List of practice Questions

Evaluate \( \int_0^{\frac{\pi}{2}} \left( e^{\sin x} - e^{\cos x} \right) \, dx \)

The minimum value for the LPP \( Z = 6x + 2y \), subject to \( 2x + y \geq 16, x \geq 6, y \geq 1 \) is

A fair coin is tossed 2 times. A person receives \( X^3 \) if he gets \( X \) number of heads. His expected gain is

If \( x = 3 \sin \theta \), \( y = 3 \cos \theta \cos \phi \), \( z = 3 \cos \theta \sin \phi \), then \( x^2 + y^2 + z^2 = \)

Evaluate \( \int_{-2}^{1} \left[ x + 1 \right] \, dx \), where \( [x] \) is the greatest integer function not greater than \( x \)

If the origin is the centroid of the triangle whose vertices are \( A(2, p, -3) \), \( B(q, -2, 5) \) and \( C(-5, 1, r) \), then

Write the statement in symbolic form 'Sandeep neither likes tea nor coffee but enjoys a soft drink'. Where \( p \): Sandeep likes tea, \( q \): Sandeep likes coffee, \( r \): Sandeep enjoys a soft drink

If \( f(x) = \frac{x^2 + 2{18} \), for \( -2<x<4 \), and 0 otherwise, is the p.d.f. of a random variable \( X \), then the value of \( P(|X|<2) \) is}

If \( a \), \( b \), \( c \) are non-negative distinct numbers and \( ai + aj + ck \), \( i + j + k \), and \( ci + cj + bk \) are coplanar vectors, then

If foci of the ellipse \( \frac{x^2}{16} + \frac{y^2}{b^2} = 1 \) (\( b^2<16 \)) and the hyperbola \( \frac{x^2}{144} - \frac{y^2}{81} = 1 \) coincide, then the value of \( b^2 \) is

The principal value of \( \sin^{-1} \left( -\frac{1}{2} \right) \) is

The separate equations of the lines represented by the equation \( 3x^2 - 2\sqrt{3} xy - 3y^2 = 0 \) are

If \( A = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}, B = \begin{bmatrix} 4 & 3 \\ 1 & 1 \end{bmatrix} \), then \( (A + B)^{-1x} = \)

The verbal statement of the same meaning, of the statement 'If the grass is green then it rains in July' is

If line \( x + y = 0 \) touches the curve \( ax^2 = 2y^2 - b \) at \( (1, -1) \), then the values of \( a \) and \( b \) are respectively

Evaluate \( \int \frac{dx}{\cos 2x + \sin^2 x} \)

Evaluate \( \sin 690^\circ \times \sec 240^\circ \)

The equation of a plane containing the lines \( \vec{r} = (i + 2j - 4k) + \lambda (2i + 3j + 6k) \) and \( \vec{r} = (i + 3j + 4k) + \mu (i + j - k) \) is

Evaluate \( 2 \tan^{-1} \left( \frac{1}{3} \right) - \tan^{-1 \left( \frac{3}{4} \right) \)}

The rate of decay of mass of a certain substance at time \( t \) is proportional to the mass at that instant. The time during which the original mass of \( m_0 \) gm will be left to \( m_1 \) gm is

If the points \( (1, 1, \lambda) \) and \( (-3, 0, 1) \) are equidistant from the plane \( 3x + 4y - 12z + 13 = 0 \), then the integer value of \( \lambda \) is

If \( x = a(1 - \cos\theta) \), \( y = a(\theta - \sin\theta) \), then \( \frac{d^2y}{dx^2} = \)

If \( A \) and \( B \) are independent events and \( P(A) = \frac{2{3} \), \( P(B) = \frac{3}{5} \), then \( P(A' \cap B) = \)}

The differential equation whose solution is \( y = c_1 \cos ax + c_2 \sin ax \) (where \( c_1 \) and \( c_2 \) are arbitrary constants) is

Evaluate \( \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx \)