KCET 2017

The value of $c$ in mean value theorem for the function $f(x)=x^2$ in $[2,4]$ is

The rate of change of volume of a sphere with respect to its surface area when the radius is 4 cm is

The function $f(x)=x^2+2 x-5$ is strictly increasing in the interval

The point on the curve $y^2=x$ where the tangent makes an angle of $\pi / 4$ with $X$-axis is

Area of the region bounded by the curve $y=\cos x, x=0$ and $x=\pi$ is

The area of the region bounded by the curve $y=x^2$ and the line $y=16$

The total number of terms in the expansion of $(x+a)^{47}-(x-a)^{47}$ after simplification is

If $\left(\frac{1+i}{1-i}\right)^m=1$, then the least positive integral value of $m$ is

\(\int\limits_{-\pi / 2}^{\pi / 2} \frac{d x}{e^{\sin x}+1}\)

$\int\limits_{-5}^5|x+2| d x$ is equal to

$\int\limits_0^{\pi / 2} \frac{\tan ^7 x}{\cot ^7 x+\tan ^7 x} d x$ is equal to

$\int\limits_{0.2}^{3.5}[x] d x$ is equal to :

\(\int\limits_0^{\pi / 2} \frac{1}{a^2 \cdot \sin ^2 x+b^2 \cdot \cos ^2 x} d x\)

General solution of differential equations
$\frac{d y}{d x}+y=1(y \neq 1)$ is

The degree of the differential equation $\left[1+\left(\frac{d y}{d x}\right)^2\right]^2=\frac{d^2 y}{d x^2}$ is

The integrating factor of the differential equation
$x \cdot \frac{d y}{d x}+2 y=x^2$ is $(x \neq 0)$

The derivative of $\cos ^{-1}\left(2 x^2-1\right)$ w.r.t. $\cos ^{-1} x$ is

$$ \text { If } y=\left|\begin{array}{ccc} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{array}\right| \text {, then } \frac{d y}{d x} \text { is equal to } $$

If $y=\log (\log x)$, then $\frac{d^2 y}{d x^2}$ is equal to

If $\sin x=\frac{2 t}{1+t^2}, \tan y+\frac{2 t}{1-t^2}$, then $\frac{d y}{d x}$ is equal to

If $y=\tan ^{-1}\left(\frac{\sin x+\cos x}{\cos x-\sin x}\right)$, then $\frac{d y}{d x}$ is equal to

The eccentricity of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ is

If $f(x)=8 x^3, g(x)=x^{1 / 3}$, then $f \circ g(x)$ is

Let $f: R \rightarrow R$ be defined by $f(x)=x^4$, then

If $|x-2| \leq 1$, then

Binary operation * on $R-\{-1\}$ defined by $a^* b=\frac{a}{b+1}$ is

The range of the function $f(x)=\sqrt{9-x^2}$ is

\(\int \sqrt{x^2+2 x+5} d x \text { is equal to }\)

$\int \frac{\cos 2 x-\cos 2 \theta}{\cos x-\cos \theta} d x$ is equal to

$\int \frac{(x+3) e^x}{(x+4)^2} d x$ is equal to

If $\tan ^{-1} x+\tan ^{-1} y=\frac{4 \pi}{5}$, then $\cot ^{-1} x+\cot ^{-1} y$ is equal to

The range of $\sec ^{-1} x$ is

If $f(x)=\left\{\begin{array}{cll}k x^2 & \text { if } & x \leq 2 \\ 3 & \text { if } & x>2\end{array}\right.$ is continuous at $x=2$, then the value of $k$ is

\(The\,\,value\,\,of\,\,\mathop {\lim }\limits_{\theta \to 0} {{1 - \cos 4\theta } \over {1 - \cos 6\theta }}\,\,is\)

The shaded region in the figure is the solution set of the inequations

If an LPP admits optimal solution at two consecutive vertices of a feasible region, then

The contrapositive statement of the statement "If $x$ is prime number, then $x$ is odd" is

If a matrix $A$ is both symmetric and skew symmetric, then

$$
\text { } \begin{aligned}
Let\,\,\,\Delta & =\left|\begin{array}{lll}
A x & x^2 & 1 \\
B y & y^2 & 1 \\
C z & z^2 & 1
\end{array}\right| \text { and }
\Delta_1 =\left|\begin{array}{ccc}
A & B & C \\
x & y & z \\
z y & z x & x y
\end{ar...

If $A$ is a square matrix of order $3 \times 3$, then $|K A|$ is equal to

If $\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|$, then $x$ is equal to

If $A=\frac{1}{\pi}\left|\begin{array}{ll}\sin ^{-1}(\pi x) & \tan ^{-1}\left(\frac{x}{\pi}\right) \\ \sin ^{-1}\left(\frac{x}{\pi}\right) & \cot ^{-1}(\pi x)\end{array}\right|$
$B=\left|\begin{array}{cc}-\cos ^{-1}(\pi x) & \tan ^{-1}\lef...

If $2\left|\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right|+\left|\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right|=\left|\begin{array}{ll}5 & 6 \\ 1 & 8\end{array}\right|$, then the value of $x$ and $y$ are

If ${ }^n C_{12}={ }^n C_8$, then $n$ is equal to

The probability distribution of $X$ is
$$ \begin{array}{|c|l|c|c|c|} \hline \boldsymbol{X} & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{P}(\boldsymbol{X}) & 0.3 & k & 2 k & 2 k \\ \hline \end{array} $$
\(\text { The value of } k \text { is }\)

A box has 100 pens of which 10 are defective. The probability that out of a sample of 5 pens drawn one by one with replacement and atmost one is defective, is

Two events $A$ and $B$ will be independent if

The area of triangle with vertices $(K, 0),(4,0)$, $(0,2)$ is 4 sq units, then value of $K$ is

$3+5+7+\ldots$ to $n$ terms is

If $A$ and $B$ are finites sets and $A \subset B$, then