List of top Questions asked in TS EAMCET

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: 

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: 

If the equation of the circle which cuts each of the circles \[ x^2 + y^2 = 4, \] \[ x^2 + y^2 - 6x - 8y + 10 = 0, \] \[ x^2 + y^2 + 2x - 4y - 2 = 0 \] at the extremities of a diameter of these circles is \[ x^2 + y^2 + 2gx + 2fy + c = 0, \] then the value of \( g + f + c \) 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: 

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 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:
If \( L \) is the line of intersection of two planes \[ x + 2y + 2z = 15 \quad \text{and} \quad x - y + z = 4 \] and the direction ratios of the line \( L \) are \( (a, b, c) \), then the value of \[ \frac{a^2 + b^2 + c^2}{b^2} \] is:

The foot of the perpendicular drawn from \( A(1,2,2) \) onto the plane \[ x + 2y + 2z - 5 = 0 \] is \( B(a, \beta, \gamma) \). If \( \pi(x,y,z) = x + 2y + 2z + 5 = 0 \) is a plane then \(-\pi(A):\pi(B) \) is: 

Two statements are given below: 

I. Milk sugar is a disaccharide of \( \alpha \)-D-galactose and \( \beta \)-D-glucose. 
II. Sucrose is a disaccharide of \( \alpha \)-D-glucose and \( \beta \)-D-fructose. The correct answer is:

If \[ y = \log \left( x - \sqrt{x^2 - 1} \right), \] then \[ (x^2 - 1)y'' + xy' + e^y + \sqrt{x^2 - 1} = \] evaluates to:
The sum of the maximum and minimum values of the function \[ f(x) = \frac{x^2 - x + 1}{x^2 + x + 1} \] is:
If the area of the region enclosed by the curve \( ay = x^2 \) and the line \( x + y = 2a \) is \( k a^3 \), then \( k \) is:
The order and degree of the differential equation \[ \frac{dy}{dx} + \left( \frac{d^2y}{dx^2} + 2 \right)^{\frac{1}{2}} + \frac{d^3y}{dx^3} + 5 = 0 \] are respectively:

If \( y = \sin x + A \cos x \) is the general solution of \[ \frac{dy}{dx} + f(x)y = \sec x, \] then an integrating factor of the differential equation is: 

Wave picture of light has failed to explain
(1) photoelectric effect
(2) interference of light
(3) diffraction of light
(4) polarization of light

 A capacitor of capacitance (4.0 ± 0.2) μF is charged to a potential of (10.0 ± 0.1) V. The charge on the capacitor is:

A body is thrown vertically upwards with a velocity of 35 ms−1 from the ground.The ratio of the speeds of the body at times 3 s and 4 s of its motion is: