MHT CET 2025 23rd April Evening Shift

If the line $a x+b y+c=0$ is normal to the curve $x y=1$, then

The sum of two nonzero numbers is 4 . The minimum value of the sum of their reciprocals is

The combined equation of the tangent and normal to the curve $x y=15$ at the point $(5,3)$ is________

The length and breadth of a rectangle are $x_{x \mathrm{~cm}}$ and $y \mathrm{~cm}$ respectively. If the length decreases at the rate of $5 \mathrm{~cm} /$ minute and the breadth increases at the rate of $3 \mathrm{~cm} /$ minute, then the ...

The area bounded by the curve $y=x^2+3, y=x, x=3$ and $y$-axis is

The equations of the tangents to the circle $x^2+y^2=36$ which are perpendicular to the line $5 x+y-2=0$ are

Argument of the complex number $z=\frac{13-5 i}{4-9 i}, i=\sqrt{-1}$ is

\(\int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\left(x^2+\log \left(\frac{\pi-x}{\pi+x}\right) \cdot \cos x\right) d x=\)

\(\int_0^1 \log \left(\frac{1}{x}-1\right) d x=\)

The differential equation whose solution is $\mathrm{A} x^2+\mathrm{B} y^2=1$, where A and B are arbitrary constants is of

Solution of $(2 y-x) \frac{d y}{d x}=1$ is

The integrating factor of $y+\frac{\mathrm{d}}{\mathrm{d} x}(x y)=x(\sin x+\log x)$ is

The rate of change of volume of spherical balloon at any instant is directly proportional to its surface area. If initially its radius is 3 cm , after 2 minutes its radius becomes 9 cm , then radius of balloon after 4 minutes is

If $\mathrm{f}(x)=3 x^3+2 x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+\mathrm{f}^{\prime \prime \prime}(3)$ then $\mathrm{f}(x)=$ __________ .

If $x=\sin \theta, y=\sin ^3 \theta$, then $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{6}$ is

If $x=\mathrm{e}^{\tan ^{-1}\left(\frac{y-x^2}{x^2}\right)}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=1$ is

The foci of the conic $25 x^2+16 y^2-150 x=175$ are

$$ \begin{aligned} & f(x)=\left\{\begin{array}{ll} 3-x, & -1 \leqslant x<0 \\ 1+\frac{5 x}{3}, & -3 \leqslant x \leqslant 2 \end{array}\right. \text { and } \\ & g(x)=\left\{\begin{aligned} -x, & -2 \leqslant x \leqslant 3 \\ x, & 0 \leqsla...

\(\int \frac{x}{1+x^4} d x=\)

\(\int \frac{(5 \sin \theta-2) \cos \theta}{\left(5-\cos ^2 \theta-4 \sin \theta\right)} d \theta=\)

\(\int \sqrt{x^2+3 x} d x=\)

If $0 \leqslant \cos ^{-1} x \leqslant \pi$ and $\frac{-\pi}{2} \leqslant \sin ^{-1} x \leqslant \frac{\pi}{2}$, then at $x=\frac{1}{5}$ the value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ is

If $\left(\cos ^{-1} x\right)^2-\left(\sin ^{-1} x\right)^2>0$, then

The function $\mathrm{f}(x)=2 x-\left|x-x^2\right|$ is

Let $\mathrm{A}=\mathop {\lim }\limits_{x \to {0^ + }}\left(1+\tan ^2 \sqrt{x}\right)^{\frac{1}{2 x}}$, then $\log _{\mathrm{e}} \mathrm{A}=$

The difference between the maximum value and minimum value of objective function $\mathrm{z}=3 x+5 y$ subject to constraints $x+3 y \leq 60$, $x+y \geq 10, x-y \geq 0, x, y \geq 0$ is

Consider the statements given by following
(A) If $4+3=8$, then $5+3=9$
(B) If $6+4=10$, then moon is flat
(C) If both (A) and (B) are true, then $5+6=17$
Then which of the following statement is correct?

Number of switches in alternative equivalent simple circuit for the circuit is (are)

If $A=\left[\begin{array}{rrr}1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0\end{array}\right], B=\operatorname{adj} A$ and $C=5 A$, then $\frac{|\operatorname{adjB}|}{|\mathrm{C}|}=$

If ${ }^{15} \mathrm{C}_4+{ }^{15} \mathrm{C}_5+{ }^{16} \mathrm{C}_6+{ }^{17} \mathrm{C}_7+{ }^{18} \mathrm{C}_8={ }^{19} \mathrm{C}_{\mathrm{r}}$, then the value of $r$ is equal to

If $X \sim B\left(6, \frac{1}{2}\right)$, then $P(|X-2| \leqslant 1)=$

If A and B are independent events such that $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right)=\frac{3}{25}$ and $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=\frac{8}{25}$, then $P(A)=$

A random variable X has following p.d.f. $\mathrm{f}(x)=\mathrm{kx}(1-x), 0 \leqslant x \leqslant 1 \quad$ and $\quad \mathrm{P}(x>\mathrm{a})=\frac{20}{27}$, then $\mathrm{a}=$

In a triangle with one of the angles $120^{\circ}$, the lengths of the sides form an A.P. If length of the greatest side is 7 m , then the area of the triangle is

Let $\mathrm{A} \equiv(0,0), \mathrm{B}(3,0), \mathrm{C}(0,-4)$ are vertices of $\triangle A B C$, then the co-ordinates of incentre of $\triangle \mathrm{ABC}$ is

In a triangle ABC with usual notations if $b \sin C(b \cos C+c \cos B)=42$, then area of triangle $\mathrm{ABC}=$

The probability distribution of a random variable X is given by$$ \begin{array}{|l|c|c|c|c|c|} \hline \mathrm{X}=x_i & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}\left(\mathrm{X}=x_i\right) & 0.4 & 0.3 & 0.1 & 0.1 & 0.1 \\ \hline \end{array} $$
...

The equation $x^2-3 x y+\lambda y^2+3 x-5 y+2=0$, where $\lambda$ is real number represents pair of lines If $\theta$ is acute angle between the lines, then $\frac{\operatorname{cosec}^2 \theta}{\sqrt{10}}=$

The angle between the line $x=\frac{y-1}{2}=\frac{z-3}{\lambda}$ and the plane $x+2 y+3 z=6$ is $\cos ^{-1} \sqrt{\frac{5}{14}}$, then the value of $\lambda$ is

If the foot of the perpendicular drawn from the origin to a plane is $\mathrm{P}(2,-1,4)$, then the equation of the plane is

The angle between the lines $x=y, z=0$ and $y=0, \mathrm{z}=0$ is

If $\theta$ is the angle between the lines whose direction cosines are given by $6 \mathrm{mn}-2 \mathrm{n} l+5 l \mathrm{~m}=0$ and $3 l+\mathrm{m}+5 \mathrm{n}=0$, then $\sin \theta=$

The line passing through the points $(a, 1,6)$ and $(3,4, \mathrm{~b})$ crosses the $y z$-plane at $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$, then the value of $(3 a+4 b)$ is

Let the plane passing through point $(2,1,-1)$ containing line joining the points $(1,3,2)$ and $(1,2,1)$ makes intercepts $\mathrm{p}, \mathrm{q}, \mathrm{r}$ on co-ordinate axes, then $\mathrm{p}+\mathrm{q}+\mathrm{r}=$

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

If $\sin \theta=\frac{1}{2}\left(x+\frac{1}{x}\right)$, then $\sin 3 \theta+\frac{1}{2}\left(x^3+\frac{1}{x^3}\right)=$

Let $\bar{a}=\hat{i}+\hat{j}, \bar{b}=2 \hat{i}-\hat{k}, \bar{c}=3 \hat{i}-\hat{j}+\hat{k}$, then vector $\overline{\mathrm{p}}$ satisfying $\overline{\mathrm{p}} \cdot \overline{\mathrm{a}}=0$ and $\overline{\mathrm{p}} \times \overline{\m...

If $\bar{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \bar{b}=\hat{i}-2 \hat{j}-2 \hat{k}, \bar{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$ and if $\overline{\mathrm{d}}$ is vector perpendicular to both $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}, \overli...

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

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors such that $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0},|\overline{\mathrm{a}}|=3,|\overline{\mathrm{~b}}|=4,|\overline{\mathrm{c}}|=5$, then $\overline{\mat...