List of top Questions asked in AP EAPCET

\(\begin{vmatrix} a+b+2c & a & b \\[0.3em] c & b+c+2c & b \\[0.3em] c & a & c+a2b \end{vmatrix}\)

If \(n\) is an integer and \(Z = \cos \theta + i \sin \theta\), \(\theta \neq (2n + 1)\) \(\frac{\pi}{2}\), then: \(\frac{1 + Z^{2n}}{1 - Z^{2n}}\) = ?

The equation \( x^4 - x^3 - 6x^2 + 4x + 8 = 0 \) has two equal roots. If \( \alpha, \beta \) are the other two roots of this equation, then \( \alpha^2 + \beta^2 = \):
Four digit numbers with all digits distinct are formed using the digits 1, 2, 3, 4, 5, 6, 7 in all possible ways. If \( p \) is the total number of numbers thus formed and \( q \) is the number of numbers greater than 3400 among them, then \( p : q = \):
If \[ \frac{x + 2}{(x^2 + 3)(x^4 + x^2)(x^2 + 2)} = \frac{Ax + B}{x^2 + 3} + \frac{Cx + D}{x^2 + 2} + \frac{Ex^3 + Fx^2 + Gx + H}{x^4 + x^2}, \] then \[ (E + F)(C + D)(A) = \]
Assertion (A): If \( A = 10^\circ, B = 16^\circ, C = 19^\circ \), then: \[ \tan(2A) \tan(2B) + \tan(2B) \tan(2C) + \tan(2C) \tan(2A) = 1. \] Reason (R): If \( A + B + C = 180^\circ \), then: \[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right) + \cot\left(\frac{C}{2}\right) = \cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right). \]
Number of solutions of the trigonometric equation \[ 2 \tan 2\theta - \cot 2\theta + 1 = 0 \quad \text{lying in the interval} \quad [0, \pi] \]
If 7 and 8 are the lengths of two sides of a triangle and \( a \) is the length of its smallest side. The angles of the triangle are in AP and \( a \) has two values \( a_1 \) and \( a_2 \) satisfying this condition. If \( a_1 < a_2 \), then \( 2a_1 + 3a_2 = \):
If \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) are position vectors of 4 points such that \( 2\vec{a} + 3\vec{b} + 5\vec{c} - 10\vec{d} = 0 \), then the ratio in which the line joining \( \vec{c} \) divides the line segment joining \( \vec{a} \) and \( \vec{b} \) is:
The angle between the planes \( \vec{r} \cdot (12\hat{i} + 4\hat{j} - 3\hat{k}) = 5 \) and \( \vec{r} \cdot (5\hat{i} + 3\hat{j} + 4\hat{k}) = 7 \) is:
The shortest distance between the skew lines \( \vec{r} = (2\hat{i} - \hat{j}) + t(\hat{i} + 2\hat{k}) \) and \( \vec{r} = (-2\hat{i} + \hat{k}) + s(\hat{i} - \hat{j} - \hat{k}) \) is:

The coefficient of variation for the frequency distribution is:

Two persons A and B throw a pair of dice alternately until one of them gets the sum of the numbers appeared on the dice as 4 and the person who gets this result first is declared as the winner. If A starts the game, then the probability that B wins the game is:
If \( X \sim B(5, p) \) is a binomial variate such that \( p(X = 3) = p(X = 4) \), then \( P(|X - 3| < 2) = \dots \)
The perimeter of the locus of the point \( P \) which divides the line segment \( QA \) internally in the ratio 1:2, where \( A = (4, 4) \) and \( Q \) lies on the circle \( x^2 + y^2 = 9 \), is:

If the pair of lines represented by $$ 3x^2 - 5xy + P y^2 = 0 $$ and $$ 6x^2 - xy - 5y^2 = 0 $$ have one line in common, then the sum of all possible values of \( P \) is: 

The pole of the straight line \[ 9x + y - 28 = 0 \] with respect to the circle \[ 2x^2 + 2y^2 - 3x + 5y - 7 = 0 \] is:

The equation of a circle which touches the straight lines $x + y = 2$, $x - y = 2$ and also touches the circle $x^2 + y^2 = 1$ is:

If the ordinates of points \( P \) and \( Q \) on the parabola \[ y^2 = 12x \] are in the ratio 1:2, then the locus of the point of intersection of the normals to the parabola at \( P \) and \( Q \) is:

The product of perpendiculars from the two foci of the ellipse $$ \frac{x^2}{9} + \frac{y^2}{25} = 1 $$ on the tangent at any point on the ellipse is: 

The value of \( c \) such that the straight line joining the points \[ (0,3) \quad \text{and} \quad (5,-2) \] is tangent to the curve \[ y = \frac{c}{x+1} \] is:
If the plane \[ x - y + z + 4 = 0 \] divides the line joining the points \[ P(2,3,-1) \quad \text{and} \quad Q(1,4,-2) \] in the ratio \( l:m \), then \( l + m \) is:
If the line with direction ratios \[ (1, a, \beta) \] is perpendicular to the line with direction ratios \[ (-1,2,1) \] and parallel to the line with direction ratios \[ (\alpha,1,\beta), \] then \( (\alpha, \beta) \) is:
The equation of the plane which bisects the angle between the planes $$ 3x - 6y + 2z + 5 = 0 \quad \text{and} \quad 4x - 12y + 3z - 3 = 0 \quad \text{which contains the origin is:} $$ 
Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}. \]