MHT CET 2025 26th April Evening Shift

If $x$ and $y$ are sides of two squares such that $y=x-x^2$, then the rate of change of area of the second square with respect to that of the first square is

The equation of the tangent to the curve $y=\mathrm{be}^{-x / \mathrm{a}}$ at the point where it crosses the Y axis is

The minimum value of $a x+b y$ where $x y=c^2$ is

The function $\mathrm{f}(x)=[x(x-2)]^2$ is increasing in the set

If the area bounded by the curve $x^2=4 y, \mathrm{X}$-axis and the line $x=4$ is divided into equal areas by the line $x=\alpha$, then the value of $\alpha$ is …

Two tangents to the circle $x^2+y^2=4$ at the points A and B meet at $\mathrm{P}(-4,0)$. Then the area of quadrilateral PAOB, where ' $O$ ' is the origin is

Let z be the complex number such that $|z|+z=3+i$ where $i=\sqrt{-1}$, then $|z|=$

\(\int\limits_0^1 x\left|x-\frac{1}{2}\right| \mathrm{d} x=\)

\(\int_0^{\frac{\pi}{2}} \frac{300 \sin x+100 \cos x}{\sin x+\cos x} d x=\ldots\)

The population of towns A and B increase at the rate proportional to their population present at that time. At the end of the year 1984, the population of both the towns was 20,000 . At the end of the year 1989, the population of town A was...

The equation of the curve passing through the point $(0,2)$ given that the sum of the ordinate and abscissa of any point exceeds the slope of the tangent to the curve at that point by 5 is

The solution of the differential equation $(1+x) \frac{\mathrm{d} y}{\mathrm{~d} x}-x y=1-x$ is

The differential equation representing the family of parabolas having vertex at the origin and axis along the positive Y -axis is

If $u=\log (\sqrt{x-1}-\sqrt{x+1})$ and $v=\sqrt{x+1}+\sqrt{x-1}$ then $\frac{d u}{d v}=\ldots$.

If $\mathrm{f}(x)=3 x^2+2 x \mathrm{f}^{\prime}(1)+\mathrm{f}^{\prime \prime}(2)$, then $\mathrm{f}(x)=\ldots \ldots .$.

The tangent to the ellipse $9 x^2+16 y^2=288$ making equal intercepts on the co-ordinate axes intersects the X -axis and the Y -axis in the points $A$ and $B$ respectively. Then $A(\triangle O A B)=$ (where O is origin)

\(\int x^2 \cos x d x=\)

\(\int \frac{\mathrm{d} x}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=\mathrm{A} x^{\frac{1}{2}}+\mathrm{B} x^{\frac{1}{3}}+\mathrm{C} x^{\frac{1}{6}}+\mathrm{D} \log \left(x^{\frac{1}{6}}+1\right)+\mathrm{k}\)
(where k is the integration constant)...

\(\int \frac{\mathrm{d} x}{\sin ^2 x \cos ^2 x}=\)

If $\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x$, then the value of $\sin x$ is

$$ \begin{array}{r}If\,\,\,\, y=\tan ^{-1}\left(\frac{1}{1+x+x^2}\right)+\tan ^{-1}\left(\frac{1}{x^2+3 x+3}\right) +\tan ^{-1}\left(\frac{1}{x^2+5 x+7}\right) \end{array} $$
then the value of $y^{\prime}(0)$ is

Considering only the principal values of the inverse trigonometric functions, the value of $\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$ is

Let $f(x)= \begin{cases}\frac{x^4-5 x^2+4}{|(x-1)(x-2)|} & , x \neq 1,2 \\ 6 & , x=1 \\ 12 & , x=2\end{cases}$
Then $\mathrm{f}(x)$ is continuous on the set

\(\mathop {\lim }\limits_{x \to 0} \frac{\mathrm{e}^{x^2}-\cos 3 x}{\sin x \log (1+2 x)}=\)

A manufacturing company produces two items, A and B. Each toy should be processed by two machines, I and II. Machine I can be operated for maximum 10 hours 40 minutes. It takes 20 minutes for an item of A and 15 minutes for B. Machine II ca...

If $p^3=q^4=r^6=t^7=s^2$, then $\log _t(p q r s)=\ldots \ldots$

If $p, q, r, s$ are statements, where, $\mathrm{p}: \mathrm{A}^2-\mathrm{B}^2=(\mathrm{A}-\mathrm{B})(\mathrm{A}+\mathrm{B}) ; \mathrm{A}, \mathrm{B}$ are matrices, $A B \neq B A$
q: $5 \leq 5$
r: ${ }^8 \mathrm{C}_1+{ }^8 \mathrm{C}_2+{ }^...

Which of the following statement is a tautology?

If $A$ is a matrix of order 2 and $I$ is the identity matrix of order 2 such that $A^2-4 A+3 I=0$ then $(A+3 I)^{-1}=$

The number of ways in which 6 boys and 4 girls can be seated around a round table such that 2 special boys and a special girl never sit together is

The probability that a non leap year selected at random will contain 52 Saturdays or 53 Sundays is

A fair $n$ faced die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$, then $\mathrm{n}=($ where $\mathrm{n} \in \mathbb{N})$

A fair coin is tossed a fixed number of times. If the probability of getting 5 tails is same as the probability of getting 7 tails, then the probability of getting 3 tails is

In a triangle $A B C$ with usual notations if $\angle A=30^{\circ}$, then the value of $\left(1+\frac{a}{c}+\frac{b}{c}\right)\left(1+\frac{c}{b}-\frac{a}{b}\right)=$

In a triangle PQR with usual notations, $\angle \mathrm{R}=\frac{\pi}{2}$. If $\tan \frac{\mathrm{P}}{2}$ and $\tan \frac{\mathrm{Q}}{2}$ are the roots of the equation $a x^2+b x+c=0(a \neq 0)$, then

If the angles $\mathrm{A}, \mathrm{B}$ and C of a triangle are in A.P. and if $\mathrm{a}, \mathrm{b}$ and c denote the length of the sides opposite to $\mathrm{A}, \mathrm{B}$ and C respectively, then the value of $\frac{a}{b} \sin 2 B+\fr...

A random variable $X$ has the following probability distribution :
$$ \begin{array}{|l|c|c|c|c|} \hline \mathrm{X}=x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(\mathrm{X}=x) & 0.1 & 0.2 & 0.3 & 0.4 \\ \hline \end{array} $$
The mean and standard d...

The acute angle between the line $4 x-2 y+13=0$ and the line which makes equal intercepts with the co-ordinate axes is

If the pair of lines $3 x^2-5 x y+\mathrm{p} y^2=0$ and $6 x^2-x y-5 y^2=0$ have one line common, then $\mathrm{p}=$

The lines $\frac{6 x-6}{18}=\frac{y+1}{3}=\frac{z-1}{5} \quad$ and $\frac{3 x+6}{12}=\frac{y-1}{3}=\frac{z+1}{2}$ are $\ldots$

The equation of the plane passing through the point $(1,1,1)$ and through the line of intersection of $x+2 y-z+1=0$ and $3 x-y-4 z+3=0$ is

The direction cosines of a normal to the plane passing through $(4,2,3),(-1,4,2)$ and $(3,2,1)$ are …..

The distance of the point $\mathrm{A}(3,-4,5)$ from the plane $2 x+5 y-6 z=16$ measured along the line $\frac{x}{2}=\frac{y}{1}=\frac{z}{-2}$ is

The line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z}{1}$ intersects the XY plane and the YZ plane at points A and B respectively. The equation of line through the points A and B is

If $0 \leq x \leq \pi$ and $81^{\sin ^2 x}+81^{\cos ^2 x}=30$ Then $x$ takes the value

The values of $x$ for which the angle between the vectors $\overline{\mathrm{a}}=2 x^2 \hat{\mathrm{i}}+4 x \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$ is obtuse, a...

The vectors $\bar{a}, \bar{b}$ and $\bar{c}$ are such that $|\overline{\mathrm{a}}|=2,|\overline{\mathrm{~b}}|=4,|\overline{\mathrm{c}}|=4$. If the projection of $\overline{\mathrm{b}}$ on $\overline{\mathrm{a}}$ is equal to projection of $...

The unit vectors perpendicular to the plane determined by the points $\mathrm{A}(1,-1,2), \mathrm{B}(2,0,-1)$, $\mathrm{C}(0,2,1)$ is

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

In the above figure, P divides AC in the ratio $3: 4$ and Q divides BC in the ratio $4: 3$. Then M divides AQ in the ratio