List of top Mathematics Questions asked in AP EAPCET

In \( \triangle ABC \), the coordinates of the vertices are \( A(1,8,4) \), \( B(0,-11,4) \), and \( C(2,-3,1) \). If \( D \) is the foot of the perpendicular drawn from \( A \) to \( BC \), then the coordinates of \( D \) are:
The combined equation of the lines passing through the point \( (3, 4) \) and each making an angle of 45° with the line \( x + y + 1 = 0 \) is:
The locus of a point which is at a distance of 2 units from the line \( 2x - 3y + 4 = 0 \) and at a distance of \( \sqrt{13} \) units from a point \( (5,0) \), is:
If the lines given by \( (x^2 + y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2 \) are perpendicular to each other, then \( \sin^2 \alpha + \tan^2 \alpha = \):
The equal sides of an isosceles triangle are given by equations \( 7x - y + 3 = 0 \) and \( x + y - 3 = 0 \). If the slope \( m \) of the third side is an integer, then \( m = \):
If eight coins are tossed simultaneously, then the probability of getting at least six heads is:
In \( \triangle ABC \), the coordinates of the vertex \( A \) are \( (-3, 1) \). If the equation of the median through \( B \) is \( 2x + y - 3 = 0 \) and the equation of the bisector of angle \( C \) is \( 7x - 4y - 1 = 0 \), then the equation of the side \( BC \) is:
If the area of the triangle formed by the lines \( y = x + c \) and \( 2x^2 + 5xy + 3y^2 = 0 \) is \( \frac{1}{20} \) sq. units, then \( c = \)
The variance of 20 observations is 5. If each one of the observations is multiplied by 2, then the variance of the resulting observations is:
If four dice are thrown simultaneously, then the probability that none of the dice shows the number 1 on its face, is:
Let \( \overline{OA} = 2\overline{i} - 3\overline{j} + \overline{k} \), \( \overline{OB} = \overline{i} - 4\overline{j} - 3\overline{k} \), and \( \overline{OC} = -3\overline{i} + \overline{j} + 2\overline{k} \) be the position vectors of three points A, B, C respectively. If G is the centroid of triangle ABC, then find: \[ BC^2 + CA^2 + AB^2 + 9(OG)^2 \]
In a game, a pair of dice is rolled 24 times. If a person wins the game by not getting 6 on both the dice in any one of the 24 rolls, then the probability that a person wins the game is:
The range of a random variable \( X \) is \( \{0, 1, 2\} \). If \( P(X=0) = 3C^3 \), \( P(X=1) = 4C-10C^2 \), and \( P(X=2) = 5C-1 \), then the value of \( C \) is:
Bag \( B_1 \) contains 4 white and 2 black balls. Bag \( B_2 \) contains 3 white and 4 black balls. A bag is chosen at random and a ball is drawn from it at random, then the probability that the ball drawn is white, is:
If \[ \frac{2x^2 + 5x + 6}{(x + 2)^3} = \frac{a}{x + 2} + \frac{b}{(x + 2)^2} + \frac{c}{(x + 2)^3}, \] then \( a \cdot b + b \cdot c + c \cdot a = \)?

\(3\mathbf{\overline{i}} - 2\mathbf{\overline{j}} - \mathbf{\overline{k}}, -2\mathbf{\overline{i}} - \mathbf{\overline{j}} + 3\mathbf{\overline{k}}, -\mathbf{\overline{i}} + 3\mathbf{\overline{j}} - 2\mathbf{\overline{k}}\) are the position vectors of the vertices \( A \), \( B \), and \( C \) of a triangle \( ABC \)respectively. If \( H \) is its orthocenter, then find \( \overline{HA} + \overline{HB} + \overline{HC} \).

If \( P_1, P_2, P_3 \) are the lengths of the altitudes drawn from the vertices \( A, B, C \) of \( \triangle ABC \) respectively, then: \[ \cos A \cdot \frac{1}{P_1} + \cos B \cdot \frac{1}{P_2} + \cos C \cdot \frac{1}{P_3} = ? \]
Let \( \pi_1 \) be the plane determined by the vectors \( \overline{\mathbf{i}} + 2\overline{\mathbf{j}} - 2\overline{\mathbf{k}} \) and \( \overline{\mathbf{i}} + 3\overline{\mathbf{j}} - 2\overline{\mathbf{k}} \). Let \( \pi_2 \) be the plane determined by the vectors \( \overline{\mathbf{j}} + 2\overline{\mathbf{k}} \) and \( 3\overline{\mathbf{k}} - 2\overline{\mathbf{i}} \). If \( \theta \) is the angle between \( \pi_1 \) and \( \pi_2 \), then \( \cos \theta = \)
If \( \overline{\mathbf{a}} = 2\overline{\mathbf{i}} + 3\overline{\mathbf{j}} + 4\overline{\mathbf{k}} \), \( \overline{\mathbf{b}} = 3\overline{\mathbf{i}} + 4\overline{\mathbf{k}} \), and \( \overline{\mathbf{c}} = 5\overline{\mathbf{i}} + 4\overline{\mathbf{k}} \) are three vectors, then a vector which is perpendicular to \( \overline{\mathbf{a}} \) and \( \overline{\mathbf{b}} \times \overline{\mathbf{c}} \) is
If \( \mathbf{a} \) and \( \mathbf{b} \) are two vectors such that \( \mathbf{a} \) is not parallel to \( \mathbf{b} \), and if \[ p = (x + 2y + 3)\mathbf{a} + (5x - y + 2)\mathbf{b}, \quad q = (2x + 3y + 5)\mathbf{a} + (x - 5y - 2)\mathbf{b} \] are two vectors such that \( p = 2q \), then find \( x - 2y \).
If \( \alpha = \log_e(2 + \sqrt{3}) \), then evaluate: \[ \frac{\cosh \alpha }{1 - \tanh \alpha} + \frac{\sinh \alpha}{1 - \coth \alpha} \]
If \( 1 + \sqrt{1 + a} = (1 + \sqrt{1 - a}) \cot \alpha \) and \( 0<a<1 \), then \( \sin 4\alpha = \)
If \( A = \frac{\pi}{24} \), then \[ \frac{\cos A + \cos 3A + \cos 5A + \cos 7A}{\sin A + \sin 3A + \sin 5A + \sin 7A} \]
If \( \sec(\theta + \alpha) \), \( \sec(\theta) \), and \( \sec(\theta - \alpha) \) are in arithmetic progression, then \( \sin^2 \theta = \)
In \( \triangle ABC \), if \[ \cot \frac{A}{2} : \cot \frac{B}{2} : \cot \frac{C}{2} = 3 : 7 : 9 \] then \( a : b : c = \) ?