List of top Mathematics Questions asked in TS EAMCET

Given the two parabolas: \[ S = y^2 - 4ax = 0, \quad S' = y^2 + ax = 0 \] where $ P(t) $ is a point on the parabola $ S' = 0 $. If $ A $ and $ B $ are the feet of the perpendiculars from $ P $ to the coordinate axes and $ AB $ is a tangent to the parabola $ S = 0 $ at the point $ Q(t_1) $, then the value of $ t_1 $ is:

Given that $ a $ and $ b $ are the semi-major and semi-minor axes of an ellipse whose axes are along the coordinate axes. If its latus rectum is of length 4 units and the distance between its foci is $ 4\sqrt{2} $, then the value of $ a^2 + b^2 $ is:

If the tangent drawn at a point $P(t)$ on the hyperbola $$ x^2 - y^2 = c^2 $$ cuts the X-axis at $T$ and the normal drawn at the same point $P$ cuts the Y-axis at $N$, then the equation of the locus of the midpoint of $TN$ is:

If the harmonic conjugate of $ P(2,3,4) $ with respect to the line segment joining the points \[ A(3,-2,2) \quad \text{and} \quad B(6,-17,-4) \] is $ Q(\alpha, \beta, \gamma) $, then the value of $ \alpha + \beta + \gamma $ is:

The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is:

The equation of the circle passing through the origin and cutting the circles $x^2 + y^2 + 6x - 15 = 0$ and $x^2 + y^2 - 8y - 10 = 0$ orthogonally is:

S = (-1,1) is the focus, $ 2x - 3y + 1 = 0 $ is the directrix corresponding to S and $ \frac{1}{2} $ is the eccentricity of an ellipse. If $ (a,b) $ is the centre of the ellipse, then $ 3a + 2b $ is:

If the line \[ 2x + by + 5 = 0 \] forms an equilateral triangle with \[ ax^2 - 96bxy + ky^2 = 0, \] then $ a + 3k $ is: \

The combined equation of a possible pair of adjacent sides of a square with area 16 square units whose centre is the point of intersection of the lines \[ x + 2y - 3 = 0 \quad \text{and} \quad 2x - y - 1 = 0 \] is: \

By shifting the origin to the point $ (h,5) $ by the translation of coordinate axes, if the equation \[ y = x^2 - 9x^2 + cx - d \] transforms to $ Y = X^2 $, then $ \left( \frac{d - c}{h} \right) $ is: \

Let $ \bar{a} $ be a vector perpendicular to the plane containing non-zero vectors $ \bar{b} $ and $ \bar{c} $. If $ \bar{a}, \bar{b}, \bar{c} $ are such that

\[ |\bar{a} + \bar{b} + \bar{c}| = \sqrt{|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2}, \]

then

\[ |(\bar{a} \times \bar{b}) \cdot (\bar{a} \times \bar{c})| = \]

A plane $ \pi_1 $ passing through the point $ 3\hat{i} - 7\hat{j} + 5\hat{k} $ is perpendicular to the vector $ \hat{i} + 2\hat{j} - 2\hat{k} $ and another plane $ \pi_2 $ passing through the point $ 2\hat{i} + 7\hat{j} - 8\hat{k} $ is perpendicular to the vector $ 3\hat{i} + 2\hat{j} + 6\hat{k} $. If $ p_1 $ and $ p_2 $ are the perpendicular distances from the origin to the planes $ \pi_1 $ and $ \pi_2 $ respectively, then $ p_1 - p_2 $ is:

Given two planes with equations $ \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + \mathbf{k}) = 5 $ and $ \mathbf{r} \cdot (2\mathbf{i} + \mathbf{j} - \mathbf{k}) = 3 $. A plane $ \pi $ passing through the line of intersection of these two planes also passes through the point (0,1,2). If the equation of $ \pi $ is $ \mathbf{r} \cdot (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = m $, determine the value of $ \frac{bc}{a^2} $:

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|.$

Let $ \bar{a}, \bar{b}, \bar{c} $ be three vectors each having $ \sqrt{2} $ magnitude such that

\[ (\bar{a}, \bar{b}) = (\bar{b}, \bar{c}) = (\bar{c}, \bar{a}) = \frac{\pi}{3}. \]

If

\[ \bar{x} = \bar{a} \times (\bar{b} \times \bar{c}) \quad \text{and} \quad \bar{y} = \bar{b} \times (\bar{c} \times \bar{a}), \]

then:

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 = $?

In a triangle $ABC$, if \[ r_1 r_2 + rr_3 = 35, \quad r_2 r_3 + rr_1 = 63, \quad r_3 r_1 + rr_2 = 45 \] then $2s$ is:

In triangle ABC, if $ A = 45^\circ $, $ C = 75^\circ $, and $ R = \sqrt{2} $, then the value of $ r $ is:

A vector of magnitude $\sqrt{2}$ units along the internal bisector of the angle between the vectors $2\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ and $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ is:

$ \vec{a}, \vec{b}, \vec{c} $ are three vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 2\sqrt{2}$, $|\vec{c}| = 5$, and $ \vec{c} $ is perpendicular to the plane of $ \vec{a} $ and $ \vec{b} $.

If the angle between the vectors $ \vec{a} $ and $ \vec{b} $ is $ \frac{\pi}{4} $, then

\[ |\vec{a} + \vec{b} + \vec{c}| = \ ? \]

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:

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

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 $3a+b+5c =$