MHT CET 2024 3rd May Morning Shift

A stone is dropped into a quiet lake and waves move in circles at speed of $8 \mathrm{~cm} / \mathrm{sec}$. At the instant when the radius of the circular wave is 12 cm . how fast is the enclosed area increasing?

If the half life of substance is 5 years, then the total amount of the substance left after 15 years, when initial amount is 64 gms is

A bullet is shot horizontally and its distance S cm at time t second is given by $\mathrm{S}=1200 \mathrm{t}-15 \mathrm{t}^2$, then the distance covered by the bullet when it comes to the rest, is

The equation of the normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

A triangular park is enclosed on two sides by a fence and on the third side a straight river bank. The two sides having fence are of same length $x$. The maximum area (in sq. units) enclosed by the park is

The area (in sq. units) of the region bounded by the curve $x^2=4 y$ and the straight line $x=4 y-2$ is

The equation of the concentric circle, with the circle $\mathrm{C}_1$ having equation $x^2+y^2-6 x-4 y-12=0$ and having double area compared to the area of $\mathrm{C}_1$, is

If $|z|=1$ and $w=\frac{z-1}{z+1}$ (where $\left.z \neq-1\right)$, then $\operatorname{Re}(w)$ is

If $\int_\limits0^{\frac{\pi}{3}} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} d \theta=1-\frac{1}{\sqrt{2}},(k>0)$, then the value of $k$ is

The differential equation $\left[\frac{1+\left(\frac{d y}{d x}\right)^2}{\left(\frac{d^2 y}{d x^2}\right)}\right]^{\frac{3}{2}}=\mathrm{kx}$ is of

The differential equation of family of circles, whose centres are on the X -axis and also touch the Y -axis is

If $y=\log \left[\mathrm{e}^{5 x}\left(\frac{3 x-4}{x+5}\right)^{\frac{4}{3}}\right]$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

 Let $f$ be a twice differentiable function such that $\mathrm{f}^{\prime \prime}(x)=-\mathrm{f}(x), \mathrm{f}^{\prime}(x)=\mathrm{g}(x)$ and $\mathrm{h}(x)=[\mathrm{f}(x)]^2+[\mathrm{g}(x)]^2$. If $\mathrm{h}(5)=1$, then $\mathrm{h}(10)$ ...

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

If $g(x)=x^2+x-1$ and (gof) $(x)=4 x^2-10 x+5$, then $\mathrm{f}(2)$ is equal to

If $\int \frac{\mathrm{d} x}{\cos ^3 x \sqrt{2 \sin 2 x}}=(\tan x)^A+C(\tan x)^B+K$, where K is a constant of integration, then the value of $5(A+B+C)$ is equal to

\(\int \frac{2 x^2-1}{\left(x^2+4\right)\left(x^2-3\right)} d x=\)

If $\quad \int(2 x+4) \sqrt{x-1} d x=a(x-1)^{5 / 2}+b(x-1)^{3 / 2}+c$ where $c$ is a constant of integration, then the value of $(2 a+b)$ is

The value of $\int \frac{(x-1) \mathrm{e}^x}{(x+1)^3} \mathrm{~d} x$ is equal to

The number of real solutions of $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$ is

Let $\mathrm{f}(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{1}{2}\right], \quad \mathrm{f}(x)$ is continuous in $\left[0, \frac{\pi}{2}\right]$, then $\mathrm{f}\left(\frac{\pi}{4}\right)$ is

Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation \((\sqrt[3]{1+a}-1) x^2+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0\) where $a>-1$ then $\lim _\limits{a \rightarrow 0^{+}} \alpha(a)$ and $\lim _\limits{a \rightarrow 0^{+}} \beta(a)$ re...

The shaded area in the figure below is the solution set for a certain linear programming problem, then the linear constraints are given by

If p and q are statements, then _________ is a contingency.

Consider the following statements
p : the switch $\mathrm{S}_1$ is closed.
q : the switch $\mathrm{S}_2$ is closed.
$r$ : the switch $\mathrm{S}_3$ is closed.
Then the switching circuit represented by the statement $(p \wedge q) \vee(\sim p...

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

The number of four letter words that can be formed using letters of the word BARRACK

A random variable x has the following probability distribution. Then value of $k$ is
_________ and $\mathrm{P}(3< x \leq 6)$ has the value

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Let $\mathrm{X} \sim \mathrm{B}\left(6, \frac{1}{2}\right)$, then $\mathrm{P}[|x-4| \leqslant 2]$ is

A person throws an unbiased die. If the number shown is even, he gains an amount equal to the number shown. If the number is odd, he loses an amount equal to the number shown. Then his expectation is ₹.

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Let X denote the random variable of number of jacks obtained in the two drawn cards. Then $P(X=1)+P(X=2)$ equals

The mean of $n$ observations is $\bar{x}$. If three observations $\mathrm{n}+1, \mathrm{n}-1,2 \mathrm{n}-1$ are added such that mean remains same, then value of $n$ is

The number of possible distinct straight lines passing through $(2,3)$ and forming a triangle with co-ordinate axes whose area is 12 sq . units are,

The line L given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line K is parallel to L and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

Equation of the plane containing the straight line $\frac{x}{3}=\frac{y}{2}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{4}=\frac{y}{3}=\frac{z}{2}$ and $\frac{x}{2}=\frac{y}{-4}=\frac{z}{3}$ is

The value of $m$, such that $\frac{x-4}{1}=\frac{y-2}{1}=\frac{2 z-m}{3}$ lies in the plane $2 x-5 y+2 z=7$, is

The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane $2 x-y+z+3=0$ is the line

Let $\mathrm{P}(2,3,6)$ be a point in space and Q be a point on the line $\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k})$. Then the value of $\mu$ for which vector $\overline{\mathrm{PQ}}$ is parallel to the plane $x-...

Let $S=\left\{x \in(-\pi, \pi) \mid x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set S is equal to

The number of roots of the equation, $(81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30$ in the interval $[0, \pi]$, is equal to

Let $2 \sin ^2 x+3 \sin x-2>0$ and $x^2-x-2<0$. ( $x$ is measured in radians). The $x$ lies in the interval

If $\theta$ and $\alpha$ are not odd multiples of $\frac{\pi}{2}$ then $\tan \theta=\tan \alpha$ implies principal solution is

If $\tan x=\frac{3}{4}$ and $\pi< x< \frac{3 \pi}{2}$, then $\cos \frac{x}{2}=$ ___________

The approximate value of $\cos \left(30^{\circ}, 30^{\prime}\right)$ is given that $1^{\circ}=0.0175^{\circ}$ and $\cos 30^{\circ}=0.8660$

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}$ and $3 \hat{i}+4 \hat{j}-12 \hat{k}$, is

If $|\bar{a}|=\sqrt{27},|\bar{b}|=7$ and $|\bar{a} \times \bar{b}|=35$, then $\bar{a} \cdot \bar{b}$ is equal to

If $\mathrm{A} \equiv(1,-1,0), \mathrm{B} \equiv(0,1,-1)$ and $\mathrm{C} \equiv(-1,0,1)$, then the unit vector $\overline{\mathrm{d}}$ such that $\overline{\mathrm{a}}$ and $\overline{\mathrm{d}}$ are perpendiculars and $\overline{\mathrm{...

Let the vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ be such that $|\overline{\mathrm{a}}|=2,|\overline{\mathrm{~b}}|=4$ and $|\bar{c}|=4$. If the projection of $\bar{b}$ on $\bar{a}$ is equal to the project...

Let $\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$. If $\overline{\mathrm{c}}$ is a vector such that $\overline{\mathrm{a}} \cdot \overline{\mathr...

Let $\bar{a}, \bar{b}, \bar{c}$ be three non-coplanar vectors and $\overline{\mathrm{p}}, \overline{\mathrm{q}}, \overline{\mathrm{r}}$ defined by the relations
$$\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}...