List of top Questions asked in AP EAPCET

If \[ f(x)= \begin{cases} \tan^{-1}x, & |x|\leq 1\\[2mm] \frac{1}{2}\left(|x|-1\right), & |x|\gt 1 \end{cases} \] then the domain of \[ \frac{d}{dx}f(x) \] is

Let \(A = (-3, -2, 7)\) and \(B = (3, 1, -2)\). Let a plane perpendicular to the line segment AB divide AB in the ratio 2:1. Then the intercept made by the plane on y-axis is

If \(d_1,d_2,d_3\) are the distances of the point \((1,2,3)\) from the \(X,Y,Z\)-coordinate axes respectively, then \(2d_2^2+d_3^2+1=\)

If \(P = (3, 12, 4)\) and \(Q\) is a point on the line \(OP\) such that \(OQ = 3\), then the sum of all the coordinates of \(Q\) is

If one of the roots of the equation \(x^2-5x-14=0\) is the length of the semi-conjugate axis of the hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] and the square of the other root is the semi-transverse axis, then the focus of the hyperbola that lies on the positive \(x\)-axis is

If the chord of contact of a point \(P\) w.r.t. the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) passes through a fixed point \((\alpha, \beta)\), then the locus of \(P\) is a straight line. The sum of the squares of the intercepts made by this line on the coordinate axes is

Let \[ E_1=\frac{x^2}{9}+\frac{y^2}{4}=1 \] and \[ E_2=\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] be two ellipses and \(R\) be a rectangle with sides parallel to the coordinate axes. Let \(E_1\) be inscribed ellipse in \(R\) and \(E_2\) be circumscribed ellipse on \(R\). If \(E_2\) passes through \((0,4)\), then

The straight line \(x \cos \alpha + y \sin \alpha = p\) cuts the circle \(x^2 + y^2 - a^2 = 0\) at A and B. Then the equation of circle having AB as diameter is

The shortest distance from the line \(3x + 4y = 25\) to the circle \(x^2 + y^2 - 6x + 8y = 0\) is

For any real number \(\lambda \neq 1\), the centre of the circle that passes through \(A(1, \lambda)\), \(B(\lambda, 1)\), and \(C(\lambda, \lambda)\) is

The equation of directrix of parabola \(x^2 + 8x + 12y + 4 = 0\) is

The ratio of the areas of the concentric circles \(x^2 + y^2 - 6x + 12y + 15 = 0\) and \(x^2 + y^2 - 6x + 12y - 15 = 0\) is

If a point \(P(\alpha, \beta)\) on the line \(y = 1\) is such that the two distinct chords drawn on \(x^2 + y^2 - \alpha x - y = 0\) from \(P\) are bisected by the x-axis, then

Suppose the pairs of straight lines \(x^2 - 2axy - y^2 = 0\) and \(x^2 - 2bxy - y^2 = 0\) are such that each pair bisects the angles between the other two. Then \(ab =\)

Given two points \(Q(3,4)\) and \(R(1,2)\). What is the point \(P(x,y)\) on the line \(2x - y - 1 = 0\) for which \(PQ + PR = QR\) holds?

Suppose the vertices of a triangle are given by \(A(0,3)\), \(B(-2,0)\), and \(C(6,1)\). For \((\alpha, \alpha + 1)\) to lie inside the triangle, \(\alpha\) should lie in the interval

For a triangle formed by \((0,0)\), \((4,0)\), and \((3,4)\), the orthocenter is

Suppose a triangle of area 27 sq. units is formed by \(18x^2 - 9xy + y^2 = 0\) and \(y = c\). Then the centroid of the triangle is

Four cards are drawn at random from a pack of 52 playing cards. The probability of getting all four cards of the same suit is

A special lottery is to be held to select a student who will live in the only deluxe room available in a hostel. 100 III year, 150 II year, and 200 I year students have applied for the room. Each III year student’s name is placed in the lottery 3 times, each II year student’s name 2 times, and each I year student’s name 1 time. The probability that a III year student gets the room is

In \(\Delta ABC\), if \(A = (1,2)\), and \(B\) and \(C\) lie on \(y = x + \alpha\) (where \(\alpha\) is variable), then the locus of the orthocenter of the triangle is

The discrete random variables \(X\) and \(Y\) are independent from one another and are defined as \(X \sim B(n_1, 0.5)\) and \(Y \sim B(n_2, 0.4)\). If the variance of both \(X\) and \(Y\) is 6, then \(\sqrt{n_1+n_2} =\)

The probability that A wakes up before the alarm rings is 0.4. Then, the mean and variance of the number of times A wakes up before the alarm rings in the next 7 days, respectively are:

A box contains 6 bottles of \(V_1\) drink, 3 bottles of \(V_2\) drink, and 4 bottles of \(V_3\) drink. If three bottles are drawn at random, find the probability that the three are not of the same variety.

One card is drawn at random from a well shuffled pack of 52 cards. The probability that the card drawn is a face card (Jack, Queen, and King only) is