List of top Mathematics Questions asked in TS EAMCET

Two cards are drawn at random one after the other with replacement from a pack of playing cards. If \( X \) is the random variable denoting the number of ace cards drawn, then the mean of the probability distribution of \( X \) is:

A rectangle is formed by the lines \[ x = 4, \quad x = -2, \quad y = 5, \quad y = -2 \] and a circle is drawn through the vertices of this rectangle. The pole of the line \[ y + 2 = 0 \] with respect to this circle is: 

The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is: 

If the equation of the circle which cuts each of the circles \[ x^2 + y^2 = 4, \] \[ x^2 + y^2 - 6x - 8y + 10 = 0, \] \[ x^2 + y^2 + 2x - 4y - 2 = 0 \] at the extremities of a diameter of these circles is \[ x^2 + y^2 + 2gx + 2fy + c = 0, \] then the value of \( g + f + c \) is:

The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is: 

Given that \( a \) and \( b \) are the semi-major and semi-minor axes of an ellipse whose axes are along the coordinate axes. If its latus rectum is of length 4 units and the distance between its foci is \( 4\sqrt{2} \), then the value of \( a^2 + b^2 \) is:
If the harmonic conjugate of \( P(2,3,4) \) with respect to the line segment joining the points \[ A(3,-2,2) \quad \text{and} \quad B(6,-17,-4) \] is \( Q(\alpha, \beta, \gamma) \), then the value of \( \alpha + \beta + \gamma \) is:
If \( L \) is the line of intersection of two planes \[ x + 2y + 2z = 15 \quad \text{and} \quad x - y + z = 4 \] and the direction ratios of the line \( L \) are \( (a, b, c) \), then the value of \[ \frac{a^2 + b^2 + c^2}{b^2} \] is:

The foot of the perpendicular drawn from \( A(1,2,2) \) onto the plane \[ x + 2y + 2z - 5 = 0 \] is \( B(a, \beta, \gamma) \). If \( \pi(x,y,z) = x + 2y + 2z + 5 = 0 \) is a plane then \(-\pi(A):\pi(B) \) is: 

If \[ y = \log \left( x - \sqrt{x^2 - 1} \right), \] then \[ (x^2 - 1)y'' + xy' + e^y + \sqrt{x^2 - 1} = \] evaluates to:
The sum of the maximum and minimum values of the function \[ f(x) = \frac{x^2 - x + 1}{x^2 + x + 1} \] is:
If the area of the region enclosed by the curve \( ay = x^2 \) and the line \( x + y = 2a \) is \( k a^3 \), then \( k \) is:
The order and degree of the differential equation \[ \frac{dy}{dx} + \left( \frac{d^2y}{dx^2} + 2 \right)^{\frac{1}{2}} + \frac{d^3y}{dx^3} + 5 = 0 \] are respectively:

If \( y = \sin x + A \cos x \) is the general solution of \[ \frac{dy}{dx} + f(x)y = \sec x, \] then an integrating factor of the differential equation is: 

Evaluate the integral: \[ I = \int \frac{1}{x^m \sqrt[m]{x^m + 1}} dx. \]
A rhombus is inscribed in the region common to the two circles \[ x^2 + y^2 - 4x - 12 = 0 \] and \[ x^2 + y^2 + 4x - 12 = 0. \] If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area (in Sq. units) of the rhombus is:
If \( m \) is the slope and \( P(\beta, \beta) \) is the midpoint of a chord of contact of the circle \[ x^2 + y^2 = 125, \] then the number of values of \( \beta \) such that \( \beta \) and \( m \) are integers is: \
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:
If \( X \sim B(6, p) \) is a binomial variate and \[ \frac{P(X=4)}{P(X=2)} = \frac{1}{9}, \] then the value of \( p \) is:
If \( z = x + iy \) satisfies the equation \[ z^2 + az + a^2 = 0, \quad a \in \mathbb{R}, \] then:
If \( \omega \) is the complex cube root of unity and \[ \left( \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} \right)^k + \left( \frac{a + b\omega + c\omega^2}{b + a\omega^2 + c\omega} \right)^2 = 2, \] then \( 2k + 1 \) is always:
If \( f(x) \) is given as: \[ f(x) = \begin{cases} \frac{8}{x^3} - 6x, & \text{if } 0 < x \leq 1 \\ \frac{x - 1}{\sqrt{x}-1}, & \text{if } x > 1 \end{cases} \] is a real-valued function, then at \( x = 1 \), \( f \) is:
\[ \lim_{n\to\infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right)\cdots\left(1+\frac{n^2}{n^2}\right)\right]^{\frac{1}{n}} = \]
Evaluate \( \int_{-2}^{2} x^4 (4 - x^2)^{\frac{7}{2}} \, dx \):
Area of the region enclosed between the curves \(y^{2}=4(x+7)\) and \(y^{2}=5(2-x)\) is