List of practice Questions

If the lines \( \ell_1 : \ell m + mn + n = 0 \), \( \ell_2 : mn + m + n = 0 \) are concurrent then

If \( y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \infty \), then

The equation of all lines having slope 2 which are tangent to the curve \( y = \frac{1}{x - 3} \), is

The function \( f(x) = (x(x - 2))^2 \) is increasing in the set

If \( a^2 x^4 + b^2 y^4 = c^4 \), then the maximum value of \( xy \) is

In the truth table for the statement \( (p \land q) \rightarrow (q \lor \neg p) \), the last column has the truth value in the following order is

If the value of mode and mean is 60 and 66 respectively, then the value of median is

Let \( R \) be the relation on the set \( \mathbb{R} \) of all real numbers, defined by \( aRb \) if \( |a - b| \leq 1 \). Then, \( R \) is

The greatest and least values of \( \left( \sin(x) \right)^2 + \left( \cos(x) \right)^2 \) are respectively

The value of \[ \frac{1}{2} \cos^{-1} \left( \cos \left( \frac{\pi}{3} - \frac{\sqrt{63}}{8} \right) \right) \] is

If \( 4a^2 + b^2 + 2c^2 + 4ab - 6ac - 3bc = 0 \), the family of lines \( ax + by + c = 0 \) is concurrent at one or the other of the two points-

A pair of tangents are drawn from the origin to the circle \( x^2 + y^2 + 20(x + y) + 20 = 0 \), then the equation of the pair of tangent are

An ellipse has OB as semi-minor axis, \( F \) and \( F' \) its foci and the angle \( \angle \text{FBF'} \) is a right angle. Then the eccentricity of the ellipse is

If the line \( 2x - 3y = k \) touches the parabola \( y^2 = 6x \), then find the value of \( k \).

S and T are the foci of an ellipse and B is an end of the minor axis. If \( \triangle \text{STB} \) is an equilateral triangle, then the eccentricity of the ellipse is

Let \( f(x) = (x^5 - 1)(x^3 + 1) \), \( g(x) = (x^2 - 1)(x^2 - x + 1) \) and let \( h(x) \) be such that \( f(x) = g(x)h(x) \). Then

In how many ways can 5 boys and 5 girls be seated at a round table so that no two girls may be together?

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

The coefficient of the middle term in the expansion of \( (2 + 3x)^4 \) is

If \( C_0, C_1, C_2, \dots \) denote the binomial coefficients in the expansion of \( (1 + x)^n \), then the value of \[ C_0 + (C_0 + C_1) + (C_0 + C_1 + C_2) + \dots + (C_0 + C_1 + C_2 + \dots + C_n) \] is

The sum of the series \[ 1 + 2^2 + 3^2 + 4^2 + \dots + 100^2 \] is

The quadratic equation whose roots are the \( x \) and \( y \) intercepts of the line passing through \( (1, 1) \) and making a triangle of area \( A \) with the co-ordinate axes is

If \( \cos \theta + \sin \theta = x \cos \theta \) and \( \sin \theta = y \cos \theta \), then \( x^2 + y^2 = \)

If \( \cos 76^\circ = \cos \left( 90^\circ - \theta \right) \), then the general value of \( \theta \) is

If the real part of \( \frac{z + 1}{z - 1} = 4 \), then the locus of the point representing \( z \) in the complex plane is