MHT CET 2021 21th September Morning Shift

A body at an unknown temperature is placed in a room which is held at a constant temperature of \(30^{\circ} \mathrm{F}\). If after 10 minutes the temperature of the body is \(0^{\circ} \mathrm{F}\) and after 20 minutes the temperature of t...

A stone is thrown into a quite lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of 4 cm/sec, then the rate of increase in its area, at the instant when its radius is 10 cm, is _________ cm$$^...

The function \(f(x)=\cot ^{-1} x+x\) is increasing in the interval.

The curves \(\frac{x^2}{a^2}+\frac{y^2}{4}=1\) and \(y^3=16 x\) intersect each other orthogonally, then \(a^2=\)

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

The equation of the circle whose centre lies on the line \(x-4 y=1\) and which passes through the points \((3,7)\) and \((5,5)\) is

The complex number with argument \(\frac{5 \pi^{\mathrm{c}}}{6}\) at a distance of 2 units from the origin is

\(\int_\limits0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1-\sin x \cos x} d x=\)

If \(f(x)=|x-1|+|x-2|+|x-3|, \forall x \in[1,4]\), then \(\int_\limits1^4 f(x) d x=\)

The general solution of the differential equation \(\left(3 x y+y^2\right) d x+\left(x^2+x y\right) d y=0\) is

The differential equation of family of circles whose centres lie on \(\mathrm{X}\)-axis is

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

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

If \(e^{-y} \cdot y=x\), then \(\frac{d y}{d x}\) is

If \(y=\operatorname{cosec}^{-1}\left[\frac{\sqrt{x}+1}{\sqrt{x}-1}\right]+\cos ^{-1}\left[\frac{\sqrt{x}-1}{\sqrt{x}+1}\right]\), then \(\frac{d y}{d x}=\)

The derivative of \((\log x)^x\) with respect to \(\log x\) is

If f(x) = 3[x] + 5{x + 1}, where [x] is greatest integer function of x and {x} is fractional part function of x, then f(\(-\)1.32) =

\(\int[1+2 \tan x(\tan x+\sec x)]^{\frac{1}{2}} d x=\)

If \(\int \frac{x^3}{\sqrt{1+x^2}} d x=a\left(1+x^2\right)^{\frac{3}{2}}+b \sqrt{1+x^2}+c\), then \(a+b=\), (where \(c\) is constant of integration)

\(\int e^{\tan x}\left(\sec ^2 x+\sec ^3 x \sin x\right) d x=\)

\(\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}\), then the values of \(x\) are

$$\begin{aligned} & \text { If the function } \mathrm{f}(\mathrm{x})=1+\sin \frac{\pi}{2}, \quad-\infty<\mathrm{x} \leq 1 \\ & =\mathrm{ax}+\mathrm{b}, \quad 1<\mathrm{x}<3 \\ & =6 \tan \frac{x \pi}{12}, \quad 3 \leq x<6 \\ \end{aligned}$$
...

\(\lim _\limits{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}=\)

The shaded figure given below is the solution set for the linear inequations. Choose the correct option.

Negation of the statement \(\forall x \in R, x^2+1=0\) is

If \(p, q\) are true statements and \(r\) is false statement, then which of the following is correct.

If $$F(\propto)=\left[\begin{array}{ccc}\cos \propto & -\sin \propto & 0 \\ \sin \propto & \cos \propto & 0 \\ 0 & 0 & 1\end{array}\right]$$, where \(\propto \in R\), then \([F(\propto)]^{-1}=\)

If $$A=\left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1\end{array}\right], \operatorname{adj} A=\left[\begin{array}{ccc}5 & x & -2 \\ 1 & 1 & 0 \\ -2 & -2 & y\end{array}\right]$$, then value of \(x+y\) is

$$\mathrm{A}^{-1}=\frac{-1}{2}\left[\begin{array}{cc}1 & -4 \\ -1 & 2\end{array}\right]$$, then \(2 A+I_2=\quad\)
where \(I_2\) is a unit matrix of order 2

The number of ways in which 8 different pearls can be arranged to form a necklace is

In a meeting \(60 \%\) of the members favour and \(40 \%\) oppose a certain proposal. A member is selected at random and we take \(\mathrm{X}=0\) if he opposed and \(\mathrm{X}=1\) if he is in favour, then \(\operatorname{Var} \mathrm{X}=\)

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs selected at random from the lot and sent to retain store, then the probability that the store will receive at most one defective bulb is

A coin is tossed and a die is thrown. The probability that the outcome will be head or a number greater than 4 or both, is

If \(\mathrm{X}\) is a random variable with p.m.f. as follows.
$$\begin{aligned}
\mathrm{P}(\mathrm{X}=\mathrm{x}) & =\frac{5}{16}, \mathrm{x}=0,1 \\
& =\frac{\mathrm{kx}}{48}, \mathrm{x}=2, \quad \text { then } \mathrm{E}(\mathrm{x})= \\
&...

For two data sets each of size 5 , the variance are given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively. The variance of the combined data set is

If the two lines given by \(a x^2+2 h x y+b y^2=0\) make inclinations \(\propto\) and \(\beta\), then \(\tan (\alpha+\beta)=\)

If the polar co-ordinates of a point are \(\left(\sqrt{2}, \frac{\pi}{4}\right)\), then its Cartesian co-ordinates are

The equation of a line passing through \((\mathrm{p} \cos \propto, \mathrm{p} \sin \propto)\) ) and making an angle \((90+\propto)\) with positive direction of \(\mathrm{X}\)-axis is

The product of the perpendicular distances from \((2,-1)\) to the pair of lines \(2 x^2-5 x y+2 y^2=0\) is

If the lines $\frac{1-x}{3}=\frac{7 y-14}{2 \lambda}=\frac{z-3}{2}$ and $\frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles, then $\lambda=$

The Cartesian equation of the plane passing through the point A(7, 8, 6) and parallel to the XY plane is

The equation of the plane passing through \((-2,2,2)\) and \((2,-2,-2)\) and perpendicular to the plane \(9 x-13 y-3 z=0\) is

\(2 \sin \left(\theta+\frac{\pi}{3}\right)=\cos \left(\theta-\frac{\pi}{6}\right)\), then \(\tan \theta=\)

If \(\frac{\cos (A+B)}{\cos (A-B)}=\frac{\sin (C+D)}{\sin (C-D)}\), then \(\tan A \tan B \tan C=\)

The distance between parallel lines
$$\begin{aligned}
& \bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}+\hat{j}-2 \hat{k}) \text { and } \\
& \bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+\hat{j}-2 \hat{k}) \text { is }
\end{align...

The vertices of triangle \(\mathrm{ABC}\) are \(\mathrm{A} \equiv(3,0,0) ; \mathrm{B} \equiv(0,0,4) ; \mathrm{C} \equiv(0,5,4)\). Find the position vector of the point in which the bisector of angle A meets B C is

In a quadrilateral PQRS, M and N are mid-points of the sides PQ and RS respectively. If \(\overline {PS} + \overline {QR} = t\overline {MN}\), then t =

If vectors \(\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \bar{b}=-\hat{i}+2 \hat{j}+\hat{k}\) and \(\bar{c}=3 \hat{i}+\hat{j}+2 \hat{k}\) are such that, \(\bar{a}+\lambda \bar{b}\) is perpendicular to \(\bar{c}\), then \(\lambda=\)

If \(\bar{a}=3 \hat{i}-5 \hat{j}, \bar{b}=6 \hat{i}+3 \hat{j}\) are two vectors and \(\bar{c}\) is a vector such that \(\bar{c}=\bar{a} \times \bar{b}\), then \(a: b\) : is

If \(|\bar{a}|=3,|\bar{b}|=4,|\bar{a}-\bar{b}|=5\), then \(|\bar{a}+\bar{b}|=\)