List of practice Questions

The minimum value of \( Z = 5x + 8y \) subject to \[ x + y \geq 5, \quad 0 \leq x \leq 4, \quad y \geq 2, \quad x \geq 0, \quad y \geq 0 \] is

If the p.m.f. of a random variable \( X \) is given by \[ P(X = x) = \frac{5}{25} \quad \text{if} \quad x = 0, 1, 2, 3, 4, 5 \] then which of the following is not true?

If the angle between the lines given by the equation \[ x^2 - 3xy + Ay^2 + 3x - 5y + 2 = 0, \quad \lambda \geq 0, \quad \text{is} \, \tan^{-1} \left( \frac{1}{3} \right), \, \text{then} \, \lambda = \]

The points of discontinuity of the function \[ f(x) = \begin{cases} \frac{1}{x - 1} & \text{if } 0 \leq x \leq 2 \\ \frac{x + 5}{x + 3} & \text{if } 2<x \leq 4 \end{cases} \] in its domain are

If \[ \frac{x}{\sqrt{1 + x^2}} + \frac{y}{\sqrt{1 + y^2}} = 0, \quad x \neq y, \text{ then } (1 + x^2)^2 \frac{dy}{dx} = \]

The integrating factor of the differential equation \[ (1 + x^2) \, dt = \left( \tan^{-1} x - t \right) \, dx \text{ is} \]

The integral \[ \int \cot x \cdot \log \left[ \log (\sin x) \right] \, dx = \]

If the equation \[ kxy + 5x + 3y + 2 = 0 \text{ represents a pair of lines, then } k = \]

The integral \[ \int_0^a \frac{x}{\sqrt{a - x}} \, dx = \]

If the vectors \( i + j + k \), \( i - j + k \) and \( 2i + 3j + mk \) are coplanar, then \( m = \)

If \( f(x) = \log(\sec x + \tan x) \), then \[ f'\left( \frac{\pi}{4} \right) = \]

The differential equation of all lines perpendicular to the line \[ 5x + 2y + 7 = 0 \text{ is} \]

If for an Arithmetic progression, 9 times the ninth term is equal to 13 times the thirteenth term, then the value of the twenty-second term is

If \[ A = \begin{pmatrix} 2 & 5 \\ 0 & 1 \\ 3 & 0 \end{pmatrix}, \quad A^{-1} = \begin{pmatrix} 3 & -1 \\ -6 & 6 \\ -5 & 2 \end{pmatrix} \] then the values of \( \alpha \) and \( \beta \) are, respectively

The rate of decay of a certain substance is directly proportional to the amount present at that instant. Initially, there are 27 grams of the substance and 3 hours later it is found that 8 grams are left, then the amount left after one more hour is

The p.d.f. of a continuous random variable \( X \) is given by \[ f(x) = \frac{x + 2}{18}, \quad \text{if} \, -2<x<4, \quad f(x) = 0, \, \text{otherwise}. \] Then \( P[ |x|<1 ] = \)

The bacteria increases at the rate proportional to the number of bacteria present. If the original number \( N \) doubles in 4 hours, then the number of bacteria in 12 hours will be

The sum of the cofactors of the elements of the second row of the matrix \[ \begin{pmatrix} 1 & 3 & 3 & 2 \\ -2 & 0 & 4 & 1 \\ 5 & 2 & 1 & 1 \end{pmatrix} \] is

If \[ \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 4, \quad \text{then} \quad \frac{dy}{dx} = \]

If \( (a, -2a), \, a>0 \) is the midpoint of a line segment intercepted between the co-ordinate axes, then the equation of the line is

The angle between the lines \[ \mathbf{r_1} = (i + 2j + 3k) + \lambda (i + j + 2k) \quad \text{and} \quad \mathbf{r_2} = (3i + k) + \lambda' (2i + j - k), \quad \lambda, \lambda' \in \mathbb{R} \] is

The differential equation obtained from the function \( y = a(x - a)^2 \) is

If \[ \int \frac{\sin \theta}{\sin^3 \theta} \, d\theta = \frac{1}{2k} \log \left| \frac{k + \tan \theta}{k - \tan \theta} \right| + c, \text{ then } k = \]

The area of the region bounded by the curve \( y = 4x^3 - 6x^2 + 4x + 1 \) and the lines \( x = 1, x = 5 \) and the X axis is

If \( f(x) = \log(\sin x) \), \( x \in \left[ \frac{\pi}{6}, \frac{5\pi}{6} \right] \), then the value of \( c \) by applying L.M.V.T. is