List of practice Questions

Let \( R \) be a reflexive relation on the finite set \( A \) having 10 elements and if \( m \) is the number of ordered pairs in \( R \), then

The range of values of \( \theta \) in the interval \( \left( 0, \pi \right) \) such that the points \( (3, 2) \) and \( (\cos \theta, \sin \theta) \) lie on the same sides of the line \( x + y - 1 = 0 \), is:

Which of the following numbers is the coefficient of \( x^{100} \) in the expansion of \[ \log_e \left( \frac{1 + x}{1 + x^2} \right), \text{where} |x| < 1? \]

The coefficient of \( x^{50} \) in the expression of \[ (1 + x)^{1000} + 2x(1 + x)^{999} + 3x^2(1 + x)^{998} + \ldots + 1001x^{1000} \]

Let \( a, b, c, d \) be non-zero numbers. If the point of intersection of the lines \[ 4ax + 2ay + c = 0 \text{and} 5bx + 2by + d = 0 \] lies in the fourth quadrant and is equidistant from the two, then the relation between \( a, b, c, d \) is

A bag contains different kinds of balls in which there are 5 yellow, 4 black, and 3 green balls. If 3 balls are drawn at random, then find the probability that no black ball is chosen.

If \[ \int f(x) \, dx = g(x), \text{ then } \int x^5 f(x^3) \, dx \] is

If \[ \lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \lim_{x \to k} \frac{x^3 - k^2}{x^2 - k^2}, \text{ then find } k \]

The graph of the function \( f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right) \) is symmetric about:

Largest value of \( \cos^2 \theta - 6 \sin \theta \cos \theta + 3 \sin^2 \theta + 2 \)

If \( \mathbf{a} \) and \( \mathbf{b} \) are vectors in space, given by \( \mathbf{a} = \frac{\hat{i} - 2\hat{j}}{\sqrt{5}} \) and \( \mathbf{b} = \frac{2\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{14}} \), then the value of \[ (2\mathbf{a} + \mathbf{b}) \cdot \left[ (\mathbf{a} \times \mathbf{b}) \times ( \mathbf{a} - 2\mathbf{b} ) \right] \] is:

The locus of the mid-point of all chords of the parabola \( y^2 = 4x \) which are drawn through its vertex is:

The value of \[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{x \sin x}{\cos^2 x} \, dx \text{ is:} \]

A computer producing factory has only two plants \(T_1\) and \(T_2\). Plant \(T_1\) produces 20% and plant \(T_2\) produces 80% of total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that \( P(\text{computer turns out to be defective given that it is produced in plant } T_1) = 10P(\text{computer turns out to be defective given that it is produced in plant } T_2)\), where \( P(E) \) denotes the probability of an event \(E\). A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant \(T_2\) is

The perimeter of a triangle \( \triangle ABC \) is 6 times the arithmetic mean of the sines of its angles. If the side \( a \) is 1, then the angle \( A \) is:

Why is chlorobenzene less reactive towards nucleophilic substitution reaction?

What is the maximum number of stars that can be packed inside the blue colour boundary including the one that is shown in the image below? The stars can be scaled but should not overlap. At least 4 points of every star should touch the blue colour boundary.

Prepare a landscape painting of any one of the following subjects on the given drawing sheet in a rectangle of 20 cm x 15 cm. The medium of the painting should be watercolour, poster colour, or any other medium. Selection of the medium and the composition of the landscape should be beautiful and attractive:

(i) Scene of a river bank

(ii) Women working in a field

(iii) In a village scene, there are some trees and a hut.

In which state is the famous temple of Khajuraho situated?

Which colour medium is most transparent?

Which of the following is a cold colour?