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Electrical Engineering QUESTION #2174

In LQR optimal control, the cost $J = \int_0^\infty (x^TQx + u^TRu)\,dt$ is minimized by $u^* = -Kx$ where $K = R^{-1}B^TP$. The positive-definite matrix $P$ satisfies:

The Algebraic Riccati Equation: $PA + A^TP - PBR^{-1}B^TP + Q = 0$✔️
The Lyapunov equation: $PA + A^TP + Q = 0$
The Sylvester equation: $AP + PA^T = -Q$
The Hamiltonian eigenvalue problem only

Correct Answer Logic:

The steady-state matrix $P$ in LQR is the unique positive-definite solution of the Algebraic Riccati Equation (ARE): $PA + A^TP - PBR^{-1}B^TP + Q = 0$, from which the optimal gain $K = R^{-1}B^TP$ is derived.

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Planning QUESTION #4981

Question 1082

What does the Location Quotient (LQ) measure in regional economic analysis?

Absolute employment in a sector
Regional specialization in an industry✔️
Total regional GDP
Population density

Correct Answer Logic:

Location Quotient formula $LQ = \frac{(e_i/e)}{(E_i/E)}$ compares a region's share of employment in sector $i$ to the national share. LQ > 1 indicates regional specialization; LQ < 1 indicates underrepresentation relative to national average.

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Information Technology QUESTION #1137

Question 1083

The process of mapping logical addresses to physical addresses is handled by:

Operating System
Memory Management Unit (MMU)✔️
Compiler
Linker

Correct Answer Logic:

The MMU is a hardware component that translates virtual (logical) addresses generated by the CPU into physical addresses in RAM.

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Mathematics QUESTION #6974

Question 1084

Find the value of $\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2 x}{1+2^x}\,dx$

$4\pi$
$\dfrac{\pi}{4}$✔️
$\dfrac{\pi}{8}$
$\dfrac{\pi}{2}$

Correct Answer Logic:

Let $I=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2x}{1+2^x}dx$. Use the property $\int_{-a}^a f(x)dx$: replace $x\to-x$:

$I=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2x}{1+2^{-x}}dx=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{2^x\sin^2x}{1+2^x}dx$

Adding: $2I=\displaystyle\int_{-\pi/2}^{\pi/2}\sin^2x\,dx=\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{1-\cos2x}{2}dx=\dfrac{\pi}{2}$

$I=\dfrac{\pi}{4}$

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General Knowledge QUESTION #183

Question 1085

The current Secretary-General of the UN is:

Ban Ki-moon
Antonio Guterres✔️
Kofi Annan
None of these

Correct Answer Logic:

Antonio Guterres has been serving since January 2017.

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Accounting & Finance QUESTION #6063

Question 1086

The classification of costs into fixed and variable categories is a key feature of which costing system?

Absorption Costing
Direct Costing
Standard Costing
Marginal Costing✔️

Correct Answer Logic:

Marginal Costing (also called variable costing) explicitly separates costs into fixed and variable components. Only variable costs are charged to products; fixed costs are treated as period costs. In Absorption Costing, both fixed and variable overheads are absorbed into product costs without explicit separation. Standard Costing focuses on pre-set cost benchmarks. Direct Costing is essentially another name for marginal/variable costing, but Marginal Costing is the most precise and standard term here.

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Boilers Mechanical Engineering QUESTION #2495

Question 1087

Hooke's Law in shear relates shear stress $\tau$ and shear strain $\gamma$ via the modulus of rigidity $G$. The relationship between $E$, $G$, and Poisson's ratio $\nu$ is:

$E = G(1 + \nu)$
$E = 2G(1 - \nu)$
$E = 2G(1 + \nu)$✔️
$E = G/(2\nu)$

Correct Answer Logic:

From elasticity theory, the three elastic constants are related by $E = 2G(1+\nu)$. For steel: $E \approx 207$ GPa, $G \approx 79.3$ GPa, $\nu \approx 0.292$.

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Logical Reasoning QUESTION #3952

Question 1088

The increase in red meat intake has become a _____ and _____ issue in public health circles.

abated...dubious✔️
skyrocketed...stressful
abounded...divisive
stabilized...newsworthy

Correct Answer Logic:

4

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Agronomy QUESTION #2028

Question 1089

In irrigation scheduling, the concept of 'allowable depletion' refers to:

The amount of water that can be applied per irrigation event
The fraction of total available soil water that can be depleted before crop stress occurs, typically 50% for most field crops✔️
The depth of irrigation water lost to evaporation
The time interval between two consecutive irrigations

Correct Answer Logic:

Allowable depletion (management allowable depletion, MAD) is the fraction of plant-available water (PAW) that can be extracted before plant stress affects yield. For most field crops, MAD = 50%, meaning irrigation is triggered when 50% of PAW is depleted to prevent yield loss.

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Paper-I (Anatomy/Physiology and Biochemistry/Microbiology) QUESTION #3458

Question 1090

'Bicarbonate' is the most important buffer in:

The stomach
The blood (extracellular fluid)✔️
The cell nucleus
The skin

Correct Answer Logic:

The $CO_2$ / $HCO_3^-$ system maintains blood $pH$.

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Tourism & Hospitality Management QUESTION #1596

Question 1091

The Pakistan Tourist Destinations course requires students to develop which practical skill through field visits?

Financial analysis of tourism businesses
Preparation of a visit report documenting tourist destinations observed✔️
Management of hotel front office operations
Designing a tourism marketing campaign

Correct Answer Logic:

The Pakistan Tourist Destinations course includes a practical component (Module 7) requiring students to conduct Visits to tourist destinations and prepare a visit report — applying classroom knowledge of tourist resources, facilities, and circuits to real-world observation and documentation.

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English QUESTION #492

Question 1092

She had left him with a calmness and a poise that accord well with his own inward emotions.

had left
calmness and a poise that accord
well with
own inward✔️

Correct Answer Logic:

Error: 'accord' should be 'accorded'

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Urdu QUESTION #1398

Question 1093

پریم چند کا اصل نام بتائیے؟

مادھو رام
امرت لال
دھنپت رائے✔️
سدرشن

Correct Answer Logic:

منشی پریم چند (1880–1936) کا اصل نام دھنپت رائے تھا۔ وہ اردو اور ہندی کے یکساں عظیم افسانہ نگار اور ناول نگار تھے۔ ان کے مشہور ناولوں میں ''گؤدان''، ''بازارِ حسن'' (اردو: ''سوزِ وطن'') شامل ہیں۔ انہیں ''افسانے کا باوا آدم'' بھی کہا جاتا ہے۔

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Mathematics QUESTION #1329

Question 1094

The wave equation $u_{xx}=\dfrac{1}{c^2}u_{tt}$ describes waves that travel at speed:

1
$c$✔️
$c^2$
$\dfrac{1}{c}$

Correct Answer Logic:

The wave equation $u_{xx}=\dfrac{1}{c^2}u_{tt}$ has solutions of the form $f(x\pm ct)$, representing waves propagating in the $\pm x$ direction with speed $c$. The constant $c$ is the wave speed (e.g., for a string, $c=\sqrt{T/\mu}$).

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