MHT CET 2024 10th May Morning Shift

A point moves along the arc of parabola $y=2 x^2$. Its abscissa increases uniformly at the rate of 2 units $/ \mathrm{sec}$. At the instant, the point is passing through ( 1,2 ), its distance from origin is increasing at the rate of

The equation of the normal to the curve $y=x \log x$, which is parallel to the line $2 x-2 y+3=0$, is

Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm ? ( 1 decimeter $=10 \mathrm{~cm}$)

Area (in sq.units) lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and the lines $x=0$ and $x=2$ is

The abscissae of the two points A and B are the roots of the equation $x^2+2 a x-b^2=0$ and their ordinates are roots of the equation $y^2+2 p y-q^2=0$. Then the equation of the circle with AB as diameter is given by

If $\mathrm{w}=\frac{-1+i \sqrt{3}}{2}$, where $\mathrm{i}=\sqrt{-1}$, then the value of $\left(3+w+3 w^2\right)^4$ is

$\int_\limits0^{\frac{\pi}{2}}|\sin x-\cos x| d x$ has the value

The general solution of $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x+y+1}{x+y-1}$ is

A radio-active substance has a half-life of h days, then its initial decay rate is given by (where radio-active substance has initial mass $\mathrm{m}_0$)

If $\cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y \sin x=6 x, 0 < x < \frac{\pi}{2}$, then general solution of the differential equation is

If \(y = {{\sin x} \over {1 + {{\cos x} \over {1 + {{\sin x} \over {1 + {{\cos x} \over {...}}}}}}}}\), then $\frac{dy}{dx}$ is given by

If $y$ is a function of $x$ and $\log (x+y)=2 x y$, then the value of $y^{\prime}(0)$ is

If $y=a x^{n+1}+b x^{-n}$, then $x^2 \frac{d^2 y}{d x^2}=$

The curve $x^4-2 x y^2+y^2+3 x-3 y=0$ cuts the X -axis at $(0,0)$ at an angle of

The domain of definition of the function $y(x)$ is given by the equation $2^x+2^y=2$, is

If, $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{e}}|\mathrm{f}(\theta)|+\mathrm{c}$ (where c is a constant of integration), then the ordered pair $(\lambda,|f(\theta)|)$ is equal t...

If $\quad \int(2 x+4) \sqrt{x-1} \mathrm{~d} x=\mathrm{a}(x-1)^{\frac{5}{2}}+\mathrm{b}(x-1)^{\frac{3}{2}}+\mathrm{c}$, (where c is a constant of integration), then the value of $a+b$ is

\(\int \frac{1+\sin (\log x)}{1+\cos (\log x)} d x=\)

\(\int \frac{\sqrt{x}}{x+1} d x=\)

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\pi$ and $x^2+y^2+z^2+k x y z=1$, then k is

The value of $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is

The numerical value of $\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right)$

The approximate value of $x^3-2 x^2+3 x+2$ at $x=2.01$ is

The values of $a$ and $b$, so that the function
$$f(x)= \begin{cases}x+\mathrm{a} \sqrt{2} \sin x & , 0 \leq x \leq \frac{\pi}{4} \\ 2 x \cot x+b & , \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ \mathrm{a} \cos 2 x-\mathrm{b} \sin x & , \frac...

The function to be maximized is given by $Z=3 x+2 y$. The feasible region for this function is the shaded region given below, then the linear constraints for this region are given by

Negation of the statement "The payment will be made if and only if the work is finished in time." is

The inverse of $p \rightarrow(q \rightarrow r)$ is logically equivalent to

If $A+B=\left[\begin{array}{cc}1 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 1\end{array}\right]$ where $A$ is symmetric and $B$ is skew-symmetric matrix, then the matrix $\left(A^{-1} B+A B^{-1}\right)$ at $\theta=\frac{\pi}{6}$ is...

Words of length 10 are formed by using the letters A, B, C, D, E, F, G, H, I, J. Let $x$ be number of such words where no letter is repeated and $y$ be number of such words where exactly two letters are repeated twice and no other letter is...

A random variable X has the following probability distribution

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In a game, 3 coins are tossed. A person is paid ₹ 100$, if he gets all heads or all tails; and he is supposed to pay ₹ 40 , if he gets one head or two heads. The amount he can expect to win/lose on an average per game in (₹) is

In a Binomial distribution consisting of 5 independent trials, probabilities of exactly 1 and 2 successes are 0.4096 and 0.2048 respectively, then the probability, of getting exactly 4 successes, is

 There are three events $\mathrm{A}, \mathrm{B}, \mathrm{C}$, one of which must and only one can happen. The odds are 8:3 against $\mathrm{A}, 5: 2$ against B and the odds against C is $43: 17 \mathrm{k}$, then value of k is

If the angles $\mathrm{A}, \mathrm{B}$ and C of a triangle ABC are in the ratio $2: 3: 7$ respectively, then the sides a, b and c are respectively in the ratio

$\mathrm{S}_1=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}, \mathrm{S}_2=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^2$ and $\mathrm{S}_3=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^3$, then the value of $\lim _\limits{n...

Mean and variance of six observations are 6 and 12 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is

Let a line intersect the co-ordinate axes in points $A$ and $B$ such that the area of the triangle $O A B$ is 12 sq. units. If the line passes through the point $(2,3)$, then the equation of the line is

$\triangle \mathrm{OAB}$ is formed by the lines $x^2-4 x y+y^2=0$ and the line $A B$. The equation of line $A B$ is $2 x+3 y-1=0$. Then the equation of the median of the triangle drawn from the origin is

If the line, $\frac{x-3}{2}=\frac{y+2}{1}=\frac{z+4}{3}$ lies in the plane, $\ell x+m y-z=9$, then $\ell^2+m^2$ is equal to

If the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3 y-\alpha z+\beta=0$, then $(\alpha, \beta)=$

Equation of the plane, through the points $(-1,2,-2)$ and $(-1,3,2)$ and perpendicular to $y \mathrm{z}$ - plane, is

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

If for certain $x, 3 \cos x \neq 2 \sin x$, then the general solution of, $\sin ^2 x-\cos 2 x=2-\sin 2 x$, is

The value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ is

If $\quad \overline{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{k}}, \overline{\mathrm{b}}=x \hat{\mathrm{i}}+\hat{\mathrm{j}}+(1-x) \hat{\mathrm{k}} \quad$ and $\overline{\mathrm{c}}=y \hat{\mathrm{i}}+x \hat{\mathrm{j}}+(1+x-y) \hat{\mathrm...

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be three vectors having magnitudes 1,1 and 2 respectively. If $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}}=\overline{0}$, then the acute angle b...

Let $\bar{A}, \bar{B}, \bar{C}$ be vectors of lengths 3 units, 4 units, 5 units respectively. let $\bar{A}$ be perpendicular to $\overline{\mathrm{B}}+\overline{\mathrm{C}}, \overline{\mathrm{B}}$ be perpendicular to $\overline{\mathrm{C}}+...

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be three non-zero vectors such that no two of them are collinear and $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=\frac{1}{3}|\overline{\mathrm{~b}}||\overline{\math...

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})) \cdot(\overline{\mathrm{a}} \times \...

If $|\overline{\mathrm{a}}|=2,|\overline{\mathrm{~b}}|=3$ and $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ are mutually perpendicular vectors, then the area of the triangle whose vertices are $0, a+2 b, a-2 b$ is