MHT CET 2020 19th October Evening Shift

The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is

The equation of the normal to the curve $2 x^2+y^2=12$ at the point $(2,2)$ is

The area of the square increases at the rate of $0.5 \mathrm{~cm}^2 / \mathrm{sec}$. The rate at which its perimeter is increasing when the side of the square is 10 cm long, is

The area of the region bounded by the curve $y=\sin x$ between $x=-\pi$ and $x=\frac{3 \pi}{2}$ is

If the sum of the mean and the variance of a binomial distribution for 5 trials is 1.8 , then $p=$

$\int_\limits0^1 \tan ^{-1}\left(\frac{2 x-1}{1+x-x^2}\right) d x=$

The c.d.f, $F(x)$ associated with p.d.f. $f(x)=3\left(1-2 x^2\right)$. If $0< x<1$ is $k\left(x-\frac{2 x^3}{k}\right)$, then value of $k$ is

\(\int_\limits0^{\frac{\pi}{2}} \frac{\sqrt[7]{\sin x}}{\sqrt[7]{\sin x}+\sqrt[7]{\cos x}} d x=\)

$\int_\limits0^1\left(1-\frac{x}{1!}+\frac{x^2}{2!}-\frac{x^3}{3!}+\ldots\right.$ upto $\left.\infty\right) e^{2 x} d x=$

The integrating factor of the differential equation $x \frac{d y}{d x}+y \log x=x^2$ is

The rate of disintegration of a radio active element at time $t$ is proportional to its mass, at the time. Then the time during which the original mass of 1.5 gm . Will disintegrate into its mass of 0.5 gm . is proportional to

The general solution of the differential equation $\left(1+y^2\right)+\left(x-e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0$ is

The order and degree of the differential equation $\left[1+\frac{1}{\left(\frac{d y}{d x}\right)^2}\right]^{\frac{5}{3}}=5 \frac{d^2 y}{d x^2}$ are respectively

If the population grows at the rate of $8 \%$ per year, then the time taken for the population to be doubled, is (Given $\log 2=0.6912$)

If $x=a \sin t-b \cos t, y=a \cos t+b \sin t$, then $y^3 \frac{d^2 y}{d x^2}+x^2+y^2=$

If $y=\sin ^{-1}\left[\frac{\sqrt{1+x}+\sqrt{1-x}}{2}\right]$, then $\frac{d y}{d x}=$

If $x^2 y^2=\sin ^{-1} \sqrt{x^2+y^2}+\cos ^{-1} \sqrt{x^2+y^2}$ then $\frac{d y}{d x}=$

The range of the function $f(x)=\frac{x-3}{5-x}, x \neq 5$ is

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

\(\int \log x \cdot(\log x+2) d x=\)

\(\int \frac{d x}{x^2+4 x+13}=\)

If $f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}$, for $x \neq \pi$ is continuous at $x=\pi$, then $f(\pi)=$

The LPP to maximize $Z=x+y$, subject to $x+y \leq 1,2 x+2 y \geq 6, x \geq 0, y \geq 0$ has

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

The statement pattern $p \wedge(q \vee \sim p)$ is equivalent to

If $A=\left[\begin{array}{ll}4 & 5 \\ 2 & 1\end{array}\right]$ and $A^2-5 A-6 I=0$, then $A^{-1}=$

The cofactors of the elements of the first column of the matrix $A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 3 & 1 & 2 \\ -1 & 1 & 2\end{array}\right]$ are

The equation of the directrix of the parabola $3 x^2=16 y$ is

The area of the triangle formed by the lines joining vertex of the parabola $x^2=12 y$ to the extremities of its latus rectum is

The odds in favour of getting sum multiple of 3 , when pair of dice are thrown is

If $X$ is a.r.v. with c.d.f $F(x)$ and its probability distribution is given by

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With usual notations, if the angles $A, B, C$ of a $\triangle A B C$ are in $A P$ and $b: c=\sqrt{3}: \sqrt{2}$

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

The quadratic equation whose roots are the numbers having arithmetic mean 34 and geometric mean 16 is

If $R=\{(a, b) / b=a-1, a \in Z, 5 < a < 9$, then the range of $R$ 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 equation of a line passing through the point $(7,-4)$ and perpendicular to the line passing through the points $(2,3)$ and $(1,-2)$ is

If the equation $3 x^2+10 x y+3 y^2+16 y+k=0$ represents a pair of lines, then the value of kis

The point $P$ lies on the line $A, B$ where $A=(2,4,5)$ and $B \equiv(1,2,3)$. If $z$ co-ordinate of point $P$ is 3 , the its $y$ co-ordinate is

A line makes angles $\alpha, \beta, \gamma$ with the co-ordinate axes and $\alpha+\beta=90^{\circ}$, then $\gamma=$

The equations of planes parallel to the plane $x+2 y+2 z+8=0$, which are at a distance of 2 units from the point $(1,1,2)$ are

The equation of a plane containing the point $(1,-1,2)$ and perpendicular to the planes $2 x+3 y-2 z=5$ and $x+2 y-3 z=8$ is

The equation of the line passing through $(1,2,3)$ and perpendicular to the lines $x-1=\frac{y+2}{2}=\frac{z+4}{4}$ and $\frac{x-1}{2}=\frac{y-2}{2}=z+3$ is

If $2 \cos ^2 \theta+3 \cos \theta=2$, then permissible value of $\cos \theta$ is

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

\(\frac{1-\sin \theta+\cos \theta}{1-\sin \theta-\cos \theta}=\)

If $A$ and $B$ are supplementary angles, then $\sin ^2 \frac{A}{2}+\sin ^2 \frac{B}{2}=$

If $\mathbf{a}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \hat{\mathbf{k}}$ and $\mathbf{c}=7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+23 \hat{\mathbf{k}}$ are three vectors, then which o...

$\mathbf{a}$ and $\mathbf{b}$ are non-collinear vectors. If $p=(2 x+1) a-b$ and $q=(x-2) a+b$ are collinear vectors, then $x=$

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors and $p=\frac{\mathbf{b} \times \mathbf{c}}{[a b c]}, q=\frac{\mathbf{c} \times \mathbf{a}}{[a b c]}, r=\frac{\mathbf{a} \times \mathbf{b}}{[a b c]}$, then $\mathbf{a} \cdot \m...