MHT CET 2019 3rd May Morning Shift

- The edge of a cube is decreasing at the rate of $0.04 \mathrm{~cm} / \mathrm{sec}$. If the edge of the cube is 10 cms , then rate of decrease of surface area of the cube is...

If $r$ is the radius of spherical balloon at time $t$ and the surface area of balloon changes at a constant rate $K$, then......

The slope of normal to the curve $x=\sqrt{t}$ and $y=t-\frac{1}{\sqrt{t}}$ at $t=4$ is ..........

If $f(x)=x+\frac{1}{x}, x \neq 0$, then local maximum and minimum values of function $f$ are respectively.......

. Area of the region bounded by $y=\cos x, x=0$, $x=\pi$ and $X$-axis is ... sq. units.

If $\omega$ is a complex cube root of unity and $A=\left[\begin{array}{ccc}\omega & 0 & 0 \\ 0 & \omega^2 & 0 \\ 0 & 0 & 1\end{array}\right]$ then $A^{-1}=\ldots$

\(\int_0^{\frac{\pi}{2}} \sqrt{\cos \theta} \cdot \sin ^3 \theta d \theta=\) ............

\(\int_{\frac{\pi}{18}}^{\frac{4 \pi}{9}} \frac{2 \sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=\ldots \ldots\)

The order of the differential equation of all circles which lie in the first quadrant and touch both the axes is......

The solution of differential equation $\left(x^2+1\right) \frac{d y}{d x}+\left(y^2+1\right)=0$ is $\ldots$

If $y=\log \left[\frac{x+\sqrt{x^2+25}}{\sqrt{x^2+25}-x}\right]$ then $\frac{d y}{d x}=\ldots \ldots$

If $y=\tan ^{-1}\left(\frac{1-\cos 3 x}{\sin 3 x}\right)$, then $\frac{d y}{d x}=$ .......

The length of the latusrectum of an ellipse is $\frac{18}{5}$ and eccentricity is $\frac{4}{5}$, then equation of the ellipse is .....

The domain of the real valued function $f(x)=\sqrt{\frac{x-2}{3-x}}$ is......

Which of the following function has period 2?

The eccentricity of the hyperbola $25 x^2-9 y^2=225$ is .......

\(\int \log x \cdot[\log (e x)]^{-2} d x=\ldots\)

If $\int \frac{1}{1-\cot x} d x=A x+B \log |\sin x-\cos x|+c$ then $A+B=\ldots \ldots$

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

Derivative of $\sin ^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right)$ with respect to $\cos ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right)$ is

$\sin \left[3 \sin ^{-1}(0.4)\right]=\ldots \ldots$

If $f(x)$ is continuous at $x=3$, where
$$\begin{aligned} f(x) & =a x+1, & \text { for } x \leq 3 \\ & =b x+3 & , \text { for } x>3 \text { then } \end{aligned}$$

$$\begin{aligned} & \text { If } f(x)=\left[\tan \left(\frac{\pi}{4}+x\right)\right]^{\frac{1}{x}}, \quad x \neq 0 \\ & =k \text {, } \qquad x=0 \text { is continuous }\\ & x=0 \end{aligned}$$ Then $k=$

The maximum value of $Z=5 x+4 y$, Subject to $y \leq 2 x, x \leq 2 y, x+y \leq 3, x \geq 0, y \geq 0$ is ........

If $z=a x+b y ; a, b>0$ subject to $x \leq 2, y \leq 2, x+y \geq 3, x \geq 0, y \geq 0$ has minimum value at $(2,1)$ only, then......

Let $a: \sim(p \wedge \sim r) \vee(\sim q \vee s)$ and $b:(p \vee s) \leftrightarrow(q \wedge r)$. If the truth values of $p$ and $q$ are true and that of $r$ and $s$ are false, then the truth values of $a$ and $b$ are respectively......

5. "If two triangles are congruent, then their areas are equal" is the given statement then the contrapositive of, the inverse of the given statement is

Which of the following statement pattern is a tautology?

If $A$ and $B$ are square matrices of order 3 such that $|A|=2,|B|=4$, then $|A(\operatorname{adj} B)|=\ldots$

Let $X$ be the number of successes in ' $n$ ' independent Bernoulli trials with probability of success $p=\frac{3}{4}$. The least value of ' $n$ ' so that $P(X \geq 1) \geq 0.9375$ is ......

The probability that three cards drawn from a pack of 52 cards, all are red is

$$\begin{aligned} &\text { The pdf of a random variable } X \text { is }\\ &\begin{aligned} f(x) & =3\left(1-2 x^2\right), & & 0< x<1 \\ & =0 & & \text { otherwise } \end{aligned} \end{aligned}$$
The $P\left(\frac{1}{4}< x<\frac{1}{3}\right...

A player tosses 2 fair coins. He wins Rs. 5 if 2 heads appear, Rs. 2 If 1 head appear and Rs. 1 if no head appears, then variance of his winning amount is

In $\triangle A B C$, with the usual notations, if $\left(\tan \frac{A}{2}\right)\left(\tan \frac{B}{2}\right)=\frac{3}{4}$ then $a+b=\ldots \ldots$

In $\triangle A B C$, with the usual notations, if $\sin B \sin C=\frac{b c}{a^2}$, then the triangle is. ...........

If $A=\left\{x \in R / x^2+5|x|+6=0\right\}$ then $n(A)=\ldots \ldots$

For a sequence $\left(t_n\right)$, if $S_n=5\left(2^n-1\right)$ then $t_n=$ .........

If $A, B, C$ are $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a GP respectively then $A^{q-r} \cdot B^{r-p} \cdot C^{p-q}=\ldots \ldots$

The joint equation of lines passing through origin and having slopes $(1+\sqrt{2})$ and $\frac{-1}{1+\sqrt{2}}$ is ..........

If sum of the slopes of the lines given by $x^2-4 p x y+8 y^2=0$ is three times their product then $p=$ ...........

The acute angle between lines $x-3=0$ and $x+y=19$ is.......

The polar co-ordinates of $P$ are $\left(2, \frac{\pi}{6}\right)$. If $Q$ is the image of $P$ about the $X$-axis then the polar co-ordinates of $Q$ are.....̣...

The vector equation of the plane $\mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})$ in scalar product form is $\mathbf{r} \cdot(3 \hat{\mathbf{i}}+2 \hat{...

The direction ratios of the normal to the plane passing through origin and the line of intersection of the planes $x+2 y+3 z=4$ and $4 x+3 y+2 z=1$ are $\ldots \ldots$

If line $\frac{2 x-4}{\lambda}=\frac{y-1}{2}=\frac{z-3}{1}$ and $\frac{x-1}{1}=\frac{3 y-1}{\lambda}=\frac{z-2}{1}$ are perpendicular to each other then $\lambda=$ ............

Which of the following can not be the direction cosines of a line?

\(\frac{1-2\left[\cos 60^{\circ}-\cos 80^{\circ}\right]}{2 \sin 10^{\circ}}=\ldots \ldots\)

If the scalar triple product of the vectors $-3 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ and $7 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ is 272 th...

$\mathbf{a}$ and $\mathbf{b}$ are non-collinear vectors. If $\mathbf{c}=(x-2) \mathbf{a}+\mathbf{b}$ and $\mathbf{d}=(2 x+1) \mathbf{a}-\mathbf{b}$ are collinear vectors, then the value of $x=\ldots \ldots$

For any non zero vector, a, b, c
$\mathbf{a} \cdot[(\mathbf{b}+\mathbf{c}) \times(\mathbf{a}+\mathbf{b}+\mathbf{c})]=$ ..........