MHT CET 2025 5th May Evening Shift

The area of the triangle formed by the co-ordinate axes and a tangent to the curve $x y=\mathrm{a}^2$ at the point $\left(x_1, y_1\right)$ is _______ sq. units (where a, $x_1$ and $y_1$ are non-zero)

A spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 \pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the bal...

The minimum value of the slope of the tangent to curve $y=x^3-3 x^2+2 x+93$ is

The approximate value of $\frac{1}{(2.002)^2}$ is

The area of the region bounded by the parabola $y^2=27 x$ and the line $x=1$ is ________ sq.units.

If one of the diameters of the circle, given by the equation $x^2+y^2-4 x+6 y-12=0$, is a chord of a circle, ' S ', whose centre is at $(-3,2)$, then the length of radius of ' S ' is _______ units.

The area of the triangle whose vertices are $i, \omega$ and $\omega^2$ is (Where $\omega$ is a complex cube root of unity other than $1, i$ is an imaginary number)__________ sq.units

\(\int_0^1 \log (x+1) d x=\)

The value of $\int_0^\pi\left|\sin ^3 x\right| \mathrm{d} x$ is

The solution of the equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{x+y+1}$ is

The solution of $\log \left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)=2 x-5 y, y(0)=0$ is

The population $p$ of the city at time $t$ is given by $\frac{\mathrm{dp}}{\mathrm{dt}}=\frac{\mathrm{p}}{2}-100$. If initial population is 100 then $\mathrm{p}=$

The integrating factor of the differential equation $x \frac{\mathrm{~d} y}{\mathrm{~d} x}+y \log x=x \cdot \mathrm{e}^x x^{-\frac{1}{2}} \log x(x>0)$ is

Derivative of
$y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots \ldots \ldots \ldots \ldots \ldots \infty}}}$ is

If $x \cdot \log _e\left(\log _e x\right)-x^2+y^2=4(y>0)$, then $\frac{d y}{d x}$ at $x=\mathrm{e}$ is

The eccentric angle of the point $\mathrm{P}(-6,2)$ of the ellipse $\frac{x^2}{48}+\frac{y^2}{16}=1$ is

The function $\mathrm{f}(x)=\sec \left[\log \left(x+\sqrt{1+x^2}\right)\right]$ is________function

\(\int \frac{d x}{\cos x(1+\cos x)}=\)

If $A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]$ where $a=7^x, b=7^{7^x}, c=7^{7^{7^x}}$ then $\int|A| d x$, (Where $|A|$ is the determinant of the matrix $A$ ) is equal to

$\int \frac{\sin 7 x}{\cos 9 x \cos 2 x} \mathrm{~d} x$ is equal to

The value of $\tan \left[2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right]$ is

\(\cot ^{-1}\left(2 \cdot 1^2\right)+\cot ^{-1}\left(2 \cdot 2^2\right)+\cot ^{-1}\left(2 \cdot 3^2\right)+\ldots \ldots \ldots \infty=\)

If $x=\tan ^{-1}\left\{\frac{\sqrt{1+t^2}-1}{t}\right\}, y=\cos ^{-1}\left\{\frac{1-t^2}{1+t^2}\right\}, \quad$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

If $\quad f(x)=\left\{\begin{array}{cc}\frac{9^x-2 \cdot 3^x+1}{\log (1+3 x) \cdot \tan 2 x} & , \text { if } x \neq 0 \\ a(\log b)^c & , \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $\mathrm{a}+\mathrm{b}+\mathrm{c}=$

Define $f(x)=\left\{\begin{array}{cl}b-a x & , \text { if } x<2 \\ 3 & , \text { if } x=2 \\ a+2 b x & , \text { if } x>2\end{array}\right.$ and if $\lim _{x \rightarrow 2} f(x)$ exists, then $\frac{a}{b}=$

The feasible region for the constraints $x-y \geq 0, x-5 y \leq-5, x \geq 0, y \geq 0$ is shown by the figure:

The negation of statement pattern $(\mathrm{p} \wedge \sim \mathrm{q}) \rightarrow(\mathrm{p} \vee \sim \mathrm{q})$ is

If p : switch $\mathrm{S}_1$ is closed, q : switch $\mathrm{S}_2$ is closed then correct interpretation from the following circuit is

If $A=\left[\begin{array}{rrr}1 & -2 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]$ then $A(I+\operatorname{adj} A)=$

The vectors $\bar{p}=\hat{i}+a \hat{j}+a^2 \hat{k}, \bar{q}=\hat{i}+b \hat{j}+b^2 \hat{k}$ and $\overline{\mathrm{r}}=\hat{\mathrm{i}}+\mathrm{c} \hat{\mathrm{j}}+\mathrm{c}^2 \hat{\mathrm{k}}$ are non-coplanar and $\left|\begin{array}{lll}...

The value of ${ }^{47} \mathrm{C}_4+\sum\limits_{\mathrm{j}=1}^5{ }^{(52-\mathrm{j})} \mathrm{C}_3$ is

If $\mathrm{X} \sim \mathrm{B}(35, \mathrm{p})$ such that $7 \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)$ then the value of $\frac{\mathrm{P}(\mathrm{X}=15)}{\mathrm{P}(\mathrm{X}=20)}$ is equal to

Two cards are drawn successively with replacement from fair playing 52 cards. let X denote number of kings obtained when two cards are drawn, then $\mathrm{E}\left(\mathrm{X}^2\right)=$

Three numbers are chosen at random from numbers 1 to 20 . The probability that they are consecutive is

A student studies for X number of hours during a randomly selected school day. The probability that X can take the values, has the following form, where k is some constant.
$$ \mathrm{P}(\mathrm{X}=x)= \begin{cases}0.2, & \text { if } x=0 \...

With usual notation, in a triangle ABC $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$, then the value of $\cos B$ is equal to

In a triangle $A B C$, with usual notations, the sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are such that they are roots of the equation $x^3-11 x^2+38 x-40=0$ then $\frac{\cos \mathrm{A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\fra...

If the line $3 x+4 y-24=0$ intersects X and Y axes in points A and B respectively then incentre of the triangle OAB where O is origin is

The equation $x^2-3 x y+2 y^2+3 x-5 y+2=0$ represents a pair of straight lines. If $\theta$ is the angle between them, then the value of $\cos \theta$ is equal to

The distance of the point $(5,3,-1)$ from the plane passing through points $(2,1,0),(3,-2,4)$ and $(1,-3,3)$ is

The equation of a line passing through the point $(-1,2,3)$ and perpendicular to the lines $\frac{x}{2}=\frac{y-1}{-3}=\frac{z+2}{-2}$ and $\frac{x+3}{-1}=\frac{y+3}{2}=\frac{z-1}{3}$ is

The co-ordinates of the point in which line joining $(1,1,1)$ and $(2,2,2)$ intersects the plane $x+y+\mathrm{z}=9$ are

The equation of plane passing through $(1,0,0)$ and $(0,1,0)$ and making an angle $45^{\circ}$ with the plane $x+y-3=0$ is

A line L is passing through points $\mathrm{A}(1,3,2)$ and $\mathrm{B}(2,2,1)$. If mirror image of point $\mathrm{P}(1,1,-1)$ in the line L is $(x, y, z)$ then $x+y+\mathrm{z}=$

The general solutions of the equation $\tan ^2 \theta+\sec 2 \theta=1$ are

If $\sin (\alpha+\beta)=1, \sin (\alpha-\beta)=\frac{1}{2}, \alpha, \beta \in\left[0, \frac{\pi}{2}\right]$, then $\tan (\alpha+2 \beta) \cdot \tan (2 \alpha+\beta)=$

If $\overline{\mathrm{c}}=5 \overline{\mathrm{a}}+6 \overline{\mathrm{~b}}$ and $3 \overline{\mathrm{c}}=\overline{\mathrm{a}}-4 \overline{\mathrm{~b}}$ then

If $\overline{\mathrm{a}}=\frac{1}{\sqrt{10}}(3 \hat{\mathrm{i}}+\hat{\mathrm{k}}), \overline{\mathrm{b}}=\frac{1}{7}(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}})$, then the value of $(\overline{\mathrm{a}}-2 \overline{\mathrm{...

Two adjacent sides of a parallelogram $A B C D$ are given by $\overline{\mathrm{AB}}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}$ and $\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$. The side ...

ABCD is a quadrilateral with $\overline{\mathrm{AB}}=\overline{\mathrm{a}}, \overline{\mathrm{AD}}=\overline{\mathrm{b}}$ and $\overline{\mathrm{AC}}=2 \overline{\mathrm{a}}+3 \overline{\mathrm{~b}}$. If its area is $\alpha$ times the area ...