List of practice Questions

The area enclosed by \( y = 3x - 5 \), \( y = 0 \), \( x = 3 \), and \( x = 5 \) is:

The order and degree of the differential equation \[ \left( 1 + 4 \frac{dy}{dx} \right)^{2/3} = 4 \frac{d^2 y}{dx^2} \] are respectively:

The solution of the differential equation \[ \frac{dy}{dx} = (4x + y + 1)^2 \] is:

The system of equations \[ 2x + y - 5 = 0, \quad x - 2y + 1 = 0, \quad 2x - 14y - a = 0 \] is consistent. Then, \( a \) is equal to:

A die is rolled twice and the sum of the numbers appearing on them is observed to be 7. What is the conditional probability that the number 2 has appeared at least once?

The locus of the mid-points of the focal chord of the parabola \( y^2 = 4ax \) is:

Find the value of \[ \sin 12^\circ \sin 48^\circ \sin 54^\circ \]

In an equilateral triangle, the inradius, circumradius, and one of the exradii are in the ratio:

Let \( p \) and \( q \) be two statements. Then, \( p \vee q \) is false if:

In how many ways 6 letters can be posted in 5 different letter boxes?

If \( A \) and \( B \) are two sets such that \( A \times B \) consists of 6 elements, find \( B \times A \):

A line making angles 45° and 60° with the positive directions of the axes of \( x \) and \( y \) makes with the positive direction of \( z \)-axis, an angle of:

Which of the following is correct?

If \( \alpha, \beta, \gamma \) are the roots of \( x^3 + ax^2 + b = 0 \), then the value of \[ \frac{\alpha \beta}{\gamma}, \quad \frac{\beta \gamma}{\alpha}, \quad \frac{\gamma \alpha}{\beta} \]

If the axes are shifted to the point \( (1, 2) \) without solution, then the equation \[ 2x^2 + 2y^2 - 4x + 4y = 0 \] becomes:

If \( f(x) = \frac{x^2}{2}, \text{for} x \leq 0, \frac{2\sin x}{x}, \text{for} x>0 \), then \( x = 0 \) is:

In a group \( (G^*) \), the equation \( x * a = b \) has a:

If the roots of the equation \( x^2 + ax + b = 0 \) are \( c \) and \( d \), then one of the roots of the equation \[ x^2 + (2c + a)x + c^2 + ac + b = 0 \] is:

The sum of the coefficients of \( (6a - 5b)^n \), where \( n \) is a positive integer, is:

Find the value of \( (7.995)^{1/3} \) correct to four decimal places.

The values of constants \( a \) and \( b \), so that \[ \lim_{x \to \infty} \left( \frac{x^2 + 1}{x + 1} - ax - b \right) = 0 \] are:

The projection of the vector \( \mathbf{i} - 2\mathbf{j} + \mathbf{k} \) on the vector \( 4\mathbf{i} - 4\mathbf{j} + 7\mathbf{k} \) is:

If \( a, b, c \) are three non-zero vectors such that \( a + b + c = 0 \) and \( m = a \cdot b + b \cdot c + c \cdot a \), then:

A line passes through \( (2, 2) \) and is perpendicular to the line \( 3x + y = 3 \). What is its y-intercept?

The number of common tangents to the circles \( x^2 + y^2 = 4 \) and \( x^2 + y^2 - 6x - 8y = 24 \) is: