MHT CET 2023 10th May Evening Shift

The value of \(\mathrm{c}\) for the function \(\mathrm{f}(x)=\log x\) on [\(1\), e] if LMVT can be applied, is

Let \(\mathrm{f}(x)=\mathrm{e}^x-x\) and \(\mathrm{g}(x)=x^2-x, \forall x \in \mathrm{R}\), then the set of all \(x \in \mathrm{R}\), where the function \(\mathrm{h}(x)=(\mathrm{fog})(x)\) is increasing is

The displacement '\(\mathrm{S}\)' of a moving particle at a time \(t\) is given by \(S=5+\frac{48}{t}+t^3\). Then its acceleration when the velocity is zero, is

If the surface area of a spherical balloon of radius \(6 \mathrm{~cm}\) is increasing at the rate \(2 \mathrm{~cm}^2 / \mathrm{sec}\), then the rate of increase in its volume in \(\mathrm{cm}^3 / \mathrm{sec}\) is

The value of \(\alpha\), so that the volume of parallelopiped formed by \(\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}\) and \(\alpha \hat{\mathrm{i}}+\hat{\mathrm{k}}\) becomes minimum, is

In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the n...

The area bounded by the \(\mathrm{X}\)-axis and the curve \(y=x(x-2)(x+1)\) is

The value of \(\frac{{ }^{10} \mathrm{C}_{\mathrm{r}}}{{ }^{11} \mathrm{C}_{\mathrm{r}}}\), when both the numerator and denominator are at their greatest values, is

The parametric equations of the circle \(x^2+y^2+2 x-4 y-4=0\) are

The argument of \(\frac{1+i \sqrt{3}}{\sqrt{3}+i}, i=\sqrt{-1}\) is

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) and \(\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) be continuous functions. Then the value of the integral $$\int_\limits{\frac{-\pi}{2}}^{\frac{\pi}{2}}[\mathrm{f}(x)+\mathrm{f}(-x)]...

\(\int_\limits 0^\pi \frac{x \tan x}{\sec x+\cos x} d x=\)

\(\int_\limits0^1 \cos ^{-1} x d x=\)

The general solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{3 x^2}{1+x^3}\right) y=\frac{1}{x^3+1}\) is

The differential equation of all circles, passing through the origin and having their centres on the \(\mathrm{X}\)-axis, is

Let \(f\) be a differentiable function such that \(\mathrm{f}(1)=2\) and \(\mathrm{f}^{\prime}(x)=\mathrm{f}(x)\), for all \(x \in \mathrm{R}\). If \(\mathrm{h}(x)=\mathrm{f}(\mathrm{f}(x))\), then \(\mathrm{h}^{\prime}(1)\) is equal to

If \(y\) is a function of \(x\) and \(\log (x+y)=2 x y\), then \(\frac{d y}{d x}\) at \(x=0\) is

If \(x=3 \tan \mathrm{t}\) and \(y=3 \sec \mathrm{t}\), then the value of \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\) at \(\mathrm{t}=\frac{\pi}{4}\) is

If \(y=\tan ^{-1}\left(\frac{\log \left(\frac{\mathrm{e}}{x^2}\right)}{\log \left(e x^2\right)}\right)+\tan ^{-1}\left(\frac{4+2 \log x}{1-8 \log x}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is

The value of \(\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}\) is

\(\int \frac{5 \tan x}{\tan x-2} \mathrm{~d} x=x+\mathrm{a} \log |\sin x-2 \cos x|+\mathrm{c},\)
(where \(c\) is a constant of integration), then the value of \(a\) is

The value of \(\int \frac{\left(x^2-1\right) d x}{x^3 \sqrt{2 x^4-2 x^2+1}}\) is

\(\int \mathrm{e}^x\left(1-\cot x+\cot ^2 x\right) \mathrm{d} x=\)

The domain of the function \(\mathrm{f}(x)=\sin ^{-1}\left(\frac{|x|+5}{x^2+1}\right)\) is \((-\infty,-a] \cup[a, \infty)\). Then a is equal to

The value of \(\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)\) is

If \(\tan ^{-1} a+\tan ^{-1} b+\tan ^{-1} c=\pi\), then which of the following statement is true?

Let \(\mathrm{S}=\left\{\mathrm{t} \in \mathrm{R} / \mathrm{f}(x)=|x-\pi|\left(\mathrm{e}^{|x|}-1\right) \sin |x|\right.\) is not differentiable at \(\mathrm{t}\}\), then \(\mathrm{S}\) is

If \(\mathrm{f}(x)=\frac{4}{x^4}\left[1-\cos \frac{x}{2}-\cos \frac{x}{4}+\cos \frac{x}{2} \cdot \cos \frac{x}{4}\right]\) is continuous at \(x=0\), then \(\mathrm{f}(0)\) is

\(\lim _\limits{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}=\)

The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of objective function \(3 x+5 y\), subject to the linear constraints given by this system of inequations is

If \(\mathrm{p}\) and \(\mathrm{q}\) are true statements and \(\mathrm{r}\) and \(\mathrm{s}\) are false statements, then the truth values of the statement patterns \((p \wedge q) \vee r\) and $$(\mathrm{p} \vee \mathrm{s}) \leftrightarrow(...

The negation of the statement pattern \(\sim s \vee(\sim r \wedge s)\) is equivalent to

If $$B=\left[\begin{array}{ccc}3 & \alpha & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3\end{array}\right]$$ is the adjoint of a \(3 \times 3\) matrix \(\mathrm{A}\) and \(|\mathrm{A}|=4\), then \(\alpha\) is equal to

The three ships namely A, B and C sail from India to Africa. If the odds in favour of the ships reaching safely are \(2: 5,3: 7\) and \(6: 11\) respectively, then probability of all of them arriving safely is

If the sum of mean and variance of a Binomial Distribution is \(\frac{15}{2}\) for 10 trials, then the variance is

In a game, 3 coins are tossed. A person is paid ₹ 7 /-, if he gets all heads or all tails; and he is supposed to pay ₹ 3 /-, if he gets one head or two heads. The amount he can expect to win on an average per game is ₹

A fair die is tossed twice in succession. If \(\mathrm{X}\) denotes the number of sixes in two tosses, then the probability distribution of \(\mathrm{X}\) is given by

In a triangle \(\mathrm{ABC}\), with usual notations, if \(\mathrm{c}=4\), then value of \((a-b)^2 \cos ^2 \frac{C}{2}+(a+b)^2 \sin ^2 \frac{C}{2}\) is

If \(x, y, z\) are in A.P. and \(\tan ^{-1} x, \tan ^{-1} y\) and \(\tan ^{-1} z\) are also in A.P., then

Mean and variance of six observations are 8 and 16 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is

The joint equation of a pair of lines passing through the origin and making an angle of \(\frac{\pi}{4}\) with the line \(3 x+2 y-8=0\) is

Two sides of a square are along the lines \(5 x-12 y+39=0\) and \(5 x-12 y+78=0\), then area of the square is

The plane through the intersection of planes \(x+y+z=1\) and \(2 x+3 y-z+4=0\) and parallel to \(\mathrm{Y}\)-axis also passes through the point

The perpendicular distance of the origin from the plane \(x-3 y+4 z-6=0\) is

Two lines \(\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}\) and \(\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4} \quad\) intersect at the point R. Then reflection of \(\mathrm{R}\) in the \(x y\)-plane has co-ordinates

If \((\bar{a} \times \bar{b}) \times \bar{c}=-5 \bar{a}+4 \bar{b}\) and \(\bar{a} \cdot \bar{b}=3\), then the value of \(\bar{a} \times(\bar{b} \times \bar{c})\) is

If \(\bar{p}=\hat{i}+\hat{j}+\hat{k}\) and \(\bar{q}=\hat{i}-2 \hat{j}+\hat{k}\). Then a vector of magnitude \(5 \sqrt{3}\) units perpendicular to the vector \(\bar{q}\) and coplanar with \(\bar{p}\) and \(\bar{q}\) is

If \(\bar{a}\) and \(\bar{b}\) are two unit vectors such that \(\bar{a}+2 \bar{b}\) and \(5 \bar{a}-4 \bar{b}\) are perpendicular to each other, then the angle between \(\bar{a}\) and \(\bar{b}\) is

If \(\overline{\mathrm{a}}=\mathrm{m} \overline{\mathrm{b}}+\mathrm{nc}\), where $$\overline{\mathrm{a}}=4 \hat{\mathrm{i}}+13 \hat{\mathrm{j}}-18 \hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}...

If the volume of tetrahedron, whose vertices are with position vectors \(\hat{i}-6 \hat{j}+10 \hat{k},-\hat{i}-3 \hat{j}+7 \hat{k}, 5 \hat{i}-\hat{j}+\lambda \hat{k}\) and \(7 \hat{i}-4 \hat{j}+7 \hat{k}\) is 11 cubic units, then value of $...