A bag contains 4 red and 6 black balls. A ball is drawn at random, its colour is noted and it is returned to the bag. Moreover, 2 additional balls of the colour drawn are put in the bag and then a ball is drawn at random. What is the probability that the second ball is red?

• (A) \( \frac{1}{3} \)
• (B) \( \frac{2}{5} \)
• (C) \( \frac{3}{5} \)
• (D) \( \frac{4}{5} \)

Find the integrating factor (I.F.) for the differential equation \[ \frac{dy}{dx} + y\sec x = \tan x . \]

• (A) \( \sec x - \tan x \)
• (B) \( \sec x + \tan x \)
• (C) \( \tan x \)
• (D) \( \sec x \)

If \( \omega \) is an imaginary cube root of unity, find the value of \( (1 + \omega - \omega^2)^7 \).

• (A) \( 128\omega^2 \)
• (B) \( -128\omega^2 \)
• (C) \( 64\omega \)
• (D) \( -64\omega^2 \)

In an LCR series circuit, the resonance frequency is \( f \). If the capacitance is made 4 times, what will be the new resonance frequency?

• (A) \( 4f \)
• (B) \( 2f \)
• (C) \( f/2 \)
• (D) \( f/4 \)

A circular coil of radius \(R\) carries a current \(I\). The magnetic field at its centre is \(B\). At what distance from the centre on the axis of the coil will the magnetic field be \(B/8\)?

• (A) \( R \)
• (B) \( 2R \)
• (C) \( \sqrt{3}R \)
• (D) \( 4R \)

In a common emitter transistor amplifier, the audio signal voltage across the collector resistance of \(2\,k\Omega\) is \(2V\). If the current amplification factor (\(\beta\)) is \(100\) and the base resistance is \(1\,k\Omega\), find the input signal voltage.

• (A) \( 0.1\,V \)
• (B) \( 0.01\,V \)
• (C) \( 0.02\,V \)
• (D) \( 0.005\,V \)

Which of the following electrolytes is most effective in the coagulation of a negative sol like \(As_2S_3\)?

• (A) \( NaCl \)
• (B) \( MgCl_2 \)
• (C) \( AlCl_3 \)
• (D) \( KCl \)

Which linkage joins the monosaccharide units in sucrose?

• (A) \( C_1 - C_4 \) glycosidic linkage
• (B) \( C_1 - C_2 \) glycosidic linkage
• (C) \( C_1 - C_6 \) glycosidic linkage
• (D) \( C_2 - C_4 \) glycosidic linkage

A compound is formed by two elements \(X\) and \(Y\). Atoms of \(Y\) make ccp and those of element \(X\) occupy all the octahedral voids. What is the formula of the compound?

• (A) \( XY_2 \)
• (B) \( X_2Y \)
• (C) \( XY \)
• (D) \( X_2Y_3 \)

Complete the series: \(285,\,253,\,221,\,189,\, ?\)

• (A) \( 165 \)
• (B) \( 157 \)
• (C) \( 169 \)
• (D) \( 145 \)