List of practice Questions

The given equation of rectangular hyperbola is \[ x^2 - y^2 = 6^2 \quad \text{(length of latus rectum is 16)} \] The asymptotes are parallel to each other

If \( (2, 7, 3) \) is one end of a diameter of the sphere \( x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0 \), then the coordinates of the other end of the diameter are

Given lines are \[ x = my + n, \, z = py + q \] and \[ x = m'y + n', \, z = p'y + q' \] Above equations can be rewritten as \[ x - n' = p', \quad m = 1, \, p = 1 \] Lines will be perpendicular if

A tetrahedron has vertices at \( O(0, 0, 0) \), \( A(1, -2, 1) \), \( B(2, 1, 1) \), and \( C(1, -1, 2) \). Then, the angle between the faces \( OAB \) and \( ABC \) is

If a line segment OP makes angles of \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \) with the X-axis and Y-axis, respectively, then the direction cosines are

If \( p, q \) are simple propositions with truth values \( T, F \), then the truth value of \( (\neg p \vee q) \wedge \neg r \) is

Let \( f(x) = 25x^{24}(1 - x)^{75} \), on the interval \( [0, 1] \), then \[ f'(x) = 25x^{24}(1 - x)^{75} - 75x^{25}(1 - x)^{74} = 25x^{24}(1 - x)^{74}[(1 - x) - 3x] \]

If \( |z| \geq 3 \), then the least value of \( |z + 4| \) is

If \( l = x^2 + y^2 + z^2 \) is a normal to the ellipse \[ \frac{x^2}{16} + \frac{y^2}{9} = 1, \] then

If \( \text{line} \, y = mx + c \) is a normal to the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \] then \( c^2 \) is equal to

Correct result is as follows: \[ P \, (\neg \, Q) \, \equiv (P \, \vee \, Q) \, \implies (P \, \vee \, Q) \, \implies \, R \, \implies \, S \]

The correct result of the equation \( 3x^2 + 2x + 7y^2 - 14y + 4 \) is

The structure \( (N_3) \) satisfies the closure property, associativity and commutativity but the identity element 0 does not belong to \( N \).

The integral \( \int \frac{dx}{\cos x + \sqrt{5} \sin x} \) equals

The differential equation \( (3x + 4y + 1) dx + (4x + 5y + 1) dy = 0 \) represents a family of

If \( \Delta r = \left| \sum_{r=1}^{n} r \right| \), then \( \sum_{r=1}^{n} \Delta r \) is equal to

If \( A, B, C \) are three events associated with a random experiment, then \( P(A \,|\, B) \) is

If \( A = \begin{pmatrix} 1 & 2
3 & 1 \end{pmatrix} \), then rank \( A \) is

The probability of getting a double six in one throw of two dice

The rate constant of the reaction \( A \rightarrow B \) at different times are given below: \[ \text{Concentration (M)} \quad \text{Time (Minutes)}
0.069 \quad 0 \quad 0.052 \quad 17 \quad 0.035 \quad 34 \quad 0.018 \quad 51 \] The rate constant of the reaction according to the correct order of reaction is

If the complex number \( z \) lies on a circle with center at the origin and radius \( \frac{1}{4} \), then the complex number \( 1 + z \) lies on a circle with radius

A compound which gives a negative test with ninhydrin, it cannot be a protein or an amino acid. As, it gives a positive test with Benedict’s solution. The compound is

The compound used for making super glue or crazy glue is

\( X' \) and \( Y' \) respectively are

In the Cannizzaro reaction given below: \[ 2Ph - CHO \xrightarrow{\text{NaOH}} Ph - CH_2OH + Ph - CO_2 \] the slowest step is