MHT CET 2023 12th May Evening Shift

A poster is to be printed on a rectangular sheet of paper of area \(18 \mathrm{~m}^2\). The margins at the top and bottom of \(75 \mathrm{~cm}\) each and at the sides \(50 \mathrm{~cm}\) each are to be left. Then the dimensions i.e. height ...

The equation of the normal to the curve \(3 x^2-y^2=8\), which is parallel to the line \(x+3 y=10\), is

If the curves \(y^2=6 x\) and \(9 x^2+b y^2=16\) intersect each other at right angle, then value of '\(b\)' is

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 __________ radians/second when...

A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 4 meter and volume is 36 cubic meters. If building of the tank costs ₹ 100 per square meter for the base and ₹ 50 per square met...

The area (in sq. units) of the smaller part of the circle \(x^2+y^2=\mathrm{a}^2\) cut off by the line \(x=\frac{\mathrm{a}}{\sqrt{2}}\) is

The circles \(x^2+y^2+2 \mathrm{a} x+\mathrm{c}=0\) and \(x^2+y^2+2 b y+c=0\) touch each other externally, if

If \(|z-2+i| \leq 2\), then the difference between the greatest and least value of \(|z|\) is ________, \((\mathrm{i}=\sqrt{-1})\)

The value of \(\int_\limits0^\pi\left|\sin x-\frac{2 x}{\pi}\right| \mathrm{d} x\) is

The solution of \(\mathrm{e}^{y-x} \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y(\sin x+\cos x)}{(1+y \log y)}\) is

Water flows from the base of rectangular tank, of depth 16 meters. The rate of flow of the water is proportional to the square root of depth at any time \(\mathrm{t}\). If depth is \(4 \mathrm{~m}\) when \(\mathrm{t}=2\) hours, then after 3...

If \((2+\sin x) \frac{\mathrm{d} y}{\mathrm{~d} x}+(y+1) \cos x=0\) and \(y(0)=1\), then \(y\left(\frac{\pi}{2}\right)\) is

If \(\mathrm{f}(x)=\sin ^{-1}\left(\frac{2 \log x}{1+(\log x)^2}\right)\), then \(\mathrm{f}^{\prime}(\mathrm{e})\) is

For \(x>1\), if \((2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}\), then \(\left(1+\log _e 2 x\right)^2 \frac{d y}{d x}\) is equal to

If \(\tan y=\frac{x \sin \alpha}{1-x \cos \alpha}\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{m}}{x^2+2 \mathrm{n} x+1}\), then \(\mathrm{m}^2+\mathrm{n}^2\) is

If \(\mathrm{f}(x)=x^2+1\) and \(\mathrm{g}(x)=\frac{1}{x}\), then the value of \(\mathrm{f}(\mathrm{g}(\mathrm{g}(\mathrm{f}(x))))\) at \(x=1\) is

\(\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} d x=\)

If \(\int \frac{\sqrt{1-x^2}}{x^4} \mathrm{~d} x=\mathrm{A}(x)\left(\sqrt{1-x^2}\right)^{\mathrm{m}}+\mathrm{c}\) for a suitable chosen integer \(\mathrm{m}\) and a function \(\mathrm{A}(x)\), where \(\mathrm{c}\) is a constant of integrati...

\(\int\left(\frac{\tan \left(\frac{1}{x}\right)}{x}\right)^2 d x=\)

\(\int \frac{1}{(x+2)(1+x)^2} d x\) has the value

The value of \(x\), for which \(\sin \left(\cot ^{-1}(x)\right)=\cos \left(\tan ^{-1}(1+x)\right)\), is

If \(\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x\), then \(x\) is

\(\lim _\limits{x \rightarrow 0} \frac{x \cot 4 x}{\sin ^2 x \cdot \cot ^2(2 x)} \text { is equal to }\)

Given $$\mathrm{f}(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2} & , \text { if } x<0 \\ \mathrm{a} & , \text { if } x=0 \\ \frac{\sqrt{x}}{\sqrt{16-\sqrt{x}-4}}, & \text { if } x>0\end{array}\right.$$
If \(\mathrm{f}(x)\) is continuous...

The maximum value of \(z=7 x+8 y\) subject to the constraints \(x+y \leq 20, y \geq 5, x \leq 10, x \geq 0, y \geq 0\) is

Let
Statement 1 : If a quadrilateral is a square, then all of its sides are equal.
Statement 2: All the sides of a quadrilateral are equal, then it is a square.

The given following circuit is equivalent to

If $$A=\left[\begin{array}{cc}2 a & -3 b \\ 3 & 2\end{array}\right]$$ and \(A \cdot \operatorname{adj} A=A A^T\), then \(2 a+3 b\) is

The number of words that can be formed by using the letters of the word CALCULATE such that each word starts and ends with a consonant, are

An irregular six faced die is thrown and the probability that, in 5 throws it will give 3 even numbers is twice the probability that it will give 2 even numbers. The number of times, in 6804 sets of 5 throws, you expect to give no even numb...

A box contains 100 tickets numbered 1 to 100 . A ticket is drawn at random from the box. Then the probability, that number on the ticket is a perfect square, is

Three fair coins with faces numbered 1 and 0 are tossed simultaneously. Then variance (X) of the probability distribution of random variable \(\mathrm{X}\), where \(\mathrm{X}\) is the sum of numbers on the upper most faces, is

The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the length of the sides of the triangle (in units) are

If two angles of \(\triangle \mathrm{ABC}\) are \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), then the ratio of the smallest and greatest sides are

In \(\triangle \mathrm{ABC}, \mathrm{m} \angle \mathrm{B}=\frac{\pi}{3}\) and \(\mathrm{m} \angle \mathrm{C}=\frac{\pi}{4}\). Let point \(\mathrm{D}\) divide \(\mathrm{BC}\) internally in the ratio \(1: 3\), then $$\frac{\sin (\angle B A D)...

The discrete random variable \(\mathrm{X}\) can take all possible integer values from 1 to \(\mathrm{k}\), each with a probability \(\frac{1}{\mathrm{k}}\), then its variance is

For 20 observations of variable $x$, if \(\sum\left(x_i-2\right)=20\) and \(\sum\left(x_i-2\right)^2=100\), then the standard deviation of variable \(x\) is

If the pair of lines given by \((x \cos \alpha+y \sin \alpha)^2=\left(x^2+y^2\right) \sin ^2 \alpha\) are perpendicular to each other, then \(\alpha\) is

If \(\mathrm{k}_{\mathrm{i}}\) are possible values of \(\mathrm{k}\) for which lines \(\mathrm{k} x+2 y+2=0,2 x+\mathrm{k} y+3=0\) and \(3 x+3 y+\mathrm{k}=0\) are concurrent, then \(\sum \mathrm{k}_{\mathrm{i}}\) has the value

The equation of the line, passing through \((1,2,3)\) and parallel to planes \(x-y+2 z=5\) and \(3 x+y+z=6\), is

The shortest distance (in units) between the lines \(\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}\) and \(\bar{r}=(2 \hat{i}-2 \hat{j}+3 \hat{k})+\lambda(\hat{i}+2 \hat{j})\) is

The length (in units) of the projection of the line segment, joining the points \((5,-1,4)\) and \((4,-1,3)\), on the plane \(x+y+z=7\) is

If the volume of tetrahedron, whose vertices are \(\mathrm{A}(1,2,3), \mathrm{B}(-3,-1,1), \mathrm{C}(2,1,3)\) and \(D(-1,2, x)\) is \(\frac{11}{6}\) cubic units, then the value of \(x\) is

Equation of plane containing the line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\) and perpendicular to the plane containing the lines \(\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\) and \(\frac{x}{4}=\frac{y}{2}=\frac{z}{3}\) is

\(\cos ^2 48^{\circ}-\sin ^2 12^{\circ}=\)
_________, if \(\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}\)

\(A, B, C, D\) are four points in a plane with position vectors \(\bar{a}, \bar{b}, \bar{c}, \bar{d}\) respectively such that \((\bar{a}-\bar{d}) \cdot(\bar{b}-\bar{c})=(\bar{b}-\bar{d}) \cdot(\bar{c}-\bar{a})=0\). The point \(D\), then is ...

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

The unit vector which is orthogonal to the vector \(3 \hat{i}+2 \hat{j}+6 \hat{k}\) and coplanar with the vectors \(2 \hat{i}+\hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+\hat{k}\) is

If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors, no two of them are collinear, \(\vec{a}+2 \vec{b}\) is collinear with \(\vec{c}, \vec{b}+3 \vec{c}\) is collinear with \(\vec{a}\), then \(\vec{a}+2 \vec{b}\) is

If

then \(|\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}| \text { is }\)