List of top Mathematics Questions asked in TS EAMCET

\(\lim_{n \to \infty} \frac{1}{n^2} \left[ e^{1/n} + 2e^{2/n} + 3e^{3/n} + \dots + 2n e^{2n/n} \right] =\)

If \(\int \frac{dx}{(x^2+9)\sqrt{x^2+16}} = \frac{1}{3\sqrt{7}} \tan^{-1} \left( K \frac{x}{\sqrt{16+x^2}} \right) + c\), then \(K =\)

\(\int \frac{x^3}{x^4 + 3x^2 + 2} dx =\)

If \(I_n = \int \frac{1}{(x^2+1)^n} dx\), then \(2n I_{n+1} - (2n-1) I_n =\)

\(\int \left( \frac{1}{x^2} + \frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} \right) dx =\)

If \(f:[a,b] \to [c,d]\) is a continuous and strictly increasing function, then \(\frac{d-c}{b-a}\) is

If a particle is moving in a straight line so that after \(t\) seconds its distance \(S\) (in cms) from a fixed point on the line is given by \(S = f(t) = t^3 - 5t^2 + 8t\) then the acceleration of the particle at \(t=5\) sec is (in cm/sec\(^2\))

\(P(5,2)\) is a point on the curve \(y=f(x)\) and \(\frac{7}{2}\) is the slope of the tangent to the curve at P. The area of the triangle formed by the tangent and the normal to the curve at P with x-axis is

The local maximum value \(l\) and local minimum value \(m\) of \(f(x) = \frac{x^2+2x+2}{x+1}\) in \(\mathbb{R} - \{-1\}\) exist at \(\alpha, \beta\) respectively, then \(\frac{l+m}{\alpha+\beta} =\)

The radius of a cone of height 9 units is changed from 2 units to 2.12 units. The exact change and approximate change in the volume of the cone are respectively

If \(y = (\sin^{-1}x)^2\), then \((1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} =\)

If \(\frac{d}{dx}\left\{ \frac{x-1}{x-\sqrt{x}} e^{2x+1} \right\} = \frac{x-1}{x-\sqrt{x}} e^{2x+1} f(x)\), then \(f(4) =\)

Consider the following statements
Assertion (A): For \(x \in \mathbb{R} - \{1\}\), \(\frac{d}{dx}\left(\tan^{-1}\left(\frac{1+x}{1-x}\right)\right) = \frac{d}{dx}(\tan^{-1}x)\)
Reason (R): For \(x<1\), \(\tan^{-1}\left(\frac{1+x}{1-x}\right) = \frac{\pi}{4} + \tan^{-1}x\),
for \(x>1\), \(\tan^{-1}\left(\frac{1+x}{1-x}\right) = -\frac{3\pi}{4} + \tan^{-1}x\)
The correct answer is

If the function \(g(x) = \begin{cases} K\sqrt{x+1} & , 0 \le x \le 3 \\ mx + 2 & , 3<x \le 5 \end{cases}\) is differentiable, then \(K + m =\)

If \(f(x) = \begin{cases} \frac{a\sin x - bx + cx^2 + x^3}{2\log(1+x) - 2x^3 + x^4} & , x \neq 0 \\ 0 & , x = 0 \end{cases}\) is continuous at \(x = 0\), then

If \(f(x) = \frac{x(a^x - 1)}{1 - \cos x}\) and \(g(x) = \frac{x(1 - a^x)}{a^x \left(\sqrt{1 - x^2} - \sqrt{1 + x^2}\right)}\), then \(\lim_{x \to 0} (f(x) - g(x)) =\)

If the points \((1, 1, \lambda)\) and \((-3, 0, 1)\) are equidistant from the plane \(3x + 4y - 12z + 13 = 0\), then the values of \(\lambda\) are

If L is a line common to the planes \(3x + 4y + 7z = 1\), \(x - y + z = 5\) then the direction ratios of the line L are

O(0,0,0), A(3,1,4), B(1,3,2) and C(0,4,-2) are the vertices of a tetrahedron. If G is the centroid of the tetrahedron and \(G_1\) is the centroid of its face ABC, then the point which divides \(GG_1\) in the ratio 1:2 is

Let \(x\) be the eccentricity of a hyperbola whose transverse axis is twice its conjugate axis. Let \(y\) be the eccentricity of another hyperbola for which the distance between the foci is 3 times the distance between its directrices. Then \(y^2 - x^2 =\)

If any tangent drawn to the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\) touches one of the circles \(x^2 + y^2 = \alpha^2\), then the range of \(\alpha\) is

The curve represented by \(\frac{x^2}{12-\alpha} + \frac{y^2}{\alpha-10} = 1\) is

The locus of a point which divides the line segment joining the focus and any point on the parabola \(y^2 = 12x\) in the ratio \(m:n\) (\(m+n \ne 0\)) is a parabola. Then the length of the latus rectum of that parabola is

For the parabola \(y = x^2 - 3x + 2\), match the items in list-1 to that of the items in list-2.
S is a focus, Z is intersection of axis and directrix, P is one end point of latus rectum, Q is the point on the parabola at which tangent is parallel to X-axis

A circle S given by \(x^2 + y^2 - 14x + 6y + 33 = 0\) cuts the X-axis at A and B (OB \(>\) OA). C is midpoint of AB. L is a line through C and having slope \((-1)\). If L is the diameter of a circle S' and also the radical axis of the circles S and S', then the equation of the circle S' is