MHT CET 2024 2nd May Morning Shift

The normal to the curve, $y(x-2)(x-3)=x+6$ at the point, where the curve intersects the Y-axis, passes through the point

The length of the longest interval, in which the function $3 \sin x-4 \sin ^3 x$ is increasing, is

If Rolle's theorem holds for the function $\mathrm{f}(x)=x^3+\mathrm{bx}{ }^2+\mathrm{ax}+5$ on $[1,3]$ with $\mathrm{c}=2+\frac{1}{\sqrt{3}}$, then the values of $a$ and $b$ respectively are

The area of the region bounded by hyperbola $x^2-y^2=9$ and its latus rectum is

If $\left(m_i, \frac{1}{m_i}\right), m_i>0, i=1,2,3,4$ are four distinct points on a circle, then the product $\mathrm{m}_1 \mathrm{~m}_2 \mathrm{~m}_3 \mathrm{~m}_4$ is equal to

If $\mathrm{a}>0$ and $\mathrm{z}=\frac{(1+\mathrm{i})^2}{\mathrm{a}-\mathrm{i}}, \mathrm{i}=\sqrt{-1}$, has magnitude $\sqrt{\frac{2}{5}}$ then $\bar{z}$ is equal to

Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then $\int_0^a f(x) g(x) d x$ is equal to

If $x \frac{\mathrm{~d} y}{\mathrm{~d} x}=y(\log y-\log x+1)$, then general solution of this equation is

A spherical metal ball at 80$^\circ$C cools in 5 minutes to 60$^\circ$C, in surrounding temperature of 20$^\circ$C, then the temperature of the ball after 20 minutes is approximately

The solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=(x-y)^2$ when $y(1)=1$ is

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

If $y=\sqrt{\frac{1-\sin ^{-1} x}{1+\sin ^{-1} x}}$, then $\left(\frac{d y}{d x}\right)$ at $x=0$ is

If $y=a \sin x+b \cos x \quad$ (where $\mathrm{a}$ and $\mathrm{b}$ are constants), then $y^2+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2$ is

For a suitable chosen real constant a, let a function $f: \mathbb{R}-\{-\mathrm{a}\} \rightarrow \mathbb{R}$ be defined by $f(x)=\frac{a-x}{a+x}$. Further suppose that for any real number $x \neq-\mathrm{a}$ and $\mathrm{f}(x) \neq-\mathrm{...

\(\int 3^{3^x} \cdot 3^x d x=\)

\(\int \log (1+x)^{1+x} \mathrm{~d} x=\)

$\int \frac{\mathrm{d} x}{3-2 \cos 2 x}=\frac{\tan ^{-1}(\mathrm{f}(x))}{\sqrt{5}}+\mathrm{c}$, (where c is a constant of integration), then $f(\pi / 4)$ has the value

\(\int\left(\frac{x+2}{x+4}\right)^2 \cdot e^x \mathrm{~d} x=\)

The value of $\sin \left(2 \cos ^{-1}\left(-\frac{3}{5}\right)\right)$ is

The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$ where $0 \leq \cos ^{-1} x \leq \pi$ and $-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$, is

Considering only the principal values of inverse function, the set
\(A=\left\{x \geq 0 / \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}\)

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

The value of $\lim _\limits{x \rightarrow 0} \frac{x}{|x|+x^2}$ is

Let K be the set of all real values of $x$, where the function $\mathrm{f}(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ is not differentiable. Then the set K is

If $\mathrm{f}(x)=\frac{1+\cos \pi x}{\pi(1-x)^2}$, for $x \neq 1$ is continuous at $x=1$, then $\mathrm{f}(1)$ is equal to

The point, at which the maximum value of $10 x+6 y$ subject to the constraints $x+y \leq 12$, $2 x+y \leq 20, x \geq 0, y \geq 0$ occurs, is

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

If statement I : If the work is not finished on time, the contractor is in trouble.
statement II : Either the work is finished on time or the contractor is in trouble.
then

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

If 3 books on Physics, 2 books on Chemistry and 4 books on Mathematics are to be arranged on a shelf so that all the Physics books are together and all the Mathematics books are together, then the number of such arrangements is

Ten bulbs are drawn successively, with replacement, from a lot containing $10 \%$ defective bulbs, then the probability that there is at least one defective bulb, is

A bag contains 4 Red and 6 Black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with 3 additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, th...

If a discrete random variable X takes values $0,1,2,3, \ldots \ldots$. with probability $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1) 5^{-x}$, where k is a constant, then $\mathrm{P}(\mathrm{X}=0)$ is

In $\triangle A B C$, with usual notations, if $\frac{1}{b+c}+\frac{1}{c+a}=\frac{3}{a+b+c}$, then $m \angle C$ is equal to

The mean of 100 observations is 50 and their standard deviation is 5 , then the sum of all squares of all the observations is

If two lines $x+(a-1) y=1 \quad$ and $2 x+a^2 y=1(a \in R-\{0,1\})$ are perpendicular, then the distance of their point of intersection from the origin is

The slopes of the lines given by $x^2+2 h x y+2 y^2=0$ are in the ratio $1: 2$, then $h$ is

The Cartesian equation of the plane, passing through the points $(3,1,1),(1,2,3)$ and $(-1,4,2)$, is

The vector equation of a line whose Cartesian equations are $y=2,4 x-3 z+5=0$ is

If the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane $\ell x+m y-z=9$, then $\ell^2+m^2$ is

The equation of the line passing through the point $(-1,3,-2)$ and perpendicular to each of the lines $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and $\frac{x+2}{-3}=\frac{y-1}{2}=\frac{z+1}{5}$, is

The principal solutions, of the equation $\sqrt{3} \sec x+2=0$, are

If $\bar{a}=(2 \hat{i}+2 \hat{j}+3 \hat{k}), \vec{b}=(-\hat{i}+2 \hat{j}+\hat{k}) \quad$ and $\bar{c}=(3 \hat{i}+\hat{j})$ such that $(\bar{a}+\lambda \bar{b})$ is perpendicular to $\bar{c}$, then the value of $\lambda$ is

For all real $x$, the vectors $C x \hat{i}-6 \hat{j}-3 \hat{k}$ and $x \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{C} x \hat{\mathrm{k}}$ make an obtuse angle with each other, then the value of C can be in

If $x_0$ is the point of local minima of $f(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$ where $\overline{\mathrm{a}}=x \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, $\overline{\mathrm{b}}=-...

$\hat{a}, \hat{b}$, and $\hat{c}$ are three unit vectors such that $\hat{a} \times(\hat{b} \times \hat{c})=\frac{\sqrt{3}}{2}(\hat{b}+\hat{c})$. If $\dot{b}$ is not parallel to $\hat{c}$, then the angle between $\hat{a}$ and $\hat{b}$ is

The scalar $\overline{\mathrm{a}} \cdot[(\overline{\mathrm{b}}+\overline{\mathrm{c}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})]$ equals

The volume of parallelopiped formed by vectors $\hat{i}+m \hat{j}+\hat{k}, \hat{j}+m \hat{k}$ and $m \hat{i}+\hat{k}$ becomes minimum when $m$ is

If the vectors $\overline{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\mathrm{mi}+\mathrm{j}+\mathrm{nk}$ are mut...

Let $\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$. Let $\overline{\mathrm{c}}$ be a vector such that $|\bar{c}-\bar{a}|=3$ and $|(\overline{\math...