MHT CET 2021 24th September Morning Shift

The velocity of a particle at time \(t\) is given by the relation \(v=6 t-\frac{t^2}{6}\). Its displacement S is zero at \(\mathrm{t}=0\), then the distance travelled in \(3 \mathrm{~sec}\) is

\(f(x)=\log |\sin x|\), where \(x \in(0, \pi)\) is strictly increasing on

The maximum area of the rectangle that can be inscribed in a circle of radius \(r\) is

Area bounded by the lines \(y=x, x=-1, x=2\) and the \(X\)-axis is

The difference between the maximum values of \({ }^6 C_r\) and \({ }^n C_r\) is 16, then \(n=\)

Equationof the chord of the circle \(x^2+y^2-4 x-10 y+25=0\) having mid-point \((1,2)\) is

The sqaure roots of the complex number \((-5-12 \mathrm{i})\) are

\(\int_\limits0^4 x[x] d x=\)
(where \([\mathrm{x}]\) denotes greatest integer function not greater than \(\mathrm{x}]\)

\(\int_\limits0^{\pi / 2} \log \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x=\)

The order of the differential equation whose solution is \(y=a \cos x+b \sin x+c e^{-x}\) is

The general solution of the differential equation \((2 y-1) d x-(2 x+3) d y=0\) is

The particular solution of the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{x+y-1}\) when \(\mathrm{x}=\frac{2}{3}\) and \(y=\frac{1}{3}\) is

The differential equation of the family of parabolas with focus at the origin and the \(X\)-axis a axis, is

If \(y=\log _{10} x+\log _x 10+\log _x x+\log _{10} 10\), then \(\frac{d y}{d x}=\)

\(\text { If } u=\cos ^3 x, v=\sin ^3 x \text {, then }\left(\frac{d v}{d u}\right)_{x=\frac{\pi}{4}} \text { is equal to }\)

If \(f(x)=\frac{x}{2 x+1}\) and \(g(x)=\frac{x}{x+1}\), then \((f \circ g)(x)=\)

If \(\int \frac{(\cos x-\sin x)}{8-\sin 2 x} d x=\frac{1}{p} \log \left[\frac{3+\sin x+\cos x}{3-\sin x-\cos x}\right]+c\), then \(p=\) (where \(\mathrm{c}\) is a constant of integration)

\(\int \cos ^3 x \cdot e^{\log (\sin x)} d x=\)

\(\int \frac{\mathrm{dx}}{32-2 \mathrm{x}^2}=\mathrm{A} \log (4-\mathrm{x})+\mathrm{B} \log (4+\mathrm{x})+\mathrm{c}\), then the values of \(\mathrm{A}\) and \(\mathrm{B}\) are respectively (where c is a constant of integration)

The value of \(\sin ^{-1}\left(\frac{-1}{2}\right)+\sin ^{-1}\left(\frac{-\sqrt{3}}{2}\right)\) is,

If \(y=\tan ^{-1}\left[\frac{\log \left(\frac{e}{x^2}\right)}{\log \left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log x}{1-6 \log x}\right]\), then \(\frac{d^2 y}{d x^2}=\)

\(\lim _\limits{x \rightarrow 1} \frac{a b^x-a^x b}{x^2-1}=\)

If the function
$$\begin{array}{rlrl} f(x) & =3 a x+b, & & \text { for } x<1 \\ & =11, & & \text { for } x=1 \\ & =5 a x-2 b, & \text { for } x>1 \end{array}$$
is continuous at \(x=1\). Then, the values of \(a\) and \(b\) are

The common region of the solutions of the inequations \(x+2 y \geq 4,2 x-y \leq 6\) and \(x, y>0\) is

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 \(\mathrm{p}\) : It is raining and \(\mathrm{q}\) : It is pleasant, then the symbolic form of "It is neither raining nor pleasant" is

The negation of \(p \wedge(q \rightarrow r)\) is

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

If $$A=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 1 & a \\ 2 & 4 & 7\end{array}\right]$$ and $$B=\left[\begin{array}{ccc}13 & 2 & b \\ -3 & -1 & 2 \\ -2 & 0 & 1\end{array}\right]$$ where matrix B is inverse of matrix A, then the value of a an...

For a \(3 \times 3\) matrix \(\mathrm{A}\), if $$\mathrm{A}(\operatorname{adj} \mathrm{A})=\left[\begin{array}{ccc}-10 & 0 & 0 \\ 0 & -10 & 2 \\ 0 & 0 & -10\end{array}\right]$$, then the value of determinant of A is

If \(\mathrm{P}(\mathrm{A})=\frac{3}{10}, \mathrm{P}(\mathrm{B})=\frac{2}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{5}\), then \(\mathrm{P}(\mathrm{A} / \mathrm{B}) \times \mathrm{P}(\mathrm{B} / \mathrm{A})=\)

A die is thrown four times. The probability of getting perfect square in at least one throw is

If the probability distribution function of a random variable X is given as

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
ove...

The probability distribution of a discrete random variable X is

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden...

If in a \(\triangle A B C\), with usual notations, \(\mathrm{a}^2, \mathrm{~b}^2, \mathrm{c}^2\) are in A.P. then \(\frac{\sin 3 B}{\sin B}=\)

In any \(\triangle A B C\), with usual notations, \(c(a \cos B-b \cos A)=\)

The arithmetic mean of marks in Mathematics for four divisions A, B, C and D were \(80,75,70\) and 72 respectively. Their standard deviations were \(12,6,8\) and 10 respectively. Then, division ______ has more uniformity.

If one of the lines given by \(k x^2+x y-y^2=0\) bisect the angle between the co-ordinate axes then the values of \(k\) are

If the angle between the lines is \(\frac{\pi^{\mathrm{C}}}{4}\) and slope of one of the lines is \(\frac{1}{2}\), then slope of the other line is

If the lines respresented by \(a x^2-b x y-y^2=0\) make angle \(\alpha\) and \(\beta\) with the positive direction of \(\mathrm{X}\)-axis, then \(\tan (\alpha+\beta)=\)

Equation of the plane passing through the point \((1,2,3)\) and parallel to the plane \(2 x+3 y-4 z=0\)

If \(\mathrm{A}=(-2,2,3), \mathrm{B}=(3,2,2), \mathrm{C}=(4,-3,5)\) and \(\mathrm{D}=(7,-5,-1)\) Then the projection of \(\overline{\mathrm{AB}}\) on \(\overline{\mathrm{CD}}\) is

The length of perpendicular drawn from the point \(2 \hat{i}-\hat{j}+5 \hat{k}\) to the line \(\overline{\mathrm{r}}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})\) is

If the lines \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-2}{1}=\frac{y+m}{2}=\frac{z-2}{1}\) intersect each other, then value of m is

If \(\mathrm{A}\) and \(\mathrm{B}\) are the foot of the perpendicular drawn from the point \(\mathrm{Q}(\mathrm{a}, \mathrm{b}, \mathrm{c})\) to the planes \(\mathrm{YZ}\) and \(\mathrm{ZX}\) respectively, then the equation of the plane th...

If \(a \sin \theta=b \cos \theta\), where \(a, b \neq 0\), then \(a\cos 2 \theta+b \sin 2 \theta=\)

If the vectors \(\vec{a}=2 \hat{i}+p \hat{j}+4 \hat{k}\) and \(\vec{b}=6 \hat{i}-9 \hat{j}+q \hat{k}\) are collinear, then \(p\) and \(q\) are

\(\vec{a}=4 \hat{i}+13 \hat{j}-18 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}+3 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}-4 \hat{k}\) are three vectors such that \(\vec{a}=x \vec{b}+y \vec{c}\), then \(x+y=\)

If \(|\vec{a}|=4,|\vec{b}|=5\), then the values of \(k\) for which \(\vec{a}+k \vec{b}\) is perpendicular to \(\vec{a}-k \vec{b}\) are

If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=\hat{j}-\hat{k}, \vec{a} \times \bar{b}=\bar{c}\) and \(\vec{a} \cdot \vec{b}=1\), then \(\vec{b}\)