MHT CET 2021 23th September Morning Shift

Function \(f(x)=e^{-1 / x}\) is strictly increasing for all \(x\) where

If \(x=-2\) and \(x=4\) are the extreme points of \(y=x^3-\alpha x^2-\beta x+5\), then

10 is divided into two parts such that the sum of double of the first and square of the other is minimum, then the numbers are respectively

The area bounded by the parabola \(y^2=4 a x\) and its latus-rectum \(x=a\) is

If a circle passes through the points \((0,0),(x, 0)\) and \((0, y)\), then the coordinates of its centre are

If \(z=x+iy\) satisfies the condition \(|z+1|=1\), then \(z\) lies on the

\(\int_\limits0^1|5 x-3| d x=\)

\(\int_0^{\pi / 2} \frac{\cos x}{3 \cos x+\sin x} d x=\)

The particular solution of differential equation \((x+y) d y+(x-y) d x=0\) at \(x=y=1\) is

The general solution of the differential equation \(\frac{d y}{d x}+\frac{y^2+y+1}{x^2+x+1}=0\) is

The general solution of the differential equation \(\frac{d x}{d t}=\frac{x \log x}{t}\) is

If the surrounding air is kept at \(25^{\circ} \mathrm{C}\) and a body cools from \(80^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) in 30 minutes, then temperature of the body after one hour will be

The general solution of the differential equation \(\frac{d y}{d x}=2^{y-x}\) is

If \(y=\log \sqrt{\tan x}\), then the value of \(\frac{d y}{d x}\) at \(x=\frac{\pi}{4}\) is

If \(\mathrm{x}=\mathrm{a}\left(\mathrm{t}-\frac{1}{\mathrm{t}}\right)\) and \(\mathrm{y}=\mathrm{b}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)\), then \(\frac{\mathrm{dy}}{\mathrm{dx}}=\)

If \(f(x)=\operatorname{cosec}^{-1}\left[\frac{10}{6 \sin \left(2^x\right)-8 \cos \left(2^x\right)}\right]\), then \(f^{\prime}(x)=\)

Range of the function \(f(x)=3+2^x+4^x\) is

\(\int \frac{10^{\frac{x}{2}}}{\sqrt{10^{-x}-10^x}} d x=\)

If \(\int \frac{\sin x}{\sin (x-\alpha)} d x=A x+B \log \sin (x-\alpha)+c\), then the value of A and B are respectively (where \(\mathrm{c}\) is a constant of integration)

\(\int e^{\left(e^x+x\right)} d x=\)

If \(\tan ^{-1}\left[\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right]=\alpha\), then the value of \(\sin 2 \alpha\) is

\(\tan \left(\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right)=\)

\(\lim _\limits{x \rightarrow 1}\left[\frac{\sqrt{x}-1}{\log x}\right]=\)

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

The maximum value of the objective function \(z=2 x+3 y\) subject to the constraints \(x+y \leq 5,2 x+y \geq 4\) and \(x \geq 0, y \geq 0\) is

S1 : If \(-\)7 is an integer, then \(\sqrt{-7}\) is a complex number
\(\mathrm{S} 2\) : \(-\)7 is not an integer or \(\sqrt{-7}\) is a complex number

Negation of the statement : \(3+6>8\) and \(2+3<6\) is

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

If $$\mathrm{A}=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$$, then \(\mathrm{A}(\operatorname{adj} \mathrm{A})=\)

If $$A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right], B=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$$ and $$X=\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right]$$ such that \(A X=B\), then t...

Out of 7 consonants and 4 vowels, the number of words consisting of 3 consonants and 2 vowels are

For the probability distribution given by following

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Rooms in a hotel are numbered from 1 to 19. Rooms are allocated at random as guests arrive. The first guest to arrive is given a room which is a prime number. The probability that the second guest to arrive is given a room which is a prime ...

A random variable X has the following probability distribution

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The probability distribution of the number of doublets in four throws of
a pair of dice is given by

In a triangle \(\mathrm{ABC}\), with usual notations \(\mathrm{a}=2, \mathrm{~b}=3, \mathrm{c}=5\), then \(\frac{\cos \mathrm{A}}{\mathrm{a}}+\frac{\cos \mathrm{B}}{\mathrm{b}}+\frac{\cos \mathrm{C}}{\mathrm{c}}=\)

140. Given that total of 16 values is 528 and sum of the squares of deviation from 33 is 9158. The variance is

If two lines represented by \(a x^2+2 h x y+b y^2=0\) makes angles \(\alpha\) and \(\beta\) with positive direction of \(\mathrm{X}\)-axis, then \(\tan (\alpha+\beta)=\)

The combined equation of a pair of lines passing through the origin and inclined at \(60^{\circ}\) and \(30=\) respectively with \(x\)-axis is

The equation of line, where length of the perpendicular segment from origin to the line is 4 and the inclination of this perpendicular segment with the positive direction of X-axis is 30\(^\circ\), is

The coordinates of the foot of the perpendicular drawn from the origin to the plane \(2 x+y-2 z=18\) are

The vector equation of the line passing through \(\mathrm{P}(1,2,3)\) and \(\mathrm{Q}(2,3,4)\) is

The distance between the parallel lines \(\frac{x-2}{3}=\frac{y-4}{5}=\frac{z-1}{2}\) and \(\frac{x-1}{3}=\frac{y+2}{5}=\frac{z+3}{2}\) is

If A(3, 2, \(-\)1), B(\(-\)2, 2, \(-\)3) and D(\(-\)2, 5, \(-\)4) are the vertices of a parallelogram, then the area of the parallelogram is

Equation of planes parallel to the plane \(x-2y+2z+4=0\) which are at a distance of one unit from the point (1, 2, 3) are

If \(3 \sin \theta=2 \sin 3 \theta\) and \(0< \theta<\pi\), then \(\sin \theta=\)

If \(\hat{a}\) is a unit vector such that \((\bar{x}-\hat{a}) \cdot(\bar{x}+\hat{a})=8\), then \(|\bar{x}|=\)

Let \(\vec{v}=2 \hat{i}+2 \hat{j}-\hat{k}\) and \(\bar{w}=\hat{i}+3 \hat{k}\). If \(\bar{u}\) is a unit vector, then the maximum value of the scalar triple product \([\bar{u} \bar{v} \bar{w}]\) is

If \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{c}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}\) and $$\overlin...

If \(3 \hat{j}, 4 \hat{k}\) and \(3 \hat{j}+4 \hat{k}\) are the position vectors of the vertices \(A, B, C\) respectively of \(\triangle A B C\), then the position vector of the point in which the bisector of \(\angle \mathrm{A}\) meets $$\...