List of top Mathematics Questions asked in TS EAMCET

Let \( \bar{n} \) be a unit vector normal to the plane \( \pi \) containing the vectors \( \bar{T} + 3\bar{k} \) and \( 2\bar{i} + \bar{j} - \bar{k} \). If this plane \( \pi \) passes through the point \( (-3,7,1) \) and \( p \) is the perpendicular distance from the origin to this plane, then \( \sqrt{p^2 + 5} \) is:

If \( \bar{a} = \bar{i} - \bar{j} + 3\bar{k} \), \( \bar{c} = -\bar{k} \) are position vectors of two points and \( \bar{b} = 2\bar{i} - \bar{j} + \lambda \bar{k} \),

\( \bar{d} = \bar{i} + \bar{j} - \bar{k} \) are two vectors, then the lines \( r = \bar{a} + t \bar{b} \), \( r = \bar{c} + s \bar{d} \) are:

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:

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

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

The real-valued function \[ f(x) = \frac{|x - a|}{x - a} \] is analyzed as follows:

If the function

\[ f(x) = \begin{cases} \frac{(e^x - 1) \sin kx}{4 \tan x}, & x \neq 0 \\ P, & x = 0 \end{cases} \]

is differentiable at \( x = 0 \), then:

If

\[ A = \{ P(\alpha, \beta) \mid \text{the tangent drawn at P to the curve } y^3 - 3xy + 2 = 0 \text{ is a horizontal line} \} \]

and

\[ B = \{ Q(a, b) \mid \text{the tangent drawn at Q to the curve } y^3 - 3xy + 2 = 0 \text{ is a vertical line} \} \]

then \( n(A) + n(B) = \)

In a \( \triangle ABC \), the sides \( b, c \) are fixed. In measuring angle \( A \), if there is an error of \( \delta A \), then the percentage error in measuring the length of the side \( a \) is:

Consider the curves \( y = f(x) \) and \( x = g(y) \), and let \( P(x,y) \) be a common point of these curves. If at \( P \), on the curve \( y = f(x) \), \[ \frac{dy}{dx} = Q(x), \] and at the same point \( P \) on the curve \( x = g(y) \), \[ \frac{dx}{dy} = -Q(x), \] then:

If Rolle's Theorem is applicable for the function:

\[ f(x) = \begin{cases} x^p \log x, & x \neq 0 \\ 0, & x = 0 \end{cases} \]

on the interval \([0,1]\), then a possible value of \( p \) is:

If \[ \int (1 + x - x^x) e^{x + x^x} dx = f(x) + c, \] then \( f(1) - f(-1) = \)

If \[ \int \sqrt{\csc x + 1} \, dx = k \tan^{-1} (f(x)) + c, \] then \[ \frac{1}{k} f\left(\frac{\pi}{6}\right) = ? \]

Evaluate the integral: \[ \frac{3}{25} \int_{0}^{25\pi} \sqrt{|\cos x - \cos^3 x|} \, dx. \]

If \( m, l, r, s, n \) are integers such that \( 9 >m >l >s >n >r >2 \) and \[ \int_{0}^{\frac{\pi}{2}} \sin^n x \cos^r x \, dx = 4 \int_{0}^{\frac{\pi}{2}} \sin^m x \cos^r x \, dx, \] \[ \int_{0}^{\frac{\pi}{2}} \sin^l x \cos^r x \, dx = 4 \int_{0}^{\frac{\pi}{2}} \sin^s x \cos^r x \, dx, \] \[ \int_{0}^{\frac{\pi}{2}} \sin^n x \cos^r x \, dx = 0, \] then the equation involving \( s, l, m, r \) is:

If \( 0 \leq x \leq \frac{\pi}{2} \), then \[ \lim\limits_{x \to a} \frac{2\cos x - 1}{2\cos x - 1} \] Options:

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:

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

The variance of the first 10 natural numbers which are multiples of 3 is:

If three numbers are randomly selected from the set \( \{1,2,3,\dots,50\} \), then the probability that they are in arithmetic progression is:

The probability that exactly 3 heads appear in six tosses of an unbiased coin, given that the first three tosses resulted in 2 or more heads, is:

A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is: \

Two cards are drawn at random one after the other with replacement from a pack of playing cards. If \( X \) is the random variable denoting the number of ace cards drawn, then the mean of the probability distribution of \( X \) is:

A rectangle is formed by the lines \[ x = 4, \quad x = -2, \quad y = 5, \quad y = -2 \] and a circle is drawn through the vertices of this rectangle. The pole of the line \[ y + 2 = 0 \] with respect to this circle is: