AP EAPCET 2022 - 5th July Morning Shift

If \(3 f(\cos x)+2 f(\sin x)=5 x\), then \(f^{\prime}(\cos x)+f^{\prime}(\sin x)=\)

If the normal drawn at a point \(P\) on the curve \(3 y=6 x-5 x^3\) passes through \((0,0)\), then the positive integral value of the abscissa of the point \(P\) is

The line joining the points \((0,3)\) and \((5,-2)\) is a tangent to the curve \(y=\frac{c}{x+1}\), then \(c=\)

If \(a, b>0\), then minimum value of \(y=\frac{b^2}{a-x}+\frac{a^2}{x}, 0< x< a\) is

The point on the curve \(y=x^2+4 x+3\) which is closest to the line \(y=3 x+2\) is

The locus of mid-points of points of intersection of \(x \cos \theta+y \sin \theta=1\) with the coordinate axes is

The radius of the circle having. \(3 x-4 y+4=0\) and \(6 x-8 y-7=0\) as its tangents is

A circle is such that \((x-2) \cos \theta+(y-2) \sin \theta=1\) touches it for all values of \(\theta\). Then, the circle is

The least distance of the point \((10,7)\) from the circle \(x^2+y^2-4 x-2 y-20=0\) is

Suppose that the \(x\)-coordinates of the points \(A\) and \(B\) satisfy \(x^2+2 x-a^2=0\) and their \(y\)-coordinates satisfy \(y^2+4 y-b^2=0\). Then, the equation of the circle with \(A B\) as its diameter is

The radical centre of the three circles \(x^2+y^2-1=0, x^2+y^2-8 x+15=0\) and \(x^2+y^2+10 y+24=0\) is

By simplifying \(i^{18}-3 i^7+i^2\left(1+i^4\right)(i)^{22}\), we get

The values of \(x\) for which \(\sin x+i \cos 2 x\) and \(\cos x-i \sin 2 x\) are conjugate to each other are

The locus of a point \(z\) satisfying \(|z|^2=\operatorname{Re}(z)\) is a circle with centre

Let \(T>0\) be a fixed number. \(f: R \rightarrow R\) is a continuous function such that \(f(x+T)=f(x), x \in R\)
If \(I=\int_\limits0^T f(x) d x\), then \(\int_\limits0^{5 T} f(2 x) d x=\)

\(\int_\limits1^3 x^n \sqrt{x^2-1} d x=6 \text {, then } n=\)

[ . ] represents greatest integer function, then \(\int_{-1}^1(x[1+\sin \pi x]+1) d x=\)

$$\begin{aligned} & \lim _{n \rightarrow \infty}\left[\frac{n}{(n+1) \sqrt{2 n+1}}+\frac{n}{(n+2) \sqrt{2(2 n+2)}}\right. \\ & \left.+\frac{n}{(n+3) \sqrt{3(2 n+3)}}+\ldots n \text { terms }\right]=\int_\limits0^1 f(x) d x \end{aligned}$$
...

The general solution of the differential equation \(\frac{d y}{d x}=\cos ^2(3 x+y)\) is \(\tan ^{-1}\left(\frac{\sqrt{3}}{2} \tan (3 x+y)\right)=f(x)\). Then, \(f(x)=\)

If the general solution of the differential equation \(\cos ^2 x \frac{d y}{d x}+y=\tan x\) is \(y=\tan x-1+C e^{-\tan x}\) satisfies \(y\left(\frac{\pi}{4}\right)=1\), then \(C=\)

Assertion (A) Order of the differential equations of a family of circles with constant radius is two.
Reason (R) An algebraic equation having two arbitrary constants is general solution of a second order differential equation.

Assertion (A) \(\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)=\frac{x^2 \sin x}{\log x}\left(\cot x+\frac{2}{x}-\frac{1}{x \log x}\right)\)
Reason (R) $$\frac{d}{d x}\left(\frac{u v}{w}\right)=\frac{u v}{w}\left[\frac{u^{\prime}}{u}+\...

If \(x=f(\theta)\) and \(y=g(\theta)\), then \(\frac{d^2 y}{d x^2}=\)

\(y=x^3-a x^2+48 x+7\) is an increasing function for all real values of \(x\), then \(a\) lies in the interval

If the angle between the straight lines joining the foci and the ends of the minor axis of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is \(90^{\circ}\), then it eccentricity

\(f(x)=\log \left(\left(\frac{2 x^2-3}{x}\right)+\sqrt{\frac{4 x^4-11 x^2+9}{|x|}}\right) \text { is }\)

Let \(f: R-\left\{\frac{-1}{2}\right\} \rightarrow R\) be defined by \(f(x)=\frac{x-2}{2 x+1}\). If \(\alpha\) and \(\beta\) satisfy the equation \(f(f(x))=-x\), then \(4\left(\alpha^2+\beta^2\right)=\)

The locus of point of intersection of tangents at the ends of normal chord of the hyperbola \(x^2-y^2=a^2\) is

If \(e_1\) and \(e_2\) are the eccentricities of the hyperbola \(16 x^2-9 y^2=1\) and its conjugate respectively. Then, \(3 e_1=\)

\(\frac{2 x^2+1}{x^3-1}=\frac{A}{x-1}+\frac{B x+C}{x^2+x+1} \Rightarrow 7 A+2 B+C=\)

\(\int \frac{3 x+4}{x^3-2 x+4} d x=\log f(x)+C \Rightarrow f(3)=\)

\(\int \frac{e^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x=\)

\(\int \frac{d x}{(x-3)^{\frac{4}{5}}(x+1)^{\frac{6}{5}}}=\)

If \(I_n=\int\left(\cos ^n x+\sin ^n x\right) d x\) and \(I_n-\frac{n-1}{n} I_{n-2} =\frac{\sin x \cos x}{n} f(x)\), then \(f(x)=\)

\(\lim _\limits{x \rightarrow-\infty} \log _e(\cosh x)+x=\)

If \(a, b\) and \(c\) are three distinct real numbers and \(\lim _\limits{x \rightarrow \infty} \frac{(b-c) x^2+(c-a) x+(a-b)}{(a-b) x^2+(b-c) x+(c-a)}=\frac{1}{2}\), then \(a+2 c=\)

\(\lim _\limits{x \rightarrow-\infty} \frac{3|x|-x}{|x|-2 x}-\lim _\limits{x \rightarrow 0} \frac{\log \left(1+x^3\right)}{\sin ^3 x}=\)

\(4^x-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2 x-1} \Rightarrow x=\)

If $$A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]$$, then \(A A^T\) is a

If \(A X=D\) represents the system of simultaneous linear equations \(x+y+z=6, 5 x-y+2 z=3\) and \(2 x+y-z=-5\), then (Adj \(A\)) \(D=\)

If $$A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$$, then \(\operatorname{det}\left(A^6+B^6\right)=\)

Let $$G(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$$. If \(x+y=0\) then \(G(x) G(y)=\)

Which of the following represents a parabola?

\(\text { If } 10{ }^n C_2=3^{n+1} C_3 \text {, then the value of } n \text { is }\)

There are 10 points in a plane, out of these 6 are collinear. If \(N\) is the total number of triangles formed by joining these points, then \(N=\)

In an examination, the maximum marks for each of three subjects is \(n\) and that for the fourth subject is \(2 n\). The number of ways in which candidates can get \(3 n\) marks is

The probability of getting a sum 9 when two dice are thrown is

If \(A\) and \(B\) are two events such that \(P(B) \neq 0\) and \(P(B) \neq 1\), then \(P(\bar{A} \mid \bar{B})\) is

Two brothers \(X\) and \(Y\) appeared for an exam. Let \(A\) be the event that \(X\) has passed the exam and \(B\) is the event that \(Y\) has passed. The probability of \(A\) is \(\frac{1}{7}\) and of \(B\) is \(\frac{2}{9}\). Then, the pr...

A bag contains 4 red and 3 black balls. A second bag contains 2 red and 3 black balls. One bag is selected at random. If from the selected bag, one ball is drawn at random, then the probability that the ball drawn is red, is