MHT CET 2023 10th May Morning Shift

An open metallic tank is to be constructed, with a square base and vertical sides, having volume 500 cubic meter. Then the dimensions of the tank, for minimum area of the sheet metal used in its construction, are

A square plate is contracting at the uniform rate \(4 \mathrm{~cm}^2 / \mathrm{sec}\), then the rate at which the perimeter is decreasing, when side of the square is \(20 \mathrm{~cm}\), is

A ladder of length \(17 \mathrm{~m}\) rests with one end against a vertical wall and the other on the level ground. If the lower end slips away at the rate of \(1 \mathrm{~m} / \mathrm{sec}\)., then when it is \(8 \mathrm{~m}\) away from th...

If the line \(a x+b y+c=0\) is a normal to the curve \(x y=1\), then

A kite is \(120 \mathrm{~m}\) high and \(130 \mathrm{~m}\) of string is out. If the kite is moving away horizontally at the rate of \(39 \mathrm{~m} / \mathrm{sec}\), then the rate at which the string is being out, is

Area of the region bounded by the curve \(y=\sqrt{49-x^2}\) and \(\mathrm{X}\)-axis is

If the circles \(x^2+y^2=9\) and \(x^2+y^2+2 \alpha x+2 y+1=0\) touch each other internally, then the value of \(\alpha^3\) is

If \(w=\frac{z}{z-\frac{1}{3} i}\) and \(|w|=1, i=\sqrt{-1}\), then \(z\) lies on

If \(\int_\limits0^{\frac{1}{2}} \frac{x^2}{\left(1-x^2\right)^{\frac{3}{2}}} \mathrm{~d} x=\frac{\mathrm{k}}{6}\), then the value of \(\mathrm{k}\) is

The population \(\mathrm{P}=\mathrm{P}(\mathrm{t})\) at time \(\mathrm{t}\) of certain species follows the differential equation \(\frac{d P}{d t}=0.5 P-450\). If \(P(0)=850\), then the time at which population becomes zero is

The differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sqrt{1-y^2}}{y}\) determines a family of circles with

General solution of the differential equation \(\cos x(1+\cos y) \mathrm{d} x-\sin y(1+\sin x) \mathrm{d} y=0\) is

If \(y=\cos ^{-1}\left(\frac{\mathrm{a}^2}{\sqrt{x^4+\mathrm{a}^4}}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is

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

If \(\mathrm{f}(x)=\mathrm{e}^x, \mathrm{~g}(x)=\sin ^{-1} x\) and \(\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x))\), then \(\frac{\mathrm{h}^{\prime}(x)}{\mathrm{h}(x)}\) is

\(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} ; \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are two functions such that \(\mathrm{f}(x)=2 x-3, \mathrm{~g}(x)=x^3+5\), then \((\mathrm{fog})^{-1}(-9)\) is

If \(\int \sqrt{\frac{x-7}{x-9}} d x=A \sqrt{x^2-16 x+63}+\log \left|(x-8)+\sqrt{x^2-16 x+63}\right|+c,\)
(where \(\mathrm{c}\) is a constant of integration) then \(\mathrm{A}\) is

\(\int \frac{1}{7-6 x-x^2} d x=\)

\(\int \frac{d x}{\sin x+\cos x}=\)

If \(\mathrm{I}=\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}\), then \(\mathrm{I}\) is

Considering only the principal values of an inverse function, the set
\(\mathrm{A}=\left\{x \geq 0 / \tan ^{-1} x+\tan ^{-1} 6 x=\frac{\pi}{4}\right\}\)

The solution of the equation \(\tan ^{-1}(1+x)+\tan ^{-1}(1-x)=\frac{\pi}{2}\) is

The value of \(\lim _\limits{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}\) is

If the function \(\mathrm{f}(x)\) is continuous in \(0 \leq x \leq \pi\), then the value of \(2 a+3 b\) is where
$$f(x)= \begin{cases}x+a \sqrt{2} \sin x & \text { if } 0 \leq x < \frac{\pi}{4} \\ 2 x \cot x+b & \text { if } \frac{\pi}{4} \...

The vertices of the feasible region for the constraints \(x+y \leq 4, x \leq 2, y \leq 1, x+y \geq 1, x, y \geq 0\) are

The logical statement \([\sim(\sim p \vee q) \vee(p \wedge r)] \wedge(\sim q \wedge r)\) is equivalent to

The given circuit is equivalent to

If $$B=\left[\begin{array}{lll}1 & \alpha & 2 \\ 1 & 2 & 2 \\ 2 & 3 & 3\end{array}\right]$$ is the adjoint of a \(3 \times 3\) matrix A and \(|A|=5\), then \(\alpha\) is equal to

A group consists of 8 boys and 5 girls, then the number of committees of 5 persons that can be formed, if committee consists of at least 2 girls and at most 2 boys, are

For a binomial variate \(\mathrm{X}\) with \(\mathrm{n}=6\) if \(P(X=4)=\frac{135}{2^{12}}\), then its variance is

The p.d.f. of a discrete random variable is defined as
$$\mathrm{f}(x)=\left\{\begin{array}{l} \mathrm{k} x^2, 0 \leq x \leq 6 \\ 0, \text { otherwise } \end{array}\right.$$
Then the value of \(F(4)\) (c.d.f) is

A player tosses 2 fair coins. He wins ₹5 if 2 heads appear, ₹ 2 if one head appears and ₹ 1 if no head appears. Then the variance of his winning amount in ₹ is :

Three critics review a book. For the three critics the odds in favor of the book are \(2: 5, 3: 4\) and \(4: 3\) respectively. The probability that the majority is in favor of the book, is given by

If one side of a triangle is double the other and the angles opposite to these sides differ by \(60^{\circ}\), then the triangle is

The variance, for first six prime numbers greater than 5, is

The number of integral values of \(\mathrm{p}\) in the domain \([-5,5]\), such that the equation \(2 x^2+4 x y-p y^2+4 x+q y+1=0\) represents pair of lines, are

The points \((1,3),(5,1)\) are opposite vertices of a diagonal of a rectangle. If the other two vertices lie on the line \(y=2 x+\mathrm{c}\), then one of the vertex on the other diagonal is

The line \(\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}\) lies in the plane \(x+3 y-\alpha z+\beta=0\), then the value of \(\alpha^2+\alpha \beta+\beta^2\) is

Let \(\mathrm{P}\) be a plane passing through the points \((2,1,0),(4,1,1)\) and \((5,0,1)\) and \(R\) be the point \((2,1,6)\). Then image of \(R\) in the plane \(P\) is

The co-ordinates of the point, where the line through \(A(3,4,1)\) and \(B(5,1,6)\) crosses the \(\mathrm{XZ}\)-plane, are

\(\mathrm{ABC}\) is a triangle in a plane with vertices \(\mathrm{A}(2,3,5), \mathrm{B}(-1,3,2)\) and \(\mathrm{C}(\lambda, 5, \mu)\). If median through \(\mathrm{A}\) is equally inclined to the co-ordinate axes, then value of $$\lambda+\mu...

The number of possible solutions of \(\sin \theta+\sin 4 \theta+\sin 7 \theta=0, \theta \in(0, \pi)\) are

The number of solutions of \(\tan x+\sec x=2 \cos x\) in \([0,2 \pi]\) are

The value of \(\tan \frac{\pi}{8}\) is

Scalar projection of the line segment joining the points \(\mathrm{A}(-2,0,3), \mathrm{B}(1,4,2)\) on the line whose direction ratios are \(6,-2,3\) is

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

The vector projection of \(\overline{\mathrm{AB}}\) on \(\overline{\mathrm{CD}}\), where \(A \equiv(2,-3,0), B \equiv(1,-4,-2), C \equiv(4,6,8)\) and \(\mathrm{D} \equiv(7,0,10)\), is

The vectors are \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \bar{b}=\hat{i}+\hat{j}\). If \(\bar{c}\) is a vector such that \(\bar{a} \cdot \bar{c}=|\bar{c}|\) and \(|\bar{c}-\bar{a}|=2 \sqrt{2}\), angle between \(\bar{a} \times \bar{b}\) and $$...

If \(\bar{a}=\hat{i}+2 \hat{j}+\hat{k}, \bar{b}=\hat{i}-\hat{j}+\hat{k}, \bar{c}=\hat{i}+\hat{j}-\hat{k}\), then a vector in the plane of \(\bar{a}\) and \(\bar{b}\), whose projection on \(\overline{\mathrm{c}}\) is \(\frac{1}{\sqrt{3}}\), ...

Let \(\bar{a}, \bar{b}, \bar{c}\) be three non-zero vectors, such that no two of them are collinear and \((\bar{a} \times \bar{b}) \times \bar{c}=\frac{1}{3}|\bar{b}||\bar{c}| \bar{a}\). If \(\theta\) is the angle between the vectors $$\bar...