AP EAPCET 2022 - 4th July Morning Shift

At any point \((x, y)\) on a curve if the length of the subnormal is \((x-1)\) and the curve passes through \((1,2)\), then the curve is a conic. A vertex of the curve is

If the curves \(y=x^3-3 x^2-8 x-4\) and \(y=3 x^2+7 x+4\) touch each other at a point \(P\), then the equation of common tangent at \(P\) is

The condition that \(f(x)=a x^3+b x^2+c x+d\) has no extreme value is

If \(x^3-2 x^2 y^2+5 x+y-5=0\), then at \((\mathrm{l}, \mathrm{l}), y^{\prime \prime}(\mathrm{l})=\)

The minimum value of \(f(x)=x+\frac{4}{x+2}\) is

The maximum value of \(f(x)=\frac{x}{1+4 x+x^2}\) is

The length of the intercept on the line \(4 x-3 y-10=0\) by the circle \(x^2+y^2-2 x+4 y-20=0\) is

The pole of the line \(\frac{x}{a}+\frac{y}{b}=1\) with respect to the circle \(x^2+y^2=c^2\) is

The locus of centers of the circles, possessing the same area and having \(3 x-4 y+4=0\) and \(6 x-8 y-7=0\) as their common tangent, is

For any two non-zero real numbers \(a\) and \(b\) if this line \(\frac{x}{a}+\frac{y}{b}=1\) is a tangent to the circle \(x^2+y^2=1\), then which of the following is true?

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