formed by using the digits $1,2,3,4,5,6,7$ without repetition, then out of these numbers, the numbers exactly divisible by 25 are

If a random variable $X$ has p.d.f. $f(x)=\left\{\begin{array}{ll}\frac{a x^2}{2}+b x & , \text { if } 1 \leqslant x \leqslant 3 \\ 0 & , \text { otherwise }\end{array}\right.$ and $f(2)=2$, then the values of $a$ and $b$ are, respectively

Three urns respectively contain 2 white and 3 black, 3 white and 2 black and 1 white and 4 black balls. If one ball is drawn from each um, then the probability that the selection contains 1 black and 2 white balls is

In a box containing 100 apples, 10 are defective. The probability that in a sample of 6 apples, 3 are defective is

Four defective oranges are accidentally mixed with sixteen good ones. Three oranges are drawn from the mixed lot. The probability distribution of defective oranges is

In a triangle ABC with usual notations if, $\cot \frac{A}{2}=\frac{b+c}{a}$, then the triangle $A B C$ is

In a triangle ABC with usual notations, if $3 \mathrm{a}=\mathrm{b}+\mathrm{c}$, then $\cot \frac{\mathrm{B}}{2} \cdot \cot \frac{\mathrm{C}}{2}=$

The lines $x+2 \mathrm{a} y+\mathrm{a}=0, x+3 \mathrm{~b} y+\mathrm{b}=0$, $x+4 c y+c=0$ are concurrent then $a, b, c$ are in

The circumradius of the triangle formed by the lines $x y+2 x+2 y+4=0$ and $x+y+2=0$ is

The line L is passing through $(1,2,3)$. The distance of any point on the line L from the line $\overline{\mathrm{r}}=(3 \lambda-1) \hat{\mathrm{i}}+(-2 \lambda+3) \hat{\mathrm{j}}+(4+\lambda) \hat{\mathrm{k}}$ is constant. Then the line L ...

The distance of the plane $\overline{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ from the origin is

If the angle between the line $x=\frac{y-1}{2}=\frac{z-3}{\lambda}$ and the plane $x+2 y+3 z=4$ is $\cos ^{-1} \sqrt{\frac{5}{14}}$, then the value of $\lambda$ is

The angle between the lines $x-3 y-4=0,4 y-z+5=0$ and $x+3 y-11=0,2 y-z+6=0$ is

If the point $(1, \alpha, \beta)$ lies on the line of the shortest distance between the lines $\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2}$ and $\frac{x+2}{-1}=\frac{y+6}{2}, \mathrm{z}=1$, then $\alpha+\beta=$

If $\frac{1}{6} \sin \theta, \cos \theta, \tan \theta$ are in G.P., then the general solution of $\theta$ is

If $\sin \mathrm{A}=\mathrm{n} \sin (\mathrm{A}+2 \mathrm{~B})$, then $\tan (\mathrm{A}+\mathrm{B})=$

The number of integral values of $p$ for which the vectors $(p+1) \hat{i}-3 \hat{j}+p \hat{k}, p \hat{i}+(p+1) \hat{j}-3 \hat{k}$ and $-3 \hat{\mathrm{i}}+\mathrm{p} \hat{\mathrm{j}}+(\mathrm{p}+1) \hat{\mathrm{k}}$ are linearly dependent v...

If $\bar{p}=2 \hat{i}+\hat{k}, \bar{q}=\hat{i}+\hat{j}+\hat{k}, \bar{r}=4 \hat{i}-3 \hat{j}+7 \hat{k}$ and a vector $\overline{\mathrm{m}}$ is such that $\overline{\mathrm{m}} \times \overline{\mathrm{q}}=\overline{\mathrm{r}} \times \overl...

If the area of parallelogram, whose diagonals are $\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\alpha \hat{\mathrm{k}}$ is $\frac{\sqrt{93}}{2}$ sq. units, then $\alpha=$

If the lengths of three vectors $\bar{a}, \bar{b}$ and $\bar{c}$ are $5,12,13$ units respectively, and each one is perpendicular to the sum of the other two, then $|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=\ldots \...

The projection of the line segment joining $\mathrm{P}(2,-1,0)$ and $\mathrm{Q}(3,2,-1)$ on the line whose direction ratios are $1,2,2$ is