List of top Questions asked in AP EAPCET- 2022

Evaluate \[ \int \frac{dx}{\sin\left(x-\frac{\pi}{3}\right)\cos x} \]

The minimum value of \[ \left(1+\frac{1}{\sin^n\alpha}\right) \left(1+\frac{1}{\cos^n\alpha}\right) \] is

The sum of the intercepts made by a tangent drawn to the curve \((x/a)^n + (y/b)^n = 2\) at \((a,b)\) on the coordinate axes is

At \(x = 0\), \(f(x) = \cos x - 1 + \frac{x^2}{2} - \frac{x^3}{3}\)

Maximum area of the rectangle inscribed in a circle of radius 10 cm is

If \(x = a \cos^3 \theta, y = a \sin^3 \theta\), then \(\frac{d^2y}{dx^2}\) at \(\theta = \pi/4\) is

The angle between the curve \(2y = e^{-x/2}\) and the y-axis is \(\tan^{-1}(k)\). Then \(k =\)

Evaluate \(\lim_{x \to 0} \frac{x^2 \log(\cos x)}{\log(1+x^2)}\)

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

If \[ y=\frac{ax+b}{cx+d} \] and \[ \frac{dx}{dy}=\frac{ad-bc}{Py^2+Qy+R} \] then \(P+Q+R=\)

If \(\lim_{n \to \infty} x_n\) exists and is finite, \(x_1 = 2\), \(x_{n+1} = \frac{a + b x_n}{b + c x_n}\) \(\forall n \in \mathbb{N}\) and \(c \gt b \gt a \gt 0\), then \(\lim_{n \to \infty} x_n =\)

The integral value of \(n\) for which \[ \lim_{x\to 0}\frac{(\cos x-1)(\cos x-e^x)}{x^n} \] is a finite non-zero real number 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 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

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

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

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

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

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

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 \(ab =\)

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