MHT CET 2023 11th May Morning Shift

Value of \(c\) satisfying the conditions and conclusions of Rolle's theorem for the function \(\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]\) is

If \(\mathrm{f}(x)=x \mathrm{e}^{x(1-x)}, x \in \mathrm{R}\), then \(\mathrm{f}(x)\) is

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

Number of common tangents to the circles \(x^2+y^2-6 x-14 y+48=0\) and \(x^2+y^2-6 x=0\) are

If \(\mathrm{z}=x+\mathrm{i} y\) and \(\mathrm{z}^{1 / 3}=\mathrm{p}+\mathrm{iq}\), where \(x, y, \mathrm{p}, \mathrm{q} \in \mathrm{R}\) and \(\mathrm{i}=\sqrt{-1}\), then value of \(\left(\frac{x}{\mathrm{p}}+\frac{y}{\mathrm{q}}\right)\)...

\(\int\limits_0^\pi \frac{d x}{4+3 \cos x}=\)

Let \(\mathrm{f}(x)=\int \frac{\sqrt{x}}{(1+x)^2} \mathrm{~d} x, x \geq 0\), then \(\mathrm{f}(3)-\mathrm{f}(1)\) is equal to

Let \(\mathrm{f}(x)\) be positive for all real \(x\). If \(\mathrm{I}_1=\int_\limits{1-\mathrm{h}}^{\mathrm{h}} x \mathrm{f}(x(1-x)) \mathrm{d} x\) and \(\mathrm{I}_2=\int_\limits{1-\mathrm{h}}^{\mathrm{h}} \mathrm{f}(x(1-x)) \mathrm{d} x\)...

\(\int_\limits{-1}^3\left(\cot ^{-1}\left(\frac{x}{x^2+1}\right)+\cot ^{-1}\left(\frac{x^2+1}{x}\right)\right) \mathrm{d} x=\)

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

A curve passes through the point \(\left(1, \frac{\pi}{6}\right)\). Let the slope of the curve at each point \((x, y)\) be \(\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0\), then, the equation of the curve is

If \(y=\log _{\sin x} \tan x\), then \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{x=\frac{\pi}{4}}\) has the value

Let \(\mathrm{f}(x)=\log (\sin x), 0 < x < \pi\) and \(\mathrm{g}(x)=\sin ^{-1}\left(\mathrm{e}^{-x}\right), x \geq 0\). If \(\alpha\) is a positive real number such that \(\mathrm{a}=(\mathrm{fog})^{\prime}(\alpha)\) and $$\mathrm{b}=(\mat...

Derivative of \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right)\) w.r.t. \(\cos ^{-1} x^2\) is

The domain of the function given by \(2^x+2^y=2\) is

\(\int \frac{\mathrm{e}^x(1+x)}{\cos ^2\left(\mathrm{e}^x \cdot x\right)} \mathrm{d} x=\)

If \(\int \frac{\mathrm{d} x}{x \sqrt{1-x^3}}=\mathrm{k} \log \left(\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right)+\mathrm{c}\), (where \(\mathrm{c}\) is a constant of integration), then value of \(\mathrm{k}\) is

\(\int \frac{\log (\cot x)}{\sin 2 x} d x=\)

If \(\alpha=3 \sin ^{-1} \frac{6}{11}\) and \(\beta=3 \cos ^{-1}\left(\frac{4}{9}\right)\), where the inverse trigonometric functions take only the principal values, then the correct option is

\(\text { The value of } \sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 3\right) \text { is }\)

The value of \(2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}\)

\(\text { If } y=\sqrt{\frac{1-\sin ^{-1}(x)}{1+\sin ^{-1}(x)}} \text {, then } \frac{\mathrm{d} y}{\mathrm{~d} x} \text { at } x=0 \text { and } y=1 \text { is }\)

\(\lim _\limits{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{2}{x^3-3 x^2+2 x}\right]\) is equal to

The left-hand derivative of \(\mathrm{f}(x)=[x] \sin (\pi x)\), at \(x=\mathrm{k}, \mathrm{k}\) is an integer and [.] is the greatest integer function, is

If $$\mathrm{f}(x)=\left\{\begin{array}{ll}\frac{\sqrt{1+\mathrm{m} x}-\sqrt{1-\mathrm{m} x}}{x}, & -1 \leq x < 0 \\ \frac{2 x+1}{x-2} & , 0 \leq x \leq 1\end{array}\right.$$ is continuous in the interval \([-1,1]\), then \(\mathrm{m}\) is ...

For the following shaded area, the linear constraints except \(x,y \ge 0\) are

The statement pattern \(\mathrm{p} \rightarrow \sim(\mathrm{p} \wedge \sim \mathrm{q})\) is equivalent to

If $$P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$$ is the adjoint of a \(3 \times 3\) matrix \(A\) and \(|A|=4\), then value of \(\alpha\) is

Let \(\omega \neq 1\) be a cube root of unity and \(S\) be the set of all non-singular matrices of the form $$\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]$$ where each of \(a, b\) and \(c\) ...

Five students are to be arranged on a platform such that the boy \(B_1\) occupies the second position and such that the girl \(G_1\) is always adjacent to the girl \(G_2\). Then, the number of such possible arrangements is

From a lot of 20 baskets, which includes 6 defective baskets, a sample of 2 baskets is drawn at random one by one without replacement. The expected value of number of defective basket is

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with these three vertices is equilateral, equals ___________.

A binomial random variable \(\mathrm{X}\) satisfies \(9. p(X=4)=p(X=2)\) when \(n=6\). Then \(p\) is equal to

If in \(\triangle \mathrm{ABC}\), with usual notations, \(a \cdot \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}\), then

In a triangle \(\mathrm{ABC}\), with usual notations, if \(\mathrm{m} \angle \mathrm{A}=60^{\circ}, \mathrm{b}=8, \mathrm{a}=6\) and \(\mathrm{B}=\sin ^{-1} x\), then \(x\) has the value

If variance of \(x_1, x_2 \ldots \ldots, x_n\) is \(\sigma_x^2\), then the variance of \(\lambda x_1, \lambda x_2, \ldots \ldots, \lambda x_{\mathrm{n}}(\lambda \neq 0)\) is

If the angle between the lines represented by the equation \(x^2+\lambda x y-y^2 \tan ^2 \theta=0\) is \(2 \theta\), then the value of \(\lambda\) is

\(\mathrm{a}\) and \(\mathrm{b}\) are the intercepts made by a line on the co-ordinate axes. If \(3 \mathrm{a}=\mathrm{b}\) and the line passes through \((1,3)\), then the equation of the line is

If the direction cosines \(l, \mathrm{~m}, \mathrm{n}\) of two lines are connected by relations \(l-5 \mathrm{~m}+3 \mathrm{n}=0\) and \(7 l^2+5 \mathrm{~m}^2-3 \mathrm{n}^2=0\), then value of \(l+\mathrm{m}+\mathrm{n}\) is

The mirror image of the point \((1,2,3)\) in a plane is \(\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)\). Thus, the point _________ lies on this plane.

A plane is parallel to two lines, whose direction ratios are \(1,0,-1\) and \(-1,1,0\) and it contains the point \((1,1,1)\). If it cuts co-ordinate axes \((\mathrm{X}, \mathrm{Y}, \mathrm{Z}\) - axes resp.) at $$\mathrm{A}, \mathrm{B}, \ma...

The lines \(\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} \quad\) and \(\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{2}\)

The vector equation of the line \(2 x+4=3 y+1=6 z-3\) is

The value of \(\tan \left(\frac{\pi}{8}\right)\) is _________.

If the volume of the parallelopiped is \(158 \mathrm{~cu}\). units whose coterminus edges are given by the vectors \(\bar{a}=(\hat{i}+\hat{j}+n \hat{k}), \bar{b}=2 \hat{i}+4 \hat{j}-n \hat{k}\) and \(\bar{c}=\hat{i}+n \hat{j}+3 \hat{k}\), w...

If \(\bar{a}, \bar{b}, \bar{c}\) are three vectors such that $$\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}}+\overline{\mathrm{c}})+\overline{\mathrm{b}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})+\overline{\mathrm{c}} \cdot(\ov...

Let \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}\) and \(\bar{b}=\hat{i}+\hat{j}\). If \(\bar{c}\) is a vector such that $$\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=|\overline{\mathrm{c}}|,|\overline{\mathrm{c}}-\overline{\mathrm{a}}|=2 \s...

If \(\quad \overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}, \quad \overline{\mathrm{b}}=2 \hat{\mathrm{j}}-\hat{\mathrm{k}} \quad\) and $$\quad \overline{\mathrm{r}} \times \overline{\mathrm{a}}=\overline{\mathrm{b}} \times \overlin...

Let \(\bar{a}, \bar{b}\) and \(\bar{c}\) be three unit vectors such that \(\bar{a} \times(\bar{b} \times \bar{c})=\frac{\sqrt{3}}{2}(\bar{b}+\bar{c})\). If \(\bar{b}\) is not parallel to \(\bar{c}\), then the angle between \(\bar{a}\) and $...

If \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are two unit vectors such that \(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}\) and \(5 \bar{a}-4 \bar{b}\) are perpendicular to each other, then the angle between \(\bar{a}\) and ...