List of top Mathematics Questions asked in TS EAMCET

Evaluate the limit: \[ \lim_{\theta \to \frac{\pi}{2}} \frac{8\tan^4\theta + 4\tan^2\theta + 5}{(3 - 2\tan\theta)^4} \]
The number of ways in which 4 different things can be distributed to 6 persons so that no person gets all the things is \;?
If \(^nC_r = C_r\) and \[ 2 \frac{C_1}{C_0} + 4 \frac{C_2}{C_1} + 6 \frac{C_3}{C_2} + \dots + 2n \frac{C_n}{C_{n-1}} = 650 \] then \[ ^nC_2 = \ ? \]
If \( \lim_{x \to 4} \frac{2x^2 + (3+2a)x + 3a{x^3 - 2x^2 - 23x + 60} = \frac{11}{9} \), then find \( \lim_{x \to a} \frac{x^2 + 9x + 20}{x^2 - x - 20} \).}
If \( \vec{b} = \mathbf{i}-\mathbf{j} +2\mathbf{k},\;\vec{c} = \mathbf{i} +2\mathbf{j} -\mathbf{k} \) are two vectors and \( \vec{a} \) is a vector such that \( \cos\angle(\vec{a},\vec{b}\times \vec{c}) = \frac{2}{\sqrt{3}} \). If \( \vec{a} \) is a unit vector, then \( \lvert \vec{a}\times(\vec{b}\times \vec{c})\rvert =\,? \)
The real-valued function \[ f(x) = \frac{|x - a|}{x - a} \] is analyzed as follows:
The equation of the straight line whose slope is \( -\frac{2}{3} \) and which divides the line segment joining \( (1,2) \) and \( (-3,5) \) in the ratio 4:3 externally is:

Solving the System of Linear Equations

If (x,y,z) = (α,β,γ) is the unique solution of the system of simultaneous linear equations:

    3x - 4y + 2z + 7 = 0,    2x + 3y - z = 10,    x - 2y - 3z = 3,    

then α = ?

Evaluate the integral: \[ \int 4\cos^2 x - 5\sin^2 x \cos x \, dx. \]

Given the function:

\[ f(x) = \frac{2x - 3}{3x - 2} \]

and if \( f_n(x) = (f \circ f \circ \ldots \circ f)(x) \) is applied \( n \) times, find \( f_{32}(x) \).

Given \((\sin \theta - \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = 5\) and \(\theta\) lies in the third quadrant, find \((\sin \theta + \cos \theta)^3\).
If \[ \frac{3x^4 - 2x^2 +1}{(x-2)^4} = A + \frac{B}{x-2} + \frac{C}{(x-2)^2} + \frac{D}{(x-2)^3} + \frac{E}{(x-2)^4}, \] then \(2A + 3B - C - D + E =\;?\)
If \(\frac{x^2}{2x^2 + 7x + 6} = \frac{Ax + B}{x^2 + a} + \frac{Cx + D}{ax^2 + 3}\), then \(A + B + C - 2D =\)
f is a real valued function satisfying the relation \(f\bigl(3x + \tfrac{1}{2x}\bigr) \;=\; 9x^2 + \tfrac{1}{4x^2}\). If \(f\bigl(x + \tfrac{1}{x}\bigr) = 1\) then \(x =\) ?
The position vectors of the vertices \( A, B, C \) of a triangle are given as: \[ \vec{A} = \hat{i} - 2\hat{j} + k, \quad \vec{B} = 2\hat{i} + \hat{j} - k, \quad \vec{C} = \hat{i} - \hat{j} - 2k \] If \( D \) and \( E \) are the midpoints of \( BC \) and \( CA \) respectively, then the unit vector along \( \overrightarrow{DE} \) is:
If are the roots of the equation \[ 2x^3 \;-\; 5x^2 \;+\; 4x \;-\; 3 \;=\; 0, \] then \[ \sum \alpha\beta(\alpha+\beta) \;=\;? \]
If the coefficients of three consecutive terms in the expansion of (1 + x)23 are in arithmetic progression, then those terms are ?
If \( T_4 \) represents the 4th term in the expansion of \( \left( 5x + \frac{7}{x} \right)^{-\frac{3}{2}} \) and \( x \not\in \left[ \frac{\sqrt{7}}{5}, \frac{\sqrt{7}}{5} \right] \), then \( \left( x^3 \cdot \sqrt{5x} \right) T_4 = \):}

If Rolle's Theorem is applicable for the function:

\[ f(x) = \begin{cases} x^p \log x, & x \neq 0 \\ 0, & x = 0 \end{cases} \]

on the interval \([0,1]\), then a possible value of \( p \) is:

If the function \( f(x) \) is given by \[ f(x) = \begin{cases} \frac{\tan(a(x-1))}{\frac{x-1}{x}}, & tif0<x<1 
\frac{x^3-125}{x^2 - 25} , & \text{if } 1 \leq x \leq 4 
\frac{b^x - 1}{x}, & \text{if } x>4 \end{cases} \] is continuous in its domain, then find \( 6a + 9b^4 \).

The number of normals that can be drawn through the point \((9,6)\) to the parabola \(y^2 = 4x\) is:
If \(\vec{a},\vec{b}\) are two vectors such that \(\lvert \vec{a}\rvert =3,\;\lvert \vec{b}\rvert =4,\;\lvert \vec{a}+\vec{b}\rvert =\sqrt{37},\;\lvert \vec{a}-\vec{b}\rvert = k,\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\theta,\) then \(\frac{4}{13}\,(k \sin \theta)^2 =\,?\)
If the normal drawn at the point \( P(9, 9) \) on the parabola \( y^2 = 9x \) meets the parabola again at \( Q(a, b) \), then \( 2a + b = \text{?} \)
S' is the focus of the ellipse \( \frac{x^2}{25} + \frac{y^2}{b^2} = 1, (b<5) \) lying on the negative X-axis and \( P(\theta) \) is a point on this ellipse. If the distance between the foci of this ellipse is 8 and \( S'P = 7 \), then \( \theta \) is:
The trigonometric equation \( \sin^{-1}x = 2\sin^{-1}a \) has a solution. Find the valid range for \( a \).