List of top Mathematics Questions asked in AP EAPCET

If \( P, Q \) and \( R \) are \( 3 \times 3 \) matrices such that} \[ P = \begin{bmatrix} 3x^2 + x + 3 & 2x^2 - x + 4 & 7x^2 + 8x + 5 \\ 5x^2 + 3x + 2 & 4x^2 - 2x - 1 & 7x^2 + 5x + 8 \\ 3x^2 + 2x + 5 & 4x^2 - x - 2 & 3x^2 + 8x + 7 \end{bmatrix} = Px^2 + Qx + R \] then det \( R = \) ?
Let \( A \subseteq \mathbb{R}, B \subseteq \mathbb{R} \) and \( f : A \to B \) be defined by \( f(x) = x^2 - 3x + 2 \). If \( f \) is a bijection, then
If \( f:\mathbb{R} \to \mathbb{R} \) is defined by \( f(x+y) = f(x) + f(y) \ \forall x, y \in \mathbb{R} \) and \( f(1) = 7 \), then \(\sum\limits_{r=1}^n f(r) =\)
Let \( A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \), where \( \theta \in \mathbb{R} \). Then \( A^n \), for any positive integer \( n \), is equal to:
Three boxes \( B_1, B_2, B_3 \) contain white, black, and red balls as follows: \[ \begin{array}{|c|c|c|c|} \hline & \textbf{White} & \textbf{Black} & \textbf{Red} \\ \hline B_1 & 2 & 1 & 2 \\ B_2 & 3 & 2 & 4 \\ B_3 & 4 & 3 & 2 \\ \hline \end{array} \] A die is thrown:
- Box \( B_1 \) is chosen if the die shows 1 or 2
- Box \( B_2 \) is chosen if the die shows 3 or 4
- Box \( B_3 \) is chosen if the die shows 5 or 6
A ball is drawn at random from the selected box. the ball drawn is red, find the probability it came from box \( B_2 \):
If \( f(x) = \begin{cases} ax^2 - bx + 2, & x<3 \\ bx - 3, & x \geq 3 \end{cases} \) is differentiable at every \( x \in \mathbb{R} \), then the area (in sq units) of the triangle formed by the line \(\frac{x}{a} + \frac{y}{b} = 1\) with the coordinate axes is:
The general solution of \( \frac{dy}{dx} = \cos^2(x - y - 1) \) is given by \( x = \):
The degree of the differential equation $\log \left( \frac{dy}{dx} \right) = \left( 2x + 3 \frac{dy}{dx} \right)^2$ is
If $y = y(x)$ is a particular solution of $\sqrt{1 - x^2} \frac{dy}{dx} + \frac{2x}{\sqrt{1 - x^2}} y = x$, $y(0) = 1$, then $y\left(\frac{1}{2}\right) =$
Given that $\frac{d}{dx} \int_{0}^{\phi(x)} f(t) dt = f(\phi(x)) \phi'(x)$. For all $x \in (0, \frac{\pi}{2})$, if $\int_{1}^{\cos x} t^2 f(t) dt = \cos 2x$, then $f\left(\frac{1}{\sqrt{2}}\right) =$
\(\int \frac{8^{1+x} + 4^{1+x}}{2^{2x}} dx =\):
If $I = \int_{1}^{3} \sqrt{3 + x + x^2} dx$, then $I$ lies in the interval
The area bounded by \( y - 1 = -|x| \) and \( y + 1 = |x| \) is:
If $[\cdot]$ denotes the greatest integer function, then $\int_{0}^{1000} e^{x - \lfloor x \rfloor} dx =$
\( \int_{-4\pi}^{4\pi} \tan^9 x \sin^6 x \cos^3 x \, dx = \)
If \( f(x) \) is a function such that \( f'(x) = \sqrt{f^2(x) - 1} \) and \( f(0) = 1 \), then \( f(1) = \):
The equation of the normal to the curve \( y = \cosh x \) drawn at the point nearest to the origin is:
If \(\int \frac{dx}{1 + \sin x} = \tan \left( \frac{x}{2} - \theta \right) + C\), then \(\theta =\):
Let \( n \in (0, \infty) \). If for all the curves \( y = x^n \log x \) for distinct values of \( n \), we have \( y = x - 1 \) as the tangent at a fixed point \((\alpha, \beta)\), then \(\alpha + \beta = \):
\( f(x) \) and \( g(x) \) are differentiable functions such that \( \frac{f(x)}{g(x)} = \) a non-zero constant. If \( \frac{f'(x)}{g'(x)} = \alpha(x) \) and \( \left( \frac{f(x)}{g(x)} \right)' = \beta(x) \), then \( \frac{\alpha(x) - \beta(x)}{\alpha(x) + \beta(x)} = \):
The locus of the point on the curve \( y = \sin x \) where the tangent drawn at that point always passes through the point \( (0, \pi) \) is:
Let \( S_n = 1 + 3x + 9x^2 + 27x^3 + \ldots + n \text{ terms} -\frac{1}{3}<x<\frac{1}{3} \). If \( f(x) = S_n \), then \( f(x) \) is discontinuous at the point \( x = \):
If all the normals drawn to the curve \( y = \frac{1 + 3x^2}{3 + x^2} \) at the points of intersection of \( y = \frac{1 + 3x^2}{3 + x^2} \) and \( y = 1 \) pass through the point \( (\alpha, \beta) \), then \( 3\alpha + 2\beta = \):
Let \([ \cdot ]\) denote the greatest integer function.
Assertion (A): \(\lim_{x \to \infty} \frac{[x]}{x} = 1\)
Reason (R): \(f(x) = x - 1\), \(g(x) = [x]\), \(h(x) = x\) and \(\lim_{x \to \infty} \frac{f(x)}{x} = \lim_{x \to \infty} \frac{h(x)}{x} = 1\):
\(A(-2, 9)\) and \(B(1, 6)\) are two points on the curve \(y = x^2 + 5\). The coordinates of the point \(C\) on the curve such that the tangent drawn at \(A\) is parallel to the chord \(BC\) is: