AP EAPCET 2022 - 4th July Morning Shift

If the tangent at the point \(P\) on the circle \(x^2+y^2+6 x+6 y=2\) meets the straight line \(5 x-2 y+6=0\) at a point \(Q\) on the \(Y\)-axis, then the length of \(P Q\) is

If \(\frac{x-1}{3+i}+\frac{y-1}{3-i}=i\), then the true statement among the following is

\(i z^3+z^2-z+i=0 \Rightarrow|z|=\)

The number of integer solutions of the equation \(|1-i|^x=2^x\) is

\(\int_0^\pi x\left(\sin ^2(\sin x)+\cos ^2(\cos x)\right) d x=\)

\(\int_0^1 a^k x^k d x=\)

Let \(\alpha\) and \(\beta(\alpha<\beta)\) are roots of \(18 x^2-9 \pi x+\pi^2=0, f(x)=x^2, g(x)=\cos x\). Then, \(\int_\alpha^\beta x(g \circ f(x)) d x=\)

If the solution of \(\frac{d y}{d x}-y \log _e 0.5=0, y(0)=1\), and \(y(x) \rightarrow k\), as \(x \rightarrow \infty\), then \(k=\)

\(y=A e^x+B e^{-2 x}\) satisfies which of the following differential equations?

If \(f(x)=\cot ^{-1}\left(\frac{x^x+x^{-x}}{2}\right)\), then \(f^{\prime}(1)=\)

The eccentric angle of a point on the ellipse \(x^2+3 y^2=6\) lying at a distance of 2 units from its centre is

A stick of length \(r\) units slides with its ends on coordinate axes. Then, the locus of the mid-point of the stick is a curve whose length is

If \(f(x)=\sqrt{2-x^2}\) and \(g(x)=\log (1-x)\) are two real valued functions, then the domain of the function \((f+g)(x)\) is

The range of the real valued function \(f(x)=\sqrt{\frac{x^2+2 x+8}{x^2+2 x+4}}\) is

Let origin be the centre, \(( \pm 3,0)\) be the foci and \(\frac{3}{2}\) be the eccentricity of a hyperbola.
Then, the line \(2 x-y-1=0\)

The locus of a variable point whose chord of contact w.r.t. the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) subtends a right angle at the origin is

The value of \(\frac{1+\tan \mathrm{h} x}{1-\tan \mathrm{h} x}\) is

Assertion (A) If \(I_n=\int \cot ^n x d x\), then
\(I_6+I_4=\frac{-\cot ^5 x}{5}\)
Reason (R) \(\int \cot ^n x d x=\frac{-\cot ^{n-1} x}{n} -\int \cot ^{n-2} x d x\)

If \(I_n=\int \tan ^n x d x\), and \(I_0+I_1+2 I_2+2 I_3+2 I_4 +I_5+I_6=\sum_\limits{k=1}^n \frac{\tan ^k x}{k}\), then \(n=\)

The parametric form of a curve is \(x=\frac{t^3}{t^2-1} y=\frac{t}{t^2-1}\), then \(\int \frac{d x}{x-3 y}=\)

\(\int \frac{e^{\cot x}}{\sin ^2 x}(2 \log \operatorname{cosec} x+\sin 2 x) d x=\)

$$\begin{aligned} & \text { If } \lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^4+4 x^2+5}}=k \\ & \lim _{x \rightarrow 0} x^4 \sin \left(\frac{1}{3 \sqrt{x}}\right)=l \text {. Then, } k+l= \end{aligned}$$

\(\lim _\limits{n \rightarrow \infty}\left(\frac{1}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right)=\)

Let \(f: R^{+} \longrightarrow R^{+}\) be a function satisfying \(f(x)-x=\lambda\) (constant), \(\forall x \in R^{+}\) and \(f(x f(y))=f(x y)+x, \forall x, y, \in R^{+}\). Then, $$\lim _\limits{x \rightarrow 0} \frac{(f(x))^{1 / 3}-1}{(f(x)...

If \(\lim _\limits{n \rightarrow \infty} x^n \log _e x=0\), then \(\log _x 12=\)

If \(f(x)=\operatorname{Max}\{3-x, 3+x, 6\}\) is not differentiable at \(x=a\), and \(x=b\), then \(|a|+|b|=\)

If \(4^x-3^{x-1 / 2}=3^{x+1 / 2}-2^{2 x-1}\), then the value of \(x\) is

For \(i=1,2,3\) and \(j=1,23\)
If \(a_i^2+b_i^2+c_i^2=1, a_i a_j+b_i b_j+c_i c_j=0, \forall i \neq j\)
and $$A=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$$, then $$\operatorname{det}\left...

If $$A=\left[\begin{array}{cc}\alpha^2 & 5 \\ 5 & -\alpha\end{array}\right]$$ and \(\operatorname{det}\left(A^{10}\right)=1024\), then \(\alpha=\)

Let $$A=\left[\begin{array}{ccc}5 & \sin ^2 \theta & \cos ^2 \theta \\ -\sin ^2 \theta & -5 & 1 \\ \cos ^2 \theta & 1 & 5\end{array}\right]$$. Then, maximum value of \(\operatorname{det}(A)\) is

If $$A=\frac{1}{7}\left[\begin{array}{ccc}3 & -2 & 6 \\ -6 & -3 & 2 \\ -2 & 6 & 3\end{array}\right]$$, then

If \(\frac{x^4+24 x^2+28}{\left(x^2+1\right)^3}=\frac{A x+B}{x^2+1}\) \(+\frac{C x+D}{\left(x^2+1\right)^2}+\frac{E x+F}{\left(x^2+1\right)^3},\) then the value of \(A+B+C+D+E+F=\)

Suppose a parabola with focus at \((0,0)\) has \(x-y+1=0\) as its tangent at the vertex. Then, the equation of its directrix is

If \(a x+b y=1\) is a normal to the parabola \(y^2=4 p x\), then the condition is

If a polygon of \(n\) sides has 560 diagonals, then \(n=\)

How many chords can be drawn through 21 points on a circle?

A person writes letters to 6 friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes?
Notation $$D_n=n!\left(\sum_\limits{i=0}^n ...

The total number of permutations of \(n\) different things taken not more than \(r\) at a time, when each thing may be repeated any number of times is

If \(x\) is chosen at random from the set \(\{1,2,3, 4\}\) and \(y\) is chosen at random from the set \(\{5,6,7\}\), then the probability that \(x y\) will be even is

The discrete random variables \(X\) and \(Y\) are independent from one another and are defined as \(X \sim B(16,0.25)\) and \(Y \sim P(2)\). Then, the sum of the variance of \(X\) and \(Y\) is

A box contains 100 balls, numbered from 1 to 100 . If 3 balls are selected one after the other at random with replacement from the box, then the probability that the sum of the three numbers on the balls selected from the box is an odd numb...

If 6 is the mean of a Poisson distribution, then \(P(X \geq 3)=\)

In a lottery, containing 35 tickets, exactly 10 tickets bear a prize. If a ticket is drawn at random, then the probability of not getting a prize is

A bag contains 7 green and 5 black balls. 3 balls are drawn at random one after the other. If the balls are not replaced, then the probability of all three balls being green is

In a \(\triangle A B C, r_1 \cot \frac{A}{2}+r_2 \cot \frac{B}{2}+r_3 \cot \frac{C}{2}=\)

If in a \(\triangle A B C, a=2, b=3\) and \(c=4\), then \(\tan (A / 2)=\)

In a \(\triangle A B C\), if \(a \neq b, \frac{a \cos A-b \cos B}{a \cos B-b \cos A}+\cos C=\)

If the angles of a \(\triangle A B C\) are in the ratio \(1: 2: 3\), then the corresponding sides are in the ratio

The number of positive real roots of the equation \(3^{x+1}+3^{-x+1}=10\) is

If \(f(x)=a x^2+b x+c\) for some \(a, b, c \in R\) with \(a+b+c=3\) and \(f(x+y)=f(x)+f(y)+x y, \forall x, y \in R\). Then, \(\sum_\limits{n=1}^{10} f(n)=\)