MHT CET 2023 11th May Evening Shift

If the function \(f\) is given by \(f(x)=x^3-3(a-2) x^2+3 a x+7\), for some \(\mathrm{a} \in \mathbb{R}\), is increasing in \((0,1]\) and decreasing in \([1,5)\), then a root of the equation \(\frac{\mathrm{f}(x)-14}{(x-1)^2}=0(x \neq 1)\) ...

If \(a\) and \(b\) are positive number such that \(a>b\), then the minimum value of \(a \sec \theta-b \tan \theta\left(0 < \theta < \frac{\pi}{2}\right)\) is

\(A\) rod \(A B, 13\) feet long moves with its ends \(A\) and \(B\) on two perpendicular lines \(O X\) and \(O Y\) respectively. When \(A\) is 5 feet from \(O\), it is moving away at the rate of \(3 \mathrm{feet} / \mathrm{sec}\). At this i...

The equation of the tangent to the curve \(y=\sqrt{9-2 x^2}\), at the point where the ordinate and abscissa are equal, is

At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production \(\mathrm{P}\) w.r.t. additional number of worker \(x\) is given by \(\frac{\mathrm{dp}}{\mathrm{d} x}=100-12 \sqrt{x}\).

If the firm empl...

The slope of the tangent to a curve \(y=\mathrm{f}(x)\) at \((x, \mathrm{f}(x))\) is \(2 x+1\). If the curve passes through the point \((1,2)\), then the area (in sq. units), bounded by the curve, the \(\mathrm{X}\)-axis and the line $$x=1$...

If a circle passes through points \((4,0)\) and \((0,2)\) and its centre lies on \(\mathrm{Y}\)-axis. If the radius of the circle is \(r\), then the value of \(r^2-r+1\) is

If \(x=\frac{5}{1-2 \mathrm{i}}, \mathrm{i}=\sqrt{-1}\), then the value of \(x^3+x^2-x+22\) is

Let \(f:[-1,2] \rightarrow[0, \infty)\) be a continuous function such that \(\mathrm{f}(x)=\mathrm{f}(1-x), \forall x \in[-1,2]\)
Let \(\mathrm{R}_1=\int_{-1}^2 x \mathrm{f}(x) \mathrm{d} x\) and \(\mathrm{R}_2\) be the area of the region b...

The solution of \(\frac{\mathrm{d} x}{\mathrm{~d} y}+\frac{x}{y}=x^2\) is

The solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{y}{x}=\sin x\) is

If \(\mathrm{f}(x)=3^x ; \mathrm{g}(x)=4^x\), then \(\frac{\mathrm{f}^{\prime}(0)-\mathrm{g}^{\prime}(0)}{1+\mathrm{f}^{\prime}(0) \mathrm{g}^{\prime}(0)}\) is

\(\text { For all real } x \text {, the minimum value of } \frac{1-x+x^2}{1+x+x^2} \text { is }\)

The set of all points, where the derivative of the functions \(\mathrm{f}(x)=\frac{x}{1+|x|}\) exists, is

If \(y=[(x+1)(2 x+1)(3 x+1) \ldots(\mathrm{n} x+1)]^{\frac{3}{2}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=0\) is

If \(\mathrm{f}(x)=\frac{3 x+4}{5 x-7}\) and \(\mathrm{g}(x)=\frac{7 x+4}{5 x-3}\), then \(\mathrm{f}(\mathrm{g}(x))=\)

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

If \(\int \frac{\cos 8 x+1}{\cot 2 x-\tan 2 x} \mathrm{~d} x=\mathrm{A} \cos 8 x+\mathrm{c}\), where \(\mathrm{c}\) is an arbitrary constant, then the value of \(\mathrm{A}\) is

The value of \(\int(1-\cos x) \cdot \operatorname{cosec}^2 x d x\) is

If \(\mathrm{I}=\int \sin (\log (x)) \mathrm{d} x\), then \(\mathrm{I}\) is given by

If \(\cos ^{-1} \sqrt{\mathrm{p}}+\cos ^{-1} \sqrt{1-\mathrm{p}}+\cos ^{-1} \sqrt{1-\mathrm{q}}=\frac{3 \pi}{4}\), then \(\mathrm{q}\) is

\(\pi+\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)\) is equal to

\(\text { If } l=\lim _\limits{x \rightarrow 0} \frac{x}{|x|+x^2} \text {, then the value of } l \text { is }\)

If $$\mathrm{f}(x)=\left\{\begin{array}{cc}\frac{x-3}{|x-3|}+\mathrm{a} & , \quad x < 3 \\ \mathrm{a}+\mathrm{b} & , \quad x=3 \\ \frac{|x-3|}{x-3}+\mathrm{b}, & x>3\end{array}\right.$$
Is continuous at \(x=3\), then the value of $$\mathrm{...

The maximum value of \(z=3 x+5 y\) subject to the constraints \(3 x+2 y \leq 18, x \leq 4, y \leq 6, x, y \geq 0\), is

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

If truth value of logical statement \((p \leftrightarrow \sim q) \rightarrow(\sim p \wedge q)\) is false, then the truth values of \(p\) and \(q\) are respectively

If $$\mathrm{A}=\left[\begin{array}{ll}\mathrm{i} & 1 \\ 1 & 0\end{array}\right]$$ where \(\mathrm{i}=\sqrt{-1}\) and \(\mathrm{B}=\mathrm{A}^{2029}\), then \(\mathrm{B}^{-1}=\)

The teacher wants to arrange 5 students on the platform such that the boy \(B_1\) occupies second position and the girls \(G_1\) and \(G_2\) are always adjacent to each other, then the number of such arrangements is

Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Then mean of number of tens is

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

If \(\mathrm{A}\) and \(\mathrm{B}\) are two events such that \(\mathrm{P}(\mathrm{A})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{1}{3}\), then the value of $$\mathrm{P}\left(\mathrm{A}^{\p...

Let a random variable \(\mathrm{X}\) have a Binomial distribution with mean 8 and variance 4. If \(\mathrm{P}(\mathrm{X} \leq 2)=\frac{\mathrm{K}}{2^{16}}\), then \(\mathrm{K}\) is

If the vertices of a triangle are \((-2,3),(6,-1)\) and \((4,3)\), then the co-ordinates of the circumcentre of the triangle are

In triangle \(\mathrm{ABC}\) with usual notations \(\mathrm{b}=\sqrt{3}, \mathrm{c}=1, \mathrm{~m} \angle \mathrm{A}=30^{\circ}\), then the largest angle of the triangle is

If the angles of a triangle are in the ratio \(4: 1: 1\), then the ratio of the longest side to its perimeter is

If both mean and variance of 50 observations \(x_1, x_2, \ldots \ldots, x_{50}\) 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

The joint equation of the lines pair of lines passing through the point \((3,-2)\) and perpendicular to the lines \(5 x^2+2 x y-3 y^2=0\) is

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

If the line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}\) meets the plane \(x+2 y+3 z=15\) at the point \(P\), then the distance of \(\mathrm{P}\) from the origin is

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

The angle between the line \(\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}\) and plane \(x-2 y-\lambda z=3\) is \(\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\), then value of \(\lambda\) is

If \(\cos x+\cos y-\cos (x+y)=\frac{3}{2}\), then

If the general solution of the equation \(\frac{\tan 3 x-1}{\tan 3 x+1}=\sqrt{3}\) is \(x=\frac{\mathrm{n} \pi}{\mathrm{p}}+\frac{7 \pi}{\mathrm{q}}, \mathrm{n}, \mathrm{p}, \mathrm{q}, \in \mathrm{Z}\), then \(\frac{p}{q}\) is

If \(\bar{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \bar{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}, \bar{c}=2 \hat{i}-\hat{j}+4 \hat{k}\), then a vector \(\overline{\mathrm{d}}\) which is parallel to vector $$\overline{\mathrm{a}} \times \overline{\mathrm{b}}...

The unit vector perpendicular to each of the vectors \(\bar{a}+\bar{b}\) and \(\bar{a}-\bar{b}\), where \(\bar{a}=\hat{i}+\hat{j}+\hat{k}\) and \(\overline{\mathrm{b}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\) is

Let \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \bar{b}=\hat{i}+\hat{j}\) and \(\bar{c}\) be a vector such that \(|\bar{c}-\bar{a}|=4,|(\bar{a} \times \bar{b}) \times \bar{c}|=3\) and the angle between \(\overline{\mathrm{c}}\) and $$\overline{\...

If the area of the parallelogram with \(\bar{a}\) and \(\bar{b}\) as two adjacent sides is \(16 \mathrm{sq}\). units, then the area of the parallelogram having \(3 \overline{\mathrm{a}}+2 \overline{\mathrm{b}}\) and $$\overline{\mathrm{a}}+...

If \(\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}\) and \(\overline{\mathrm{c}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) are such tha...

If \(\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) and $$\overline{\mathrm{c}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\beta \hat{...