List of practice Questions

The centres of a set of circles, each of radius 3, lie on the circle \( x^2 + y^2 = 25 \). The locus of any point in the set is

A tower \( A \) leans towards west making an angle \( \theta \) with the vertical. The angular elevation of \( B \), the topmost point of the tower is \( \beta \) as observed from a point \( C \) at a distance \( d' \) from \( B \). If the angular elevation of \( B \) from point \( D \) due east of \( C \) is the same and \( 2d \) from \( C \), then \( \theta \) can be given as

\( \theta \) and \( \gamma \) are the roots of the equation \( x^2 - \alpha x + \beta = 0 \) and if \( \theta + \gamma = \alpha \), then what is the value of \( \theta^2 + \gamma^2 \)?

The angle of intersection of the circles \( x^2 + y^2 - 8x - 9 = 0 \) and \( x^2 + y^2 + 2x - 4y - 11 = 0 \) is

Which of the following is the correct expansion of the series \[ \sum_{n=0}^{\infty} \left( \binom{C}{n} \right) \left( \frac{3}{5} \right)^n \left( \frac{2}{5} \right)^{n+1} \]

If two events A and B. If odds against A are 2:1 and those in favour of \( A \cup B \) are 3:1, then

The value of \( 2 \tan^{-1} x - \left( \text{cosec} \, \tan^{-1} x - \tan \, \cot \, x \right) \)

The proposition \( \neg (p \iff q) \) is equivalent to

If truth values of \( p \) be F and \( q \) be T, then truth value of \( \neg(p \vee q) \) is

The rate of change of the surface area of a sphere of radius \( r \), when the radius is increasing at the rate of \( 2 \, \text{cm/s} \), is proportional to

If \( N \) denote the set of all natural numbers and \( R \) the relation on \( N \times N \) defined by \( (a, b) R (c, d) \), if \( a(b + c) = b(a + d) \), then \( R \) is

What is the degeneracy of the level of H-atom that has energy \( \frac{R_H}{9} \)?

Match the following and choose the correct option given below. Compound/Type \quad Use A. Dry ice 1. Anti-knocking compound B. Semiconductor 2. Electronic diode or triode C. Solder 3. Joining circuits D. TEL 4. Refrigerant for preserving food

The relation \( R \) defined on set \( A = \{x : |x|<3, x \in \mathbb{R} \} \) by \( R = \{(x, y): y = |x|\} \) is

The solution of the differential equation \[ \frac{dy}{dx} = \frac{y}{f(x)} - y^2 \] is

If \( a \), \( b \), and \( c \) are in AP, then determinant \[ \begin{vmatrix} x+2 & x+3 & x+4
x+4 & x+5 & x+6
x+7 & x+8 & x+9 \end{vmatrix} \]

An organic compound \( B \) is formed by the reaction of ethyl magnesium iodide with a substance \( A \), followed by treatment with dilute aqueous acid. Compound \( B \) does not react with PCC or PDC in dichloromethane. Which of the following is a possible compound for \( A \)?

\( \text{CH}_3\text{CH}_2\text{CH}_2\text{COCH}_3 \) reacts with \( \text{CH}_3 \text{MgBr} \) (one mole) followed by treatment with \( \text{H}_2 \text{O} \). The compound \( A \) formed in this reaction is

For the cell reaction \( 2\text{Cu}^{2+} + \text{Co} \rightarrow 2\text{Co}^{2+} + \text{Cu} \), \( E^\circ_{\text{cell}} \) is 1.89V. If \( E^\circ_{\text{Co}^{2+}/\text{Co}} \) is -0.28V, what is the value of \( E^\circ_{\text{Cu}^{2+}/\text{Cu}} \)?

A constant current of 30 A is passed through an aqueous solution of NaCl for a time of 1.00 h. What is the volume of Cl\(_2\) gas at STP produced?

Consider the following reaction:

The reaction of zinc with \( \text{Cu}^{2+} \) produces the following. Entropy change \( \Delta S \) is given by \( 96.5 J \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \).

What transition in the hydrogen spectrum would have the same wavelength as the Balmer transition, \( n = 4 \) to \( n = 2 \) of He\(^+\) spectrum?

Select the correct statements(s)

The product Y is