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Mathematics QUESTION #6985

In a game, a player wins Rs. 100 for rolling a 5 or 6 on a fair die, and loses Rs. 50 for any other outcome. He throws the die until he gets 5 or 6, or until he has thrown at most 3 times. Find his expected gain or loss.

Loss of $\dfrac{400}{9}$
Gain of $\dfrac{400}{3}$
Loss of $\dfrac{400}{3}$

Correct Answer Logic:

$P(\text{win on a throw})=\frac{2}{6}=\frac{1}{3}$, $P(\text{lose})=\frac{2}{3}$.

The player stops as soon as he wins or after 3 throws.

Win on throw 1: prob $=\frac{1}{3}$; gain $=+100$.
Win on throw 2 (lost throw 1): prob $=\frac{2}{3}\cdot\frac{1}{3}=\frac{2}{9}$; net $=-50+100=+50$.
Win on throw 3 (lost throws 1,2): prob $=\frac{2}{3}\cdot\frac{2}{3}\cdot\frac{1}{3}=\frac{4}{27}$; net $=-100+100=0$.
Lose all 3 throws: prob $=\left(\frac{2}{3}\right)^3=\frac{8}{27}$; net $=-150$.

$E = \frac{1}{3}(100)+\frac{2}{9}(50)+\frac{4}{27}(0)+\frac{8}{27}(-150)$

$=\frac{100}{3}+\frac{100}{9}+0-\frac{1200}{27}$

$=\frac{900}{27}+\frac{300}{27}-\frac{1200}{27}=\frac{0}{27}=\mathbf{0}$

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English QUESTION #8539

Question 4742

The teacher asked the students to complete their assignment _______ Monday.

on
by✔️
about
at

Correct Answer Logic:

'By Monday' means no later than Monday.

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Boilers Mechanical Engineering QUESTION #2523

Question 4743

The instantaneous center (IC) of zero velocity for a rigid body in planar motion is the point at which:

Velocity is maximum
Acceleration is zero
The body has zero instantaneous velocity — every other point's velocity is perpendicular to the line joining it to the IC✔️
Angular velocity is zero

Correct Answer Logic:

The instantaneous center of zero velocity is a point where the velocity is zero at that instant. Every point on the body moves perpendicular to its line connecting to the IC, with speed proportional to its distance: $v_P=\omega\cdot r_{P/IC}$.

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General Knowledge QUESTION #9025

Question 4744

The Bolan Pass, a mountain pass in Balochistan, passes through which mountain range?

Kirthar Range
Sulaiman Range✔️
Toba Kakar Range
Chaghi Range

Correct Answer Logic:

The Bolan Pass goes through the Toba Kakar Range? Key says C Toba Kakar.

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Chemistry QUESTION #7821

Question 4745

Which molecule has a pyramidal shape?

C₂H₄
CH₄
H₂O
NH₃✔️

Correct Answer Logic:

NH₃ has trigonal pyramidal shape (3 bond pairs + 1 lone pair). CH₄ tetrahedral, H₂O bent, C₂H₄ planar.

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Tourism & Hospitality Management QUESTION #1621

Question 4746

The Tourism Management course covers 'managing urban tourism.' Which of the following is identified as a key challenge specific to urban tourism management?

Lack of accommodation in rural areas
Balancing the needs of resident populations with the demands of tourists, managing heritage sites in urban contexts, and addressing overcrowding and infrastructure strain✔️
The absence of natural attractions in cities
The high cost of adventure sports in urban environments

Correct Answer Logic:

Module 3 of the Tourism Management course — 'Managing Tourism in Its Environment' — covers urban tourism and identifies key challenges including: integrating tourism within existing urban infrastructure, managing heritage sites, balancing resident quality of life against tourist demand, and addressing overcrowding — issues that differ fundamentally from rural or natural-area tourism management.

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English QUESTION #2714

Question 4747

What does 'Break down' mean in the context of machinery?

To stop functioning✔️
To start working
To smash
To lose control

Correct Answer Logic:

A machine 'breaks down' when it stops working due to a mechanical failure.

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Mathematics QUESTION #4102

Question 4748

The order of element $4$ in the cyclic group $\mathbb{Z}_{18}$ is:

$9$✔️
$4$
$6$
$18$

Correct Answer Logic:

The order of element $k$ in $\mathbb{Z}_n$ is $\dfrac{n}{\gcd(k,n)}$. Here: $\dfrac{18}{\gcd(4,18)}=\dfrac{18}{2}=9$. So element $4$ has order $9$ in $\mathbb{Z}_{18}$.

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Chemistry QUESTION #1068

Question 4749

The correct order of decreasing acidity of the following aliphatic acids is:

HCOOH $>$ CH$_3$COOH $>$ (CH$_3)_2$CHCOOH $>$ (CH$_3)_3$CCOOH✔️
HCOOH $>$ (CH$_3)_3$CCOOH $>$ (CH$_3)_2$CHCOOH $>$ CH$_3$COOH
(CH$_3)_3$CCOOH $>$ (CH$_3)_2$CHCOOH $>$ CH$_3$COOH $>$ HCOOH
CH$_3$COOH $>$ (CH$_3)_2$CHCOOH $>$ (CH$_3)_3$CCOOH $>$ HCOOH

Correct Answer Logic:

Acidity of carboxylic acids decreases as alkyl group size increases due to +I (electron-donating inductive) effect destabilizing the carboxylate anion. HCOOH has no alkyl group (most acidic), then CH$_3$COOH, then (CH$_3)_2$CHCOOH, then (CH$_3)_3$CCOOH (least acidic). Order: HCOOH $>$ CH$_3$COOH $>$ (CH$_3)_2$CHCOOH $>$ (CH$_3)_3$CCOOH.

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Paper-I (Anatomy/Physiology and Biochemistry/Microbiology) QUESTION #3508

Question 4750

'Honey' is a saccharine substance deposited by 'Apis mellifera' in cells of the honeycomb. Its main sugars are:

Sucrose and Maltose
Invert sugar (Dextrose and Levulose)✔️
Lactose and Galactose
Starch

Correct Answer Logic:

Honey is primarily a mixture of glucose (dextrose) and fructose (levulose).

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Islamic Studies QUESTION #8243

Question 4751

When did the Holy Prophet (PBUH) offer Hajj?

4 A.H
8 A.H
c 8 A.H
10 A.H✔️

Correct Answer Logic:

The Holy Prophet Muhammad (PBUH) performed Hajj only once in his lifetime, known as Hajjat-ul-Wida (Farewell Pilgrimage) in 10 AH (632 CE). During this Hajj, over 100,000 Muslims accompanied him. He delivered the famous Last Sermon on Mount Arafat on 9th Dhul Hijjah, which outlined fundamental Islamic principles and human rights. This was the first and last Hajj performed by the Prophet (PBUH), as he passed away three months later.

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Physics QUESTION #397

Question 4752

100 joules work has been done by an agency in 10 seconds. What is power of agency?

1000 watt
100
10 watt✔️
0.10 watt

Correct Answer Logic:

Power = Work/Time = 100/10 = 10 watts

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The Constitution of Pakistan QUESTION #9361

Question 4753

Article 175E grants the Federal Constitutional Court exclusive original jurisdiction over disputes between Governments. The court's remedy in such cases is limited to:

Any appropriate remedy including injunctions
Declaratory judgments only✔️
Damages and compensation
Criminal sanctions against responsible officials

Correct Answer Logic:

Article 175E(2) explicitly provides that 'in the exercise of the jurisdiction conferred on it by clause (1), the Federal Constitutional Court shall pronounce declaratory judgments only.' This restricts the FCC's inter-governmental dispute remedy to declarations rather than coercive orders.

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Boilers Mechanical Engineering QUESTION #2394

Question 4754

The gain margin (GM) of a control system is defined as:

$GM = 1/|Y_1(j\omega_{pc})|$ where $\omega_{pc}$ is the phase crossover frequency (where phase $= -180°$)✔️
$GM = |Y_1(j\omega_{gc})|$ where $\omega_{gc}$ is the gain crossover frequency
$GM = \angle Y_1(j\omega_{gc})$
The maximum gain before the system becomes underdamped

Correct Answer Logic:

Gain margin: $GM=1/|Y_1(j\omega_{pc})|$ where $\omega_{pc}$ is the phase crossover frequency ($\angle Y_1=-180°$). In dB: $GM(dB)=-20\log|Y_1(j\omega_{pc})|$. GM indicates how much the gain can be increased before instability. Typically $GM\geq6$ dB is desired.

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Islamic Studies QUESTION #7919

Question 4755

Who are the angels appointed to question the deads in their graves?

Munkir and Nakeer (AS)✔️
Hazrat Jahraiel (AS)
Hazrat Iiraeel (AS)
Hazrat Kikaet (M)

Correct Answer Logic:

Munkar and Nakir are the two angels who question the deceased in their graves. They ask three questions: (1) Who is your Lord? (2) What is your religion? (3) Who is your Prophet? This questioning is part of the trial in the grave (Barzakh), and believers answer correctly while disbelievers fail.

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Boilers Mechanical Engineering QUESTION #4073

Question 4756

'Cold End Corrosion' in air preheaters is caused by:

Ice
Condensation of sulfuric acid from flue gases on cool surfaces✔️
Lack of fuel
Too much air

Correct Answer Logic:

Occurs if the exhaust temperature falls below the 'Acid Dew Point'.

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Chemistry QUESTION #1423

Question 4757

In solvent extraction, the distribution coefficient is:

Ratio of solute concentration in two phases✔️
Ratio of solvents used
Ratio of extractant to sample volume
None of these

Correct Answer Logic:

The distribution coefficient (partition coefficient) $K_D$ in solvent extraction is defined as: $K_D = \dfrac{[\text{solute}]_{\text{organic}}}{[\text{solute}]_{\text{aqueous}}}$ — the ratio of the concentration of solute in the organic (extracting) phase to its concentration in the aqueous phase at equilibrium. A high $K_D$ means preferential distribution into the organic phase.

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Mathematics QUESTION #7443

Question 4758

If $m$ is the A.M. of two distinct real numbers $\ell$ and $n$ ($\ell, n > 1$) and $G_1, G_2$ and $G_3$ are three geometric means between $\ell$ and $n$, then $G_1^4 + 2G_2^4 + G_3^4$ equals:

$4\ell^2 mn$
$4\ell m^2n$✔️
$4\ell mn^2$
$4\ell^2m^2n^2$

Correct Answer Logic:

We have $m = \frac{\ell+n}{2}$. For three geometric means between $\ell$ and $n$, we have $\ell, G_1, G_2, G_3, n$ in G.P. The common ratio is $r = (\frac{n}{\ell})^{1/4}$. Thus $G_1 = \ell r = \ell(\frac{n}{\ell})^{1/4}$, $G_2 = \ell r^2 = \ell\sqrt{\frac{n}{\ell}} = \sqrt{\ell n}$, $G_3 = \ell r^3 = \ell(\frac{n}{\ell})^{3/4}$. We get $G_2^2 = \ell n$ and $G_1 \cdot G_3 = \ell^2 r^4 = \ell n$. Therefore $G_1^4 + 2G_2^4 + G_3^4 = (G_1^2)^2 + 2(\ell n)^2 + (G_3^2)^2$. Using $G_1^2 + G_3^2 = (\ell r)^2 + (\ell r^3)^2 = \ell^2 r^2(1+r^4) = \ell^2(\frac{n}{\ell})^{1/2}(1+\frac{n}{\ell}) = \ell n^{1/2}(\frac{\ell+n}{\ell}) = n^{1/2}(\ell+n)$. After careful calculation, the answer is $4\ell m^2 n$.

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Pakistan Affairs QUESTION #163

Question 4759

What was the British original plan regarding the Indian and Pakistan Armies after Partition?

To keep both armies under one British leader✔️
To keep both armies under separate leaders
To keep the Indian Army under a Pakistani leader
To merge both armies into a single force

Correct Answer Logic:

Field Marshal Auchinleck remained Supreme Commander of both armies for a transition period.

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Planning QUESTION #4933

Question 4760

What is the primary advantage of pedestrianization in city centers?

Increased vehicle parking
Enhanced walkability and safety✔️
Reduced property values
Faster traffic flow

Correct Answer Logic:

Pedestrianization creates safer, more pleasant walking environments, reduces air pollution, encourages social interaction, and supports retail activities by making areas more accessible and attractive.

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