KCET 2022

The function \(f(x)=\log (1+x)-\frac{2 x}{2+x}\) is increasing on

The coordinates of the point on the \(\sqrt{x}+\sqrt{y}=6\) at which the tangent is equally inclined to the axes is

The function \(f(x)=4 \sin ^3 x-6 \sin ^2 x +12 \sin x+100\) is strictly

Area of the region bounded by the curve \(y=\tan x\), the \(X\)-axis and line \(x=\frac{\pi}{3}\) is

If \(3 x+i(4 x-y)=6-i\) where \(x\) and \(y\) are real numbers, then the values of \(x\) and \(y\) are respectively,

If \([x]\) is the greatest integer function not greater than \(x\), then \(\int_\limits0^8[x] d x\) is equal to

\(\int_0^{\pi / 2} \sqrt{\sin \theta} \cos ^3 \theta d \theta\) is equal to

\(\int_0^1 \frac{x e^x}{(2+x)^3} d x\) is equal to

Evaluate \(\int_\limits2^3 x^2 d x\) as the limit of a sum

\(\int_0^{\pi / 2} \frac{\cos x \sin x}{1+\sin x} d x\) is equal to

If \(\frac{d y}{d x}+\frac{y}{x}=x^2\), then \(2 y(2)-y(1)=\)

The solution of the differential equation \(\frac{d y}{d x}=(x+y)^2\) is

If \(y(x)\) is the solution of differential equation \(x \log x \frac{d y}{d x}+y=2 x \log x, y(e)\) is equal to

The sum of the degree and order of the differential equation \(\left(l+y_1^2\right)^{2 / 3}=y_2\) is

If \(y=\left(1+x^2\right) \tan ^{-1} x-x\), then \(\frac{d y}{d x}\) is

If \(x=e^\theta \sin \theta, y=e^\theta \cos \theta\) where \(\theta\) is a parameter, then \(\frac{d y}{d x}\) at \((1,1)\) is equal to

If \(y=e^{\sqrt{x \sqrt{x} \sqrt{x}}...,} x >1\), then \(\frac{d^2 y}{d x^2}\) at \(x=\log _e 3\) is

If \(f(1)=1, f^{\prime}(l)=3\), then the derivative of \(f(f(f(x)))+(f(x))^2\) at \(x=1\) is

If \(y=x^{\sin x}+(\sin x)^x\), then \(\frac{d y}{d x}\) at \(x=\frac{\pi}{2}\) is

If \(e^y+x y=e\) the ordered pair \(\left(\frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)\) at \(x=0\) is equal to

The domain of the function \(f(x)=\frac{1}{\log _{10}(1-x)}+\sqrt{x+2}\) is

If \(f: R \rightarrow R\) be defined by
$$f(x)=\left\{\begin{array}{llc} 2 x: & x>3 \\ x^2: & 1< x \leq 3 \\ 3 x: & x \leq 1 \end{array}\right.$$
then \(f(-1)+f(2)+f(4)\) is

\(\int \frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha} d x\) is equal to

If \(\int \frac{d x}{(x+2)\left(x^2+1\right)}=a \log \left|1+x^2\right|+b \tan ^{-1} x +\frac{1}{5} \log |x+2|+c,\) then

Domain \(\cos ^{-1}[x]\) is, where [ ] denotes a greatest integer function

If $$f(x)=\left\{\begin{array}{cc}x^2-1, & 0< x<2 \\ 2 x+3, & 2 \leq x<3\end{array}\right.$$,
the quadratic equation whose roots are \(\lim _\limits{x \rightarrow 2^{-}} f(x)\) and \(\lim _\limits{x \rightarrow 2^{+}} f(x)\) is

\(\lim _\limits{y \rightarrow 0} \frac{\sqrt{3+y^3}-\sqrt{3}}{y^3}=\)

The corner points of the feasible region of an LPP are \((0,2),(3,0),(6,0),(6,8)\) and \((0,5)\), then the minimum value of \(z=4 x+6 y\) occurs at

A dietician has to develop a special diet using two foods \(X\) and \(Y\). Each packet (containing \(30 \mathrm{~g}\) ) of food. \(X\) contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each pack...

If \(A\) is a matrix of order \(3 \times 3\), then \(\left(A^2\right)^{-1}\) is equal to

If $$A=\left[\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right]$$, then the inverse of the matrix \(A^3\) is

If \(A\) is a skew symmetric matrix, then A\(^{2021}\) is

If $$A=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]$$, then \((a I+b A)^n\) is (where \(I\) is the identify matrix of order 2)

If \(A\) is a \(3 \times 3\) matrix such that \(|5 \cdot \operatorname{adj} A|=5\), then \(|A|\) is equal to

If there are two values of '\(a\)' which makes determinant
$$\Delta=\left|\begin{array}{ccc} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a \end{array}\right|=86$$
Then, the sum of these number is

If $$A_n=\left[\begin{array}{cc}1-n & n \\ n & 1-n\end{array}\right]$$, then
\(\left|A_1\right|+\left|A_2\right|+\ldots .\left|A_{2021}\right|=\)

If all permutations of the letters of the word MASK are arranged in the order as in dictionary with or without meaning, which one of the following is 19th word?

Find the mean number of heads in three tosses of a fair coin.

If \(A\) and \(B\) are two events such that \(P(A)=\frac{1}{2}, P(B)=\frac{1}{2}\) and \(P(A \mid B)=\frac{1}{4}\), then \(P\left(A^{\prime} \cap B^{\prime}\right)\) is

A pandemic has been spreading all over the world. The probabilities are 0.7 that there will be a lockdown, 0.8 that the pandemic is controlled in one month if there is a lockdown and 0.3 that it is controlled in one month if there is no loc...

If \(A\) and \(B\) are two independent events such that \(P(\bar{A})=0.75, P(A \cup B)=0.65\) and \(P(B)=x\), then find the value of \(x\).

If \(A=\{1,2,3, \ldots, 10\}\), then number of subsets of \(A\) containing only odd numbers is

If \(a_1, a_2, a_3, \ldots, a_{10}\) is a geometric progression and \(\frac{a_3}{a_1}=25\), then \(\frac{a_9}{a_5}\) equals

If the set \(x\) contains 7 elements and set \(y\) contains 8 elements, then the number of bijections from \(x\) to \(y\) is

Suppose that the number of elements in set \(A\) is \(p\), the number of elements in set \(B\) is \(q\) and the number of elements in \(A \times B\) is 7, then \(p^2+q^2=\)

Let the relation \(R\) is defined in \(N\) by \(a R b\), if \(3 a+2 b=27\) then \(R\) is

If the standard deviation of the numbers \(-1, 0,1, k\) is \(\sqrt{5}\) where \(k>0\), then \(k\) is equal to

If the straight line \(2 x-3 y+17=0\) is perpendicular to the line passing through the points \((7,17)\) and \((15, \beta)\), then \(\beta\) equals

If the vertices of a triangle are \((-2,6),(3,-6)\) and \((1,5)\), then the area of the triangle is

The octant in which the point (2, \(-\)4, \(-7\)) lies is