Algebra - General Aptitude - General Aptitude (GA) Previous Year Questions

1

2014 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2014

If \( y = 5x^2 + 3 \), then the tangent at \( x = 0 \), \( y = 3 \)

A

passes through \(x = 0 \), \(y = 0 \)

B

has a slope of +1

C

is parallel to the x-axis

D

has a slope of −1

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2

2015 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2015

An operator for a spin-\( \frac{1}{2} \) particle is given by \( \hat{A} = \lambda \vec{\sigma} \cdot \vec{B} \), where \( \vec{B} = \frac{B}{\sqrt{2}} (\hat{x} + \hat{y}) \), \( \vec{\sigma} \) denotes Pauli matrices and \( \lambda \) is a constant. The eigenvalues of \( \hat{A} \) are

A

\(\pm \lambda B / \sqrt{2} \)

B

\(\pm \lambda B \)

C

\(0, \lambda B \)

D

\(0, -\lambda B \)

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3

2015 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2015

In an inertial frame \( S \), two events \( A \) and \( B \) take place at \( (ct_A = 0, \vec{r}_A = 0) \) and \( (ct_B = 0, \vec{r}_B = 2\hat{y}) \), respectively. The times at which these events take place in a frame \( S' \) moving with a velocity \( 0.6c \hat{y} \) with respect to \( S \) are given by

A

\(ct'_A = 0; ct'_B = -3/2 \)

B

\(ct'_A = 0; ct'_B = 0 \)

C

\(ct'_A = 0; ct'_B = 3/2 \)

D

\(ct'_A = 0; ct'_B = 1/2 \)

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4

2015 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2015

A particle with rest mass \(M\) is at rest and decays into two particles of equal rest masses \(\frac{3}{10}M\) which move along the \(z\) axis. Their velocities are given by

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5

2015 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2015

The entropy of a gas containing \( N \) particles enclosed in a volume \( V \) is given by \( S = Nk_B \ln \left( \frac{aVE^{3/2}}{N^{5/2}} \right) \), where \( E \) is the total energy, \( a \) is a constant and \( k_B \) is the Boltzmann constant. The chemical potential \( \mu \) of the system at a temperature \( T \) is given by

A

\(\mu = -k_B T \left[ \ln \left( \frac{aVE^{3/2}}{N^{5/2}} \right) - \frac{5}{2} \right] \)

B

\(\mu = -k_B T \left[ \ln \left( \frac{aVE^{3/2}}{N^{5/2}} \right) - \frac{3}{2} \right] \)

C

\(\mu = -k_B T \left[ \ln \left( \frac{aVE^{3/2}}{N^{5/2}} \right) - \frac{5}{2} \right] \)

D

\(\mu = -k_B T \left[ \ln \left( \frac{aVE^{3/2}}{N^{5/2}} \right) - \frac{3}{2} \right] \)

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6

2016 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2016

The binary operation \(\Box\) is defined as \(a \Box b = ab+(a+b)\), where \(a\) and \(b\) are any two real numbers. The value of the identity element of this operation, defined as the number \(x\) such that \(a \Box x = a\), for any \(a\), is _____.

A

0

B

1

C

2

D

10

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7

2018 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2018

For \(0 \leq x \leq 2\pi\), \(\sin x\) and \(\cos x\) are both decreasing functions in the interval _______.

A

\(\left(0, \frac{\pi}{2}\right)\)

B

\(\left(\frac{\pi}{2}, \pi\right)\)

C

\(\left(\pi, \frac{3\pi}{2}\right)\)

D

\(\left(\frac{3\pi}{2}, 2\pi\right)\)

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8

2021 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2021

\( \oplus \) and \( \odot \) are two operators on numbers \( p \) and \( q \) such that \( p \odot q = p - q \), and \( p \oplus q = p \times q \). Then, \( (9 \odot (6 \oplus 7)) \odot (7 \oplus (6 \odot 5)) = \)

A

40

B

-26

C

-33

D

-40

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9

2024 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2024

The real variables \(x, y, z\), and the real constants \(p, q, r\) satisfy
\[\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}\]
Given that the denominators are non-zero, the value of \(px + qy + rz\) is

A

0

B

1

C

pqr

D

p^2 + q^2 + r^2

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10

2025 · General Aptitude (GA) · General Aptitude · Algebra

Physics (PH) 2025

The sum of the following infinite series is:
\[ \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \cdots \]

A

\(\pi \)

B

\(1 + e \)

C

\(e - 1 \)

D

\(e \)

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