MHT CET 2021 20th September Evening Shift

The surface area of a spherical balloon is increasing at the rate \(2 \mathrm{~cm}^2 / \mathrm{sec}\). Then rate of increase in the volume of the balloon is , when the radius of the balloon is \(6 \mathrm{~cm}\).

If \(f(x)=2x^3-15x^2-144x-7\), then \(f(x)\) is strictly decreasing in

The equation of tangent to the circle \(x^2+y^2=64\) at the point \(\mathrm{P\left(\frac{2\pi}{3}\right)}\) is

Water is being poured at the rate of 36 m\(^3\)/min. into a cylindrical vessel, whose circular base is of radius 3 m. Then the wate level in the cylinder is rising at the rate of

The area bounded by the parabola \(y=x^2\) and the line \(y=x\) is

If \(\omega\) is complex cube root of unity and \((1+\omega)^7=A+B\omega\), then values of A and B are, respectively

If \(2 f(x)-3 f\left(\frac{1}{x}\right)=x\), then \(\int_\limits1^e f(x) d x=\)

If \(\int_\limits2^e\left[\frac{1}{\log x}-\frac{1}{(\log x)^2}\right] d x=a+\frac{b}{\log 2}\), then

If \(\mathrm{m}\) is order and \(\mathrm{n}\) is degree of the differential equation \(y=\frac{d p}{d x}+\sqrt{a^2 p^2-b^2}\), where \(p=\frac{d y}{d x}\), then the value of \(m+n\) is

The general solution of the differential equation \(\cos x \cdot \sin y d x+\sin x \cdot \cos y d y=0\) is

The differential equation of an ellipse whose major axis is twice its minor axis, is

The general solution of \(\left(x \frac{d y}{d x}-y\right) \sin \frac{y}{x}=x^3 e^x\) is

The population of a city increases at a rate proportional to the population at that time. If the population of the city increase from 20 lakhs to 40 lakhs in 30 years, then after another 15 years the population is

\(y=\sqrt{e^{\sqrt{x}}}\), then \(\frac{d y}{d x}=\)

If \(y=\sin ^{-1}\left[\cos \sqrt{\frac{1+x}{2}}\right]+x^x\), then \(\frac{d y}{d x}\) at \(x=1\) is

If \(x=a(t+\sin t), y=a(1-\cos t)\), then \(\frac{d y}{d x}=\)

Let \(A=[a, b, c, d], B=[1,2,3]\). Relation \(R_1, R_2, R_3, R_4\) are as follows :
$$\begin{aligned}
& R_1=[(\mathrm{a}, 1),(\mathrm{b}, 2),(\mathrm{c}, 1),(\mathrm{d}, 2)] \\
& \mathrm{R}_2=[(\mathrm{a}, 1),(\mathrm{b}, 1),(\mathrm{c}, 1)...

\(\int \sec ^4 x \cdot \tan ^4 x d x=\frac{\tan ^m x}{m}+\frac{\tan ^n x}{n}+c\) (where c is constant of integration), then m + n =

\(\int \operatorname{cosec}(x-a) \operatorname{cosec} x d x=\)

\(\int \frac{2 x^2-1}{x^4-x^2-20} d x=\)

\(\sin ^{-1}\left[\sin \left(-600^{\circ}\right)\right]+\cot ^{-1}(-\sqrt{3})=\)

\(\cos ^{-1}\left(\frac{4}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)=\)

If \(a=\lim _\limits{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{n^2}\) and \(b=\lim _\limits{n \rightarrow \infty} \frac{1^2+2^2+3^2+\ldots+n^2}{n^3}\), then

If \(f(x) = {{{4^{x - \pi }} + {4^{x - \pi }} - 2} \over {{{(x - \pi )}^2}}}\), for \(x \ne \pi\), is continuous at \(x=\pi\), then k =

The solution set for the system of linear inequations \(x+y \geq 1 ; 7 x+9 y \leq 63 ; y \leq 5 ; x \leq 6, x \geq 0\) and \(y \geq 0\) is represented graphically in the figure. What is the correct option?

p : It rains today
q : I am going to school
r : I will meet my friend
s : I will go to watch a movie.
Then symbolic form of the statement "If it does not rain today or I won't go to school, then I will meet my friend and I will go to watch ...

Negation of \((p \wedge q) \rightarrow(\sim p \vee r)\) is

The co-factors of the elements of second column of $$\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4\end{array}\right]$$ are

If $$A^{-1}=\left[\begin{array}{lll}3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 5 & 5\end{array}\right]$$, then \(A=\)

If $$A^{-1}=\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]$$ and $$B^{-1}=\left[\begin{array}{cc}1 & 0 \\ -3 & 1\end{array}\right]$$, then \((A B)^{-1}=\)

If \(\frac{n !}{2 !(n-2) !}\) and \(\frac{n !}{4 !(n-4) !}\) are in the ratio \(2: 1\), then \(n=\)

The p.m.f. of a random variable X is \(\mathrm{P(X = x) = {1 \over {{2^5}}}\left( {_x^5} \right),x = 0,1,2,3,4,5}=0\) then

The variance of the following probability distribution is,

If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, then probability of a success is

Two unbiased dice are thrown. Then the probability that neither a doublet nor a total of 10 will appear is

If the variance of the numbers \(2,3,11\) and \(x\) is \(\frac{49}{4}\), then the values of \(x\) are

The x-intercept of a line passing through the points \(\left(\frac{-1}{2}, 1\right)\) and (1, 2) is :

If the equation \(3x^2-kxy-3y^2=0\) represents the bisectors of angles between the lines \(x^2-3xy-4y^2=0\), then value of k is

The joint equation of pair of lines through the origin and making an equilateral triangle with the line \(y=3\) is

The Cartesian equation of the line passing through the points A(2, 2, 1) and B(1, 3, 0) is

The Cartesian equation of the plane \(\overline{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})\) is

The equation of the plane that contains the line of intersection of the planes. \(x+2 y+3 z-4=0\) and \(2 x+y-z+5=0\) and is perpendicular to the plane \(5 x+3 y-6 z+8=0\) is

The vector equation of the line whose Cartesian equations are y = 2 and 4x \(-\) 3z + 5 = 0 is

The principal solutions of \(\sqrt{3} \sec x+2=0\) are

If \(\cos x=\frac{24}{25}\) and \(x\) lięs in first quadrant, then \(\sin \frac{x}{2}+\cos \frac{x}{2}=\)

\(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are vectors such that \(|\overline{\mathrm{a}}|=5,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=3\) and each is perpendicular to the sum of the other two, then $$|\...

If \([\bar{a} \bar{b} \bar{c}]=4\), then the volume (in cubic units) of the parallelopiped with \(\bar{a}+2 \bar{b}, \bar{b}+2 \bar{c}\) and \(\overline{\mathrm{c}}+2 \overline{\mathrm{a}}\) as coterminal edges, is

\(\overline{\mathrm{a}}, \overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are three vectors such that \(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}\) and $$|\overline{\mathrm{a}}|=3,|\overline{\mathrm...

If area of the parallelogram with \(\bar{a}\) and \(\bar{b}\) as two adjacent sides is 20 square units, then the area of the parallelogram having \(3 \overline{\mathrm{a}}+\overline{\mathrm{b}}\) and $$2 \overline{\mathrm{a}}+3 \overline{\m...

If \(\bar{r}=-4 \hat{i}-6 \hat{j}-2 \hat{k}\) is a linear combination of the vectors \(\bar{a}=-\hat{i}+4 \hat{j}+3 \hat{k}\) and \(\bar{b}=-8 \hat{i}-\hat{j}+3 \hat{k}\), then