List of top Questions asked in AP EAPCET- 2025

A mass of \( 6 \times 10^{24} \) kg is to be compressed in the form of a solid sphere such that the escape velocity from its surface is \( 3 \times 10^5 \) m/s. The radius of the sphere is:

A body of mass \( M \) is moving with a uniform speed \( v \) on a frictionless horizontal surface under the influence of two forces \( F_1 \) and \( F_2 \) as shown in the figure. The net power of the system is:

Evaluate the integral: \[ I = \int_{\pi/6}^{\pi/3} \cos^{-4} x \, dx \]

Evaluate the integral: \[ I = \int_0^{3\pi/2} \frac{\cos^5 x}{\cos^3 x+\sin^3 x}dx \]

The differential equation for which \( y^2 = 4a(x + a) \) (where \( a \) is a parameter) is the general solution is:

If the lengths of the tangent, subtangent, normal, and subnormal for the curve \( y = x^2 + x - 1 \) at the point \( (1,1) \) are \( a, b, c, \) and \( d \) respectively, then their increasing order is:

Evaluate the integral: \[ \int \frac{x+1}{x^3 - 1}dx \]

Evaluate the integral: \[ \int \frac{x^4-16x^2+2x+8}{x^3-4x^2+2}dx \]

Evaluate the integral: \[ \int \frac{\sec^2 x}{(\sec x+\tan x)^{5/2}}dx \]

Evaluate the integral: \[ \int \operatorname{Cos}^{-1} \left( \frac{1-x^2}{1+x^2} \right) dx \]

If \( y = \log(\sec(\tan^{-1}x)) \) for \( x>0 \), then \( \frac{dy}{dx} \) at \( x = 1 \) is:

If \( x = \sqrt{2}e^t(\sin t - \cos t) \) and \( y = \sqrt{2}e^t(\sin t + \cos t) \), then \( \left[ \frac{d^2y}{dx^2} \right]_{t=\frac{\pi}{4}} \) is:

P and Q are the ends of a diameter of the circle \( x^2+y^2=a^2(a>\frac{1}{\sqrt{2}}) \). \( s \) and \( t \) are the lengths of the perpendiculars drawn from P and Q onto the line \( x+y=1 \) respectively. When the product \( st \) is maximum, the greater value among \( s, t \) is:

If \( \theta \) is the acute angle between the tangents drawn from the point \( (1,1) \) to the hyperbola \( 4x^2-5y^2-20=0 \), then \( \tan\theta \) is:

If \( A(2,-1,1) \), \( B(2,5,1) \) and \( C(0,-2,3) \) are the vertices of a triangle, and \( D \) is the point of intersection of the side \( BC \) and the internal angular bisector of angle \( A \), then \( AD = \):

A plane \( \pi \) given by \( ax+by+11z+d = 0 \) is perpendicular to the planes \( 2x-3y+z=4 \), \( 3x+y-z=5 \), and the perpendicular distance from the origin to the plane \( \pi \) is \( \sqrt{6} \) units. If all the intercepts made by the plane \( \pi \) on the coordinate axes are positive, then \( d = \):

If the equation of the polar of the point \( (\alpha, -1) \) with respect to the circle \( x^2+y^2-4x-6y-12=0 \) is \( y = \beta \), then \( 4(\alpha+\beta) = \):

If \( \theta \) is the angle between the tangents drawn from the point \( (-1, -1) \) to the circle \( x^2+y^2-4x-6y+c=0 \) and \( \cos\theta = -\frac{7}{25} \), then the radius of the circle is:

If the power of the point \( (1,6) \) with respect to the circle \( x^2+y^2+4x-6y-a=0 \) is \( -16 \), then \( a \) is:

The radius of the circle passing through the points of intersection of the circles \( x^2+y^2+2x+4y+1=0 \), \( x^2+y^2-2x-4y-4=0 \), and intersecting the circle \( x^2+y^2=6 \) orthogonally is:

If the tangents drawn from a point \( P \) to the ellipse \( 4x^2+9y^2-16x+54y+61=0 \) are perpendicular, then the locus of \( P \) is:

When the axes are rotated through an angle \( \theta \) about the origin in the anticlockwise direction and then translated to the new origin (2, -2), if the transformed equation of \( x^2+y^2=4 \) is \( X^2+Y^2+aX+bY+c=0 \), then \( a+b+c= \):

If the perpendicular distances from the points \( (2,3) \), \( (4,a) \), and \( (\alpha, \beta) \) onto the line \( 3x + 4y - 3 = 0 \) are equal and \( 4\alpha - 3\beta + 1 = 0 \), then the sum of all possible values of \( \alpha \), \( \alpha \) and \( \beta \) is:

The equation of the base of an equilateral triangle is \( x + y = 2 \) and its opposite vertex is \( (2,1) \). If \( m_1, m_2 \) are the slopes of the other two sides and the length of its side is \( a \), then \( |m_1 - m_2| + a\sqrt{2} = \):

From a point \( P(-4, 0) \), two tangents are drawn to the circle \( x^2+y^2-4x-6y-12=0 \) touching the circle at \( A \) and \( B \). If the equation of the circle passing through \( P, A, B \) is \( x^2+y^2+2gx+2fy+c=0 \), then \( (g,f) = \):