MHT CET 2023 13th May Morning Shift

If \(\mathrm{f}(x)=x^3+\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}\) and \(0<\mathrm{b}^2<\mathrm{c}\), then in \((-\infty, \infty)\)

The slope of the normal to the curve \(x=\sqrt{t}\) and \(y=t-\frac{1}{\sqrt{t}}\) at \(t=4\) is

Values of \(c\) as per Rolle's theorem for \(f(x)=\sin x+\cos x+6\) on \([0,2 \pi]\) are

The area (in sq. units) of the region bounded by curves \(y=3 x+1, y=4 x+1\) and \(x=3\) is

The parametric equations of the curve \(x^2+y^2+a x+b y=0\) are

The value of \(\frac{\mathrm{i}^{248}+\mathrm{i}^{246}+\mathrm{i}^{244}+\mathrm{i}^{242}+\mathrm{i}^{240}}{\mathrm{i}^{249}+\mathrm{i}^{247}+\mathrm{i}^{245}+\mathrm{i}^{243}+\mathrm{i}^{241}}, (\mathrm{i}=\sqrt{-1})\) is

If $$\mathrm{f}(x)=\left\{\begin{array}{ll}\mathrm{e}^{\cos x} \sin x & , \text { for }|x| \leq 2 \\ 2, & \text { otherwise }\end{array}\right.$$, then \(\int_\limits{-2}^3 \mathrm{f}(x) \mathrm{d} x\) is equal to

The solution of the differential equation \(\mathrm{e}^{-x}(y+1) \mathrm{d} y+\left(\cos ^2 x-\sin 2 x\right) y \mathrm{~d} x=0\) at \(x=0\), \(y=1\) is

Rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant and it is found that the number doubles in 6 hours. The number of bacteria becomes ________ times at the end of 18 hours.

The particular solution of differential equation \(\mathrm{e}^{\frac{d y}{d x}}=(x+1), y(0)=3\) is

A right circular cone has height \(9 \mathrm{~cm}\) and radius of base \(5 \mathrm{~cm}\). It is inverted and water is poured into it. If at any instant, the water level rises at the rate $$\frac{\pi}{\mathrm{A}} \mathrm{cm} / \mathrm{sec}$...

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

If \(y=\tan ^{-1}\left(\frac{4 \sin 2 x}{\cos 2 x-6 \sin ^2 x}\right)\), then \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\) at \(x=0\) is

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function such that \(\mathrm{f}(x)=x^3+x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+6, x \in \mathrm{R}\), then \(\mathrm{f}(2)\) is

If \(3 \mathrm{f}(x)-\mathrm{f}\left(\frac{1}{x}\right)=8 \log _2 x^3, x>0\), then \(\mathrm{f}(2), \mathrm{f}(4)\), \(f(8)\) are in

If \(\mathrm{I}=\int \frac{2 x-7}{\sqrt{3 x-2}} \mathrm{~d} x\), then \(\mathrm{I}\) is given by

\(\int \frac{\log \left(x^2+a^2\right)}{x^2} d x=\)

If \(\int x^5 e^{-4 x^3} \mathrm{~d} x=\frac{1}{48} \mathrm{e}^{-4 x^3} \mathrm{f}(x)+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration, then \(\mathrm{f}(x)\) is given by

If \(\mathrm{f}(x)=\int \frac{x^2 \mathrm{~d} x}{\left(1+x^2\right)\left(1+\sqrt{1+x^2}\right)}\) and \(\mathrm{f}(0)=0\), then \(\mathrm{f}(1)\) is

\(x, y, z\) are in G.P. and \(\tan ^{-1} x, \tan ^{-1} y, \tan ^{-1} z\) are in A.P., then

The function \(\mathrm{f}(\mathrm{t})=\frac{1}{\mathrm{t}^2+\mathrm{t}-2}\) where \(\mathrm{t}=\frac{1}{x-1}\) is discontinuous at

If \(\mathrm{f}(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7\), then \(\lim _\limits{\alpha \rightarrow 0} \frac{f(1-\alpha)-f(1)}{\alpha^3+3 \alpha}=\)

If feasible region is as shown in the figure, then related inequalities are

The expression \((p \wedge \sim q) \vee q \vee(\sim p \wedge q)\) is equivalent to

Negation of inverse of the following statement pattern \((p \wedge q) \rightarrow(p \vee \sim q)\) is

If $$A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$$ and \(A_{i j}\) is a cofactor of \(a_{i j}\) then the value of \(a_{21} A_{21}+a_{22} A_{22}+a_{23} A_{23}\) is

Five persons \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\) and \(\mathrm{E}\) are seated in a circular arrangement. If each of them is given a cap of one of the three colours red, blue and green, then the number of ways of distributing...

A random variable \(X\) has the following probability distribution

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Let \(\mathrm{X}\) be random variable having Binomial distribution \(B(7, p)\). If \(P[X=3]=5 P[X=4]\), then variance of \(\mathrm{X}\) is

If a continuous random variable \(\mathrm{X}\) has probability density function \(\mathrm{f}(x)\) given by
$$f(x)=\left\{\begin{array}{cl}
a x & , \text { if } 0 \leq x<1 \\
a & , \text { if } 1 \leq x<2 \\
3 a-a x & , \text { if } 2 \leq x...

A card is drawn at random from a well shuffled pack of 52 cards. The probability that it is black card or face card is

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the sides of the triangle (in units) are

In \(\triangle A B C\) with usual notation, \(\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\) and \(a=\frac{1}{\sqrt{6}}\), then the area of triangle is _______ sq. units.

Angles of a triangle are in the ratio \(4: 1: 1\). Then the ratio of its greatest side to its perimeter is

The raw data \(x_1, x_2, \ldots \ldots, x_{\mathrm{n}}\) is an A.P. with common difference \(\mathrm{d}\) and first term \(0, \bar{x}\) and \(\sigma^2\) are mean and variance of \(x_{\mathrm{i}}, \mathrm{i}=1,2, \ldots \ldots \mathrm{n}\), ...

If the angle between the lines given by \(x^2-3 x y+\lambda y^2+3 x-5 y+2=0 ; \lambda \geq 0\) is \(\tan ^{-1}\left(\frac{1}{3}\right)\), then the value of \(\lambda\) is

The base of an equilateral triangle is represented by the equation \(2 x-y-1=0\) and its vertex is \((1,2)\), then the length (in units) of the side of the triangle is

A line is drawn through the point \((1,2)\) to meet the co-ordinate axes at \(\mathrm{P}\) and \(\mathrm{Q}\) such that it forms a \(\triangle \mathrm{OPQ}\), where \(\mathrm{O}\) is the origin. If the area of \(\triangle \mathrm{OPQ}\) is ...

A line drawn from the point \(\mathrm{A}(1,3,2)\) parallel to the line \(\frac{x}{2}=\frac{y}{4}=\frac{z}{1}\), intersects the plane \(3 x+y+2 z=5\) in point \(\mathrm{B}\), then co-ordinates of point \(\mathrm{B}\) are

A line \(\mathrm{L}_1\) passes through the point, whose p. v. (position vector) \(3 \hat{i}\), is parallel to the vector \(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\). Another line \(\mathrm{L}_2\) passes through the point having ...

The equation of the line passing through the point \((-1,3,-2)\) and perpendicular to each of the lines \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) and \(\frac{x+2}{-3}=\frac{y-1}{2}=\frac{z+1}{5}\) is

If \(A(1,4,2)\) and \(C(5,-7,1)\) are two vertices of triangle \(A B C\) and \(G\left(\frac{4}{3}, 0, \frac{-2}{3}\right)\) is centroid of the triangle \(A B C\), then the mid point of side \(B C\) is

The distance of the point \((-1,-5,-10)\) from the point of intersection of the line \(\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}\) and the plane \(x-y+z=5\) is

The general solution of the equation \(3 \sec ^2 \theta=2 \operatorname{cosec} \theta\) is

The value of $$\begin{aligned} \cos \left(18^{\circ}-\mathrm{A}\right) \cdot \cos ( & \left.18^{\circ}+\mathrm{A}\right) \\ & -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left(72^{\circ}+\mathrm{A}\right) \text { is }\end{aligned}$$

The scalar product of vectors \(\overline{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and a unit vector along the sum of vectors \(\bar{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}\) and $$\bar{c}=\lambda \hat{i}+2 \hat{j}-3 \hat...

If \(\hat{\mathrm{a}}\) and \(\hat{\mathrm{b}}\) are unit vectors and \(\overline{\mathrm{c}}=\hat{\mathrm{b}}-(\hat{\mathrm{a}} \times \overline{\mathrm{c}})\), then minimum value of \([\hat{a} \hat{b} \bar{c}]\) is

If \(\bar{a}=2 \hat{i}+3 \hat{j}-4 \hat{k}\) and \(\bar{b}=\hat{i}-\hat{j}-\hat{k}\), then the projection of \(\bar{b}\) in the direction of \(\bar{a}\) is

A vector \(\bar{a}\) has components 1 and \(2 p\) with respect to a rectangular Cartesian system. This system is rotated through a certain angle about origin in the counter clock wise sense. If, with respect to the new system, \(\bar{a}\) h...

If \(\theta\) is angle between the vectors \(\bar{a}\) and \(\bar{b}\) where \(|\bar{a}|=4,|\bar{b}|=3\) and \(\theta \in\left(\frac{\pi}{4}, \frac{\pi}{3}\right)\), then $$|(\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})|^2+4(\bar{a} \cdot \bar{...