BITSAT 2025 Question Paper 29th May Shift 1 (Avalable): Download Solutions with Answer Key

Question 1:

If one root of the quadratic equation \( ax^2 + bx + c = 0 \) is double the other, then what is the correct relation among the coefficients?

• (A) \( b^2 = 8ac \)
• (B) \( b^2 = 4ac \)
• (C) \( b^2 = \frac{9ac}{2} \)
• (D) \( b^2 = 2ac \)

Evaluate the integral \( \int_0^1 \frac{\ln(1 + x)}{1 + x^2} \, dx \)

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

If \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} - \hat{j} + 2\hat{k} \), then find the angle \( \theta \) between \( \vec{a} \) and \( \vec{b} \).

• (A) \( \cos^{-1}\left(\frac{3}{\sqrt{30}}\right) \)
• (B) \( \cos^{-1}\left(\frac{5}{\sqrt{30}}\right) \)
• (C) \( \cos^{-1}\left(\frac{6}{\sqrt{30}}\right) \)
• (D) \( \cos^{-1}\left(\frac{7}{\sqrt{30}}\right) \)

If \( z = x + iy \) is a complex number such that \( |z - 1| = |z + 1| \), then the locus of \( z \) represents:

• (A) A circle with center at origin
• (B) The real axis
• (C) The imaginary axis
• (D) A line parallel to the x-axis

Two numbers are selected at random (without replacement) from the first 6 natural numbers. What is the probability that the difference of the numbers is less than 3?

• (A) \( \frac{1}{3} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{3}{5} \)
• (D) \( \frac{5}{15} \)

Solve the inequality: \( \log_2(x^2 - 5x + 6) > 1 \)

• (A) \( x \in (2,3) \cup (3, \infty) \)
• (B) \( x \in (0,1) \cup (4, \infty)) \)
• (C) \( x \in (0,2) \cup (2,3) \)
• (D) \( x \in (1,2) \cup (3, \infty) \)

If \( A = \begin{vmatrix} 1 & a & a^2
1 & b & b^2
1 & c & c^2
\end{vmatrix} \), then the value of \( A \) is:

• (A) \( (a - b)(b - c)(c - a) \)
• (B) \( (a - b)(b - c)(a - c) \)
• (C) \( (a + b)(b + c)(c + a) \)
• (D) \( (b - a)(c - b)(c - a) \)

If \( \tan A + \tan B + \tan C = \tan A \tan B \tan C \), where \( A + B + C = \pi \), then what is the value of \( \tan A \tan B + \tan B \tan C + \tan C \tan A \)?

• (A) 1
• (B) 0
• (C) 2
• (D) Cannot be determined

A uniformly charged ring of radius \( R \) carries total charge \( Q \). Find the electric field at a point on the axis at a distance \( x = \frac{R}{\sqrt{2}} \) from the center.

• (A) \( \frac{1}{4\pi\varepsilon_0} \cdot \frac{Qx}{(R^2 + x^2)^{3/2}} \)
• (B) \( \frac{1}{4\pi\varepsilon_0} \cdot \frac{QR}{(R^2 + x^2)^{3/2}} \)
• (C) \( \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{R^2} \)
• (D) \( \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{(2R^2)^{3/2}} \)

Light of wavelength \( 400\, nm \) falls on a metal with work function \( \phi = 2.0\, eV \). If the intensity of the light is doubled, what happens to the maximum kinetic energy of the emitted photoelectrons?

• (A) It doubles
• (B) It becomes zero
• (C) It increases by a factor of \( \sqrt{2} \)
• (D) It remains the same

A disc of moment of inertia \( I \) is rotating with angular velocity \( \omega \). A ring of the same mass and radius, initially at rest, is gently placed coaxially on top of the disc. What is the final angular velocity of the system?

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

A damped harmonic oscillator has an amplitude that reduces to half in 10 seconds. What will be the amplitude after 30 seconds?

• (A) \( \frac{1}{4} \) of original amplitude
• (B) \( \frac{1}{8} \) of original amplitude
• (C) \( \frac{1}{16} \) of original amplitude
• (D) \( \frac{1}{2} \) of original amplitude

An ideal gas undergoes an adiabatic expansion from volume \( V \) to \( 2V \). If the initial temperature is \( T \), what is the final temperature? (Assume the ratio of specific heats \( \gamma = \frac{5}{3} \))

• (A) \( T \)
• (B) \( \frac{T}{2} \)
• (C) \( \frac{T}{2^{2/3}} \)
• (D) \( \frac{T}{2^{5/3}} \)

A buffer solution is prepared by mixing 0.1 mol of acetic acid (\( \mathrm{p}K_a = 4.74 \)) and 0.2 mol of sodium acetate in 1 L solution. What is the pH of the buffer?

• (A) 4.44
• (B) 5.04
• (C) 4.74
• (D) 5.74

Which of the following coordination compounds shows linkage isomerism?

• (A) \( [Co(NH_3)_5Cl]Cl_2 \)
• (B) \( [Co(NH_3)_5(NO_2)]Cl_2 \)
• (C) \( [Fe(CN)_6]^{3-} \)
• (D) \( [Cr(H_2O)_6]Cl_3 \)

Which of the following compounds undergoes electrophilic substitution most readily?

• (A) Nitrobenzene
• (B) Toluene
• (C) Benzene
• (D) Benzoic acid

A first-order reaction is 25% complete in 30 minutes. How much time will it take for the reaction to be 75% complete?

• (A) 90 min
• (B) 60 min
• (C) 120 min
• (D) 150 min

Choose the word that is closest in meaning to “esoteric”.

• (A) Obvious
• (B) Mysterious
• (C) Commonplace
• (D) Confidential

Identify the correct version of the sentence:

"Hardly had he entered the room when he was hearing the explosion."

• (A) Hardly had he entered the room when he heard the explosion.
• (B) Hardly he had entered the room when he heard the explosion.
• (C) He hardly entered the room when he was hearing the explosion.
• (D) No correction needed.

The scientist’s theory was initially met with _________, but later gained widespread acclaim after consistent experimental validation.

• (A) skepticism
• (B) celebration
• (C) compliance
• (D) ignorance

Choose the correct meaning of the idiom “to throw in the towel”.

• (A) To start a new challenge
• (B) To refuse help
• (C) To admit defeat
• (D) To criticize someone openly

Rearrange the following parts to form a meaningful sentence:

P. technological advancement
Q. has led to
R. in many fields
S. a significant leap

• (A) P Q S R
• (B) P R S Q
• (C) Q P R S
• (D) P Q R S