List of top Mathematics Questions asked in AP EAPCET

If $ P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e $ is a polynomial such that: \[ P(0) = 1, \quad P(1) = 2, \quad P(2) = 5, \] \[ P(3) = 10, \quad P(4) = 17, \] then find the value of $ P(5) $=

The area of the region enclosed by the curves \[ 3x^2 - y^2 - 2xy + 4x + 1 = 0 \] and \[ 3x^2 - y^2 - 2xy + 6x + 2y = 0 \] is:

A line $ L $ passing through the point $ (2,0) $ makes an angle $ 60^\circ $ with the line $ 2x - y + 3 = 0 $. If $ L $ makes an acute angle with the positive X-axis in the anticlockwise direction, then the Y-intercept of the line $ L $ is?

If the slope of one line of the pair of lines $ 2x^2 + hxy + 6y^2 = 0 $ is thrice the slope of the other line, then $ h $ = ?

If the equation of the pair of straight lines passing through the point $ (1, 1) $ and perpendicular to the pair of lines $ 3x^2 + 11xy - 4y^2 = 0 $ is $ ax^2 + 2hxy + by^2 + 2gx + 2fy + 12 = 0 $, then find $ 2(a + h - b - g + f - 12) = ? $

Equation of the circle having its centre on the line $ 2x + y + 3 = 0 $ and having the lines $ 3x + 4y - 18 = 0 $ and $ 3x + 4y + 2 = 0 $ as tangents is:

If power of a point $ (4,2) $ with respect to the circle $ x^2 + y^2 - 2x + 6y + a^2 - 16 = 0 $ is 9, then the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is

Let $ a $ be an integer multiple of 8. If $ S $ is the set of all possible values of $ a $ such that the line $ 6x + 8y + a = 0 $ intersects the circle $ x^2 + y^2 - 4x - 6y + 9 = 0 $ at two distinct points, then the number of elements in $ S $ is:

If the circles $ x^2 + y^2 - 8x - 8y + 28 = 0 $ and $ x^2 + y^2 - 8x - 6y + 25 - a^2 = 0 $ have only one common tangent, then $ a $ is:

If the equation of the circle passing through the points of intersection of the circles \[ x^2 - 2x + y^2 - 4y - 4 = 0, \quad x^2 + y^2 + 4y - 4 = 0 \] and the point $ (3,3) $ is given by \[ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, \] then $ 3(\alpha + \beta + \gamma) $ is:

A common tangent to the circle $ x^2 + y^2 = 9 $ and the parabola $ y^2 = 8x $ is

Let $ F $ and $ F' $ be the foci of the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ (where $ b<2 $), and let $ B $ be one end of the minor axis. If the area of the triangle $ FBF' $ is $ \sqrt{3} $ sq. units, then the eccentricity of the ellipse is:

If a circle of radius 4 cm passes through the foci of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ and is concentric with the hyperbola, then the eccentricity of the conjugate hyperbola of that hyperbola is:

If a tangent to the hyperbola $ x^2 - \frac{y^2}{3} = 1 $ is also a tangent to the parabola $ y^2 = 8x $, then the equation of such tangent with the positive slope is:

If $ A(1,0,2) $, $ B(2,1,0) $, $ C(2,-5,3) $, and $ D(0,3,2) $ are four points and the point of intersection of the lines $ AB $ and $ CD $ is $ P(a,b,c) $, then $ a + b + c = ? $

The direction cosines of two lines are connected by the relations $ 1 + m - n = 0 $ and $ lm - 2mn + nl = 0 $. If $ \theta $ is the acute angle between those lines, then $ \cos \theta = $ ?

The distance from a point $ (1,1,1) $ to a variable plane $\pi$ is 12 units and the points of intersections of the plane with X, Y, Z-axes are $ A, B, C $ respectively. If the point of intersection of the planes through the points $ A, B, C $ and parallel to the coordinate planes is $ P $, then the equation of the locus of $ P $ is:

Evaluate the limit: \[ \lim_{x \to 0} \frac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2 + x^5 + x^6}}{x^4} =\]

Evaluate the limit: \[ \lim_{x \to 1} \frac{\sqrt{x} - 1}{(\cos^{-1} x)^2} =\]

If the solution of the system of simultaneous linear equations: $$ x + y - z = 6, $$ $$ 3x + 2y - z = 5, $$ $$ 2x - y - 2z + 3 = 0 $$ is $ x = \alpha, y = \beta, z = \gamma $, then $ \alpha + \beta = ?$

If the ratio of the terms equidistant from the middle term in the expansion of $(1 + x)^{12}$ is $\frac{1}{256}$, then the sum of all the terms of the expansion $(1 + x)^{12}$ 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 $

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:

Evaluate the integral: \[ \int \frac{3x^9 + 7x^8}{(x^2 + 2x + 5x^9)^2} \,dx= \]

If two numbers $x$ and $y$ are chosen one after the other at random with replacement from the set of numbers $ \{1, 2, 3, \ldots, 10\} $, then the probability that $ |x^2 - y^2| $ is divisible by 6 is: