Question papers
GATE 2025 - Q1
Consider the matrix A below:$$\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&\alpha&\beta \\0&0&0&\gamma \end{bmatrix}$$ For which of the following combinations of $\alpha$, $\beta$, and $\gamma$, is the rank of the matrix A atleast three?
(i) $\alpha=0$ and $\beta = \gamma \neq 0$
(ii) $\alpha=\beta = \gamma = 0$
(iii) $\alpha \neq 0$ and $\beta = \gamma = 0$
(iv) $\alpha=\beta = \gamma \neq 0$
a) only (i), (iii) and (iv)
b) only (iv)
c) only (ii)
d) only (i) and (iii)
Correct Answer: A
Rank of A has to be atleast three means rank can be $3$ or more ($4$).
(i) $\alpha = 0$, $\beta \neq 0$, $\gamma \neq 0$ and $\beta =\gamma$.$$\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&0&\beta \\0&0&0&\beta \end{bmatrix}$$ $R_4\leftarrow R_4-R_3$, we will get $$\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&0&\beta \\0&0&0&0 \end{bmatrix}$$$$\rho(A)=3$$
(ii) $\alpha = 0$, $\beta = 0$ and $\gamma = 0$.$$\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&0&0 \\0&0&0&0 \end{bmatrix}$$ $$\rho(A)=2$$
(iii) $\alpha \neq 0$, $\beta = 0$, $\gamma = 0$.$$\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&\alpha&0 \\0&0&0&0 \end{bmatrix}$$ Exchanging column $3$ and column $4$ , we will get $$\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&0&\alpha \\0&0&0&0 \end{bmatrix}$$$$\rho(A)=3$$
(iv) $\alpha \neq 0$, $\beta \neq 0$, $\gamma \neq 0$ and $\alpha=\beta=\gamma$.$$\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&\alpha &\alpha \\0&0&0&\alpha \end{bmatrix}$$$$\rho(A)=4$$
Hence (i), (iii) and (iv) are correct.
GATE 2025 - Q2
Consider a part of an electrical network as shown below. Some node voltages and the current flowing through the $3\Omega$ resistor are as indicated. The voltage (in volts) at node X is
a) $20/3$
b) $32/3$
c) $22/3$
d) $2/3$
Step $1$: Voltage across $3\Omega$ resistor is $$V_{3\Omega}=1×3=3V$$
Step $2$: Voltage at node A, $$9-V_{A}=3V$$$$V_{A}=9-3=6V$$
Step $3$: Now finding the current flowing through $2\;\Omega$ and $1\;\Omega$ will be$$I=\frac{8-6}{3}=2/3$$
Step $4$: Voltage across $2\Omega$ resistor is $$V_{2\Omega}=2/3×2=4/3V$$
Step $5$: Voltage at node X, $$8-V_{X}=4/3$$$$V_{X}=8-4/3=20/3$$
GATE 2025 - Q3
Consider an additive white Gaussian noise (AWGN) channel with bandwidth W and noise spectral density $\frac{N_0}{2}$
a) $103\;nJ$
b) $60\;nJ$
c) $40\;nJ$
d) $20\;nJ$
Correct Answer: Option D
One coulomb of charge equals
$$-1C=6.24×10^{18}\;e^-$$
‘n’ coulombs of charge equals,$$-nC=n×6.24×10^{18}\;e^-$$
Hence, $-3.941\;C$ equals, $$=3.941×6.24×10^{18}\;e^-$$ $$=24.59×10^{18}\;e^-$$ $$=2.459×10^{19}\;e^-$$
RPSC Lecturer - Q4
🌾If a vector field P is solenoidal, which of this is true?
a) $\oint_s P.ds=0$
b) $\oint_l P.dl=0$
c) $\nabla× (P)=0$
d) $\nabla×P\neq0$
Correct Answer: Option A and C
Unit of charge is coloumb.
Since $I=Q/t$ and $Q=It$, the unit of charge is also ampere-seconds
RPSC Lecturer - Q5
🌾The boundary condition valid at the boundary between two dielectrics $1$ and $2$ is
a) $E_{t1}=E_{t2}$
b) $E_1=E_2$
c) $D_{n1}=D_{n2}$
d) both A and C
Correct Answer: Option B
Current flow is nothing but the rate of flow of charge $$I=\frac{Q}{t}$$
RPSC Lecturer - Q6
🌾The magnetic field at any point on the axis of a current carrying circular coil will be
a) perpendicular to the axis
b) parallel to the axis
c) at an angle $45^{\circ}$ with the axis
d) zero
Correct Answer: Option B
Current flow is nothing but the rate of flow of charge $$I=\frac{Q}{t}$$
RPSC Lecturer - Q7
🌾Which of these statements is not characteristic of a static magnetic field?
a) It is conservative
b) It is solenoidal
c) It has no sink or sources
d) Magnetic flux lines are always closed
Correct Answer: Option B
Charge of an electron is $-1.602×10^{-19}\;C$
RPSC Lecturer - Q8
🌾Which of the following is zero?
a) grad div A
b) div gradient V
c) div curl A
d) curl curl A
Correct Answer: Option B
Charge of an electron is $-1.602×10^{-19}\;C$