KCET 2023

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

The distance '\(s\)' in meters travelled by a particle in '\(t\)' seconds is given by \(s=\frac{2 t^3}{3}-18 t+\frac{5}{3}\). The acceleration when the particle comes to rest is :

A particle moves along the curve \(\frac{x^2}{16}+\frac{y^2}{4}=1\). When the rate of change of abscissa is 4 times that of its ordinate, then the quadrant in which the particle lies is

An enemy fighter jet is flying along the curve, given by \(y=x^2+2\). A soldier is placed at \((3,2)\) wants to shoot down the jet when it is nearest to him. Then, the nearest distance is

A circular plate of radius \(5 \mathrm{~cm}\) is heated. Due to expansion, its radius increase at the rate of \(0.05 \mathrm{~cm} / \mathrm{s}\). The rate at which its area is increasing when the radius is \(5.2 \mathrm{~cm}\) is

In the interval \((0, \pi / 2)\) area lying between the curves \(y=\tan x\) and \(y=\cot x\) and the \(X\)-axis is

The area of the region bounded by the line \(y=x+1\) and the lines \(x=3\) and \(x=5\) is

If \(n\) is even and the middle term in the expansion of \(\left(x^2+\frac{1}{x}\right)^n\) is \(924 x^6\), then \(n\) is equal to

The modulus of the complex number \(\frac{(1+i)^2(1+3 i)}{(2-6 i)(2-2 i)}\) is

\(\int\limits_2^8 \frac{5^{\sqrt{10-x}}}{5^{\sqrt{x}}+5^{\sqrt{10-x}}} d x \text { is equals to :}\)

\(\int_{-2}^0\left(x^3+3 x^2+3 x+3+(x+1) \cos (x+1)\right) d x\) is equals to

\(\int\limits_0^\pi \frac{x \tan x}{\sec x \cdot \operatorname{cosec} x} d x\) is equals to

If a curve passes through the point \((1,1)\) and at any point \((x, y)\) on the curve, the product of the slope of its tangent and \(x\) coordinate of the point is equal to the \(y\) coordinate of the point, then the curve also passes thro...

The degree of the differential equation \(1+\left(\frac{d y}{d x}\right)^2+\left(\frac{d^2 y}{d x^2}\right)^2=\sqrt[3]{\frac{d^2 y}{d x^2}+1}\) is

If \(y=a \sin x+b \cos x\), then \(y^2+\left(\frac{d y}{d x}\right)^2\) is a

If \(f(x)=1+n x+\frac{n(n-1)}{2} x^2+\frac{n(n-1)(n-2)}{6} x^3+\ldots+x^n\), then \(f^n(1)\) is equal to :

If \(f(x)\) and \(g(x)\) are two functions with \(g(x)=x-\frac{1}{x}\) and \(f \circ g(x)=x^3-\frac{1}{x^3}\), then \(f^{\prime}(x)\) is equals to

If \(f(x)=a x+b\), where \(a\) and \(b\) are integers, \(f(-1)=-5\) and \(f(3)=3\), then \(a\) and \(b\) are respectively

\(f: R \rightarrow R\) and \(g:[0, \infty) \rightarrow R\) defined by \(f(x)=x^2\) and \(g(x)=\sqrt{x}\). Which one of the following is not true?

Let \(f: R \rightarrow R\) be defined by \(f(x)=3 x^2-5\) and \(g: R \rightarrow R\) by \(g(x)=\frac{x}{x^2+1}\), then \(g \circ f\) is

Let \(f(x)=\sin 2 x+\cos 2 x\) and \(g(x)=x^2-1\) then \(g(f(x))\) is invertible in the domain

If the function is \(f(x)=\frac{1}{x+2}\), then the point of discontinuity of the composite function \(y=f(f(x))\) is

The distance between the foci of a hyperbola is 16 and its eccentricity is \(\sqrt{2}\). Its equation is

\(\int \sqrt{\operatorname{cosec} x-\sin x} d x\) is equals to

\(\int \sqrt{5-2 x+x^2} d x\) is equals to

\(\int \frac{1}{1+3 \sin ^2 x+8 \cos ^2 x} d x\) is equals to

If \(\sin ^{-1}\left(\frac{2 a}{1+a^2}\right)+\cos ^{-1}\left(\frac{1-a^2}{1+a^2}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\) where \(a, x \in(0,1)\), then the value of \(x\) is

The value of \(\cot ^{-1}\left[\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right]\), where \(x \in\left(0, \frac{\pi}{4}\right)\) is

If \(\lim _\limits{x \rightarrow 0} \frac{\sin (2+x)-\sin (2-x)}{x}=A \cos B\), then the values of \(A\) and \(B\) respectively are

The function \(f(x)=\cot x\) is discontinuous on every point of the set

The shaded region in the figure given is the solution of which of the inequations?

The value of
\(e^{\log _{10} \tan 1^{\circ}+\log _{10} \tan 2^{\circ}+\log _{10} \tan 3^{\circ}+\ldots+\log _{10} \tan 89^{\circ}}\)
is

The contrapositive of the statement.
"If two lines do not intersect in the same plane, then they are parallel." is

$$\text { The value of }\left|\begin{array}{ccc}
\sin ^2 14^{\circ} & \sin ^2 66^{\circ} & \tan 135^{\circ} \\
\sin ^2 66^{\circ} & \tan 135^{\circ} & \sin ^2 14^{\circ} \\
\tan 135^{\circ} & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}
\end{arr...

If $$x\left[\begin{array}{l}3 \\ 2\end{array}\right]+y\left[\begin{array}{r}1 \\ -1\end{array}\right]=\left[\begin{array}{l}15 \\ 5\end{array}\right]$$, then the value of \(x\) and \(y\) are

If \(A\) and \(B\) are two matrices, such that \(A B=B\) and \(B A=A\), then \(A^2+B^2\) equals to

If $$A=\left[\begin{array}{cc}2-k & 2 \\ 1 & 3-k\end{array}\right]$$ is singular matrix, then the value of \(5 k-k^2\) is equal to

If $$\Delta=\left|\begin{array}{ccc}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{array}\right|$$ and $$\Delta_1=\left|\begin{array}{ccc}1 & 1 & 1 \\ b c & c a & a b \\ a & b & c\end{array}\right|$$, then

If $$A=\left[\begin{array}{cc}1 & \tan \alpha / 2 \\ -\tan \alpha / 2 & 1\end{array}\right]$$ and \(A B=I\), then \(B\) is equal to

Ten chairs are numbered as 1 to 10. Three women and two men wish to occupy one chair each. First the women choose the chairs marked 1 to 6 , then the men choose the chairs from the remaining. The number of possible ways is

A bag contains \(2 n+1\) coins. It is known that \(n\) of these coins have head on both sides whereas, the other \(n+1\) coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is $$\frac{31}...

Let \(A=\{x, y, z, u\}\) and \(B=\{a, b\}\). A function \(f: A \rightarrow B\) is selected randomly. The probability that the function is an onto function is

If \(A\) and \(B\) are events, such that \(P(A)=\frac{1}{4}, P(A / B)=\frac{1}{2}\) and \(P(B / A)=\frac{2}{3}\), then \(P(B)\) is

The area of a triangle with vertices \((-3,0)\), \((3,0)\) and \((0, k)\) is 9 sq units, find the value of \(k\) is

Given that \(a, b\) and \(x\) are real numbers and \(a < b, x < 0\), then

If \(p\left(\frac{1}{q}+\frac{1}{r}\right), q\left(\frac{1}{r}+\frac{1}{p}\right), r\left(\frac{1}{p}+\frac{1}{q}\right)\) are in \(\mathrm{AP}\), then \(p, q, r\)

\(n\)th term of the series \(1+\frac{3}{7}+\frac{5}{7^2}+\frac{1}{7^2}+\ldots\) is

Which of the following is an empty set?

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

The mean of 100 observations is 50 and their standard deviation is 5. Then, the sum of squares of all observations is