List of practice Questions

If the A.M. and G.M. of the roots of a quadratic equation in \( x \) are \( p \) and \( q \) respectively, then its equation is
If \( A = \left[ \begin{array}{cc} 1 & -1 \\ 7 & 0 \end{array} \right] \), \( |A| = \left| \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right| \), and \( A^2 = 8A + kI \), then the value of \( k \) is
If \( 2f(x) = f'(x) \) and \( f(0) = 3 \), then the value of \( f(2) \) is
If \( A + B + C = 180^\circ \), then the value of \( \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right) + \tan \left( \frac{B}{2} \right) \tan \left( \frac{C}{2} \right) + \tan \left( \frac{C}{2} \right) \tan \left( \frac{A}{2} \right) \) is
The angle between the two lines \( \frac{x-4}{1} = \frac{y+4}{2} = \frac{z+1}{2} \) and \( \frac{x+1}{2} = \frac{y+3}{2} = \frac{z-4}{-1} \) is
Evaluate \( \int e^{\tan^{-1} x} \left( 1 + \frac{x}{1 + x^2} \right) dx \)
The co-ordinates of the mid-point of the chord cut off by the line \( 2x - 5y + 18 = 0 \) by the circle \( x^2 + y^2 - 6x + 2y - 54 = 0 \) are
The differential equation of the family of curves \( y = e^{x} (A \cos x + B \sin x) \), where \( A \) and \( B \) are arbitrary constants, is
The area of the region bounded by the curve \( y = x^2 + 1 \), the lines \( x = 1 \), \( x = 2 \), and the x-axis is
For the probability distribution of \( X \) given below, the variance of \( X \) is
If \( n(X) = 700 \), \( n(A) = 200 \), \( n(B) = 300 \), \( n(A \cap B) = 100 \), where \( X \) is the universal set and \( A \) and \( B \) are subsets of \( X \), then \( n(A' \cap B') = \)
Evaluate \( \int 7^{7^{7x}} 7^{7x} dx = \)
The optimal solution of the L.P.P. \( Z = 8x + 3y \) subject to the constraints \( x + y \leq 3 \), \( 4x + y \leq 6 \), \( x \geq 0 \), \( y \geq 0 \) is
The distance of the point \( (2, -1, 0) \) from the plane \( 2x + y + 2z + 8 = 0 \) is
Evaluate the integral \( \int e^x \left( \frac{1 - x}{1 + x^2} \right)^2 dx = \)
Simplify \( \sqrt{2} + \sqrt{2} + 2 \cos 4\theta = \)
The approximate value of the function \( f(x) = x^3 + 5x^2 - 7x + 10 \) at \( x = 1 \) is
If the sum of the slopes of the pair of lines given by \( 4x^2 + 2hxy - 7y^2 = 0 \) is equal to the product of the slopes, then \( h \) is
If \( \sin(y + z - x) \), \( \sin(z + x - y) \), and \( \sin(x + y - z) \) are in A.P., then
The solution of the differential equation \( \log \left( \frac{dy}{dx} \right) = 9x - 6y + 6 \) is (given that \( y = 1 \) when \( x = 0 \))
If \( f(x) = 2x^2 + bx + c \), \( f(0) = 3 \) and \( f(2) = 1 \), then \( (f \circ f)(1) = \)
A line makes angles \( \alpha \), \( \beta \), \( \gamma \) with the co-ordinate axes, then \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma \) is equal to
If \( y = \tan^{-1} \left( \frac{x - \sqrt{1 - x^2}}{x + \sqrt{1 - x^2}} \right) \), then \( \frac{dy}{dx} \) is
The number of solutions of the equation \( \tan x + \sec x = 2 \cos x \) lying in the interval \( [0, 2\pi] \) is
If \( CP \) and \( CD \) is a pair of semi-conjugate diameters of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), then \( CP^2 + CD^2 = \)