BITSAT 2025 Question Paper 29th May Shift 2 (Available): Download Solutions with Answer Key

Evaluate the integral: \[ \int_0^{\pi/4} \frac{\ln(1 + \tan x)}{\cos x \sin x} \, dx \]

• (A) \( \frac{\pi}{4} \ln 2 \)
• (B) \( \frac{\pi}{8} \ln 2 \)
• (C) \( \ln 2 \)
• (D) \( \frac{1}{2} \ln 2 \)

If a point \( P(x, y) \) satisfies the condition that its distance from the point \( (3, -2) \) is equal to its distance from the line \( y = 2x + 1 \), then the locus of point \( P \) is:

• (A) A parabola
• (B) A circle
• (C) A straight line
• (D) A pair of straight lines

Let the function \( f(x) = \sqrt{\log_e(1 - x^2)} \). Then the domain of \( f(x) \) is:

• (A) \( (-1, 0) \cup (0, 1) \)
• (B) \( (-1, 1) \)
• (C) \( (-1, 1) \setminus \{0\} \)
• (D) \( \left(-1, -\frac{1}{\sqrt{e}} \right) \cup \left( \frac{1}{\sqrt{e}}, 1 \right) \)

Evaluate the sum: \[ \sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)} \]

• (A) \( \frac{1}{4} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{1}{6} \)
• (D) \( \frac{1}{3} \)

Let \( f(x) = |x^2 - 4x + 3| + |x^2 - 5x + 6| \). The minimum value of \( f(x) \) is:

• (A) 2
• (B) 3
• (C) 1
• (D) 0

Let vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be such that \[ \mathbf{a} = \hat{i} + 2\hat{j} - \hat{k}, \quad \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k}, \quad \mathbf{c} = \hat{i} + \hat{j} + \hat{k} \]
Then the volume of the parallelepiped formed by these vectors is:

A box contains 5 red balls and 4 green balls. Two balls are drawn one after another without replacement. What is the probability that the second ball is green, given that the first ball drawn was red?

• (A) \(\dfrac{1}{2}\)
• (B) \(\dfrac{5}{18}\)
• (C) \(\dfrac{2}{5}\)
• (D) \(\dfrac{4}{8}\)

Evaluate: \[ \lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{x} \]

• (A) 1
• (B) 0
• (C) \(\infty\)
• (D) 0.5

A particle moves along the x-axis under a force \( F(x) = 6x^2 \) N.
The work done by this force in moving the particle from \( x = 1 \, m \) to \( x = 2 \, m \) is:

• (A) 14 J
• (B) 18 J
• (C) 24 J
• (D) 28 J

A Carnot engine operates between temperatures of \( 600 \, K \) and \( 300 \, K \).
If it absorbs \( 900 \, J \) of heat from the source, how much work does it perform?

• (A) 300 J
• (B) 450 J
• (C) 600 J
• (D) 150 J

Light of wavelength \( 400 \, nm \) falls on a metal surface with a work function of \( 2.0 \, eV \).
(Planck’s constant \( h = 6.63 \times 10^{-34} \, Js, \, c = 3 \times 10^8 \, m/s, \, 1 \, eV = 1.6 \times 10^{-19} \, J \))
Find the maximum kinetic energy of emitted photoelectrons.

• (A) 1.1 eV
• (B) 2.1 eV
• (C) 0.1 eV
• (D) 0.8 eV

Two capacitors \( C_1 = 4\mu F \) and \( C_2 = 6\mu F \) are connected in series across a 60 V battery.
The potential difference across \( C_2 \) is:

• (A) 24 V
• (B) 36 V
• (C) 40 V
• (D) 20 V

A proton enters a uniform magnetic field \(\vec{B} = 0.5 \hat{k} \ T\) with velocity \(\vec{v} = 10^6 \hat{i} \ m/s\). The magnitude of the magnetic force on the proton is:

• (A) \(8.0 \times 10^{-14}\) N
• (B) \(1.6 \times 10^{-13}\) N
• (C) \(8.0 \times 10^{-13}\) N
• (D) 0

The pH of a 0.01 M solution of a weak acid HA is 4. Calculate its dissociation constant (Ka).

• (A) \(1.0 \times 10^{-6}\)
• (B) \(1.0 \times 10^{-8}\)
• (C) \(1.0 \times 10^{-4}\)
• (D) \(1.0 \times 10^{-5}\)

The complex \([Cr(NH_3)_4Cl_2]^+\) shows how many geometrical isomers?

• (A) 1
• (B) 2
• (C) 3
• (D) 4

The standard electrode potential for \(Zn^{2+}/Zn\) is \(-0.76\) V and for \(Cu^{2+}/Cu\) is \(+0.34\) V. The EMF of the cell

\[ Zn|Zn^{2+}||Cu^{2+}|Cu \]

is:

• (A) 1.10 V
• (B) 0.76 V
• (C) -0.42 V
• (D) 0.34 V

The acidic character of the oxides increases in the order:

• (A) Na\(_2\)O \(<\) MgO \(<\) Al\(_2\)O\(_3\) \(<\) SiO\(_2\) \(<\) P\(_2\)O\(_5\)
• (B) Na\(_2\)O \(<\) Al\(_2\)O\(_3\) \(<\) MgO \(<\) SiO\(_2\) \(<\) P\(_2\)O\(_5\)
• (C) P\(_2\)O\(_5\) \(<\) SiO\(_2\) \(<\) Al\(_2\)O\(_3\) \(<\) MgO \(<\) Na\(_2\)O
• (D) Al\(_2\)O\(_3\) \(<\) SiO\(_2\) \(<\) MgO \(<\) P\(_2\)O\(_5\) \(<\) Na\(_2\)O

A first-order reaction is 25% complete in 30 minutes. What is its half-life?