AP EAPCET 2022 - 4th July Evening Shift

Two particles \(P\) and \(Q\) located at the points \(P\left(t, t^3-16 t-3\right), Q\left(t+1, t^3-6 t-6\right)\) are moving in a plane, the minimum distance between the points in their motion is

The number of those tangents to the curve \(y^2-2 x^3-4 y+8=0\) which pass through the point \((1,2)\) is

If the straight line \(x \cos \alpha+y \sin \alpha=p\) touches the curve \(\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2\) at the point \((a, b)\) on it and \(\frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}\), then \(k=\)

Condition that 2 curves \(y^2=4 a x, x y=c^2\) cut orthogonally is

A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is

The least value of \(n\) so that \({ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3\)

For any real number \(t\), the point \(\left(\frac{8 t}{1+t^2}, \frac{4\left(1-t^2\right)}{1+t^2}\right)\) lies on a / an

A circle has its centre in the first quadrant and passes through \((2,3)\). If this circle makes intercepts of length 3 and 4 respectively on \(x=2\) and \(y=3\), its equation is

The image of the point \((3,4)\) with respect to the radical axis of the circles \(x^2+y^2+8 x+2 y+10=0\) and \(x^2+y^2+7 x+3 y+10=0\) is

The area of the circle passing through the points \((5, \pm 2),(1,2)\) is

The ratio of the largest and shortest distances from the point \((2,-7)\) to the circle \(x^2+y^2-14 x-10 y-151=0\) is

\(\sum_\limits{k=0}^{440} i^k=x+i y \Rightarrow x^{100}+x^{99} y+x^{242} y^2+x^{97} y^3=\)

Multiplicative inverse of the complex number \((\sin \theta, \cos \theta)\) is

If \(e^{i \theta}=\operatorname{cis} \theta\), then \(\sum_\limits{n=0}^{\infty} \frac{\cos (n \theta)}{2^n}=\)

If \(f(x)=\max \{\sin x, \cos x\}\) and \(g(x)=\min \{\sin x, \cos x\}\), then \(\int_0^\pi f(x) d x+\int_0^\pi g(x) d x=\)

\(\int_0^{\pi / 4} e^{\tan ^2 \theta} \sin ^2 \theta \tan \theta d \theta=\)

If \(I_n=\int_0^{\pi / 4} \tan ^n x d x\), then \(\frac{1}{I_2+I_4}+\frac{1}{I_3+I_5}+\frac{1}{I_4+I_6}=\)

\(\int_{\pi / 4}^{5 \pi / 4}(|\cos t| \sin t+|\sin t| \cos t) d t=\)

If \(l\) and \(m\) are order and degree of a differential equation of all the straight lines at constant distance of \(P\) units from the origin, then \(l m^2+l^2 m=\)

By eliminating the arbitrary constants from \(y=(a+b) \sin (x+c)-d e^{x+e+f}\), then differential equation has order of

If \(2 x-y+C \log (|x-2 y-4|)=k\) is the general solution of \(\frac{d y}{d x}=\frac{2 x-4 y-5}{x-2 y+2}\), then \(C=\)

If \(x \neq 0\) and \(f(x)\) satisfies \(8 f(x)+6 f(1 / x) =x+5\), then \(\frac{d}{d x}\left(x^2 f(x)\right)\) at \(x=1\) is

The focal distances of the point \(\left(\frac{4}{\sqrt{5}}, \frac{3}{\sqrt{5}}\right)\) on the ellipse \(\frac{x^2}{4}+\frac{y^2}{9}=1\) are

The domain of the real valued function \(f(x)=\sin \left(\log \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right.\) is

If the vertices and foci of a hyperbola are respectively \(( \pm 3,0)\) and \(( \pm 4,0)\), then the parametric equations of that hyperbola are

If the normal to the rectangular hyperbola \(x^2-y^2=1\) at the point \(P(\pi / 4)\) meets the curve again at \(Q(\theta)\), then \(\sec ^2 \theta+\tan \theta=\)

If
\(\int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x}) d x=e^{\sqrt{x}}[A x+B \sqrt{x}+C]+K\) then \(A+B+C=\)

If \(f(x)=\int x^2 \cos ^2 x\left(2 x \tan ^2 x-2 x-6 \tan x\right) d x\) and \(f(0)=\pi\), then \(f(x)=\)

If \(\int \frac{1+\sqrt{\tan x}}{\sin 2 x} d x=A \log \tan x+B \tan x+C\), then \(4 A-2 B=\)

\(\int \frac{1+\tan x \tan (x+a)}{\tan x \tan (x+a)} d x=\)

If \([\cdot]\) denotes greatest integer function, then \(\lim _\limits{x \rightarrow \frac{-3}{5}} \frac{1}{\dot{x}}\left[\frac{-1}{x}\right]=\)

\(\frac{d}{d x}\left(\lim _{x \rightarrow 2} \frac{1}{y-2}\left(\frac{1}{x}-\frac{1}{x+y-2}\right)\right)=\)

If $$f(x)=\left\{\begin{array}{cc}\frac{x^2 \log (\cos x)}{\log (1+x)} & , \quad x \neq 0 \\ 0 & , x=0\end{array}\right.$$, then at \(x=0, f(x)\) is

Let $$f(x)=\left\{\begin{array}{cl}\frac{1}{|x|}, & \text { for }|x|>1 \\ a x^2+b, & \text { for }|x| \leq 1\end{array}\right.$$. If \(\lim _\limits{x \rightarrow 1^{+}} f(x)\) and \(\lim _\limits{x \rightarrow 1^{-}} f(x)\) exist, then the...

If \(l, m(l< m)\) are roots of \(a x^2+b x+c=0\), then
\(\lim _\limits{x \rightarrow \alpha} \frac{\left|a x^2+b x+c\right|}{a x^2+b x+c}=\)

\(\left\{x \in R / \frac{\sqrt{|x|^2-2|x|-8}}{\log \left(2-x-x^2\right)}\right.\) is a real number \(\}=\)

If \(4+6\left(e^{2 x}+1\right) \tanh x=11 \cosh x+11 \sinh x\) then \(x=\)

If $$A=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]$$, then \(\left(A^T\right)^2+(12 A)^T=\)

Let $$A=\left[\begin{array}{ccc}-2 & x & 1 \\ x & 1 & 1 \\ 2 & 3 & -1\end{array}\right]$$. If the roots of the equation \(\operatorname{det} A=0\) are \(l, m\) then \(l^3-m^3=\)

If \(a, b, c\) are respectively the 5 th, 8 th, 13 th terms of an arithmetic progression, then $$\left|\begin{array}{ccc}a & 5 & 1 \\ b & 8 & 1 \\ c & 13 & 1\end{array}\right|=$$

If $$A=\left[\begin{array}{ccc}1 & 0 & 0 \\ a & -1 & 0 \\ b & c & 1\end{array}\right]$$ is such that \(A^2=I\), then

Suppose a parabola passes through \((0,4),(1,9)\) and \((4,5)\) and has its axis parallel to the \(Y\)-axis. Then, the equation of the parabola is

If a set \(A\) has \(m\)-elements and the set \(B\) has \(n\)-elements, then the number of injections from \(A\) to \(B\) is

In how many ways can the letters of the word "MULTIPLE" be arranged keeping the position of the vowels fixed?

A natural number \(n\) such that \(n!\) ends in exactly 1000 zeroes is

For two events \(A\) and \(B\), a true statement among the following is

In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked randomly. The probability that it is neither red nor green is

Five digit numbers are formed by using digits \(1,2,3,4\) and 5 without repetition. Then, the probability that the randomly chosen number is divisible by 4 is

Which of the following is not a property of a Binomial distribution?

In a Binomial distribution \(B(n, p)\), if the mean and variance are 15 and 10 respectively, then the value of the parameter \(n\) is