List of practice Questions

A signal which can be green or red with probability \( \frac{4}{5} \) and \( \frac{1}{5} \) respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is \( \frac{3}{4} \). If the signal received at station B is given, then the probability that the original signal is green is:

The value of the determinant \( \Delta = \begin{vmatrix} 1 & 4 & 3 \\ 0 & 12 & 9 \\ 1 & 2 & 2 \end{vmatrix} \) is:

If the equations \( x + ay = 0 \), \( 2x + y + az = 0 \), \( ax + y + 2z = 0 \) have non-trivial solutions, then \( a = \):

If the I.F. of the differential equation \[ \frac{dy}{dx} + 5y = \cos x \sin x e^{Adx}, \text{ then } A = \]

What is the length of the projection of \( 3\hat{i} + 4\hat{j} + 5\hat{k} \) on the xy-plane?

If \[ \int \frac{3x + 1}{(x-3)(x-5)} \, dx = \int \frac{-5}{(x-3)} \, dx + \int \frac{B}{(x-5)} \, dx, \] then the value of \( B \) is:

The vector equation of the symmetrical form of equation of straight line \( \mathbf{r} = (3\hat{i} + 7\hat{j} + 2\hat{k}) + \lambda (5\hat{i} + 4\hat{j} - 6\hat{k}) \) is:

Let the line \( \frac{x^2}{2} - \frac{y^2}{1} = 1 \) lie in the plane \( x + 3y - oz + \beta = 0 \). Then \( \beta \) equals:

The principal value of \( \sin^{-1} \left( \sin \frac{5\pi}{3} \right) \) is:

If \( \left[ 1 \quad x \quad 1 \right] \) is equal to \( \left[ 1 \quad 3 \quad 2 \right] \), then \( x \) is:

If \( A = \begin{pmatrix} -7 & -4 \\ 7 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 4 & 1 \\ 2 & 7 \end{pmatrix} \), then which statement is true?

If the parabola \( y^2 = 4ax \) passes through the point \( (1, -2) \), then the tangent at this point is:

The number of points of discontinuity of the function \( f(x) = x - [x] \) in the interval \( (0, 7) \) are:

A football is inflated by pumping air in it. When it acquires spherical shape its radius increases at the rate of \( 0.02 \, \text{cm/s} \). The rate of increase of its volume when the radius is \( 10 \, \text{cm} \) is:

The interval in which the function \( f(x) = \frac{4x^2 + 1}{x} \) is decreasing is:

The eccentricity of the ellipse whose major axis is three times the minor axis is:

The equation of the hyperbola with vertices \( (3, 0), (-3, 0) \) and semi-latus rectum 4 is given by:

Amplitude of \( \frac{1 + \sqrt{3}i}{\sqrt{3} + 1} \) is:

The value of \[ \lim_{x \to 0} \frac{x^3 \cot x}{1 - \cos x} \] is:

The connective in the statement: \[ "2 + 7>9 \, \text{or} \, 2 + 7<9" \] is:

The number of ways in which 3 prizes can be distributed to 4 children, so that no child gets all the three prizes, are:

If \( A \) and \( B \) are events such that \( P(A) = 0.42 \), \( P(B) = 0.48 \), and \( P(A \cap B) = 0.16 \), then: I. \( P(\text{not } A) = 0.58 \)
II. \( P(\text{not } B) = 0.52 \)
III. \( P(A \cup B) = 0.47 \)

The focus of the curve \( y^2 + 4x - 6y + 13 = 0 \) is:

Which one of the following statements is correct?

In case of nitrogen, \( \text{NCl}_3 \) is possible but not \( \text{NCl}_5 \), while in case of phosphorus, \( \text{PCl}_3 \) as well as \( \text{PCl}_5 \) is possible. It is due to: