MHT CET 2021 23rd September Evening Shift

The radius of a circular plate is increasing at the rate of \(0.01 \mathrm{~cm} / \mathrm{sec}\), when the radius is \(12 \mathrm{~cm}\). Then the rate at which the area increases is

If \(x=a(\theta+\sin \theta)\) and \(y=a(1-\cos \theta)\) then \(\left(\frac{d^2 y}{d x^2}\right)_{at~ \theta=\pi / 2}=\)

The equation of tangent to the curve \(y=\sqrt{2} \sin \left(2 x+\frac{\pi}{4}\right)\) at \(x=\frac{\pi}{4}\), is

The area of the region included between the parabolas \(y^2=8 x\) and \(x^2=8 y\), is

If \({ }^{11} \mathrm{C}_4+{ }^{11} \mathrm{C}_5+{ }^{12} \mathrm{C}_6+{ }^{13} \mathrm{C}_7={ }^{14} \mathrm{C}_5\), then value of \(\mathrm{r}\) is

The equation of common tangent to the circles \(x^2+y^2-4 x+10 y+20=0\) and \(x^2+y^2+8 x-6 y-24=0\) is

If amplitude of \((z-2-3 i)\) is \(\frac{3 \pi}{4}\), then locus of \(z\) is (where \(z=x+i y\))

\(\int_\limits1^3\left[\tan ^{-1}\left(\frac{x}{x^2-1}\right)+\tan ^{-1}\left(\frac{x^2-1}{x}\right)\right] d x=\)

\(\int_\limits0^\pi \frac{1}{4+3 \cos x} d x=\)

Radium decomposes at the rate proportional to the amount present at any time. If \(\mathrm{P} \%\) of amount disappears in one year, then amount of radium left after 2 years is

The particular solution of the differential equation \(y(1+\log x) \frac{d x}{d y}-x \log x=0\) when \(x=e, y=e^2\) is

The differential equation obtained by eliminating A and B from \(y=A \cos \omega t+B \sin \omega t\)

The order and degree of the differential equation \(\frac{d^2 y}{d x^2}=\sqrt{\frac{d y}{d x}}\) are respectively

The general solution of the differential equation \(\cos (x+y) \frac{d y}{d x}=1\) is

If \(y=x \tan y\), then \(\frac{d y}{d x}=\)

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

The derivative of the function \(\cot ^{-1}\left[(\cos 2 x)^{1 / 2}\right]\) at \(x=\pi / 6\) is

The domain of the function \(\log _{10}\left(x^2-5 x+6\right)\) is

\(\int \sec ^{-1} x d x=\)

\(\int \frac{d x}{\cos x \sqrt{\cos 2 x}}=\)

If \(\int \frac{\sqrt{x}}{x(x+1)} d x=k \tan ^{-1} m+c\), (where c is constant of integration), then

If \(\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\), where \(x>0\), then \(x=\)

$$\begin{aligned}
& \text { If the function given by} \mathrm{f}(\mathrm{x}) \\
& =-2 \sin \mathrm{x} \quad-\pi \leq \mathrm{x}<-(\pi / 2) \\
& =a \sin x+b \quad-(\pi / 2)< x<(\pi / 2) \\
& =\cos x \quad(\pi / 2) \leq x \leq \pi \\
\end{ali...

The minimum value of the objective function \(z=4 x+6 y\) subject to \(x+2 y \geq 80,3 x+y \geq 75, x, y \geq 0\) is

The negation of inverse of \(\sim \mathrm{p} \rightarrow \mathrm{q}\) is

"If two triangles are congruent, then their areas are equal." is the given statement, then the contrapositive of the inverse of the given statement is
(Where \(\mathrm{p}\) : Two triangles are congruent, \(\mathrm{q}\) : Their areas are equ...

Which of the following matrices are invertible?
$$\begin{aligned}
& \mathrm{A}=\left[\begin{array}{cc}
2 & 3 \\
10 & 15
\end{array}\right], \mathrm{B}=\left[\begin{array}{ccc}
1 & 2 & 3 \\
2 & -1 & 3 \\
1 & 2 & 3
\end{array}\right], \mathrm...

If $$A=\left[\begin{array}{ccc}5 & 6 & 3 \\ -4 & 3 & 2 \\ -4 & -7 & 3\end{array}\right]$$, then cofactors of all elements of second row are respectively.

A man is known to speck truth 3 out of 4 times. He throws a die and reports that it is 6. Then the probability that it is actually 6 is

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on
one face is

If the function defined by \(f(x)=K(x-x^2)\) if \(0 < x < 1=0\), otherwise is the p.d.f. of a r.v.X, then the value of \(P\left(X<\frac{1}{2}\right)\) is

A fair coin is tossed 100 times. The probability of getting a head for even number of times is

Let two cards are drawn at random from a pack of 52 playing cards. Let X be the number of aces obtained. Then the value of E(X) is

The area of the triangle \(\mathrm{ABC}\) is \(10 \sqrt{3} \mathrm{~cm}^2\), angle \(\mathrm{B}\) is \(60^{\circ}\) and its perimeter is \(20 \mathrm{~cm}\), then \(\ell(\mathrm{AC})=\)

If \(\mathrm{G}(\overline{\mathrm{g}}), \mathrm{H}(\overline{\mathrm{h}})\) and \(\mathrm{P}(\overline{\mathrm{p}})\) are respectively centroid, orthocenter and circumcentre of a triangle and $$\mathrm{x} \overline{\mathrm{p}}+\mathrm{y} \o...

With usual notations in \(\triangle\)ABC, if \(\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}\), then \(a^2, b^2, c^2\) are in

If the standard deviation of data is 12 and mean is 72, then coefficient of variation is

The joint equation of pair of lines through the origin and having slopes \((1+\sqrt{2})\) and \(\frac{1}{(1+\sqrt{2})}\) is

If \(4 a b=3 h^2\), then the ratio of slopes of the lines represented by \(a x^2+2 h x y+b y^2=0\) is

The distance between the lines \(3 x+4 y=9\) and \(6 x+8 y=15\) is

If the slopes of the lines given by the equation \(a x^2+2 h x y+b y^2=0\) are in the ratio \(5: 3\), then the ratio \(h^2: a b=\)

The Cartesian equation of a plane which passes through the points \(\mathrm{A}(2,2,2)\) and making equal nonzero intercepts on the co-ordinate axes is

The co-ordinates of the foot of the perpendicular drawn from the point \(2 \hat{i}-\hat{j}+5 \hat{k}\) to the line \(\vec{r}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})\) are

The principal solutions of \(\cot x=\sqrt{3}\) are

If \(\sin (y+z-x), \sin (z+x-y)\) and \(\sin (x+y-z)\) are in AP, then

The vectors \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) are the sides of a triangle \(\mathrm{ABC}\). The length of the...

If \(\bar{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=-\hat{i}+2 \hat{j}-4 \hat{k}\) and \(\bar{c}=\hat{i}+\hat{j}-2 \hat{k}\), then \((\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c})=\)

If \(\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\bar{c}=3 \hat{i}+\lambda \hat{j}+5 \hat{k}\) are coplanar, then \(\lambda\) is the root of the equation

The projection of \(\bar{a}=\hat{i}-2 \hat{j}+\hat{k}\) on \(\bar{b}=2 \hat{i}-\hat{j}+\hat{k}\) is

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