MHT CET 2025 23rd April Morning Shift

The point on the curve $4 y^2-4 y+2 x-1=0$ at which the tangent becomes parallel to Y -axis is

The equation of motion of the particle is $\mathrm{s}=\mathrm{at}^2+\mathrm{bt}+\mathrm{c}$. If the displacement after 1 second is 20 m , velocity after 2 seconds is $30 \mathrm{~m} /$ seconds and the acceleration is $10 \mathrm{~m} /$ seco...

A particle moves along a curve $y=\frac{2 x^3-1}{3}$. The points on the curve at which the $y$ co-ordinate is changing 18 times the $x$ co-ordinate are

The maximum value of $x^{2 / 3}+(x-2)^{2 / 3}$ is

The area bounded by the curve $x^2=8 y$ and the straight line $x-8 y+2=0$ is

Let the circle with centre at origin pass through the vertices of an equilateral triangle ABC . If $A \equiv(2,4)$, then the length of the median through A is

If $x=-2+\sqrt{-3}$, then the value of $2 x^4+5 x^3+7 x^2-x+38$ is equal to

The value of $\int_0^1 \tan ^{-1}\left(1-x+x^2\right) \mathrm{d} x$ is

\(\int_3^5 \frac{\sqrt{x} \mathrm{~d} x}{\sqrt{8-x}+\sqrt{x}}=\)

The order and degree of the differential equation $\sqrt{\frac{\mathrm{d} y}{\mathrm{~d} x}}-4 \frac{\mathrm{~d} y}{\mathrm{~d} x}-7 x=0$ is respectively

The rate at which a substance cools in moving air, is proportional to the difference between the temperature of the substance and that of air. The temperature of air is 290 K and the substance cools from 370 K to 330 K in 10 minutes. Then t...

If $x \frac{\mathrm{~d} y}{\mathrm{~d} x}=y(\log y-\log x+1)$, then the solution of the equation is

The differential equation of all circles touching the Y -axis at the origin and centre on the X -axis is

If $x=a \sin 2 t(1+\cos 2 t), y=b \cos 2 t(1-\cos 2 t)$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

If $y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\ldots \ldots \ldots \ldots \ldots \ldots . \infty}}}}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}=$

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

If $\mathrm{f}(x)=3 x+10, \mathrm{~g}(x)=x^2-1$, then $(\mathrm{fog})^{-1}(x)=$

If the tangent at the point $(2 \sec \theta, 3 \tan \theta)$ to the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$ is parallel to $3 x-y+4=0$, then the value of $\theta$ is

\(\int \frac{\mathrm{d} x}{\mathrm{e}^x-1}=\)

\(\int \frac{\mathrm{d} x}{\sqrt{x}+x}=\)

\(\int\left(\frac{x-3}{x^2+9}\right)^2 d x=\)

The value of $\tan ^2\left(\sec ^{-1} 4\right)+\cot ^2\left(\operatorname{cosec}^{-1} 3\right)$ is

\(\mathop {\lim }\limits_{x \to 2} \frac{x+3 x^2+5 x^3+7 x^4-166}{x-2}=\)

If $\mathrm{f}(x)=\frac{10^x+7^x-14^x-5^x}{1-\cos x}, x \neq 0$ is continuous at $x=0$, then the value of $\mathrm{f}(0)$ is

The graph with correct feasible region of L.P.P. for the constraints $2 x+y \leqslant 10, y \leqslant x, y \leqslant 2, x, y \geqslant 0$ is …

Which of the following is the negation of the statement " For all M>0, there exist $x \in \mathrm{~s}$ such that $x \geqslant \mathrm{M}^{\prime \prime}$

The contrapositive of the statement $\sim p \vee(q \wedge \sim r)$ is

If A and B are non-singular matrices of order 2 such that $\quad(A B)^{-1}=\frac{1}{6}\left[\begin{array}{cc}-1 & -3 \\ 2 & 3\end{array}\right] \quad$ and $A^{-1}=\frac{1}{3}\left[\begin{array}{cc}4 & 3 \\ -1 & 0\end{array}\right]$ then $B^...

A family consisting of a mother, father and their 8 children ( 4 boys and 4 girls) are to be seated at a round table in a party. How many ways can this be done if the mother and father sit together and the males and females alternate?

A fair coin is tossed 100 times. The chance of getting a head even number of times is

In a game a man wins $Rs \,\, 40$ if he gets 5 or 6 on a throw of a fair die and loses ₹ 20 for getting any other number on the die. If he decides to throw the die either till he gets a five or six or to a maximum of three throws, then his ...

Bag I contains 3 red and 2 green balls and Bag II contains 5 red and 3 green balls. A ball is drawn from one of the bag at random and it is found to be green. Then the probability that it is drawn from Bag I is

With usual notations, in $\triangle \mathrm{ABC}$, the lengths of two sides are 10 cm and 9 cm respectively. If angles $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are in A.P. then perimeter of $\triangle \mathrm{ABC}$ is

In a triangle ABC with usual notations, $\cot \frac{\mathrm{A}}{2}+\cot \frac{\mathrm{B}}{2}+\cot \frac{\mathrm{C}}{2}=$

In a triangle ABC , with usual notations, if $\mathrm{a}=5, \mathrm{~b}=4, \cos (\mathrm{~A}-\mathrm{B})=\frac{31}{32}$, then $\mathrm{c}=$

In $\triangle \mathrm{ABC}$, with usual notations, if $\cos \frac{B}{2}=\sqrt{\frac{c+a}{2 a}}$, then $a^2=$

Let mean and standard deviation of probability distribution
$$ \begin{array}{|c|c|c|c|c|} \hline \mathrm{X}=x & -3 & 0 & 1 & \alpha \\ \hline \mathrm{P}(\mathrm{X}=x) & \frac{1}{4} & \mathrm{~K} & \frac{1}{4} & \frac{1}{3} \\ \hline \end{ar...

The joint equation of two lines passing through $(-2,3)$ and parallel to the bisectors of the angle between the co-ordinate axes is

The distance of the point $(1,2)$ from the line $x+y=0$ measured parallel to the line $3 x-y=2$ is

The equation of the line passing through the point of intersection of $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ and also through the point ( $2,1,-2$ ) is

The equation of the plane containing the line $\frac{x+1}{2}=\frac{y+2}{1}=\frac{z-2}{3}$ and the point $(1,-1,3)$ is

The distance of the point $\mathrm{P}(3,4,4)$ from the point of intersection of the line joining the points $\mathrm{Q}(3,-4,-5), \mathrm{R}(2,-3,1)$ and the plane $2 x+y+z=7$ is

The equation of the plane containing the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and perpendicular to the plane containing the lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{1}$ and $\frac{x}{3}=\frac{y}{2}=\frac{z}{1}$ is

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}$ intersect, then the value of k is

If $\sin \mathrm{A}+\sin \mathrm{B}=x$ and $\cos \mathrm{A}+\cos \mathrm{B}=y$, then $\sin (A+B)=$

Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}$ and $\bar{c}=\hat{j}-\hat{k}$ be three vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=1$. If...

Let $\bar{a}=2 \hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-\hat{k}$ and vector $\bar{c}$ be coplanar. If $\bar{c}$ is perpendicular to $\bar{a}$, then $\bar{c}$ is

The area of the rectangle having vertices $\mathrm{P}, \quad \mathrm{Q}, \quad \mathrm{R}, \quad \mathrm{S}$ with position vectors $-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \ha...

If $\bar{a}, \bar{b}, \bar{c}$ are three vectors such that $|\overrightarrow{\mathrm{a}}|=\sqrt{31}, 4|\overrightarrow{\mathrm{~b}}|=|\overrightarrow{\mathrm{c}}|=2$ and $2(\overline{\mathrm{a}} \times \overline{\mathrm{b}})=3(\overline{\ma...

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}, \overline{\mathrm{d}}$ are vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overlin...