List of top Questions asked in AP EAPCET- 2022

If \[ \cos\frac{y}{x}=A\log x+C \] is the general solution of \[ \left(x\sin\frac{y}{x}\right)dy=\left(y\sin\frac{y}{x}-x\right)dx, \] then \(A=\)

A player kicks a football at an angle \(30^\circ\) with the horizontal with an initial speed \(30\,\text{ms}^{-1}\). A second player standing at a distance of \(21\sqrt{3}\,\text{m}\) from the first and in the direction of kick, starts running to catch the ball, at the same instant as kicked by first player. The minimum speed of second player to catch the ball before it hits the ground is

In a hypothetical ring shaped satellite rotating in space, artificial gravity can be achieved using centripetal force. If the satellite has radius \(10\,\text{m}\), then to achieve centripetal acceleration at a point on circumference as \(10\,\text{ms}^{-2}\), its angular speed is

If \[ \int_0^\pi \frac{dx}{1+2\sin^2x}=k, \] then greatest integer less than or equal to \(k\) is

Evaluate \[ \int_0^{\pi^2/4}\left(2\sin\sqrt{x}+\sqrt{x}\cos\sqrt{x}\right)\,dx \]

\(a,b,c,d\) are real numbers. The general solution of \[ \frac{dy}{dx}=\frac{ax+b}{cy+d} \] represents a family of straight lines, when

Assertion (A): \[ \int_{2}^{e}\left(\frac{1}{\log_e x}-\frac{1}{(\log_e x)^2}\right)\,dx = e-2\log_2 e \] Reason (R): \[ \int_{a}^{b} e^x\left(f(x)+f'(x)\right)\,dx = e^bf(b)-e^af(a) \]

If \(f\) is a continuous function and \(f(x+T)=f(x), \forall x\in R\), it is given that \[ \int_0^{NT} f(t)\,dt=N\int_0^T f(t)\,dt \] \((N\) is a natural number). Then \[ \int_0^{50\pi}\sqrt{1-\cos 2x}\,dx= \]

Evaluate \[ \int \frac{2\cos x+1}{(2+\cos x)^2}\,dx-\frac{\sin x}{2+\cos x} \]

If \[ \int \frac{1}{\cos 4x \cos 2x}\,dx = \frac{1}{4\sqrt{2}}\log\left|\frac{1+f(x)}{1-f(x)}\right| -\frac{1}{2}\log g(x)+C, \] then \[ g\left(\frac{\pi}{6}\right)-\sqrt{2}f\left(\frac{\pi}{6}\right)= \]

\(a,b,c,d,e,f,g,h\) are distinct elements in the set \[ \{-7,-5,-3,-2,2,4,6,13\} \] The minimum value of \[ (a+b+c+d)^2+(e+f+g+h)^2 \] is

For a particle moving on a straight line, it is observed that the distance \(S\) at time \(t\) is given by \[ S=6t-\frac{t^3}{2} \] The maximum velocity during the motion is

Evaluate \[ \int \sqrt{1+2\cot x(\cot x+\cosec x)}\,dx \]

Evaluate \[ \int \frac{\cos^4x}{\left(\sin^2x+\sin^{-3}x\cos^5x\right)^3}\,dx \]

The normal to the curve \[ x=a(1+\cos\theta), \quad y=a\sin\theta \] at \(\theta\) always passes through a fixed point, then the fixed point is

Two sides of a triangle are given. If the area of the triangle is maximum, then the angle between the given sides is

The angle between the tangents drawn at \((0,0)\) to the curves \[ y^3-x^2y+5y-2x=0 \] and \[ x^4-x^3y^2+5x+2y=0 \] is

If \[ x\sin(\alpha+y)=\sin y \] and \[ y=\frac{m}{x^2+2nx+1}, \] then \(m^2=\)

\([\,]\) represents the greatest integer function. At \(x=-1\), \[ \frac{d}{dx}\sin \pi [x] \] is

Let \[ f(x)= \begin{cases} |x|, & -\infty\lt x\lt 2 \\ |2x-4|, & 2\leq x\leq 20 \end{cases} \] \(x=a\) is a point where \(f(x)\) is continuous but not differentiable and \(x=b\) is a point where \(f(x)\) is not differentiable \((a\neq b)\). Then \(a+b=\)

The value of \[ \lim_{n\to \infty}\sum_{r=1}^{n}\cot^{-1}\left(r^2+\frac{3}{4}\right) \] is

If \((l_1,m_1,n_1)\) and \((l_2,m_2,n_2)\) are the direction cosines of two lines satisfying the relations \[ l^2+mn-6n^2=0 \] and \[ 2l-m+3n=0, \] then \[ |l_1l_2|+|m_1m_2|= \]

If \(n\gt 0\) and \[ \lim_{x\to 0}\frac{((a-n)nx-\tan x)\sin nx}{x^2}=0, \] then minimum value of \(a\) is

The value of \[ \lim_{x\to 0}\left(\frac{e^x-1}{x}\right)^{\frac{1+x-e^x}{x}} \] is

If \((3,4,-7)\) is the foot of the perpendicular drawn from the point \((-2,3,6)\) to the plane \(\pi\), then the sum of the intercepts made by the plane \(\pi\) on the \(x\)- and \(y\)-axes is