MHT CET 2025 19th April Morning Shift

In the mean value theorem, $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$, if $\mathrm{a}=0, \mathrm{~b}=\frac{1}{2}$ and $\mathrm{f}(x)=x(x-1)(x-2)$, then the value of $c$ is

A population $p(t)$ of 1000 bacteria introduced into a nutrient medium grows according to the relation $\mathrm{p}(\mathrm{t})=1000+\frac{1000 \mathrm{t}}{100+\mathrm{t}^2}$. The maximum size of this bacterial population is

By dropping a stone in a quiet lake, a wave in the form of circle is generated. The radius of the circular wave increases at the rate of $2.1 \mathrm{~cm} / \mathrm{sec}$. Then the rate of increase of the enclosed circular region, when the ...

The angle between the curves $x y=6$ and $x^2 y=12$ is

The ratio of the areas bounded by the curves $y=\cos x$ and $y=\cos 2 x$ between $x=0, x=\frac{\pi}{3}$ and X -axis is

The number of common tangents that can be drawn to the circles $x^2+y^2-6 x=0$ and $x^2+y^2+6 x+2 y+1=0$ is __________

$$\begin{aligned} & \mathrm{f}(x)=(\cos x+\mathrm{i} \sin x) \cdot(\cos 3 x+\mathrm{i} \sin 3 x) \cdots {[\cos (2 \mathrm{n}-1) x+\mathrm{i} \sin (2 \mathrm{n}-1) x] \mathrm{n} \in \mathbb{N}} \end{aligned}$$
Then $\mathrm{f}^{\prime \prim...

The modulus of the square root of the conjugate of $-7+24 \sqrt{-1}$ is __________

The value of $\int_1^4 \log [x] \mathrm{d} x$, where $[x]$ is the greatest integer function less than or equal to $x$ is equal to

\(\int_1^e \frac{\mathrm{e}^x}{x}(1+x \log x) \mathrm{d} x=\)

The solution of the differential equation $x \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}=1$ at $x=y=1$ with $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$ at $x=1$, is

In a bank, the principal increases continuously at a rate of $x \%$ per year. Then the rate $x$, if ₹$100$ double itself in 10 years, is ( $\log 2=0.6931$)

If $y=y(x)$ satisfies $\left(\frac{2+\sin x}{1+y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=-\cos x$ such that $y(0)=2$, then $y\left(\frac{\pi}{2}\right)$ is equal to

The order and degree of differential equation of all tangent lines to the parabola $x^2=4 y$ is respectively.

The derivative of $\tan ^{-1}\left(\sqrt{1+x^2}-1\right)$ is

For $\mathrm{n} \in \mathbb{N}$ if $y=\mathrm{a} x^{\mathrm{n}+1}+\mathrm{b} x^{-\mathrm{n}}$, then $x^2 \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}=$

An ellipse has OB as semi-minor axis, S and $\mathrm{S}^{\prime}$ are foci and angle SBS' is a right angle. Then the eccentricity of the ellipse is

\(\int \frac{\mathrm{d} x}{2 \mathrm{e}^{2 x}+3 \mathrm{e}^x+1}=\)

\(\int \frac{\mathrm{e}^{2030 \log x}-\mathrm{e}^{2029 \log x}}{\mathrm{e}^{2028 \log x}-\mathrm{e}^{2027 \log x}} \mathrm{~d} x=\ldots\)

 \(\int \frac{\sin 2 x}{(a+b \cos x)^2} d x=\)

If $4 \sin ^{-1} x+\cos ^{-1} x=\pi$ then $x=$

The sum to infinite terms of the series $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{2}{9}\right)+\ldots \ldots . .+\tan ^{-1}\left(\frac{2^{n-1}}{1+2^{2 n-1}}\right)+\ldots \ldots$. is

If $\mathrm{f}(x)$ is continuous at point $x=0$ where

$$ f(x)=\left\{\begin{array}{cc} \frac{3 \sin x+5 \tan x}{\mathrm{a}^x-1} & , x<0 \\ \frac{2}{\log 2} & , x=0 \\ \frac{8 x+2 x \cos x}{\mathrm{~b}^x-1} & , x>0 \end{array}\right. $$

th...

$\lim _\limits{x \rightarrow 3} \frac{(84-x)^{\frac{1}{4}}-3}{x-3}$ is

The feasible region represented by the given constraints $2 x+3 y \geq 12,-x+y \leq 3, x \leq 4, y \geq 3$ is denoted by

If $x+\log _{15}\left(5+3^x\right)=x \log _{15} 5+\log _{15} 24, \quad$ then $x=$ _________

Consider the three statements $\mathrm{p}: \forall \mathrm{n} \in \mathbb{N}, 10 \mathrm{n}-3$ is a prime number, when n is not divisible by 3.
$\mathrm{q}: \frac{2}{\sqrt{3}}, \frac{-2}{\sqrt{3}}, \frac{-1}{\sqrt{3}}$ are the direction cos...

The statement pattern $[(p \rightarrow q) \wedge \sim q] \rightarrow r$ is a tautology when $r$ is equivalent to

If $A=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$,
where $A_{21}, A_{22}, A_{23}$ are cofactors of $a_{21}, a_{22}, a_{23}$ respectively, then the value of $\mathrm...

The number of ways, in which 6 boys and 5 girls can sit at a round table, if no two girls are to sit together, is

If a random variable $X$ has the p.d.f. $f(x)=\left\{\begin{array}{cc}\frac{\mathrm{k}}{x^2+1} & , \text { if } 0< x< \infty \\ 0 & , \text { otherwise }\end{array}\right.$ then c.d.f. of X is

If a random variable $X$ follows the Binomial distribution $\mathrm{B}(33, \mathrm{p})$ such that $3 \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)$, then the variance of X is

The probability distribution of a discrete random variable X is

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A box contains 9 tickets numbered 1 to 9 both inclusive. If 3 tickets are drawn from the box one at a time, then the probability that they are alternatively either {odd, even, odd} or {even, odd, even} is

The smallest angle of the triangle whose sides are $6+\sqrt{12}, \sqrt{48}, \sqrt{24}$ is

In a triangle $A B C$, with usual notations, if $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$ Then $\cos \mathrm{A}: \cos \mathrm{B}: \cos \mathrm{C}$ is

The ratios of sides in a triangle ABC are $5: 12: 13$ and its area is 270 . Then sides of the triangle are

If $m_1$ and $m_2$ are the slopes of the lines represented by $a x^2+2 h x y+b y^2=0$ satisfying the condition $16 \mathrm{~h}^2=25 \mathrm{ab}$, then ............ .

The point of intersection of the diagonals of the rectangle whose sides are contained in the lines $x=8, x=10, y=11$ and $y=12$ is

The distance of the point $(-3,2,3)$ from the line passing through $(4,6,-2)$ and having direction ratios $-1,2,3$ is $\qquad$ units.

If the lines $\frac{3-x}{2}=\frac{5 y-2}{3 \lambda+1}=5-\mathrm{z}$ and $\frac{x+2}{-1}=\frac{1-3 y}{7}=\frac{4-z}{2 \mu}$ are at right angles, then $7 \lambda-10 \mu=$

If the sum of the squares of the distance of the point $\mathrm{P}(x, y, \mathrm{z})$ from the co-ordinate axes is 242 , then the distance of the point P from the origin is units.

If the directed line makes an angle $45^{\circ}$ and $60^{\circ}$ with the X and Y -axes respectively, then the obtuse angle $\theta$ made by the line with the Z -axis is

If the points $\mathrm{A}(2-x, 2,2), \mathrm{B}(2,2-y, 2)$, $\mathrm{C}(2,2,2-\mathrm{z})$ and $\mathrm{D}(1,1,1)$ are coplanar, then the locus of point $\mathrm{P}(x, y, \mathrm{z})$ is

If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2 x-y+\sqrt{\lambda} z+4=0$ is such that $\sin \theta=\frac{1}{3}$, then $\lambda+1=$

The Cartesian equation of plane through $\mathrm{A}(7,8,6)$ and parallel to the XY plane is

A plane passes through $(1,-2,1)$ and is perpendicular to the planes $2 x-2 y+z=0$ and $x-y+2 z=4$. The distance of the point $(1,2,2)$ from this plane is ________ units.

If $3 \sin \alpha=5 \sin \beta$, then $\tan \left(\frac{\alpha+\beta}{2}\right)+\tan \left(\frac{\alpha-\beta}{2}\right)=$

If $\left[\begin{array}{lll}2 \bar{p}-3 \bar{r} & \bar{q} & \bar{s}\end{array}\right]+\left[\begin{array}{lll}3 \bar{p}+2 \bar{q} & \bar{r} & \bar{s}\end{array}\right]=m\left[\begin{array}{lll}\bar{p} & \bar{r} & \bar{s}\end{array}\right] +...

The volume of the tetrahedron whose co-terminus edges are $\bar{a}, \bar{b}, \bar{c}$ is 12 cubic units. If the scalar projection of $\bar{a}$ on $\bar{b} \times \bar{c}$ is 4 , then $|\overline{\mathrm{b}} \times \overline{\mathrm{c}}|=$