MHT CET 2021 21th September Evening Shift

The abscissa of the points, where the tangent to the curve \(y=x^3-3 x^2-9 x+5\) is parallel to \(X\) axis are

For all real \(x\), the minimum value of the function \(f(x)=\frac{1-x+x^2}{1+x+x^2}\) is

The function \(f(x)=\log (1+x)-\frac{2 x}{2+x}\) is increasing on

The area bounded by the parabola \(y^2=x\), the straight line \(y=4\) and \(Y\) axis is

The equation of circle with centre at \((2,-3)\) and the circumference \(10 \pi\) units is

If \(z(2-i)=(3+i)\), then \(z^{38}=\), ( where \(z=x+i y\))

\(\int\limits_5^{10} \frac{d x}{(x-1)(x-2)}=\)

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

\(\text{I} : y^{\prime}=\frac{y+x}{x} ; \quad \text { II }: y^{\prime}=\frac{x^2+y}{x^3} ; \quad \text { III }: y^{\prime}=\frac{2 x y}{y^2-x^2}\)
S1 : Differential equations given by I and II are homogeneous differential equations.
S2 : Di...

The differential equation of the family of circles touching \(y\)-axis at the origin is

The general solution of the differential equation. \(\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d x-\left[\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)\right] d y=0\) is

If the half life period of a substance is 5 years, then the total amount of the substance left after 15 years, when initial amount is 64 gms is

If \(m\) is order and \(n\) is degree of the differential equation \(\left(\frac{d^2 y}{d x^2}\right)^5+4 \frac{\left(\frac{d^2 y}{d x^2}\right)}{\left(\frac{d^3 y}{d x^3}\right)}+\left(\frac{d^3 y}{d x^3}\right)=x^2\) then

If y = 2 sin x + 3 cos x and y + A\(\mathrm{\frac{d^2y}{dx^2}}\) = B, then the values of A, B are respectively

If \(y = {\tan ^{ - 1}}\left\{ {{{a\cos x - b\sin x} \over {b\cos x + a\sin x}}} \right\}\), then \({{dy} \over {dx}}\)

\(\int \frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}} d x=\)

\(\int[\sin |\log x|+\cos |\log x|] d x=\)

If \(\int {{{5\tan x} \over {\tan x - 2}}dx = x + a\log |\sin x - 2\cos x| + c}\), then a = (Where c is constant of integration)

If \(y=\tan ^{-1}\left[\frac{1}{1+x+x^2}\right]+\tan ^{-1}\left[\frac{1}{x^2+3 x+3}\right], x>0\), then \(\frac{d y}{d x}=\)

If \(\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)=\sin ^{-1} \alpha\), then \(\alpha=\)

$$\begin{aligned} & \text { } f(x)=\frac{\sqrt{1+p x}-\sqrt{1-p x}}{x} \text {, if } 1 \leq x<0 \\ & =\frac{2 x+1}{x-2} \quad \text {, if } 0 \leq x \leq 1 \\ \end{aligned}$$
is continuous in the interval \([-1,1]\), then \(p=\)

If \(\lim _\limits{x \rightarrow 5} \frac{x^k-5^k}{x-5}=500\), then the value of \(k\), where \(k \in N\) is

The objective function \(z=4 x+5 y\) subjective to \(2 x+y \geq 7 ; 2 x+3 y \leq 15 ; y \leq 3, x \geq 0 ; y \geq 0\) has minimum value at the point.

The logical statement (p \(\to\) q) \(\wedge\) (q \(\to\) ~p) is equivalent to

If p \(\to\) (~p \(\vee\) q) is false, then the truth values of p and q are, respectively

If $$A=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$$, then \(\operatorname{adj} A=\)

If $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1\end{array}\right]$$ and $$A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -8 & 6 & 2 c \\ 5 & -3 & 1\end{array}\right]$$, then values of a and c are respectively

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

For a set of five true or false questions, no student has written the all correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is

an urn contains 9 balls of which 3 are red, 4 are blue and 2 are green. Three balls are drawn at random from the urn. The probability that the three balls have difference colours is

It is observed that \(25 \%\) of the cases related to child labour reported to the police station are solved. If 6 new cases are reported, then the probability that at least 5 of them will be solved is

With usual notations, perimeter of a triangle \(A B C\) is 6 times the arithmetic mean of sine of its angles. If \(\mathrm{a}=1\), then measure of angle \(\mathrm{A}=\)

Let A = {10, 11, 12, 14, 26} and let f : A \(\to\) N be such that f(a) = highest prime factor of a, where a \(\in\) A, then range of f =

The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2 and 6, then the other two are

For X ~ B(n, p), if p = 0.6, E(X) = 6, then Var(X) =

A bakerman sells 5 types of cakes. Profit due to sale of each type of cake is respectively ₹ 2.5, ₹ 3 , ₹ 1.5 and ₹ 1. The demands for these cakes are \(20 \%, 5 \%, 10 \%, 50 \%\) and respectively, then the expected profit per cake is

If \(\mathrm{(m+3 n)(3 m+n)=4 h^2}\), then the acute angle between the lines represented by \(\mathrm{m x^2+2 h x y+n y^2=0}\) is

If \(\mathrm{p}\) is the length of the perpendicular from origin to the line whose intercepts on the axes are a and \(b\), then \(\frac{1}{a^2}+\frac{1}{b^2}=\)

If the lines \(x^2-4xy+y^2=0\) make angles \(\alpha\) and \(\beta\) with positive direction X-axis, then \(\cot^2\alpha+\cot^2\beta=\)

The direction cosines \(\ell, \mathrm{m}, \mathrm{n}\) of the line \(\frac{\mathrm{x}+2}{2}=\frac{2 \mathrm{y}-5}{3} ; \mathrm{z}=-1\) are

Equation of the plane passing through the point (2, 0, 5) and parallel to the vectors \(\widehat i - \widehat j + \widehat k\) and \(3\widehat i + 2\widehat j - \widehat k\) is

The co-ordinates of the point \(\mathrm{P} \equiv(1,2,3)\) and \(\mathrm{O} \equiv(0,0,0)\), then the direction cosines of \(\overline{\mathrm{OP}}\) are

The equation of the plane containing the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\) and the point \((0,7,-7)\) is

The equation of a line passing through \((3,-1,2)\) and perpendicular to the lines \(\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-2 \hat{j}+\hat{k})\) and \(\bar{r}=(2 \hat{i}+\hat{j}-3 \hat{k})+\mu(\hat{i}-2 \hat{j}+2 \hat{k})\) is

The area of the parallelogram with vertices A(1, 2, 3), B(1, 3, a), C(3, 8, 6) and D(3, 7, 3) is \(\sqrt{265}\) sq. units, then a =

The number of solutions of cos2\(\theta\) = sin\(\theta\) in (0, 2\(\pi\)) are

If \(\theta+\phi=\alpha\) and \(\tan \theta=k \tan \phi(\) where \(K>1)\), then the value of \(\sin (\theta-\phi)\) is

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

If \({\pi \over 2} < \theta < \pi\) and \(|\overline a | = 5,|\overline b | = 13,|\overline a \times \overline b | = 25\), then the value of \(\overline a \,.\,\overline b\) is

If \(|\bar{a} \times \bar{b}|^2+(\bar{a} \cdot \bar{b})^2=144\) and \(|\bar{a}|=4\), then \(|\bar{b}|=\)