MHT CET 2025 20th April Evening Shift

If the curve $y=a x^2-6 x+b$ passes through $(0,4)$ and has its tangent parallel to the X-axis at $x=\frac{3}{2}$, then the values of $a$ and $b$ respectively are

If $2 \mathrm{f}(x)+3 \mathrm{f}\left(\frac{1}{x}\right)=x^2+1, x \neq 0$ and $y=5 x^2 \mathrm{f}(x)$, then $y$ is strictly increasing in

A manufacturer produces $x$ items per week at a total cost of ₹ $\left(x^2+78 x+2500\right)$. The price per unit is given by $8 x=600-\mathrm{p}$ where ' p ' is the price of each unit. Then the maximum profit obtained is

The number of integral values of $k$ for which $x^2+y^2+\mathrm{k} x+(1-\mathrm{k}) y+5=0$ represents a circle whose radius cannot exceeds 5 , are

The value of $\frac{(\cos \theta+i \sin \theta)^4}{(\sin \theta+i \cos \theta)^5}=$ where $\mathrm{i}=\sqrt{-1}$

\(\int_0^{\frac{\pi}{2}} \frac{d x}{1+(\cot x)^{101}}=\)

\(\int_0^3 \frac{d x}{(x+2) \sqrt{x+1}}=\)

The solution of $\left(1+y^2\right)+\left(x-\mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0$ is

The particular solution of the differential equation $\cos \left(\frac{d y}{d x}\right)=7, y=1$ at $x=0$ is

If the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x}{y}=\frac{\mathrm{a}}{y}$ where a is constant, represents a family of circles then the radius of the circle is $\qquad$

The rate of reduction of a persons assets is proportional to the square root of the existing assets. The assets reduced from 25 lakhs to 6.25 lakhs in 2 years. This rate of reduction of his assets will make him bankrupt in

If $\mathrm{a}\left(4+x^2\right)=x$ and $y-x^3=\mathrm{a}^2$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=1$ is $\qquad$

If $x^y+y^x=\mathrm{a}^{\mathrm{b}}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=1, y=2$ is

If $u=\frac{\tan ^{-1} x}{\tan ^{-1} x+1}$ and $v=\tan ^{-1}\left(\tan ^{-1} x\right)$ then $\frac{d u}{d v}=$

AOB is the positive quadrant of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ in which $\mathrm{OA}=5, \mathrm{OB}=3$. The area between the arc AB and the chord AB of the ellipse in sq. units is

The equations of two ellipses are $\frac{x^2}{4}+\frac{y^2}{2}=1$ and $\frac{x^2}{36}+\frac{y^2}{\mathrm{~b}^2}=1$. If the product of their eccentricities is $\frac{\sqrt{2}}{3}$, then the product of the length of the major axis and minor a...

\(\int \log (2+x)^{2+x} d x=\)

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

\(\int \frac{\mathrm{e}^{\tan ^{-1} 2 x}}{1+4 x^2}=\)

If $2 \tan ^{-1}(\cos x)=\tan ^{-1}(2 \operatorname{cosec} x)$, then the value of $x$ is

$\mathop {\lim }\limits_{x \to 0} \frac{\left(7^x-1\right)^4}{\tan \left(\frac{x}{\mathrm{k}}\right) \cdot \log \left(1+\frac{x^2}{3}\right) \cdot \sin 4 x}=3(\log 7)^3$, then $\mathrm{k}=$

$$ f(x)= \begin{cases}{\left[x^2\right]-\left[-x^2\right],} & x \neq 3 \\ k & , x=3\end{cases} $$
is continuous at $x=3$, then $\mathrm{k}=$ where $[\cdot]$ is greatest integer function

A scholarship amount is given by $\mathrm{z}=550 x+300 y$ and is to be distributed among $x$ boys and $y$ girls. From the graph given below the maximum amount of scholarship is __________

The negation of $(p \wedge \sim q) \rightarrow(p \vee \sim q)$ is

The equivalent statement of "If three vertices of a triangle are represented by cube roots of unity, then the triangle is an equilateral triangle" is

If $\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 4 \\ 1 & 3 & 4\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}12 \\ 15 \\ 13\end{array}\right]$, then the value of $x^2+y^2+z^2=$

4 red balls and 5 green balls are selected from $n$ balls. If the sum of both the selections is greater than ${ }^{n+1} C_4$ then the value of $n$ is equal to

A fair coin is tossed 99 times. If X is the number of times head occur then $\mathrm{P}[\mathrm{X}=\mathrm{r}]$ is maximum when $\mathrm{r}=$

A random variable $X$ takes the values $0,1,2,3$, $\qquad$ with probability
$\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{5}\right)^x$, where k is a constant.
Then $\mathrm{P}(\mathrm{X}=0)$ is

For $\mathrm{k}=1,2,3$ the box $\mathrm{B}_{\mathrm{k}}$ contains k red balls and $(k+1)$ white balls. Let $P\left(B_1\right)=\frac{1}{2}, P\left(B_2\right)=\frac{1}{3}$ and $\mathrm{P}\left(\mathrm{B}_3\right)=\frac{1}{6} . \mathrm{A}$ box...

Two numbers are selected at random, without replacement from the first 6 positive integers. Let $X$ denote the larger of the two numbers. Then $\mathrm{E}(\mathrm{X})=$

In a triangle ABC with usual notations if $\mathrm{a}=13$, $b=14, c=15$ Then $\sin A=$

In a triangle $A B C$, with usual notations, $3 \mathrm{~b}=\mathrm{a}+\mathrm{c}$, then $\cot \frac{\mathrm{A}}{2} \cdot \cot \frac{\mathrm{C}}{2}=$

In a triangle ABC , the sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are such that they are the roots of the equation $x^3-11 x^2+38 x-40=0$ Then
\(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\)

In a triangle ABC with usual notations if $|\overline{\mathrm{BC}}|=8,|\overline{\mathrm{CA}}|=7,|\overline{\mathrm{AB}}|=10$ then the projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{AC}}$ is

If $\mathrm{f}(x)=2 x^3+\mathrm{m} x^2-13 x+\mathrm{n}$ and 2,3 are the roots of the equation $\mathrm{f}(x)=0$ then the value of $4 m+5 n$ is

If the equation $\mathrm{k} x y+10 x+8 y+16=0$ represents a pair of lines, then

If the equation of the median through vertex $\mathrm{A}(3, \mathrm{k})$ of $\triangle \mathrm{ABC}$ with vertices $\mathrm{B}(2,1)$ and $\mathrm{C}(-4,5)$ is $x+4 y=\mathrm{p}$, then $\mathrm{k}=$ where p and k are constants

The co-ordinates of the point where the line joining the points $(2,-3,1)$ and $(3,-4,-5)$ and intersects the plane $2 x+y+z=7$ are

If the line $\frac{x+1}{3}=\frac{y-k}{7}=\frac{z-4}{8}$ lies in the plane $2 x+\mathrm{p} y+7 z-41=0$ which is perpendicular to the plane $x+4 y-2 z+13=0$ then $\mathrm{k}=$

The equation of the plane passing through the point of intersection of the planes $2 x-y+z-3=0$ and $4 x-3 y+5 z+9=0$ and parallel to the line $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z-3}{5}$ is $\alpha x+\beta y+\gamma z+d=0$ Then $\alpha+\beta...

The distance of the point $(2,4,0)$ from the point of intersection of the lines $\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}$ and $\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}$ is

The possible values of $\theta \in(0, \pi)$ such that $\sin \theta+\sin (4 \theta)+\sin (7 \theta)=0$ are

If $3 \sin 2 \theta=2 \sin 3 \theta$ and $0<\theta<\pi$, then the value of $\sin \theta$ is equal to

The approximate value of $\cos \left(59^{\circ} 30^{\prime}\right)$ is (given $1^{\circ}=0.0175^{\mathrm{c}}, \sin 60^{\circ}=0.8660$ )

If the projection of $\bar{a}$ on $\bar{b}+\bar{c}$ is twice the projection of $\bar{b}+\bar{c}$ on $\bar{a}$ also if $|\bar{b}|=2 \sqrt{2},|\bar{c}|=4$ and the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $\frac{\pi...

If $\bar{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\bar{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$ then the value of $(2 \overline{\mathrm{a}}-\overline{\mathrm{b}}) \cdot[(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \time...

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors and $\theta$ is the angle between them, then $\tan \frac{\theta}{2}=$

If the points $\mathrm{A}(1,1,2), \mathrm{B}(2,1, \mathrm{p}), \mathrm{C}(1,0,3)$ and $D(2,2,0)$ are coplanar then the value of $p$ is

If $\bar{a}=\hat{i}+\hat{j}, \bar{b}=2 \hat{i}-\hat{k}$ then the point of intersection of the lines $\overline{\mathrm{r}} \times \overline{\mathrm{a}}=\overline{\mathrm{b}} \times \overline{\mathrm{a}}$ and $\overline{\mathrm{r}} \times \o...