GATE EE 2015 Set 2

The sentence requires two verbs to complete the thought. The first blank needs a past tense verb related to forgetting, and the second blank needs a verb phrase indicating a lack of knowledge on how to compensate. Let's analyze the options:(A) "completely forgot --- don't just know": "don't just know" is present tense, which doesn't fit the context of a past event (forgetting a birthday).
(B) "forgot completely --- don't just know": Similar to (A), "don't just know" is present tense.
(C) "completely forgot --- just don't know": This option uses "completely forgot" for the first blank, which is grammatically correct for a past event. For the second blank, "just don't know" is also grammatically correct and fits the meaning of not knowing how to make amends.
(D) "forgot completely --- just don't know": Similar to (B), "just don't know" is present tense.Upon re-evaluating the options, there seems to be a typo in the provided options (B), (C), and (D) in the OCR, where they all appear to be "don't just know" or "just don't know". Assuming the intent was to have a past tense for the second part as well, or a present tense that expresses a current state of not knowing how to fix a past mistake, let's re-examine the common phrasing. The phrase "just don't know" can be used to express a current state of uncertainty about how to proceed after a past event. Therefore, "completely forgot" (past tense) and "just don't know" (present tense, expressing current uncertainty) can work together.Let's assume the options are intended to be distinct and the provided OCR for (C) is the correct one. The structure "We completely forgot our friend's birthday and we just don't know how to make it up to him" is grammatically sound and conveys the intended meaning. The first part describes a past action (forgetting), and the second part describes a current state of not knowing how to rectify it.The final answer is $\boxed{C}$

The word "personnel" refers to the people employed in an organization or service. Let's examine each option:(A) "The industrialist had a personnel jet." This is incorrect. A jet belonging to a person would be a "personal" jet, not a "personnel" jet.
(B) "I write my experience in my personnel diary." This is incorrect. A diary for one's own use is a "personal" diary, not a "personnel" diary.
(C) "All personnel are being given the day off." This is correct. "Personnel" here refers to all the employees, and it is used correctly in this context.
(D) "Being religious is a personnel aspect." This is incorrect. A characteristic of an individual is a "personal" aspect, not a "personnel" aspect.Therefore, option (C) uses the word "personnel" correctly.The final answer is $\boxed{C}$

Let's analyze the given options:(A) "Fabric" refers to cloth or other material produced by weaving or knitting fibers.
(B) "Textile" is a broader term for any cloth or fabric, especially one produced by weaving or knitting.
(C) "Fibre" refers to a thread or filament from which a vegetable tissue, mineral substance, or textile is formed.
(D) "Apparel" is a generic term for clothing or garments.A skirt, a pair of trousers, and a shirt are all items of clothing. Therefore, "apparel" is the most appropriate generic term that includes these items.The final answer is $\boxed{D}$

Let $w$ be the weight of one pole.From Statement (I): $\frac{1}{4} w = 5 \implies w = 20$ Kg. The total weight of 10 poles is $10 \times 20 = 200$ Kg. Thus, Statement I alone is sufficient.From Statement (II): Let the total weight of 10 poles be $10w$. According to the statement, $10w = 2w + 160 \implies 8w = 160 \implies w = 20$ kg. The total weight of 10 poles is $10 \times 20 = 200$ kg. Thus, Statement II alone is sufficient.Since either statement alone is sufficient to answer the question, the correct option is (C).

The function is $f(x) = 1 - |x|$ defined on the interval $[-1, 1]$.
The absolute value function $|x|$ is always $\geq 0$ and reaches its minimum value of $0$ at $x = 0$.
Since $f(x) = 1 - |x|$, the function $f(x)$ will be at its maximum when the subtracted term $|x|$ is at its minimum.
At $x = 0$, $f(0) = 1 - |0| = 1$.
Thus, the function attains its maximum value of $1$ at $x = 0$. The pair is $(0, 1)$.

Analysis of the options:
(A) This sentence is grammatically correct and uses appropriate vocabulary.
(B) "Information" is an uncountable noun. Therefore, it should be "there was no needed information" instead of "there were no needed information".
(C) "Real" is an adjective. To modify the adjective "useful", the adverb "really" should be used ("not really useful").
(D) The construction "would not had been" is grammatically incorrect. The correct modal perfect form is "would not have been".Therefore, option (A) is the most suitable sentence.

Let the length of $PS$ be $x$. The area of $\triangle PQR$ is equal to the sum of the areas of $\triangle PQS$ and $\triangle PSR$.Area of $\triangle PQR = \frac{1}{2} qr \sin(120^\circ) = \frac{1}{2} qr \frac{\sqrt{3}}{2}$
Area of $\triangle PQS = \frac{1}{2} r x \sin(60^\circ) = \frac{1}{2} r x \frac{\sqrt{3}}{2}$
Area of $\triangle PSR = \frac{1}{2} q x \sin(60^\circ) = \frac{1}{2} q x \frac{\sqrt{3}}{2}$Equating the areas:$$\frac{1}{2} qr \frac{\sqrt{3}}{2} = \frac{1}{2} r x \frac{\sqrt{3}}{2} + \frac{1}{2} q x \frac{\sqrt{3}}{2}$$$$qr = rx + qx$$$$qr = x(r + q)$$$$x = \frac{qr}{q+r}$$Thus, the length of $PS$ is $\frac{qr}{q+r}$, which corresponds to option (B).

First, we evaluate the inner function $h(2, 5, 7, 3)$:
$p=2, q=5, r=7, s=3$
$p \times q = 10$
$r \times s = 21$
Since $(r \times s) > (p \times q)$, $h = \text{remainder of } 21 / 10 = 1$.Now we evaluate $fg(1, 4, 6, 8)$:
Based on the definition of $fgh$, $fg(p, q, r, s) = f(p, q, r, s) \times g(p, q, r, s)$.
$f(1, 4, 6, 8) = \max(1, 4, 6, 8) = 8$
$g(1, 4, 6, 8) = \min(1, 4, 6, 8) = 1$
$fg(1, 4, 6, 8) = 8 \times 1 = 8$.The final value is 8.

Let the arithmetic sequence be $a, a+d, a+2d, a+3d, a+4d$.I. $2a, 2(a+d), 2(a+2d), 2(a+3d), 2(a+4d) \implies 2a, 2a+2d, 2a+4d, 2a+6d, 2a+8d$. This is an arithmetic sequence with common difference $2d$.II. $a-3, (a+d)-3, (a+2d)-3, (a+3d)-3, (a+4d)-3 \implies a-3, a-3+d, a-3+2d, a-3+3d, a-3+4d$. This is an arithmetic sequence with common difference $d$.III. $a^2, (a+d)^2, (a+2d)^2, (a+3d)^2, (a+4d)^2$. The difference between consecutive terms is not constant (e.g., $(a+d)^2 - a^2 = 2ad + d^2$, while $(a+2d)^2 - (a+d)^2 = 2ad + 3d^2$). Thus, it is not an arithmetic sequence unless $d=0$.Therefore, only I and II are always in arithmetic sequence. The correct option is (B).

Let the coordinates of O be $(0, 0)$.
N is southeast of O by 1 km. Southeast direction is at an angle of $-45^\circ$ from the positive x-axis.
Coordinates of N: $(1 \cos(-45^\circ), 1 \sin(-45^\circ)) = (\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}) \approx (0.707, -0.707)$.
M is north of N at a distance of 4 km.
Coordinates of M: $(0.707, -0.707 + 4) = (0.707, 3.293)$.
P is south of O at a distance of 2 km.
Coordinates of P: $(0, -2)$.
The distance between M and P is:
$d = \sqrt{(0.707 - 0)^2 + (3.293 - (-2))^2} = \sqrt{0.707^2 + 5.293^2} = \sqrt{0.5 + 28.016} = \sqrt{28.516} \approx 5.34$ km.

Statement (B) is true because the sum of two differentiable functions is always differentiable.
Statement (A) is false: $f(z) = z$ is differentiable, but $g(z) = \bar{z}$ and $h(z) = z - \bar{z}$ are not.
Statement (C) is false: Continuity does not imply differentiability (e.g., $f(z) = |z|$).
Statement (D) is false: $f(z) = z$ is differentiable, but its real part $u(x,y) = x$ is not differentiable as a complex function because it doesn't satisfy the Cauchy-Riemann equations.

For a system of $n$ linear equations in $n$ unknowns $Ax = b$:

• P: A unique solution exists if and only if $A$ is non-singular ($\det(A) \neq 0$).
• Q: The equations are linearly independent if and only if the rows of $A$ are linearly independent, which means $A$ is non-singular.
• R: All eigenvalues of $A$ are nonzero if and only if their product (the determinant) is nonzero, which means $A$ is non-singular.
• S: The determinant of $A$ is nonzero means $A$ is non-singular.

Since all four statements are equivalent to $A$ being non-singular, $P \equiv Q \equiv R \equiv S$.

The matching is based on fundamental theorems in vector calculus and complex analysis:Thus, the correct matching is P-4, Q-1, R-3, S-2, which is option (B).

Given $f(t) = 2\sqrt{\frac{t}{\pi}} = \frac{2}{\sqrt{\pi}} t^{1/2}$ and its Laplace transform $L\{f(t)\} = s^{-3/2}$.
We need to find the Laplace transform of $g(t) = \sqrt{\frac{1}{\pi t}} = \frac{1}{\sqrt{\pi}} t^{-1/2}$.
Observe the relationship between $f(t)$ and $g(t)$:
$f'(t) = \frac{d}{dt} \left( \frac{2}{\sqrt{\pi}} t^{1/2} \right) = \frac{2}{\sqrt{\pi}} \cdot \frac{1}{2} t^{-1/2} = \frac{1}{\sqrt{\pi}} t^{-1/2} = g(t)$.
Using the property of Laplace transforms for derivatives:
$L\{f'(t)\} = sL\{f(t)\} - f(0)$
Since $f(0) = 2\sqrt{0/\pi} = 0$, we have:
$L\{g(t)\} = s \cdot (s^{-3/2}) - 0 = s^{1 - 3/2} = s^{-1/2}$.
Therefore, the correct option is (B).

The matching is based on the operating principles of electrical instruments:

• P. Permanent magnet moving coil (PMMC): These instruments respond to the average value of current. In AC, the average value is zero, so they are used for DC Only (P-1).
• Q. Moving iron (MI) connected through current transformer (CT): While MI instruments can measure both AC and DC, current transformers (CTs) operate on the principle of electromagnetic induction, which requires a time-varying flux. Thus, CTs do not work on DC, making the combination suitable for AC Only (Q-2).
• R. Rectifier instruments: These convert AC to DC to be measured by a PMMC. They can be used for both AC and DC measurements (R-3).
• S. Electrodynamometer: These instruments use the interaction of magnetic fields from two coils. Since the current in both coils reverses simultaneously in AC, the torque remains unidirectional. They are used for both AC and DC (S-3).

Matching: P-1, Q-2, R-3, S-3. This corresponds to option (C).

Total power $P = W_1 + W_2 = 5$ kW.
Power factor $\cos \phi = 0.707 \implies \phi = 45^\circ$.
In the two-wattmeter method, the relationship between the readings and the phase angle is given by $\tan \phi = \sqrt{3} \frac{W_1 - W_2}{W_1 + W_2}$.
Substituting the values:
$\tan 45^\circ = 1 = \sqrt{3} \frac{W_1 - W_2}{5} \implies W_1 - W_2 = \frac{5}{\sqrt{3}} \approx 2.8867$ kW.
We have two equations:
1) $W_1 + W_2 = 5$
2) $W_1 - W_2 = 2.8867$
Adding (1) and (2): $2W_1 = 7.8867 \implies W_1 = 3.943$ kW.
Subtracting (2) from (1): $2W_2 = 2.1133 \implies W_2 = 1.057$ kW.
Thus, the readings are approximately 3.94 kW and 1.06 kW.

The output voltage of the capacitive divider is given by the formula: $$V_{out} = V_{bus} \left( \frac{C_1}{C_1 + C_2} \right)$$
To find the maximum possible value of $V_{out}$, we must maximize the ratio $\frac{C_1}{C_1 + C_2}$. This occurs when the numerator $C_1$ is at its maximum value and the denominator is minimized, which happens when $C_2$ is at its minimum value.
Given nominal values and tolerances:
$C_1 = 1 \mu\text{F} \pm 10\% \implies C_{1,max} = 1.1 \mu\text{F}$
$C_2 = 9 \mu\text{F} \pm 10\% \implies C_{2,min} = 9 \times 0.9 = 8.1 \mu\text{F}$
Substituting these into the formula: $$V_{out,max} = 100 \text{ kV} \times \frac{1.1}{1.1 + 8.1} = 100 \times \frac{1.1}{9.2} \approx 11.9565 \text{ kV}$$
The specified range for the answer is 11.75 to 12.25.

The circuit shown is a Villard voltage doubler. The input voltage is $V_{in} = 100 \sin(100\pi t)$, which has a peak value $V_m = 100$ V.
In a voltage doubler, the output voltage across the load resistor $R$ is approximately $2V_m$ if the time constant $RC$ is significantly larger than the period $T$ of the input signal.
Given: $\omega RC = 100\pi RC = 50$.
The period $T = \frac{2\pi}{\omega} = \frac{2\pi}{100\pi} = 0.02$ s.
The time constant $RC = \frac{50}{100\pi} \approx 0.159$ s.
Since $RC \approx 8T$, the capacitors maintain a nearly constant voltage. The steady-state output voltage is approximately $2V_m = 2 \times 100 = 200$ V.
Therefore, the average voltage across $R$ is nearest to 200 V.

Boundary conditions at the interface of two dielectrics ($y=0$):Given $\epsilon_1 = 2$, $\epsilon_2 = 5$, and $E_{y1} = 4$.
$2 \times 4 = 5 \times E_{y2} \Rightarrow E_{y2} = \frac{8}{5} = 1.6$.
Thus, the electric field in region 2 is $\vec{E}_2 = 3\hat{a}_x + 1.6\hat{a}_y + 2\hat{a}_z$.

The magnetic flux through the loop is $\Phi = \vec{B} \cdot \vec{A} = BA \cos(\omega t)$.
The induced voltage is $V = -\frac{d\Phi}{dt} = BA\omega \sin(\omega t)$.
The peak value is $V_{peak} = BA\omega$.
Given:
$H = 10^7$ A/m $\Rightarrow B = \mu_0 H = (4\pi \times 10^{-7}) \times 10^7 = 4\pi$ T.
$A = \pi r^2 = \pi (1)^2 = \pi$ m$^2$.
$\omega = 60$ rpm $= 60 \times \frac{2\pi}{60} = 2\pi$ rad/s.
$V_{peak} = (4\pi) \times (\pi) \times (2\pi) = 8\pi^3 \approx 8 \times 31.006 = 248.05$ V.

The circuit is an inverting amplifier with a feedback impedance $Z_f$ and input resistance $R_{in}$.
$Z_{in} = R_{in} = 1\text{ k}\Omega$.
$Z_f = R_f \parallel \frac{1}{sC_f} = \frac{R_f}{1 + sR_fC_f}$, where $R_f = 1\text{ k}\Omega$ and $C_f = 0.1\text{ }\mu\text{F}$.
The transfer function is $H(s) = \frac{V_o(s)}{V_i(s)} = -\frac{Z_f}{Z_{in}} = -\frac{R_f/R_{in}}{1 + sR_fC_f} = -\frac{1}{1 + s(10^3 \times 0.1 \times 10^{-6})} = -\frac{1}{1 + s(10^{-4})}$.
For $V_i = 2 \sin(2\pi \times 2000t)$, the angular frequency is $\omega = 4000\pi$ rad/s.
The magnitude of the gain at this frequency is:
$|H(j\omega)| = \frac{1}{\sqrt{1 + (\omega \times 10^{-4})^2}} = \frac{1}{\sqrt{1 + (4000\pi \times 10^{-4})^2}} = \frac{1}{\sqrt{1 + (0.4\pi)^2}} \approx \frac{1}{\sqrt{1 + 1.579}} \approx \frac{1}{1.606} \approx 0.6227$.
The amplitude of the output voltage is $V_{o,peak} = V_{i,peak} \times |H(j\omega)| = 2 \times 0.6227 = 1.2454$ V.

In active mode, the collector current $I_C$ is given by $I_C = \beta I_B$. The base current $I_B$ is determined by the base-emitter loop: $I_B = \frac{V_{CC} - V_{BE}}{R_B}$. Since $V_{CC}$, $V_{BE}$, and $R_B$ remain unchanged when $R_C$ is replaced by $R'_C$, the base current $I_B$ and consequently the collector current $I_C$ remain constant.Case 1: With $R_C$, $V_C = 2\text{ V}$.
$V_C = V_{CC} - I_C R_C$
$2 = 10 - I_C R_C \implies I_C R_C = 8$Case 2: With $R'_C$, $V_C = 4\text{ V}$.
$V_C = V_{CC} - I_C R'_C$
$4 = 10 - I_C R'_C \implies I_C R'_C = 6$Taking the ratio:$$\frac{I_C R'_C}{I_C R_C} = \frac{6}{8} = 0.75$$$$\frac{R'_C}{R_C} = 0.75$$The answer is $0.75$.

The given Sum of Products (SOP) expression is:
$F = ABC + \bar{A}\bar{B}C + A\bar{B}C + \bar{A}BC + \bar{A}\bar{B}\bar{C}$Identifying the minterms:

• $ABC \implies m_7$ (111)
• $\bar{A}\bar{B}C \implies m_1$ (001)
• $A\bar{B}C \implies m_5$ (101)
• $\bar{A}BC \implies m_3$ (011)
• $\bar{A}\bar{B}\bar{C} \implies m_0$ (000)

So, $F(A, B, C) = \sum m(0, 1, 3, 5, 7)$.The complement of the function is the sum of the missing minterms:
$\bar{F}(A, B, C) = \sum m(2, 4, 6)$The Product of Sums (POS) form is the product of the corresponding maxterms:
$F(A, B, C) = \prod M(2, 4, 6) = M_2 \cdot M_4 \cdot M_6$Converting minterms to maxterms:

• $m_2 = \bar{A}B\bar{C} \implies M_2 = (A + \bar{B} + C)$
• $m_4 = A\bar{B}\bar{C} \implies M_4 = (\bar{A} + B + C)$
• $m_6 = AB\bar{C} \implies M_6 = (\bar{A} + \bar{B} + C)$

Thus, $F = (A + \bar{B} + C)(\bar{A} + B + C)(\bar{A} + \bar{B} + C)$. This matches option (A).