List of top Mathematics Questions asked in AP EAPCET

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 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
The image of a point \( (2, -1) \) with respect to the line \( x - y + 1 = 0 \) 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 \( \dfrac{1}{a^2} + \dfrac{1}{b^2} = \, ? \)
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 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 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
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:
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
Let \( X \) be the random variable taking values \( 1, 2, \dots, n \) for a fixed positive integer \( n \). If \( P(X = k) = \frac{1}{n} \) for \( 1 \leq k \leq n \), then the variance of \( X \) is:
The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are \( (1, 2) \) and \( (4, 5) \), 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
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:
If \( \sum\limits_{i=1}^{9} (x_i - 5) = 9 \) and \( \sum\limits_{i=1}^{9} (x_i - 5)^2 = 45 \), then the standard deviation of the nine observations \( x_1, x_2, \ldots, x_9 \) is
In \( \triangle ABC \), if \( \sin^2 B = \sin A \) and \( 2\cos^2 A = 3\cos^2 B \), then the triangle is:
Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is \( \frac{1}{4} \) and the probability that the second student gets qualified in the same exam is \( \frac{2}{5} \), then the probability that at least one of them gets qualified in that exam is
Let \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \), \( \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k} \), and \( \vec{c} = \hat{i} - 2\hat{j} + 3\hat{k} \) be three vectors. If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b} \) and \( \vec{r} . \vec{c} = 18 \), then the magnitude of the orthogonal projection of \( 4\hat{i} + 3\hat{j} - \hat{k} \) on \( \vec{r} \) is:
The set of all real values of \( c \) so that the angle between the vectors
\( \vec{a} = c\hat{i} - 6\hat{j} + 3\hat{k} \) and \( \vec{b} = x\hat{i} + 2\hat{j} + 2c\hat{k} \) is an obtuse angle for all real \( x \), is:
If the position vectors of A, B, C, D are \( \vec{A} = \hat{i} + 2\hat{j} + 2\hat{k}, \vec{B} = 2\hat{i} - \hat{j}, \vec{C} = \hat{i} + \hat{j} + 3\hat{k}, \vec{D} = 4\hat{j} + 5\hat{k} \), then the quadrilateral ABCD is a:
If in \( \triangle ABC \), \( B = 45^\circ \), \( a = 2(\sqrt{3} + 1) \) and area of \( \triangle ABC \) is \( 6 + 2\sqrt{3} \) sq. units, then the side \( b = \ ? \)
If \( \sinh^{-1}(2) + \sinh^{-1}(3) = \alpha \), then \( \sinh\alpha = \) ?
In \( \triangle ABC \), if A, B, C are in arithmetic progression, then \[ \sqrt{a^2 - ac + c^2} . \cos\left(\frac{A - C}{2}\right) =\ ? \]
The equation \[ \cos^{-1}(1 - x) - 2 \cos^{-1} x = \frac{\pi}{2} \] has: