List of top Mathematics Questions asked in TS EAMCET

The pole of the line \(x - 5y - 7 = 0\) with respect to the circle \(S \equiv x^2 + y^2 - 2x - 2y + 1 = 0\) is \(P(a,b)\). If \(C\) is the centre of the circle \(S = 0\) then \(PC =\):
P and Q are the extremities of a focal chord of the parabola \( y^2 = 4ax \). If \( P = (9,9) \) and \( Q = (p, q) \), then \( p - q \) is:
If the distance from a variable point \( P \) to the point \( (4,3) \) is equal to the perpendicular distance from \( P \) to the line \( x + 2y - 1 = 0 \), then the equation of the locus of the point \( P \) is:
If \( Q \) and \( R \) are the images of the point \( P(2,3) \) with respect to the lines \( x - y + 2 = 0 \) and \( 2x + y - 2 = 0 \) respectively, then \( Q \) and \( R \) lie on
If the line \( 4x - 3y + p = 0 \) (where \( p + 3>0 \)) touches the circle \( x^2 + y^2 - 4x + 6y + 4 = 0 \) at the point \( (h,k) \), then the value of \( h - 2k \) is:
P and Q are the points of trisection of the line segment joining the points \( (3, -7) \) and \( (-5,3) \). If line PQ subtends a right angle at a variable point R, then the locus of R is:
If a circle passing through the point \((1,1)\) cuts the circles \(x^2 + y^2 + 4x - 5 = 0\) and \(x^2 + y^2 - 4x + 3 = 0\) orthogonally, then the center of that circle is:
If \( (h, k) \) is the internal center of similitude of the circles \( x^2 + y^2 + 2x - 6y + 1 = 0 \) and \( x^2 + y^2 - 4x + 2y + 4 = 0 \), then find \( 4h \).
The centroid of a variable triangle ABC is at the distance of 5 units from the origin. If A = (2,3) and B = (3,2), then the locus of C is:
If \( (4,3) \) and \( (1,-2) \) are the endpoints of a diagonal of a square, then the equation of one of its sides is:
(a,b) is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation \(2x^2 - 3xy + 4y^2 + 5y - 6 = 0\). If the angle by which the axes are to be rotated in positive direction about the origin to remove the \(xy\)-term from the equation \(ax^2 + 23abxy + by^2 = 0\) is \( \theta \), then \(2\theta =\)
The area of the parallelogram formed by the lines \( L_1: \lambda x + 4y + 2 = 0 \), \( L_2: 3x + 4y - 3 = 0 \), \( L_3: 2x + \mu y + 6 = 0 \), and \( L_4: 2x + y + 3 = 0 \), where \( L_1 \) is parallel to \( L_2 \) and \( L_3 \) is parallel to \( L_4 \), is

The length of the normal drawn at \( t = \frac{\pi}{4} \) on the curve \( x = 2(\cos 2t + t \sin 2t) \), \( y = 4(\sin 2t + t \cos 2t) \) is: 

If \( y = a \cos 3x + b e^{-x} \), then \( y'(3\sin 3x - \cos 3x) = \):

If the real-valued function

\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]

is not defined for all \( x \in (-\infty, a] \cup (b, \infty) \), then what is \( 3^a + b^2 \)?

The point of intersection of two tangents drawn to the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{4} = 1 \] lie on the circle \[ x^2 + y^2 = 5. \] If these tangents are perpendicular to each other, then \( a \) is:
A circle \( S \) passes through the points of intersection of the circles \( x^2 + y^2 - 2x - 3 = 0 \) and \( x^2 + y^2 - 2y = 0 \). If \( x + y + 1 = 0 \) is a tangent to the circle \( S \), then the equation of \( S \) is:
If \( (1,1) \) is the vertex and \( x + y + 1 = 0 \) is the directrix of a parabola. If \( (a, b) \) is its focus and \( (c, d) \) is the point of intersection of the directrix and the axis of the parabola, then \( a + b + c + d \) is:
$$ x^5 + 4x^4 - 13x^3 - 52x^2 + 36x + 144 = 0 $$
If \(\theta\) is the angle between the vectors \(4\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \( \mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\), then \(\sin 2\theta\) is:
Given the position vectors of points \(A\) and \(B\) as \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \(C\) divides \(AB\) in the ratio 3:2. If \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) is the position vector of point \(D\), find the unit vector in the direction of \( \mathbf{CD} \):

In a triangle \(ABC\), if

\[ (a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = a^2 + b^2, \]

then \( \cos A \) is:

If \[ \frac{1}{x^4 + x^2 + 1} = \frac{Ax + B}{x^2 + ax + 1} + \frac{Cx + D}{x^2 - ax + 1} \] then \( A + B - C + D = ? \)
If \( \theta \) is the acute angle between the curves \( y^2 = x \) and \( x^2 + y^2 = 2 \), then \( \tan \theta \) is:

Among the 5 married couples, if the names of 5 men are matched with the names of their wives randomly, then the probability that no man is matched with the name of his own wife is ?