List of top Mathematics Questions asked in AP EAPCET

The domain of the derivative of the function \( f(x) = \dfrac{x}{1 + |x|} \) is
Evaluate the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{2\sqrt{2} - \left(\cos x + \sin x\right)^3}{1 - \sin 2x} \]
Let \([x]\) denote the greatest integer less than or equal to \(x\). Then \[ \lim_{x \to 2^+} \left( \frac{[x]^3}{3} - \left[ \frac{x^3}{3} \right] \right) \]
The circumradius of the triangle formed by the points \( (2, -1, 1) \), \( (1, -3, -5) \), and \( (3, -4, -4) \) is
Let \( A(2, 3, 5), B(-1, 3, 2), C(\lambda, 5, \mu) \) be the vertices of \( \triangle ABC \). If the median through the vertex \( A \) is equally inclined to the coordinate axes, then
Equation of the plane passing through the origin and perpendicular to the planes \( x + 2y - z = 1 \) and \( 3x - 4y + z = 5 \) is
Circles are drawn through the point \( (2, 0) \) to cut intercepts of length 5 units on the X-axis. If their centre lies in the first quadrant, then their equation is
Assertion (A): The length of the latus rectum of an ellipse is 4. The focus and its corresponding directrix are respectively \( (1, -2) \) and the line \( 3x + 4y - 15 = 0 \). Then its eccentricity is \( \dfrac{1}{2} \). Reason (R): Length of the perpendicular drawn from focus of an ellipse to its corresponding directrix is \( \dfrac{a(1 - e^2)}{e} \)
Then which one of the following is correct?
If the locus of a point that divides a chord of slope 2 of the parabola \( y^2 = 4x \) internally in the ratio 1 : 2 is a parabola, then its vertex is
If the eccentricity of the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] passing through the point \( (4, 6) \) is 2, then the equation of the tangent to this hyperbola at (4, 6) is
A hyperbola passes through the point \( P(\sqrt{2}, \sqrt{3}) \) and has foci at \( (\pm 2, 0) \). Then the point that lies on the tangent drawn to this hyperbola at \( P \) is
If the lines \( 3x - 4y + 4 = 0 \) and \( 6x - 8y - 7 = 0 \) are the tangents to the same circle, then the area of that circle (in sq. units) is
If the smallest circle through the points of intersection of \( x^2 + y^2 = a^2 \) and \( x \cos \alpha + y \sin \alpha = p \), \( 0<p<a \), is \[ x^2 + y^2 - a^2 + \lambda(x \cos \alpha + y \sin \alpha - p) = 0 \] then \( \lambda = \)
A circle is drawn with its centre at the focus of the parabola \( y^2 = 2px \) such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is
If one of the lines given by the pair of lines \( 3x^2 - 2y^2 + axy = 0 \) is making an angle \( 60^\circ \) with the x-axis, then \( a = \)
A circle touches both the coordinate axes and the straight line \( L \equiv 4x + 3y - 6 = 0 \) in the first quadrant. If this circle lies below the line \( L = 0 \), then the equation of that circle is
A straight line passing through the origin \( O \) meets the parallel lines \( 4x + 2y = 9 \) and \( 2x + y + 6 = 0 \) at the points \( P \) and \( Q \) respectively. Then the point \( O \) divides the line segment \( PQ \) in the ratio
The coordinate axes are rotated about the origin in the counterclockwise direction through an angle \( 60^\circ \). If \( a \) and \( b \) are the intercepts made on the new axes by a straight line whose equation referred to the original axes is \( x + y = 1 \), then \( \dfrac{1}{a^2} + \dfrac{1}{b^2} = \, ? \)
The image of a point \( (2, -1) \) with respect to the line \( x - y + 1 = 0 \) is
If a straight line is at a distance of 10 units from the origin and the perpendicular drawn from the origin to it makes an angle \( \frac{\pi}{4} \) with the negative X-axis in the negative direction, then the equation of that line is
The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are \( (1, 2) \) and \( (4, 5) \), is:
On every evening, a student either watches TV or reads a book. The probability of watching TV is \( \frac{4}{5} \). If he watches TV, the probability that he will fall asleep is \( \frac{3}{4} \), and it is \( \frac{1}{4} \) when he reads a book. If the student is found to be asleep on an evening, the probability that he watched the TV is:
A bag P contains 4 red and 5 black balls, another bag Q contains 3 red and 6 black balls. If one ball is drawn at random from bag P and two balls are drawn from bag Q, then the probability that out of the three balls drawn two are black and one is red, is
For three events \( A, B, \) and \( C \) of a sample space, if \[ P(\text{exactly one of A or B occurs}) = P(\text{exactly one of B or C occurs}) = P(\text{exactly one of C or A occurs}) = \frac{1}{4} \] and the probability that all three events occur simultaneously is \( \frac{1}{16} \), then the probability that at least one of the events occurs is
A radar system can detect an enemy plane in one out of ten consecutive scans. The probability that it can detect an enemy plane at least twice in four consecutive scans is: