List of top Mathematics Questions asked in AP EAPCET

Let \( L \) be the line passing through the points \( i\hat{i} - 9\hat{k} \) and \( 7j\hat{k} \), and \( \pi \) be the plane passing through the point \( 6i + j \) and perpendicular to the vector \( i + j + k \). If \( \theta \) is the angle between \( L \) and \( \pi \), then \( \sin \theta = \):
If \( \vec{OA} = 2i - j + k \), \( \vec{OB} = -3i - k \), and \( \vec{OC} = -2i + 2j - 3k \), then a unit vector perpendicular to the plane containing \( A, B, C \) is:
Let \( \vec{b} = 3i - 2j + k \) and \( \vec{c} = i - j - k \) be two vectors. If \( \vec{a} \) is a vector such that \( \vec{a} + \vec{b} + \vec{c} = 0 \), then \( | \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} | \) is:
Assertion (A): The function \( f(x) = \begin{cases} 1 - \cos x, & x<0 \\ \sin x, & x \geq 0 \end{cases} \) is continuous at \(x = 0\). Reason (R): \(\lim_{x \to 0} \sin x = 0\)
Which one of the following is a homogeneous differential equation?
Evaluate the integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin x \cdot \sin^2 x \cdot \cos x \, dx \]
The particular solution of the differential equation \[ (1 + y^2) \, dx - xy \, dy = 0, \quad y(1) = 0 \quad \text{represents:} \]
If \( c \) and \( d \) are arbitrary constants, then \[ y = e^{2x} \left( \cosh \sqrt{2} x + d \sinh \sqrt{2} x \right) \] is the general solution of the differential equation:
Evaluate the following integral: \[ \int (x^3 + x^2 m + x^m) (2x^{2m} + 3x^m + 6x^m) \, dx \]
The area (in square units) bounded by \[ x = 4, \quad y = -4, \quad \text{and} \quad y = x \quad \text{is:} \]
If \( a \in \mathbb{Z}^+ \), \([x]\) is the greatest integer not more than \( x \) and \[ \int_0^a \left\lfloor x \right\rfloor \, dx = 127, \text{ then } a = ? \]
The positive value of \( x \) satisfying the equation \[ \int_0^x (1 - t) \, dt = \frac{1}{2} \] is:
If \( (a^2 - 1)x + ay + (3 - a) = 0 \) is a normal to the curve \( xy = 1 \), then the interval in which ‘a’ lies is:
The maximum value of \( x^4 y^4 \) when \( a^2 x^4 + b^2 y^4 = c^6 \) is:
If \( \int f(x)\,dx = \Psi(x) \), then \( \int x^5 f(x^3)\,dx = \):
If \[ \int \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R}, \text{ then} \] \[ \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R} \]
The function \( f(x) = x^2 + \dfrac{54}{x} \)
Evaluate: \[ \int e^{\sin x} \cdot \frac{x \cos^3 x - \sin x}{\cos^2 x} \, dx \]
If \( f'(x) = a \cos x + b \sin x \), \( f'(0) = 4 \), \( f(0) = 3 \), and \( f\left( \frac{\pi}{2} \right) = 5 \), then \( f(x) = \)
Let \[ f(x) = \begin{cases} \frac{5e^{|x|} + 2}{3 - e^{|x|}}, & x \ne 0 \\ 0, & x = 0 \end{cases} \] Then at \( x = 0 \), the function \( f(x) \) is:
If \( f(x) = \begin{vmatrix} 1 & 6 + x & 36 + x^2 \\ 0 & x - 3 & 3x^2 - 27 \\ 0 & 2x - 4 & 8x^2 - 32 \end{vmatrix} \), then \( \lim\limits_{x \to 1} \frac{f(x)}{f(-x)} \) is:
Let \( f(x) \) be differentiable on \( \mathbb{R} \) and \( f'(m) \ne 0, m \in \mathbb{R} \). If \( \lim\limits_{x \to m} \frac{x f(m) - m f(x)}{x - m} + f'(m) = f(m) \), then \( m = \):
The ordinates of the points on the curve \( y = \tan^{-1}(\sin(\sqrt{x})) \), \( 0 \leq x \leq 8\pi^2 \), at which the tangent is parallel to the X-axis are:
Let \( f(x) \) be a differentiable function such that \( f(1) = 2 \), \( f(2) = 6 \), and \[ f(x+y) = f(x) + kxy + \frac{4}{3} y^2, \quad \forall x, y \in \mathbb{R}. \] Then \( f(x) = \) ?
The integrating factor of the linear differential equation \( \frac{dy}{dx} = \frac{1}{4x + 3y} \) is