List of top Questions asked in AP EAPCET- 2022

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): \[ \int_{a}^{b} e^x\left(f(x)+f'(x)\right)\,dx = e^bf(b)-e^af(a) \]

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

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

\(a,b,c,d,e,f,g,h\) 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

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

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 \[ g\left(\frac{\pi}{6}\right)-\sqrt{2}f\left(\frac{\pi}{6}\right)= \]

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

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

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

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 \(a+b=\)

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

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

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

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

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

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

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 \((3,4,-7)\) 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

Let the centroid of a triangle formed by the points \(A(4,x,1)\), \(B(y,-5,2)\), and \(C(7,8,3)\) be \(G(3,5,2)\), and \(CG\) meet \(AB\) in \(F\). Then \(F=\)

The radical centre of the circles \[ x^2+y^2+3x+2y+1=0, \] \[ x^2+y^2-x+6y+5=0 \] and \[ x^2+y^2+5x-8y+15=0 \] is

Let \[ x^2+y^2=16 \] be the equation of the auxiliary circle of a hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] and let \((4\sqrt{2},3)\) be a point on the hyperbola. Then the eccentricity of the hyperbola is

The values of \(\lambda\), for which \((\lambda,\lambda-2)\) lies inside the ellipse \[ 4x^2+9y^2=36 \] and outside the parabola \[ y^2=x, \] satisfy

The length of the latus rectum of the parabola \[ (x-2)^2+(y-3)^2=\frac{1}{25}(3x-4y+7)^2 \] is

The distance between the tangent lines to the hyperbola \[ x^2-2y^2=18 \] which are perpendicular to the line \(y=x\) is