MHT CET 2019 2nd May Evening Shift

Using differentiation, approximate value of $f(x)=x^2-2 x+1$ at $x=2.99$ is ............

A particle moves so that $x=2+27 t-t^3$. The direction of motion reverses after moving a distance of ....... units.

The function $f(x)=x^3-3 x$ is ............

The area of the region enclosed between pair of the lines $x y=0$ and the lines $x y+5 x-4 y-20=0$, is .............

The equation of the circle concentric with the circle $x^2+y^2-6 x-4 y-12=0$ and touching the $Y$-axis is ............

The intercept on the line $y=x$ by the circle $x^2+y^2-2 x=0$ is $A B$. The equation of the circle with $A B$ as a diameter is .............

\(\int_\limits a^b \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a+b-x}} d x=\ldots \ldots\)

\(\int_0^1 x(1-x)^5 d x=\ldots \ldots\)

If $\int_0^a \sqrt{\frac{a-x}{x}} d x=\frac{K}{2}$, then $K=\ldots .$.

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

The solution of the differential equation $y d x-x d y=x y d x$ is ......

The solution of the differential equation $\frac{d \theta}{d t}=-k\left(\theta-\theta_0\right)$ where $k$ is constant, is .............

If $x=\sin \theta, y=\sin ^3 \theta$ then $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{2}$ is ............

If $x^y=e^{x-y}$, then $\frac{d y}{d x}$ at $x=1$ is ...........

If $f(x)=3 x+6, g(x)=4 x+k$ and $f \circ g(x)=g \circ f(x)$ then $k=$

If \(\int \frac{\cos x-\sin x}{8-\sin 2 x} d x=\frac{1}{p} \log \left[\frac{3+\sin x+\cos x}{3-\sin x-\cos x}\right]+c,\) then $p=$ .............

$$\begin{aligned} & \text { If } \int \tan (x-\alpha) \tan (x+\alpha) \cdot \tan 2 x d x \\ & =p \log |\sec 2 x|+q \log |\sec (x+\alpha)| \\ & +r \log |\sec (x-\alpha)|+c \text { then } p+q+r=\ldots \ldots \ldots \end{aligned}$$

\(\int \frac{x^2+1}{x^4-x^2+1} d x=\ldots \ldots\)

If $4 \sin ^{-1} x+6 \cos ^{-1} x=3 \pi$ then $x=$ ............

If the function $f(x)=\frac{\left(e^{k x}-1\right) \tan k x}{4 x^2}, x \neq 0$
\(\qquad \qquad=16 \qquad x=0\)
is continuous at $x=0$, then $k=\ldots \ldots$

If $f(x)=[x]$, where $[x]$ is the greatest integer not greater than $x$, then $f^{\prime}\left(1^{+}\right)=$ ...........

If function
$$\begin{aligned} f(x) & =x-\frac{|x|}{x}, x<0 \\ & =x+\frac{|x|}{x}, x>0 \\ & =1, \quad x=0, \text { then } \end{aligned}$$

For L.P.P, maximize $z=4 x_1+2 x_2$ subject to $3 x_1+2 x_2 \geq 9, x_1-x_2 \leq 3, x_1 \geq 0, x_2 \geq 0$ has

The maximum value of $z=6 x+8 y$ subject to $x-y \geq 0, x+3 y \leq 12, x \geq 0, y \geq 0$ is $\ldots \ldots$.

If $p$ and $q$ are true and $r$ and $s$ are false statements, then which of the following is true?

The negation of " $\forall, n \in N, n+7>6$ " is .............

Which of the following statements is contingency?

If $A$ is non-singular matrix and $(A+I)(A-I)=0$ then $A+A^{-1}=$ .............

If $A=\left[\begin{array}{cc}1+2 i & i \\ -i & 1-2 i\end{array}\right]$, where $i=\sqrt{-1}$, then $A(\operatorname{adj} A)=\ldots$

In a bionomial distribution, mean is 18 and variance is 12 then $p=$ ...........

The p.d.f of a random variable $x$ is given by
$$\begin{aligned} & f(x)=\frac{1}{4 a}, \quad 00) \\ & =0 \text {, otherwise } \end{aligned}$$
and $P\left(x<\frac{3 a}{2}\right)=k P\left(x>\frac{5 a}{2}\right)$ then $k=$ ..............

If three dices are thrown then the probability that the sum of the numbers on their uppermost faces to be atleast 5 is

If $R$ is the circum radius of $\triangle A B C$, then $A(\triangle A B C)=\ldots \ldots$

In $\triangle A B C$, if $\tan A+\tan B+\tan C=6$ and $\tan A \cdot \tan B=2$ then $\tan C=$ ...........

In $\triangle A B C$; with usual notations,
\(\frac{b \sin B-c \sin C}{\sin (B-C)}=\ldots \ldots\)

If $\sum_{r=1}^n(2 r+1)=440$, then $n=$ ...............

If the sum of an infinite GP be 9 and sum of first two terms be 5 then their common ratio is ..........

If $A=\{x \mid x \in N, x$ is a prime number less than $12\}$ and $B=\{x \mid x \in N, x$ is a factor of 10$\}$, then $A \cap B=$ .............

If the standard deviation of the random variable $X$ is $\sqrt{3 p q}$ and mean is $3 p$ then $E\left(x^2\right)=\ldots \ldots$

If lines represented by \(\left(1+\sin ^2 \theta\right) x^2+2 h x y+2 \sin \theta y^2=0, \theta \in[0,2 \pi]\) are perpendicular to each other then $\theta=$ ...........

If $(-\sqrt{2}, \sqrt{2})$ are cartesian co-ordinates of the point, then its polar co-ordinates are .........

The $y$-intercept of the line passing through $A(6,1)$ and perpendicular to the line $x-2 y=4$ is ...........

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}$ intersect each other, then $\lambda=\ldots \ldots$

Equations of planes parallel to the plane $x-2 y+2 z+4=0$ which are at a distance of one unit from the point $(1,2,3)$ are ............

If the foot of the perpendicular drawn from the point $(0,0,0)$ to the plane is $(4,-2,-5)$ then the equation of the plane is .............

If $P(6,10,10), Q(1,0,-5), R(6,-10, \lambda)$ are vertices of a triangle right angled at $Q$, then value of $\lambda$ is ............

The value of $\sin 18^{\circ}$ is $\qquad$

If $A, B, C$ and $D$ are
$(3,7,4),(5,-2,-3),(-4,5,6)$ and $(1,2,3)$ respectively, then the volume of the parallelopiped with $A B, A C$ and $A D$ as the co-terminus edges, is .......... cubic units.

If the vectors $x \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+y \hat{\mathbf{j}}-z \hat{\mathbf{k}}$ are collinear then the value of $\frac{x y^2}{z}$ is equal

Which of the following is not equal to $\mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})$ ?