MHT CET 2023 14th May Evening Shift

The function \(\mathrm{f}(x)=x^3-6 x^2+9 x+2\) has maximum value when \(x\) is

If \(y=4 x-5\) is a tangent to the curve \(y^2=\mathrm{p} x^3+\mathrm{q}\) at \((2,3)\), then \(\mathrm{p}-\mathrm{q}\) is

The diagonal of a square is changing at the rate of \(0.5 \mathrm{~cm} / \mathrm{sec}\). Then the rate of change of area when the area is \(400 \mathrm{~cm}^2\) is equal to

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

The area bounded by the curve \(y=|x-2|, x=1, x=3\) and \(X\)-axis is

The centre of the circle whose radius is 3 units and touching internally the circle \(x^2+y^2-4 x-6 y-12=0\) at the point \((-1,-1)\) is

Let \(z \in C\) with \(\operatorname{Im}(z)=10\) and it satisfies \(\frac{2 z-n}{2 z+n}=2 i-1, i=\sqrt{-1}\) for some natural number \(\mathrm{n}\), then

If \(I_n=\int_\limits0^{\frac{\pi}{4}} \tan ^n \theta d \theta\), then \(I_{12}+I_{10}=\)

If the slope of the tangent of the curve at any point is equal to \(-y+\mathrm{e}^{-x}\), then the equation of the curve passing through origin is

If a body cools from \(80^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) in the room temperature of \(25^{\circ} \mathrm{C}\) in 30 minutes, then the temperature of the body after 1 hour is

The differential equation representing the family of curves \(y^2=2 \mathrm{c}(x+\sqrt{\mathrm{c}})\), where \(\mathrm{c}\) is a positive parameter, is of

If \(x=\sqrt{\mathrm{e}^{\sin ^{-1} t}}\) and \(y=\sqrt{\mathrm{e}^{\cos ^{-1} t}}\), then \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\) is

If \(\mathrm{f}^{\prime}(x)=\sin (\log x)\) and \(y=\mathrm{f}\left(\frac{2 x+3}{3-2 x}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=1\) is

Let \(\mathrm{P}(x)\) be a polynomial of degree 2, with \(\mathrm{P}(2)=-1, \mathrm{P}^{\prime}(2)=0, \mathrm{P}^{\prime \prime}(2)=2\), then \(\mathrm{P}(1.001)\) is

If \(y=\sqrt{(x-\sin x)+\sqrt{(x-\sin x)+\sqrt{(x-\sin x) \ldots.}}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\)

If \(\mathrm{f}(x)=\frac{2 x-3}{3 x-4}, x \neq \frac{4}{3}\), then the value of \(\mathrm{f}^{-1}(x)\) is

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

If \(\int \frac{x^2}{\sqrt{1-x}} \mathrm{~d} x=\mathrm{p} \sqrt{1-x}\left(3 x^2+4 x+8\right)+\mathrm{c}\) where \(\mathrm{c}\) is a constant of integration, then the value of \(p\) is

\(\int \frac{\mathrm{d} x}{\cot ^2 x-1}=\frac{1}{\mathrm{~A}} \log |\sec 2 x+\tan 2 x|-\frac{x}{\mathrm{~B}}+\mathrm{c}\), (where \(\mathrm{c}\) is constant of integration), then \(\mathrm{A}+\mathrm{B}=\)

If \(I=\int \frac{d x}{\sin (x-a) \sin (x-b)}\), then I is given by

If \(\sum_\limits{r=1}^{50} \tan ^{-1} \frac{1}{2 r^2}=p\) then \(\tan p\) is

If \(\cos ^{-1} x-\cos ^{-1} \frac{y}{3}=\alpha\), where \(-1 \leq x \leq 1$, $-3 \leq y \leq 3, x \leq \frac{y}{3}\), then for all \(x, y, 9 x^2-6 x y \cos \alpha+y^2\) is equal to

The value of \(\tan ^{-1}\left(\frac{1}{8}\right)+\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{5}\right)\) is

Given \(0 \leq x \leq \frac{1}{2}\), then the value of \(\tan \left(\sin ^{-1}\left(\frac{x}{\sqrt{2}}+\frac{\sqrt{1-x^2}}{\sqrt{2}}\right)-\sin ^{-1} x\right)\) is

The function $\mathrm{f}$ defined on \(\left(-\frac{1}{3}, \frac{1}{3}\right)\) by
$$\mathrm{f}(x)=\left\{\begin{array}{cc}
\frac{1}{x} \log \left(\frac{1+3 x}{1-2 x}\right) & , \quad x \neq 0 \\
\mathrm{k} & , \quad x=0
\end{array}\right.$...

If \(f(a)=2, f^{\prime}(a)=1, g(a)=-1, g^{\prime}(a)=2\), then as \(x\) approaches a, \(\frac{\mathrm{g}(x) \mathrm{f}(\mathrm{a})-\mathrm{g}(\mathrm{a}) \mathrm{f}(x)}{(x-\mathrm{a})}\) approaches

Let \(\mathrm{f}(x)=5-|x-2|\) and \(\mathrm{g}(x)=|x+1|, x \in \mathrm{R}\) If \(\mathrm{f}(x)\) attains maximum value at \(\alpha\) and \(\mathrm{g}(x)\) attains minimum value at \(\beta\), then $$\lim _\limits{x \rightarrow-\alpha \beta} ...

The shaded region in the following figure represents the solution set for a certain linear programming problem. Then linear constraints for this region are given by

If the statement \(\mathrm{p} \leftrightarrow(\mathrm{q} \rightarrow \mathrm{p})\) is false, then true statement/statement pattern is

The statement \([\mathrm{p} \wedge(\mathrm{q} \vee \mathrm{r})] \vee[\sim \mathrm{r} \wedge \sim \mathrm{q} \wedge \mathrm{p}]\) is equivalent to

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

If in a regular polygon, the number of diagonals are 54, then the number of sides of the polygon are

A fair die with numbers 1 to 6 on their faces is thrown. Let \(\mathrm{X}\) denote the number of factors of the number, on the uppermost face, then the probability distribution of \(\mathrm{X}\) is

The p.m.f. of a random variable \(\mathrm{X}\) is $$\mathrm{P}(x)=\left\{\begin{array}{cl}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \mathrm{n} \\ 0, & \text { otherwise }\end{array}\right.$$, then \(\mathrm{E}(\mathrm{X})\) is

There are 6 positive and 8 negative numbers. From these four numbers are chosen at random and multiplied. Then the probability, that the product is a negative number, is

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs are selected at random from the lot and are sent to retail store. Then the probability that the store will receive at most one defective bulb is

In a triangle \(\mathrm{A B C, m \angle A, m \angle B, m \angle C}\) are in A.P. and lengths of two larger sides are 10 units, 9 units respectively, then the length (in units) of the third side is

If both mean and variance of 50 observations \(x_1, x_2, \ldots, x_{50}\) are equal to 16 and 256 respectively, then mean of \(\left(x_1-5\right)^2,\left(x_2-5\right)^2, \ldots \ldots,\left(x_{50}-5\right)^2\) is

Let \(P \equiv(-3,0), Q \equiv(0,0)\) and \(R \equiv(3,3 \sqrt{3})\) be three points. Then the equation of the bisector of the angle \(\mathrm{PQR}\) is

The centroid of the triangle formed by the lines \(x+3 y=10\) and \(6 x^2+x y-y^2=0\) is

The mirror image of \(\mathrm{P}(2,4,-1)\) in the plane \(x-y+2 z-2=0\) is \((\mathrm{a}, \mathrm{b}, \mathrm{c})\), then the value of \(a+b+c\) is

If the lines \(\frac{x-\mathrm{k}}{2}=\frac{y+1}{3}=\frac{\mathrm{z}-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\frac{9}{2}}{2}=\frac{\mathrm{z}}{1}\) intersect, then the value of \(\mathrm{k}\) is

A vector parallel to the line of intersection of the planes \(\bar{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=1\) and \(\bar{r} \cdot(\hat{i}+4 \hat{j}-2 \hat{k})=2\) is

The length of the perpendicular drawn from the point \((1,2,3)\) to the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\) is

If \(\mathrm{a} \cos 2 \theta+\mathrm{b} \sin 2 \theta=\mathrm{c}\) has \(\alpha\) and \(\beta\) as its roots, then the value of \(\tan \alpha+\tan \beta\) is

Let \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}\) and \(\bar{b}=\hat{i}+\hat{j}\). If \(\bar{c}\) is a vector such that \(\bar{a} \cdot \bar{c}=|\bar{c}|,|\bar{c}-\bar{a}|=2 \sqrt{2}\) and the angle between \(\bar{a} \times \bar{b}\) and $$\bar{c...

If \(|\bar{a}|=2,|\bar{b}|=3,|\bar{c}|=5\) and each of the angles between the vectors \(\bar{a}\) and \(\bar{b}, \bar{b}\) and \(\bar{c}\), \(\bar{c}\) and \(\bar{a}\) is \(60^{\circ}\), then the value of \(|\bar{a}+\bar{b}+\bar{c}|\) is

Let \(\overline{\mathrm{u}}, \overline{\mathrm{v}}\) and \(\overline{\mathrm{w}}\) be the vectors such that \(|\overline{\mathrm{u}}|=1; |\bar{v}|=2 ;|\bar{w}|=3\). If the projection of \(\bar{v}\) along \(\bar{u}\) is equal to that of $$\o...

Let \(\bar{a}=\hat{i}+2 \hat{j}-\hat{k}\) and \(\bar{b}=\hat{i}+\hat{j}-\hat{k}\) be two vectors. If \(\bar{c}\) is a vector such that \(\bar{b} \times \bar{c}=\bar{b} \times \bar{a}\) and $$\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}...

If \(|\vec{a}|=\sqrt{3} ;|\vec{b}|=5 ; \bar{b} \cdot \bar{c}=10\), angle between \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) is \(\frac{\pi}{3}, \overline{\mathrm{a}}\) is perpendicular to $$\overline{\mathrm{b}} \times \overlin...