List of top Questions asked in TS EAMCET

If \( A(1, 2, -3) \), \( B(2, 3, -1) \), and \( C(3, 1, 1) \) are the vertices of triangle \( \Delta ABC \), then find \( \left| \frac{\cos A}{\cos B} \right|.\)
If \( a, b, c \) are the intercepts made on X, Y, Z-axes respectively by the plane passing through the points \( (1, 0, -2) \), \( (3, -1, 2) \), and \( (0, -3, 4) \), then \( 3a + 4b + 7c = \)?
Given the position vectors of points \(A\) and \(B\) as \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \(C\) divides \(AB\) in the ratio 3:2. If \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) is the position vector of point \(D\), find the unit vector in the direction of \( \mathbf{CD} \):
A unit vector \( \mathbf{e} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is coplanar with the vectors \( \mathbf{i} - 3\mathbf{j} + 5\mathbf{k} \) and \( 3\mathbf{i} + \mathbf{j} - 5\mathbf{k} \) and is perpendicular to the vector \( \mathbf{i} + \mathbf{j} + \mathbf{k} \). Calculate \(2a^2 + 3b^2 + 4c^2\):
P and Q are the points of trisection of the line segment AB. If the position vectors of A and B are \( 2\hat{i} - 5\hat{j} + 3\hat{k} \) and \( 4\hat{i} + \hat{j} - 6\hat{k} \) respectively, then the position vector of the point that divides PQ in the ratio 2:3 is:
Given vectors \( \mathbf{a} = \mathbf{i} + \mathbf{j} - 2\mathbf{k} \), \( \mathbf{b} = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \), and \( \mathbf{c} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} \). If \( \mathbf{d} \) is a normal to the plane of \( \mathbf{a} \) and \( \mathbf{b} \) and satisfies \( \mathbf{d} \cdot \mathbf{c} = 2 \), find the magnitude of \( \mathbf{d} \):

Given the vectors:

\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} \]

\[ \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k} \]

where

\[ \mathbf{a} \times \mathbf{c} = \mathbf{b} \]

\[ \mathbf{a} \cdot \mathbf{x} = 3 \]

Find:

\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]

If \( \mathbf{a} = 4\hat{i} + 5\hat{j} - 3\hat{k} \) and \( \mathbf{b} = 6\hat{i} - 2\hat{j} - 2\hat{k} \) are two vectors, then the magnitude of the component of \( \mathbf{b} \) parallel to \( \mathbf{a} \) is:
If \( \mathbf{a} = 2\overline{i} - \overline{j}, \overline{b} = 2\overline{j} - \overline{k}, \overline{c} = 2\overline{k} - \overline{i} \) are three vectors and \( \overline{d} \) is a unit vector perpendicular to \( \overline{c} \), If \( \overline{a}, \overline{b}, \overline{d} \) are coplanar vectors, then \( |\overline{d} \cdot \overline{b}| = \):
If \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors and the points \[ \lambda \vec{a} + 3\vec{b} - \vec{c}, \quad \vec{a} - \lambda \vec{b} + 3\vec{c}, \quad 3\vec{a} + 4\vec{b} - \lambda \vec{c}, \quad \vec{a} - 6\vec{b} + 6\vec{c} \] are coplanar, then one of the values of \( \lambda \) is:
A plane \(\pi\) passes through the points \( (5,1,2) \), \( (3,-4,6) \), and \( (7,0,-1) \). If \( p \) is the perpendicular distance from the origin to the plane \(\pi\) and \([l, m, n]\) are the direction cosines of a normal to the plane \(\pi\), then \([3l + 2m + 5n] =\)
The foot of the perpendicular drawn from the point \((-2, -1, 3)\) to a plane is \((1, 0, -2)\). If \(a, b, c\) are the intercepts made by the plane \(\pi\) on \(X, Y, Z\)-axes respectively, then \(3a+b+5c =\)
If \( f(x) \) is given as: \[ f(x) = \begin{cases} \frac{8}{x^3} - 6x, & \text{if } 0 < x \leq 1 \\ \frac{x - 1}{\sqrt{x}-1}, & \text{if } x > 1 \end{cases} \] is a real-valued function, then at \( x = 1 \), \( f \) is:
If \[ 2x^2 - 3xy + 4y^2 + 2x - 3y + 4 = 0 \] then \[ \left( \frac{dy}{dx} \right)_{(3,2)} = \]
If \[ x = \frac{9t^2}{1 + t^4}, \quad y = \frac{16t^2}{1 - t^4} \] then \[ \frac{dy}{dx} = \]
If \( f(x) = (2x-1)(3x+2)(4x-3) \) is a real-valued function defined on \(\left[\frac{1}{2}, \frac{3}{4}\right]\), then the value(s) of 'c' as defined in the statement of Rolle's theorem is:
The general solution of the differential equation \((3x^2 - 2xy)dy + (y^2 - 2xy)dx = 0\) is

If the interval in which the real-valued function \[ f(x) = \log\left(\frac{1+x}{1-x}\right) - 2x - \frac{x^{3}}{1-x^{2}} \] is decreasing in \( (a,b) \), where \( |b-a| \) is maximum, then {a}⁄{b} =

Find \( \frac{dy}{dx} \) for the given function:

\[ y = \tan^{-1} \left( \frac{\sin^3(2x) - 3x^2 \sin(2x)}{3x \sin(2x) - x^3} \right). \]

For a given function \( y = f(x) \), \( \delta y \) denotes the actual error in \( y \) corresponding to actual error \( \delta x \) in \( x \) and \( dy \) denotes the approximate value of \( \delta y \). If \( y = f(x) = 2x^2 - 3x + 4 \) and \( \delta x = 0.02 \), then the value of \( \delta y - dy \) when \( x = 5 \) is: \vspace{0.5cm}

If \( m \) and \( M \) are respectively the absolute minimum and absolute maximum values of a function \( f(x) = 2x^3 + 9x^2 + 12x + 1 \) defined on \([-3,0]\), then \( m + M \) is: 

If the equation of the curve which passes through the point \( (1,1) \) satisfies the differential equation: \[ \frac{dy}{dx} = \frac{2x - 5y + 3}{5x + 2y - 3}, \] then the equation of the curve is: \vspace{0.5cm}
The vertical angle of a right circular cone is \( 60^\circ \). If water is being poured into the cone at the rate of \( \frac{1}{\sqrt{3}} \) m\(^3\)/min, then the rate (m/min) at which the radius of the water level is increasing when the height of the water level is 3m is:
Evaluate \( \int_{-2}^{2} x^4 (4 - x^2)^{\frac{7}{2}} \, dx \):
The maximum interval in which the slopes of the tangents drawn to the curve \[ y = x^4 + 5x^3 + 9x^2 + 6x + 2 \] increase is: