MHT CET 2023 9th May Morning Shift

The value of \(c\) of Lagrange's mean value theorem for \(f(x)=\sqrt{25-x^2}\) on \([1,5]\) is

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

The maximum value of xy when x + 2y = 8 is

An object is moving in the clockwise direction around the unit circle \(x^2+y^2=1\). As it passes through the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), its \(y\)-co-ordinate is decreasing at the rate of 3 units per sec. The ra...

A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is \(3 \mathrm{~mm}\) and 1 hour later it reduces to \(2 \mathrm{~mm}\), then the expression for the radius \(R\) of the raindrop at any ti...

The area (in sq. units) of the region \(\mathrm{A}=\left\{(x, y) / \frac{y^2}{2} \leq x \leq y+4\right\}\) is

Two tangents to the circle \(x^2+y^2=4\) at the points \(\mathrm{A}\) and \(\mathrm{B}\) meet at the point \(\mathrm{P}(-4,0)\). Then the area of the quadrilateral \(\mathrm{PAOB}, \mathrm{O}\) being the origin, is

If \(Z_1=4 i^{40}-5 i^{35}+6 i^{17}+2, Z_2=-1+i\), where \(i=\sqrt{-1}\), then \(\left|Z_1+Z_2\right|=\)

If \(\mathrm{f}(x)\) is a function satisfying \(\mathrm{f}^{\prime}(x)=\mathrm{f}(x)\) with \(\mathrm{f}(0)=1\) and \(\mathrm{g}(x)\) is a function that satisfies \(\mathrm{f}(x)+\mathrm{g}(x)=x^2\). Then the value of the integral $$\int_\l...

The differential equation of all parabolas, whose axes are parallel to \(\mathrm{Y}\)-axis, is

The particular solution of the differential equation \(\left(1+y^2\right) \mathrm{d} x-x y \mathrm{~d} y=0\) at \(x=1, y=0\), represents

The rate of change of \(\sqrt{x^2+16}\) with respect to \(\frac{x}{x-1}\) at \(x=5\) is

If \(x^2+y^2=\mathrm{t}+\frac{1}{\mathrm{t}}\) and \(x^4+y^4=\mathrm{t}^2+\frac{1}{\mathrm{t}^2}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to

If \(\mathrm{f}(1)=1, \mathrm{f}^{\prime}(1)=3\), then the derivative of \(\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2\) at \(x=1\) is

The derivative of \(\mathrm{f}(\sec x)\) with respect to \(g(\tan x)\) at \(x=\frac{\pi}{4}\), where \(f^{\prime}(\sqrt{2})=4\) and \(g^{\prime}(1)=2\), is

\(\mathrm{f}: \mathbb{R}-\left(-\frac{3}{5}\right) \rightarrow \mathbb{R}\) is defined by \(f(x)=\frac{3 x-2}{5 x+3}\), then \(f \circ f(1)\) is

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

\(\int \frac{\mathrm{e}^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] \mathrm{d} x, x > 0=\)

If \(I=\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=P \cos x+Q \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|\) (where \(c\) is a constant of integration), then values of \(\mathrm{P}\) and \(\mathrm{Q}\) are respectively

\(\int \frac{1}{\sin (x-a) \sin x} d x=\)

If \(\sin ^{-1} x+\cos ^{-1} y=\frac{3 \pi}{10}\), then the value of \(\cos ^{-1} x+\sin ^{-1} y\) is

The value of \(\cot \left(\sum_\limits{n=1}^{23} \cot ^{-1}\left(1+\sum_\limits{k=1}^n 2 k\right)\right)\) 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

\(\lim _\limits{x \rightarrow \frac{\pi}{2}} \frac{\cot x-\cos x}{(\pi-2 x)^3}\) equals

The number of discontinuities of the greatest integer function \(\mathrm{f}(x)=[x], x \in\left(-\frac{7}{2}, 100\right)\)

If feasible region is as shown in the figure, then the related inequalities are

The approximate value of \(\log _{10} 998\) is (given that \(\log _{10} \mathrm{e}=0.4343\) )

If truth values of statements \(\mathrm{p}, \mathrm{q}\) are true, and \(\mathrm{r}\), \(s\) are false, then the truth values of the following statement patterns are respectively
$$\begin{aligned}
& \mathrm{a}: \sim(\mathrm{p} \wedge \sim \...

The negation of the statement \((p \wedge q) \rightarrow(\sim p \vee r)\) is

If $$A=\left[\begin{array}{cc}2 & -1 \\ -1 & 3\end{array}\right]$$, then the inverse of \(\left(2 A^2+5 A\right)\) is

If at the end of certain meeting, everyone had shaken hands with everyone else, it was found that 45 handshakes were exchanged, then the number of members present at the meeting, are

In a Binomial distribution with \(\mathrm{n}=4\), if \(2 \mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=2)\), then the variance is

\(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are three events, one of which must and only one can happen. The odds in favor of \(\mathrm{A}\) are \(4: 6\), the odds against \(B\) are \(7: 3\). Thus, odds against \(\mathrm{C}\) are

The probability mass function of random variable X is given by
$$P[X=r]=\left\{\begin{array}{ll} \frac{{ }^n C_r}{32}, & n, r \in \mathbb{N} \\ 0, & \text { otherwise } \end{array} \text {, then } P[X \leq 2]=\right.$$

Three fair coins numbered 1 and 0 are tossed simultaneously. Then variance Var (X) of the probability distribution of random variable \(\mathrm{X}\), where \(\mathrm{X}\) is the sum of numbers on the uppermost faces, is

Two sides of a triangle are \(\sqrt{3}+1\) and \(\sqrt{3}-1\) and the included angle is \(60^{\circ}\), then the difference of the remaining angles is

The standard deviation of the following distribution

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If the distance between the parallel lines given by the equation \(x^2+4 x y+p y^2+3 x+q y-4=0\) is \(\lambda\), then \(\lambda^2=\)

The distance of a point \((2,5)\) from the line \(3 x+y+4=0\) measured along the line \(L_1\) and \(L_1\) are same. If slope of line \(L_1\) is \(\frac{3}{4}\), then slope of the line \(\mathrm{L}_2\) is

The foot of the perpendicular drawn from the origin to the plane is \((4,-2,5)\), then the Cartesian equation of the plane is

A vector \(\overrightarrow{\mathrm{n}}\) is inclined to \(\mathrm{X}\)-axis at \(45^{\circ}\), \(\mathrm{Y}\)-axis at \(60^{\circ}\) and at an acute angle to Z-axis If \(\overrightarrow{\mathrm{n}}\) is normal to a plane passing through the...

If the Cartesian equation of a line is \(6 x-2=3 y+1=2 z-2\), then the vector equation of the line is

The number of solutions in \([0,2 \pi]\) of the equation \(16^{\sin ^2 x}+16^{\cos ^2 x}=10\) is

If \(\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}\), then \(\cos ^2 48^{\circ}-\sin ^2 12^{\circ}\) has the value

If two vertices of a triangle are \(\mathrm{A}(3,1,4)\) and \(\mathrm{B}(-4,5,-3)\) and the centroid of the triangle is \(G(-1,2,1)\), then the third vertex \(C\) of the triangle is

Let two non-collinear vectors \(\hat{a}\) and \(\hat{b}\) form an acute angle. A point \(\mathrm{P}\) moves, so that at any time \(t\) the position vector \(\overline{\mathrm{OP}}\), where \(\mathrm{O}\) is origin, is given by $$\hat{a} \si...

The distance of the point having position vector \(\hat{i}-2 \hat{j}-6 \hat{k}\), from the straight line passing through the point \((2,-3,-4)\) and parallel to the vector \(6 \hat{i}+3 \hat{j}-4 \hat{k}\) is units.

The scalar product of the vector \(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\) with a unit vector along the sum of the vectors \(2 \hat{i}+4 \hat{j}-5 \hat{k}\) and \(\lambda \hat{i}+2 \hat{j}+3 \hat{k}\) is equal to 1 , then value...

If \([(\bar{a}+2 \bar{b}+3 \bar{c}) \times(\bar{b}+2 \bar{c}+3 \bar{a})] \cdot(\bar{c}+2 \bar{a}+3 \bar{b})=54\) then the value of $$\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$$ is

The volume of parallelopiped, whose coterminous edges are given by \(\overline{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \vec{v}=\hat{i}+\hat{j}+3 \hat{k}, \bar{w}=2 \hat{i}+\hat{j}+\hat{k}\) is 1 cu. units. If...