List of top Questions asked in TS EAMCET

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:
The equations \[ 7x + y - 24 = 0 \quad \text{and} \quad x + 7y - 24 = 0 \] represent the equal sides of an isosceles triangle. If the third side passes through \( (-1,1) \), then a possible equation for the third side is: \
The slope of one of the pair of lines \(2x^2 + hxy + 6y^2 = 0\) is three times the slope of the other line, \(h = ?\)
If the quadratic equation \( 3x^2 + (2k + 1)x - 5k = 0 \) has real and equal roots, then the value of \( k \) such that \( -\frac{1}{2}<k<0 \) is:
Evaluate the integral: \[ \int \frac{(\sin^4x + 2\cos^2x - 1) \cos x}{(1 + \sin x)^6} \, dx \]
The position vector of the point of intersection of the line joining the points \( \mathbf{i} - \mathbf{j} + \mathbf{k} \) and the line joining the points \( 2\mathbf{i} + \mathbf{j} - 6\mathbf{k} \), \( 3\mathbf{i} - \mathbf{j} - 7\mathbf{k} \) is:
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:
The equation of the pair of transverse common tangents drawn to the circles \( x^2 + y^2 + 2x + 2y + 1 = 0 \) and \( x^2 + y^2 - 2x - 2y + 1 = 0 \) is:
Evaluate the integral: \[ \int (\log x)^3 \, dx \]
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:

What is the percentage of carbon in the product ‘X’ formed in the following reaction?

The variance of the first 10 natural numbers which are multiples of 3 is:
A particle of mass 4 mg is executing simple harmonic motion along x-axis with an angular frequency of 40 rad/s. If the potential energy of the particle is \( V(x) = a + bx^2 \), where \( V(x) \) is in joule and \( x \) is in meter, then the value of \( b \) is:
Evaluate the integral: \[ \int \frac{2\cos 3x - 3\sin 3x}{\cos 3x + 2\sin 3x} dx. \]
A right circular cone is inscribed in a sphere of radius 3 units. If the volume of the cone is maximum, then the semi-vertical angle of the cone is:
If \( \theta \) is the acute angle between the curves \( y^2 = x \) and \( x^2 + y^2 = 2 \), then \( \tan \theta \) is:

In a time ‘’t", the amplitude of vibrations of a damped oscillator becomes half of its initial value, then the mechanical energy of the oscillator decreases by:

The number of normals that can be drawn through the point \((9,6)\) to the parabola \(y^2 = 4x\) is:
If \( \alpha,\,\beta,\,\gamma \) are the roots of the equation \[ \begin{vmatrix} x & 2 & 2\\ 2 & x & 2\\ 2 & 2 & x \end{vmatrix} = 0 \]
If you are using eyeglasses of power \( 2D \), your near point is:

Study the following and pick up the correct combinations: 

Statement I: Acid rains are due to pollution of sulphur dioxide and nitrogen oxides.
Statement II: Incinerators are used to dispose of electronic wastes.
If \( A(1,2) \), \( B(2,1) \) are two vertices of an acute angled triangle and \( S(0,0) \) is its circumcenter, then the angle subtended by \( AB \) at the third vertex is
Observe the following f-block elements: Eu (Z = 63); Pu (Z = 94); Cf (Z = 98); Sm (Z = 62); Gd (Z = 64); Cm (Z = 96) How many of the above have half-filled f-orbitals in their ground state?
The power of a point \( (2,0) \) with respect to a circle \( S \) is \(-4\) and the length of the tangent drawn from the point \( (1,1) \) to \( S \) is \( 2 \). If the circle \( S \) passes through the point \( (-1,-1) \), then the radius of the circle \( S \) is: