List of practice Questions

A and B are events such that \( P(A \cup B) = 3/4 \), \( P(A) = 1/4 \), and \( P(A \cap B) = 2/3 \). Then the probability that \( P(A \cap B) \) is
Value of \[ \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sin x + \cos x} \, dx \]
The range of the function \[ f(x) = \frac{1}{2 - \cos 3x} \] is
The area bounded by \( y = |x| \), \( y = 0 \) and \( |x| = \frac{1}{2} \) will be:
The value of \( x \) obtained from the equation \[ \frac{x + \alpha}{\gamma} = \frac{x + \beta}{\alpha} = \frac{x + \gamma}{\beta} \] will be
The solution of the differential equation \[ \log x \frac{dy}{dx} + x = \sin 2x \] is
\[ \lim_{x \to \infty} \frac{x^2}{3x - 3} = ? \]
If \( \vec{a} \times \vec{b} \) and \( \vec{c} \times \vec{d} \) are perpendicular, then which of the following is always true?
A value of \( c \) for which conclusion of Mean Value Theorem holds for the function \( f(x) = \log x \) on the interval [1, 3] is
Negation of the proposition: If we control population growth, we prosper is
The equation \( z^2 + (2 - 3i)z + (2 + 3i) = 4 = 0 \) represents a circle of radius
The function \( f(x) = \sin x - kx - c \), where \( k \) and \( c \) are constants, decreases always when
Equation \( \frac{1}{8} \sin^3 x = \cos 6x \) represents
The acceleration of a sphere falling through a liquid is \( (30 - 3y) \, \text{cm/s}^2 \) where \( y \) is speed in cm/s. The maximum possible velocity of the sphere and the time when it is achieved are
A straight line parallel to the line \( 2x + y - 5 = 0 \) is also a tangent to the curve \( y^2 = 4x + 5 \). Then the point of contact is
\( p \)-cresol reacts with chloroform in alkaline medium to give the compound A which adds hydrogen cyanide to form the compound B. The latter on acidic hydrolysis gives chiral carboxylic acid. The structure of the carboxylic acid is
Which of the following has maximum number of lone pairs associated with Xe?
Which of the following statements is not true regarding \( (+) \) Lactose?
If one strand of DNA has the sequence \[ \text{ATGCCTGA}, \text{the sequence in the complimentary strand would be} \]
The starting reagents needed to make the azo compound shown below
\( \sin\left( \sin 5 \right) = x^2 - 4x \) holds if
The optically active tartaric acid is named as D - (+) tartaric acid because it has a positive
Consider the reaction: \[ \text{N}_2 (g) + 3 \text{H}_2 (g) \rightarrow 2 \text{NH}_3 (g) \] carried out at constant temperature and pressure. If \( \Delta H \) and \( \Delta U \) are the enthalpy and internal energy changes for the reaction, which of the following expressions is true?
What is D in the following sequence of reactions?
Knowing that the chemistry of lanthanoids (Ln) is dominated by its +3 oxidation state, which of the following statements is incorrect?