List of top Questions asked in AP EAMCET

A car travels with a speed of \(40\ \text{km h}^{-1}\). Rain drops are falling at a constant speed vertically. The traces of the rain on the side windows of the car make an angle of \(30^\circ\) with the vertical. The magnitude of the velocity of the rain with respect to the car is

If \(N_A\), \(N_B\) and \(N_C\) are the number of significant figures in \(A=0.001204\text{ m}\), \(B=43120000\text{ m}\) and \(C=1.200\text{ m}\) respectively then

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

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

A projectile with speed \(50\ \text{m s}^{-1}\) is thrown at an angle of \(60^\circ\) with the horizontal. The maximum height that can be reached is
\[ \text{Acceleration due to gravity}=10\ \text{m s}^{-2} \]

A car covers a distance at speed of \(60\text{ km h}^{-1}\). It returns and comes back to the original point moving at a speed of \(V\). If the average speed for the round trip is \(48\text{ km h}^{-1}\), then the magnitude of \(V\) is

If the solution of \[ \frac{dy}{dx}-y\log_e0.5=0,\quad y(0)=1, \] and \(y(x)\to k\), as \(x\to\infty\) then \(k=\)

The value of
\[ \int_0^1 a^k x^k\,dx \] is

Let \(\alpha\) and \(\beta\), \((\alpha\lt \beta)\), are roots of \(18x^2-9\pi x+\pi^2=0\), \(f(x)=x^2\), \(g(x)=\cos x\). Then \(\displaystyle \int_{\alpha}^{\beta} x(gof(x))dx=\)

\[ \int_{0}^{\pi} x\left(\sin^2(\sin x)+\cos^2(\cos x)\right)dx= \]

\[ \lim_{n\to\infty} \left( \frac{1}{1+n^5} + \frac{2^4}{2^5+n^5} + \frac{3^4}{3^5+n^5} +\cdots+ \frac{n^4}{n^5+n^5} \right) = \]

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

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

The condition that \(f(x)=ax^3+bx^2+cx+d\) has no extreme value is

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

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

Evaluate
\[ \int \frac{e^{\cot x}}{\sin^2 x}\left(2\log\cosec x+\sin2x\right)\,dx \]

If \(ax+by=1\) is a normal to the parabola
\[ y^2=4px, \] then the condition is

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

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

If
\[ x^3-2x^2y^2+5x+y-5=0, \] then at \((1,1)\), \(y''(1)=\)

If
\[ f(x)=\max\{3-x,\;3+x,\;6\} \] is not differentiable at \(x=a\), and \(x=b\), then \(|a|+|b|=\)

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

Let \(f:\mathbb{R}^+\to\mathbb{R}^+\) be a function satisfying
\[ f(x)-x=\lambda \text{ constant},\quad \forall x\in\mathbb{R}^+ \] and
\[ f(f(y))=f(xy)+x,\quad \forall x,y\in\mathbb{R}^+. \] Then
\[ \lim_{x\to0}\frac{(f(x))^{1/3}-1}{(f(x))^{1/2}-1}= \]

If
\[ \lim_{x\to0}\frac{|x|}{\sqrt{x^4+4x^2+5}}=k, \] and
\[ \lim_{x\to0}x^4\sin\left(\frac{1}{3\sqrt{x}}\right)=l, \] then \(k+l=\)