MHT CET 2023 12th May Morning Shift

The angle between the tangents to the curves \(y=2 x^2\) and \(x=2 y^2\) at \((1,1)\) is

The function \(\mathrm{f}(x)=\sin ^4 x+\cos ^4 x\) is increasing in

A ladder 5 meters long rests against a vertical wall. If its top slides downwards at the rate of \(10 \mathrm{~cm} / \mathrm{s}\), then the angle between the ladder and the floor is decreasing at the rate of ________ rad./s when it's lower ...

The area of the region bounded by the curves \(y=\mathrm{e}^x, y=\log x\) and lines \(x=1, x=2\) is

If \(\lambda\) is the perpendicular distance of a point \(\mathrm{P}\) on the circle \(x^2+y^2+2 x+2 y-3=0\), from the line \(2 x+y+13=0\), then maximum possible value of \(\lambda\) is

If \(a > 0\) and \(z=\frac{(1+i)^2}{a+i},(i=\sqrt{-1})\) has magnitude \(\frac{2}{\sqrt{5}}\), then \(\bar{z}\) is equal to

\(\int_\limits0^4|2 x-5| d x=\)

The decay rate of radio active material at any time \(t\) is proportional to its mass at that time. The mass is 27 grams when \(t=0\). After three hours it was found that 8 grams are left. Then the substance left after one more hour is

The differential equation \(\cos (x+y) \mathrm{d} y=\mathrm{d} x\) has the general solution given by

If \(\frac{\mathrm{d} y}{\mathrm{~d} x}=y+3\) and \(y(0)=2\), then \(y(\log 2)=\)

The approximate value of \(\sin \left(60^{\circ} 0^{\prime} 10^{\prime \prime}\right)\) is (given that \(\sqrt{3}=1.732,1^{\circ}=0.0175^{\circ}\) )

The derivative of \(\mathrm{f}(\tan x)\) w.r.t. \(\mathrm{g}(\sec x)\) at \(x=\frac{\pi}{4}\) where \(\mathrm{f}^{\prime}(1)=2\) and \(\mathrm{g}^{\prime}(\sqrt{2})=4\) is

If \(x=-1\) and \(x=2\) are extreme points of \(\mathrm{f}(x)=\alpha \log x+\beta x^2+x, \alpha\) and \(\beta\) are constants, then the value of \(\alpha^2+2 \beta\) is

\(\text { If } \log (x+y)=2 x y \text {, then } \frac{\mathrm{d} y}{\mathrm{~d} x} \text { at } x=0 \text { is }\)

\(y=(1+x)\left(1+x^2\right)\left(1+x^4\right) \ldots \ldots \ldots\left(1+x^{2 n}\right)\),
then the value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=0\) is

If \(\mathrm{g}(x)=1+\sqrt{x}\) and \(\mathrm{f}(\mathrm{g}(x))=3+2 \sqrt{x}+x\) then \(\mathrm{f}(\mathrm{f}(x))\) is

\(\int \frac{\operatorname{cosec} x d x}{\cos ^2\left(1+\log \tan \frac{x}{2}\right)}=\)

The integral \(\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^5 x+\cos ^3 x \sin ^2 x+\sin ^3 x \cos ^2 x+\cos ^5 x\right)^2} \mathrm{~d} x\) is equal to

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

If \(\int \cos ^{\frac{3}{5}} x \cdot \sin ^3 x d x=\frac{-1}{m} \cos ^m x+\frac{1}{n} \cos ^n x+c\), (where \(\mathrm{c}\) is the constant of integration), then \((\mathrm{m}, \mathrm{n})=\)

If \(x=\operatorname{cosec}\left(\tan ^{-1}\left(\cos \left(\cot ^{-1}\left(\sec \left(\sin ^{-1} a\right)\right)\right)\right)\right), \mathrm{a} \in[0,1]\)

The value of \(\sin \left(\cot ^{-1} x\right)\) is

The values of \(a\) and \(b\), so that the function
$$f(x)=\left\{\begin{array}{l}
x+a \sqrt{2} \sin x, 0 \leq x \leq \frac{\pi}{4} \\
2 x \cot x+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\
a \cos 2 x-b \sin x, \frac{\pi}{2} < x \leq \pi
...

\(\lim _\limits{x \rightarrow 0} \frac{\cos 7 x^{\circ}-\cos 2 x^{\circ}}{x^2}\) is

For a feasible region OCDBO given below, the maximum value of the objective function \(z=3 x+4 y\) is

The inverse of the statement "If the surface area increase, then the pressure decreases.", is

The contrapositive of "If \(x\) and \(y\) are integers such that \(x y\) is odd, then both \(x\) and \(y\) are odd" is

If the matrix $$\mathrm{A}=\left[\begin{array}{cc}1 & 2 \\ -5 & 1\end{array}\right]$$ and \(\mathrm{A}^{-1}=x \mathrm{~A}+y \mathrm{I}\), when \(I\) is a unit matrix of order 2 , then the value of \(2 x+3 y\) is

If \(\mathrm{T}_{\mathrm{n}}\) denotes the number of triangles which can be formed using the vertices of regular polygon of \(\mathrm{n}\) sides and \(T_{n+1}-T_n=21\), then \(\mathrm{n}=\)

The p.m.f of random variate \(\mathrm{X}\) is $$P(X)= \begin{cases}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \ldots, \mathrm{n} \\ 0, & \text { otherwise }\end{cases}$$ Then \(\mathrm{E}(\mathrm{X})=\)

An experiment succeeds twice as often as it fails. Then the probability, that in the next 6 trials there will be atleast 4 successes, is

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then the probability distribution of number of jacks is

\(\mathrm{A}\) and \(\mathrm{B}\) are independent events with \(\mathrm{P}(\mathrm{A})=\frac{1}{4}\) and \(\mathrm{P}(\mathrm{A} \cup \mathrm{B})=2 \mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})\), then \(\mathrm{P}(\mathrm{B})\) is

In a triangle, the sum of lengths of two sides is \(x\) and the product of the lengths of the same two sides is \(y\). If \(x^2-\mathrm{c}^2=y\), where \(\mathrm{c}\) is the length of the third side of the triangle, then the circumradius of...

If the variance of the numbers \(-1,0,1, \mathrm{k}\) is 5, where \(\mathrm{k} > 0\), then \(\mathrm{k}\) is equal to

The co-ordinates of the points on the line \(2 x-y=5\) which are the distance of 1 unit from the line \(3 x+4 y=5\) are

Let \(\mathrm{PQR}\) be a right angled isosceles triangle, right angled at \(\mathrm{P}(2,1)\). If the equation of the line \(\mathrm{QR}\) is \(2 x+y=3\), then the equation representing the pair of lines \(P Q\) and \(P R\) is

The centroid of tetrahedron with vertices at \(\mathrm{A}(-1,2,3), \mathrm{B}(3,-2,1), \mathrm{C}(2,1,3)\) and \(\mathrm{D}(-1,-2,4)\) is

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 the co-ordinate axes at \(\mathrm{A}, \mathrm{B}, \mathrm{C}\), then the volume of the tetrahedron $$\ma...

The equation of the plane through \((-1,1,2)\) whose normal makes equal acute angles with co-ordinate axes is

The distance of the point \(\mathrm{P}(-2,4,-5)\) from the line \(\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}\) is

If the line \(\frac{1-x}{3}=\frac{7 y-14}{2 p}=\frac{z-3}{2}\) and \(\frac{7-7 x}{3 \mathrm{p}}=\frac{y-5}{1}=\frac{6-\mathrm{z}}{5}\) are at right angles, then \(\mathrm{p}=\)

The solution set of \(8 \cos ^2 \theta+14 \cos \theta+5=0\), in the interval \([0,2 \pi]\), is

If general solution of \(\cos ^2 \theta-2 \sin \theta+\frac{1}{4}=0\) is \(\theta=\frac{\mathrm{n} \pi}{\mathrm{A}}+(-1)^{\mathrm{n}} \frac{\pi}{\mathrm{B}}, \mathrm{n} \in \mathrm{Z}\), then \(\mathrm{A}+\mathrm{B}\) has the

If \(\tan \theta=\frac{\sin \alpha-\cos \alpha}{\sin \alpha+\cos \alpha}, 0 \leq \alpha \leq \frac{\pi}{2}\), then the value of \(\cos 2 \theta\) is

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are three vectors, \(|\overline{\mathrm{a}}|=2,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=1, |\bar{b} \times \bar{c}|=\sqrt{15}\) and $$\bar{b}=2 \bar{c}+\lam...

Two adjacent sides of a parallelogram \(\mathrm{ABCD}\) are given by \(\overline{A B}=2 \hat{i}+10 \hat{j}+11 \hat{k}\) and \(\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\). The side \(\mathrm{AD}\) is rota...

If the area of the triangle with vertices \((1,2,0)\), \((1,0,2)\) and \((0, x, 1)\) is \(\sqrt{6}\) square units, then the value of $x$ is

Let \(\overline{\mathrm{A}}\) be a vector parallel to line of intersection of planes \(P_1\) and \(P_2\) through origin. \(P_1\) is parallel to the vectors \(2 \hat{j}+3 \hat{k}\) and \(4 \hat{j}-3 \hat{k}\) and \(P_2\) is parallel to $$\ha...

\(\overline{\mathrm{u}}, \overline{\mathrm{v}}, \overline{\mathrm{w}}\) are three vectors such that \(|\overline{\mathrm{u}}|=1, |\bar{v}|=2,|\bar{w}|=3\). If the projection of \(\bar{v}\) along \(\bar{u}\) is equal to projection of $$\bar{...