MHT CET 2021 22th September Morning Shift

The point on the curve \(y^2=2(x-3)\) at which the normal is parallel to the line \(y-2 x+1=0\) is

A sperical snow ball is forming so that its volume is increasing at the rate of \(8 \mathrm{~cm}^3 / \mathrm{sec}\). Find the rate of increase of radius when radius is \(2 \mathrm{~cm}\).

The area bounded between the curve \(x^2=y\) and the line \(y=4 x\) is

The equation of a circle that passes through the origin and cut off intercepts \(-2\) and 3 on the \(\mathrm{X}\)-axis and \(\mathrm{Y}\)-axis respectively is

If \(x=1+2 i\), then the value of \(x^3+7 x^2-x+16\) is

\(\int\limits_{ - \pi }^\pi {{{x\sin x} \over {1 + {{\cos }^2}x}}dx = }\)

The value of \(\int\limits_0^1 {{{\tan }^{ - 1}}\left( {{{2x - 1} \over {1 + x - {x^2}}}} \right)dx}\) is

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

The general solution of \(\sin ^{-1}\left(\frac{d y}{d x}\right)=x+y\) is

Solution of the differential equation \(\mathrm{y'=\frac{(x^2+y^2)}{xy}}\), where y(1) = \(-\)2 is given by

The differential equation of all family of lines \(y=m x+\frac{4}{m}\) obtained by eliminating the arbitrary constant \(\mathrm{m}\) is

If \(y^2=a x^2+b x+c\), where \(a, b, c\) are constants, then \(y^3 \frac{d^2 y}{d x^2}\) is equal to

\(x=\frac{1-t^2}{1+t^2}\) and \(y=\frac{2 a t}{1+t^2}\), then \(\frac{d y}{d x}=\)

A rectangle of maximum area is inscribed in an ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\), then its dimensions are

The domain of the function \(f(x)=\sqrt{x-1}+\sqrt{6-x}\) is

\(\int {{e^x}\left( {{{x - 1} \over {{x^2}}}} \right)dx = }\)

\(\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x=\quad(\text { where }|x| < 1)\)

\(\int \frac{\sec ^8 x}{\operatorname{cosec} x} d x=\)

\(\tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)+\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\)

\(y=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), 0 \leq x < \frac{\pi}{2}\), then \(\frac{d y}{d x}\) at \(x=\frac{\pi}{6}\) is

Let
\(f(x)\matrix{ { = |x| + 3,} & {if\,x \le - 3} \cr { = - 2x,} & {if\, - 3 < x < 3} \cr { = 6x - 2,} & {if\,x \ge 3} \cr }\), then

\(\lim _\limits{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^2}=\)

The maximum value of \(z=10 x+25 y\) subject to \(0 \leq x \leq 3,0 \leq y \leq 3, x+y \leq 5\) occurs at the point.

Bismath has half life period of 5 days. A sample originally has a mass of 1000 mg, then the mass of Bismath after 30 days is

If statements \(\mathrm{p}\) and \(\mathrm{q}\) are true and \(\mathrm{r}\) and \(\mathrm{s}\) are false, then truth values of \(\sim(\mathrm{p} \rightarrow \mathrm{q}) \leftrightarrow(\mathrm{r} \wedge \mathrm{s})\) and $$(\sim \mathrm{p} ...

The expression \([(p \wedge \sim q) \vee q] \vee(\sim p \wedge q)\) is equivalent to

If $$A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1\end{array}\right]$$, then the value of determinant of \(A^{-1}\) is

If \(A = \left[ {\matrix{ k & 2 \cr { - 2} & { - k} \cr } } \right]\), then A\(^{-1}\) does not exists if k =

The sum of three numbers is 6. Thrice the third number when added to the first number gives 7. On adding three times first number to the sum of second and third number we get 12. The product of these numbers is

A polygon has 44 diagonals. Then the number of sides of the polygon are

For two events \(\mathrm{A}\) and \(\mathrm{B}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5}{6}, \mathrm{P}(\mathrm{A})=\frac{1}{6}, \mathrm{P}(\mathrm{B})=\frac{2}{3}\), then \(\mathrm{A}\) and \(\mathrm{B}\) are

A random variable X has following distribution

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A coin is tossed three times. If X denotes the absolute difference between the number of heads and the number of tails, then P (X = 1) =

A random variable X \(\sim\) B (n, p), if values of mean and variance of X are 18 and 12 respectively, then n =

In \(\triangle A B C\), with usual notations, \(2 a b \sin \frac{1}{2}(A+B-C)=\)

If in \(\Delta\)ABC, with usual notations, the angles are in A.P., then \(\mathrm{\frac{a}{c}}\) sin 2 C + \(\mathrm{\frac{c}{a}}\) sin 2 A =

If the variance of the data 2, 4, 5, 6, 8, 17 is 23.33, then the variance of 4, 8, 10, 12, 16, 34 will be

If slope of one of the lines ax\(^2\) + 2hxy + by\(^2\) = 0 is twice that of the other, then h\(^2\) : ab is

Area of the triangle formed by the lines \(y^2-9 x y+18 x^2=0\) and \(y=9\) is

The equation of perpendicular bisector of the line segment joining \(A(-2,3)\) and \(B(6,-5)\) is

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

The plane \(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1\) cuts the \(X\)-axis at A, Y-axis at B and Z-axis at C, then the area of \(\triangle \mathrm{ABC}=\)

If a plane meets the axes \(\mathrm{X}, \mathrm{Y}, \mathrm{Z}\) in \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) respectively such that centroid of \(\triangle \mathrm{ABC}\) is \((1,2,3)\), then the equation of the plane is

The shortest distance between lines \(\bar{r}=(2 \hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})\) and \(\bar{r}=(\hat{r}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+\hat{j}-5 \hat{k})\) is

\(\tan 3 \mathrm{~A} \cdot \tan 2 \mathrm{~A} \cdot \tan \mathrm{A}=\)

The position vector of the point of inersection of the medians of a triangle, whose vertices are \(A(1,2,3), B(1,0,3)\) and \(C(4,1,-3)\) is

The area of the parallelogram whose diagonals are represented by the vectors \(\bar{a}=3 \hat{i}-\hat{j}-2 \hat{k}\) and \(\bar{b}=-\hat{i}+3 \hat{j}-3 \hat{k}\) is

If \(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}\) with \(|\overline{\mathrm{a}}|=3,|\overline{\mathrm{b}}|=5\) and \(|\overline{\mathrm{c}}|=7\), then angle between \(\overline{\mathrm{a}}\) and $$\overli...

If \(|\bar{u}|=2\) and \(\bar{u}\) makes angles of \(60^{\circ}\) and \(120^{\circ}\) with axes \(\mathrm{OX}\) and \(\mathrm{OY}\) in the origin, then \(\bar{u}=\)

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are mutually perpendicular vectors having magnitudes \(1,2,3\) respectively, then $$\left[\begin{array}{lll}\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\ma...