MHT CET 2025 26th April Morning Shift

A manufacturer sells $x$ items at a price of rupees $\left(6-\frac{x}{40}\right)$ each. The cost price of $x$ items is ₹ $\left(\frac{x}{5}+193\right)$. The maximum profit in ₹ __________ is

The equation of the tangent to the curve $\left(1+x^2\right) y=2-x$, where it crosses the X -axis, is

If $y=\alpha \log x+\beta x^3-x$ has extreme values at $x=-1$ and $x=1$, then $\alpha$ and $\beta$ are respectively

The area of smaller part between the circle $x^2+y^2=4$ and the line $x=1$ is _________ sq.units.

The least distance of the point $\mathrm{A}(10,7)$ from the circle $x^2+y^2-4 x-2 y-20=0$ is length of seg AM . If $\mathrm{MM}^{\prime}$ is the diameter of the circle, then the lengths of AM and $\mathrm{AM}^{\prime}$ are respectively ____...

The modulus of the square root of the complex number $6+8 \mathrm{i}$ (where $\mathrm{i}=\sqrt{-1}$ ) is

$\int_{\frac{1}{2}}^2 \frac{1}{x} \operatorname{cosec}^{101}\left(x-\frac{1}{x}\right) \mathrm{d} x=$

\(\int_0^1 \tan ^{-1} x d x=\)

The equation of a curve passing through $(1,0)$ and having slope of tangent at any point $(x, y)$ of the curve as $\frac{y-1}{x^2+x}$ is

The general solution of the differential equation $\frac{d y}{d x}=\cot x \cdot \cot y$ is

The differential equation which represents the family of curves $y=c_1 e^{c_2 x}$, where $c_1, c_2$ are arbitrary constants is

If $\mathrm{f}(x)=\frac{\sin ^2 x}{1+\cot x}+\frac{\cos ^2 x}{1+\tan x}$, then the value of $\mathrm{f}^{\prime}\left(\frac{\pi}{6}\right)$ is equal to

If $\mathrm{f}(x)=\sqrt{1+\cos ^2\left(x^2\right)}$, then $\mathrm{f}^{\prime}\left(\frac{\sqrt{\pi}}{2}\right)$ is

If $[x]^2-5[x]+6=0$, where [.] denotes the greatest integer function, then

If $\int \frac{\left(x^4+1\right)}{x\left(x^2+1\right)^2} d x=A \log |x|+\frac{B}{1+x^2}+c$, then $\mathrm{A}-\mathrm{B}$ is (where c is the constant of integration)

\(\int \frac{\sqrt{\tan x}}{\sin x \cdot \cos x} d x=\)

If $\int \frac{\mathrm{d} x}{x^4+5 x^2+4}=\mathrm{A} \tan ^{-1} x+\mathrm{B} \tan ^{-1} \frac{x}{2}+\mathrm{c}$ where c is a constant of integration, then

If $y=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), 0 \leqslant x<\frac{\pi}{2}$, then $y^{\prime}\left(\frac{\pi}{6}\right)=$

If $\sin ^{-1}(4 x)+\sin ^{-1}(4 \sqrt{3} x)=-\frac{\pi}{2}$, then the value of $x$ is

The number of positive integral solutions of $\tan ^{-1} x+\cos ^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)$ are

\(\mathop {\lim }\limits_{x \to \infty } \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}}=\)

If the function $f(x)=\left\{\begin{array}{cl}\frac{\cos a x-\cos b x}{\cos c x-\cos b x} & , \text { if } x \neq 0 \\ -1 & , \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $\mathrm{a}^2, \mathrm{~b}^2, \mathrm{c}^2$ are i...

The solution for minimizing the function $\mathrm{z}=x+y$ under an L.P.P. with constraints $x+y \geq 2, x+2 y \leq 8, y \leq 3, x, y \geq 0$ is

The money invested in a company is compounded continuously. If $Rs\, 400$ invested today becomes ₹800 in 6 years, then at the end of 30 years, it will become (in ₹)

If the truth value of the expression $[(p \vee q) \wedge(q \rightarrow r) \wedge(\sim r)] \rightarrow(p \wedge q)$ is False, then truth values of $p, q, r$ are respectively.

Consider statements $\mathrm{p}: \mathrm{S}_1$ is closed; $\mathrm{q}: \mathrm{S}_2$ is closed; $\mathrm{r}: \mathrm{S}_3$ is closed. The simplified equivalent circuit diagram and its logical statement for the switching circuit is respectiv...

Matrix A is non-singular matrix and $(A-3 I)(A-5 I)=0$, then $\frac{15}{8} A^{-1}=\ldots \ldots$

The equation of the directrix of the parabola $y^2+4 y+4 x+2=0$ is

The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4 x$, is

The number of ways in which a team of 11 players can be formed out of 25 players, if 6 out of them are always to be included and 5 of them are always to be excluded, is

The following is p.d.f. of continuous random variable X
$$ \mathrm{f}(x)= \begin{cases}\frac{x}{8} & , \text { if } 0 < x < 4 \\ 0 & , \text { otherwise }\end{cases} $$
Then $F(0.5), F(1.7)$ and $F(5)$ is respectively

The probability that a person is not a sportsperson is $\frac{1}{6}$. Then the probability that out of the 6 members of the family, 5 are sportspersons is

The cumulative distribution function of a discrete random variable X is
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X}=x & -4 & -2 & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \mathrm{~F}(\mathrm{X}=x) & 0.1 & 0.3 & 0.5 & 0.65 & 0.75 & 0.85 ...

A box contains 8 red and $x$ number of green balls. 3 balls are drawn at random, if the probability that 3 balls being red is $\frac{7}{15}$, then number of green balls is…

In a triangle ABC , with usual notations, $(\mathrm{a}+\mathrm{b}+\mathrm{c})(\mathrm{a}+\mathrm{b}-\mathrm{c})=3 \mathrm{ab}$, then $\angle \mathrm{C}=$

In a triangle $A B C$, with usual notations, $\cot \left(\frac{A+B}{2}\right) \cdot \tan \left(\frac{A-B}{2}\right)=$

In a triangle $A B C$, with usual notations, if $a=5$, $\mathrm{b}=7 \sin \mathrm{~A}=\frac{3}{4}$, then total number of triangles possible are

The circumcenter of the triangle formed by lines $x y+2 x+2 y+4=0$ and $x+y+2=0$ is

The line MN whose equation is $x-y-2=0$ cuts the X -axis at M and co-ordinates of N are $(4,2)$. The line MN is rotated about M through $45^{\circ}$ in anticlockwise direction. The equation of the line MN in the new position is

The angle between lines whose direction cosines satisfy the equation $l+m+n=0$ and $l^2-\mathrm{m}^2-\mathrm{n}^2=0$, is

If the plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{6}=1$ cuts the co-ordinate axes at points $A, B, C$ respectively, then area of the triangle ABC is

If the foot of the perpendicular drawn from the origin to a plane is $\mathrm{P}(-1,-1,2)$, then equation of the plane is

A triangle ABC is formed by $\mathrm{A}(1,-1,0)$, $B(3,5,3), C(-11,-5,6)$. The equation of internal angle bisector of angle $A$ is

The mirror image of the point $\mathrm{P}(-1,2,-4)$ in the plane $x-y-2 z+1=0$ is

If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$, then $\sin \left(\frac{\pi}{4}+\theta\right)=$

If $\bar{a}, \bar{b}, \bar{c}$ are three coplanar vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2, \overline{\mathrm{~b}} \cdot \overline{\mathrm{c}}=8$, the angle between $\overline{\mathrm{b}}$ and $\overline{\math...

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be vectors of magnitude 2,3 and 4 respectively. If $\bar{a}$ is perpendicular to $(\overline{\mathrm{b}}+\overline{\mathrm{c}}), \overline{\mathrm{b}}$ is perpen...

Let $\quad \bar{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=3 \hat{i}-\hat{j}+\beta \hat{k} \quad$ and $\bar{c}=\hat{i}+2 \hat{j}-2 \hat{k}$ where $\alpha, \beta \in \mathbb{R}$, be three vectors. If the projection of $\bar{a}$ on $\bar{c}...

The volume of tetrahedron with co-terminus edges $\bar{a}, \bar{b}, \bar{c}$ is $\frac{64}{3}$ cubic units, then volume of parallelopiped considering co-terminus edges given by the vectors $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$...

If $\quad \overline{\mathrm{a}}=\lambda x \hat{\mathrm{i}}+y \hat{\mathrm{j}}+4 z \hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=y \hat{\mathrm{i}}+x \hat{\mathrm{j}}+3 y \hat{\mathrm{k}}$, $\overline{\mathrm{c}}=-z \hat{\mathrm{i}}-2 z \hat...