KCET 2024

The length of a rectangle is five times the breadth. If the minimum perimeter of the rectangle is 180 cm , then

The maximum volume of the right circular cone with slant height 6 units is

For the function $f(x)=x^3-6 x^2+12 x-3$; $x=2$ is

The value of $C$ in $(0,2)$ satisfying the mean value theorem for the function $f(x)=x(x-1)^2, x \in[0,2]$ is equal to

The function $x^x ; x>0$ is strictly increasing at

If $f(x)=x e^{x(1-x)}$, then $f(x)$ is

The area of the region bounded by the line $y=x$ and the curve $y=x^3$ is

The area of the region bounded by the line $y=3 x$ and the curve $y=x^2$ sq units is

The value of ${ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4$ is

In the expansion of $(1+x)^n$ $\frac{C_1}{C_0}+2 \frac{C_2}{C_1}+3 \frac{C_3}{2}+\ldots+n \frac{C_n}{C_{n-1}}$ is equal to

If in two circles, arcs of the same length subtend angles $30^{\circ}$ and $78^{\circ}$ at the centre, then the ratio of their radii is

The real value of ' $\alpha$ ' for which $\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$ is purely real is

$\int\limits_1^5(|x-3|+|1-x|) d x=$

$\int_{-\pi}^\pi\left(1-x^2\right) \sin x \cdot \cos ^2 x d x$ is

The family of curves whose $x$ and $y$ intercepts of a tangent at any point are respectively double the $x$ and $y$ coordinates of that point is

The solution of $e^{d y / d x}=x+1, y(0)=3$ is

If $y=2 x^{3 x}$, then $d y / d x$ at $x=1$ is

$\frac{d}{d x}\left[\cos ^2\left(\cot ^{-1} \sqrt{\frac{2+x}{2-x}}\right)\right]$ is

If $[x]^2-5[x]+6=0$, where $[x]$ denotes the greatest integer function, then

Let the function satisfy the equation $f(x+y)=f(x) f(y)$ for all $x, y \in R$, where $f(0) \neq 0$. If $f(5)=3$ and $f^{\prime}(0)=2$, then $f^{\prime}(5)$ is

Let $f: R \rightarrow R$ be defined by $f(x)=x^2+1$. Then, the pre images of 17 and $-$3 , respectively are

Let $(g \circ f)(x)=\sin x$ and $f \circ g(x)=(\sin \sqrt{x})^2$. Then,

\(\int \frac{\sin x}{3+4 \cos ^2 x} d x\)

\(\int \frac{1}{x\left[6(\log x)^2+7 \log x+2\right]} d x \text { is }\)

$\int \frac{\sin \frac{3 x}{2}}{\sin \frac{x}{2}} d x$ is

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$, then $x(y+z)+y(z+x)+z(x+y)$ equals to

If $2 \sin ^{-1} x-3 \cos ^{-1} x=4, x \in[-1,1]$, then $2 \sin ^{-1} x+3 \cos ^{-1} x$ is equal to

Let $f: R \rightarrow R$ be given $f(x)=\tan x$. Then, $f^{-1}(1)$ is

$\lim \limits_{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \cos x-1}{\cot x-1}$ is equal to

Let $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x & 2 x \\ \sin x & x & x\end{array}\right|$. Then, $\lim _\limits{x \rightarrow 0} \frac{f(x)}{x^2}$ is

\(\lim _\limits{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots+\frac{1}{5 n}\right)=\)

The function $f(x)=|\cos x|$ is

Corner points of the feasible region for an LPP are $(0,2),(3,0),(6,0),(6,8)$ and $(0,5)$. Let $Z=4 x+6 y$ be the objective function. The minimum value of $z$ occurs at

Which one of the following observations is correct for the features of logarithm function to any base $b>1$ ?

The negation of the statement "For every real number $x ; x^2+5$ is positive" is

If $A=\left|\begin{array}{cc}x & 1 \\ 1 & x\end{array}\right|$ and $B=\left|\begin{array}{ccc}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{array}\right|$, then $\frac{d B}{d x}$ is

If $A=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)$, then $A^{10}$ is equal to

If $A$ is a square matrix, such that $A^2=A$, then $(I+A)^3$ is equal to

If $f(x)=\left|\begin{array}{ccc}x-3 & 2 x^2-18 & 2 x^3-81 \\ x-5 & 2 x^2-50 & 4 x^3-500 \\ 1 & 2 & 3\end{array}\right|$, then $f(\mathrm{l}) \cdot f(3)+f(3) \cdot f(5)+f(5) \cdot f(\mathrm{l})$ is

If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ $\operatorname{matrix} A$ and $|A|=4$, then $\alpha$ is equal to

The equation of parabola whose focus is $(6,0)$ and directrix is $x=-6$ is

A die is thrown 10 times. The probability that an odd number will come up at least once is

A random variable $X$ has the following probability distribution:

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If a random variable $X$ follows the binomial distribution with parameters $n=5, p$ and $P(X=2)=9 P(X=3)$, then $p$ is equal to

If $\triangle A B C$ is right angled at $C$, then the value of $\tan A+\tan B$ is

If $A M$ and GM of roots of a quadratic equation are 5 and 4 , respectively, then the quadratic equation is

If $S_n$ stands for sum to $n$-terms of a GP with $a$ as the first term and $r$ as the common ratio, then $S_n: S_{2 n}$ is

Two finite sets have $m$ and $n$ elements respectively. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of $m$ and $n$, respectively are

Let $A=\{2,3,4,5, \ldots, 16,17,18\}$. Let $R$ be the relation on the set $A$ of ordered pairs of positive integers defined by $(a, b) R(c, d)$ if and only if $a d=b c$ for all $(a, b),(c, d)$ in $A \times A$. Then, the number of ordered pa...

Let $a, b, c, d$ and $e$ be the observations with mean $m$ and standard deviation $S$. The standard deviation of the observations $a+k$, $b+k, c+k, d+k$ and $e+k$ is