List of top Questions asked in AP EAPCET

If the function $y = g(x)$ represents the slopes of the tangents drawn to the curve $y = 3x^3 - 5x^2 - 12x^2 + 18x - 3$ strictly increasing then the domain of $g(x)$ is
Identify the correct option from the following:

If a discrete random variable \(X\) has the probability distribution \[ P(X = x) = k \frac{2^{2x+1}}{(2x+1)!}, \quad x=0,1,2,\ldots, \] then find \(k\).

Consider the following functions
I) $f(x) = \left| \frac{1}{2 - x}, x<\frac{1}{2} \right|$
II) $f(x) = \left| \frac{1}{(2 - x)^2}, x \neq 2 \right|$
III) $f(x) = |x|$
IV) $f(x) = |x|$
Then on $[0, 1]$, Lagrange's mean value theorem is applicable to the functions Identify the correct option from the following:

If the transformed equation of the equation \[ 2x^2 + 3xy - 2y^2 - 17x + 6y + 8 = 0 \] after translating the coordinate axes to a new origin \((\alpha, \beta)\) is \[ aX^2 + 2h XY + bY^2 + c = 0, \] then find \(3\alpha + c\).

If \(630^\circ<\theta<810^\circ\) and \(\tan \theta = -\frac{7}{24}\), then find \(\cos \left(\frac{\theta}{4}\right)\).

The following equilibrium is established at STP:
\[ B_2 (g) \rightleftharpoons 2B(g) \] Atoms of B occupy 20% of total volume at STP. The total pressure of the system is 1 bar. What is its \(K_p\)? (STP volume = 22.7 L)

Let \( H(x) = 3x^4 + 6x^3 - 2x^2 + 1 \) and \( g(x) \) be a linear polynomial. If \[ \frac{H(x)}{(x-1)(x+1)(x-2)} = f(x) + \frac{g(x)}{(x-1)(x+1)(x-2)}, \] then find \( H(-1) + 2H(2) - 3H(1) \).

The value of $c$ of Rolle's theorem for the function $f(x) = 2 \sin x + \sin 2x$ in the interval $[0, \pi]$ is
Identify the correct option from the following:

In \(\triangle ABC\), the sum of the lengths of two sides is \(x\) and the product of those lengths is \(y\). If \(c\) is the length of its third side and \(x^2 - c^2 = y\), then the circumradius of that triangle is:

Evaluate \[ \tan^{-1} \frac{\sqrt{8} - 2\sqrt{15}}{\sqrt{15} + 1} + \tan^{-1} \frac{1}{\sqrt{5}} \]

Number of solutions of the equation \[ 2 \sin^2 \theta - 3 \cos^2 \theta = \sin \theta \cos \theta \] in the interval \((- \pi, \pi)\) is:

Evaluate \[ \sum_{k=0}^{12} \sin\left( (k+1) \frac{\pi}{6} + \frac{\pi}{4} \right) \sin \left( \frac{k \pi}{6} + \frac{\pi}{4} \right) \]

The differential equation corresponding to the family of parabolas whose axis is along $x = 1$ is
Identify the correct option from the following:

If the harmonic mean of the roots of the equation \[ \sqrt{2} x^2 - b x + \left( 8 - 2\sqrt{5} \right) = 0 \] is 4, then the value of \( b \) is:

Two cards are drawn at random from a pack of 52 playing cards. If both the cards drawn are found to be black in colour, then the probability that at least one of them is a face card is:

For \(\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), if \(2\cos \theta + \sin \theta = 1\), and \(7\cos \theta + 6 \sin \theta = k\), then the possible values of \(k\) are:

The slope of the non-vertical tangent drawn from the point $(3,4)$ to the circle $x^2 + y^2 = 9$ is:

If water flows with a velocity of 20 cm/s in a pipe of radius 2 cm, then the flow is (The coefficient of viscosity of water is \(10^{-3} \, \mathrm{kg\,m^{-1}s^{-1}}\) and density of water is \(10^{3} \, \mathrm{kg\,m^{-3}}\))

Problem: From a point \( P \) on the circle \( x^2 + y^2 = 4 \), two tangents are drawn to the circle \( x^2 + y^2 - 6x - 6y + 14 = 0 \). If \( A \) and \( B \) are the points of contact of those lines, then the locus of the center of the circle passing through the points \( P \), \( A \), and \( B \) is: Identify the correct option from the following: