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Mathematics QUESTION #6945

Let $A$ be a $2\times2$ symmetric matrix with integer entries. If the sum of the diagonal elements of $A^2$ equals 1, find the number of such possible matrices.

6✔️
1
4
12

Correct Answer Logic:

Let $A=\begin{pmatrix}a&b\\b&c\end{pmatrix}$. Then $A^2=\begin{pmatrix}a^2+b^2&ab+bc\\ab+bc&b^2+c^2\end{pmatrix}$.

Sum of diagonal of $A^2 = a^2+2b^2+c^2 = 1$.

Since $a,b,c$ are integers and $a^2+2b^2+c^2=1\geq0$: $b=0$ (since $2b^2\leq1$ with integer $b$ forces $b=0$), then $a^2+c^2=1$.

Solutions: $(a,c)\in\{(1,0),(−1,0),(0,1),(0,−1)\}$ and $b=0$: 4 matrices. But we should check $b=\pm1$: $2(1)=2>1$, not possible. Hmm — that gives only 4. But checking $b=0$, $a^2+c^2=1$: integer solutions are $(\pm1,0)$ and $(0,\pm1)$ — exactly 4 pairs, each with $b=0$. So $\mathbf{4}$ matrices... but wait: $b$ can also vary if we allow it. Total = $\mathbf{4}$ matrices.

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Math (Module-1) QUESTION #5459

Question 9222

$\frac{x^2}{25} = 36$ What is a solution to the given equation?

6
30✔️
450
900

Correct Answer Logic:

Multiply both sides by $25$: $x^2 = 900$, which gives $x = 30$ or $x = -30$.

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Maths QUESTION #4534

Question 9223

If $a > 0$ and $b > 0$ with $a < b$, then $\frac{1}{a}$:

$< \frac{1}{b}$
> $\frac{1}{b}$✔️
$= \frac{1}{b}$
Cannot determine

Correct Answer Logic:

Taking reciprocals of positive numbers reverses inequality. Since $a < b$, we have $\frac{1}{a} > \frac{1}{b}$

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Chemistry QUESTION #7164

Question 9224

In the redox titration of oxalate ions with permanganate in an acidic medium, how many electrons are involved in the production of a single molecule of $CO_{2}$?

$5$
$2$
$10$
$1$✔️

Correct Answer Logic:

The half-reaction for oxalate is: $C_{2}O_{4}^{2-} \rightarrow 2CO_{2} + 2e^{-}$. In this process, 2 electrons are released for every two molecules of $CO_{2}$ produced. Therefore, for one molecule of $CO_{2}$, the number of electrons involved is $2/2 = 1$.

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MCQs Section QUESTION #5118

Question 9225

شاعرِ شباب کس شاعر کو کہا جاتا ہے؟

حسرت موہانی
آغا حشر کاشمیری
جوش ملیح آبادی✔️
ان میں سے کوئی نہیں

Correct Answer Logic:

جوش ملیح آبادی کو 'شاعرِ شباب' کہا جاتا ہے کیونکہ ان کی شاعری میں جوانی اور انقلابی جذبات کا بیان ملتا ہے۔

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Nursing QUESTION #9198

Question 9226

A Charge Nurse reads a study reporting Sensitivity = 95% and Specificity = 60% for a new screening tool. What does this mean?

The tool correctly identifies 95% of true negatives and 60% of true positives
The tool correctly identifies 95% of true positives and 60% of true negatives✔️
The tool has a 95% positive predictive value and 60% negative predictive value
The tool is 95% accurate overall with a 60% error rate

Correct Answer Logic:

Correct Answer: B

Sensitivity and Specificity Definitions:

Measure	Formula	Meaning
Sensitivity	$\frac{TP}{TP + FN}$	% of true positives correctly identified (rules OUT disease if negative — SnNOut)
Specificity	$\frac{TN}{TN + FP}$	% of true negatives correctly identified (rules IN disease if positive — SpPIn)

In this case:

95% sensitivity: The tool correctly identifies 95% of patients who have the condition
60% specificity: Only 60% of those without the condition are correctly identified as negative — meaning a high false positive rate (40%)

A high-sensitivity, low-specificity tool is useful for screening where missing a case is dangerous.

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Agronomy QUESTION #2088

Question 9227

Soil health is conceptually distinct from soil fertility because:

Soil health refers only to chemical nutrient status while fertility includes biological indicators
Soil health encompasses the integrated capacity of soil to function as a living ecosystem — sustaining plant and animal productivity, maintaining or enhancing water and air quality, and promoting plant and animal health — including biological (microbial biomass, earthworms), physical (aggregate stability), and chemical (pH, CEC) dimensions✔️
Soil fertility is a broader concept than soil health
Soil health is measured only through crop yield response to fertilizer application

Correct Answer Logic:

Soil health (soil quality) is a holistic concept integrating biological, physical, and chemical soil functions. Soil fertility is a subset referring specifically to nutrient supply capacity. Key soil health indicators include: microbial biomass carbon, soil respiration, aggregate stability, infiltration rate, organic carbon content, earthworm populations, and compaction resistance — measured through the Soil Management Assessment Framework (SMAF) or Cornell CASH.

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Geography QUESTION #6309

Question 9228

In diffusion of innovation theory, which adopter category represents the critical mass for widespread acceptance?

Innovators (2.5%)
Early Adopters (13.5%)
Early Majority (34%)✔️
Late Majority (34%)

Correct Answer Logic:

The Early Majority (34%) represents the critical mass or 'tipping point' in Rogers' diffusion model. Once this pragmatic group adopts an innovation, widespread acceptance becomes inevitable. They adopt just before the average member of society.

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Physics QUESTION #1178

Question 9229

In Lorentz transformation, motion along __________ axis remains unchanged (zero contraction/dilation).

One axis
Two axes✔️
Three axes
None of these

Correct Answer Logic:

In the standard Lorentz transformation, motion is assumed along one axis (typically the x-axis). The dimensions perpendicular to the direction of motion (y and z axes) are unchanged — there is no length contraction in those directions. Only the x-dimension contracts and time dilates. So two axes (y and z) remain unaffected.

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CISA QUESTION #1730

Question 9230

What is the key distinction between an organisational policy and a procedure?

Compliance with a policy is optional, whereas following a procedure is compulsory.
A procedure offers discretionary advice for decision-making, while a policy defines binding requirements.
A policy is a high-level authoritative document mandating compliance; a procedure specifies the mandatory steps through which compliance is achieved.✔️
A policy is a mid-level advisory document issued in the absence of a standard; the procedure suggests recommended actions.

Correct Answer Logic:

A policy is a high-level directive, signed by a person of authority, that mandates a required outcome or standard of behaviour — compliance is not optional. A procedure is the lower-level, step-by-step operational document that prescribes exactly how the policy requirements are to be fulfilled in practice. Both are mandatory. This hierarchy — policy, standard, procedure, guideline — is fundamental to governance frameworks.

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Math (Module-1) QUESTION #5457

Question 9231

A group of students voted on five after-school activities. The bar graph (not shown) displays the number of students who chose each activity. Based on the graph, how many students chose activity 3?

25
39
48✔️
50

Correct Answer Logic:

The bar for activity 3 reaches 48 on the vertical axis.

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Paper-II (Pharmacognosy and Pharmaceutics-I) QUESTION #3603

Question 9232

'Fennel' (Saunf) belongs to which family?

Lamiaceae
Umbelliferae (Apiaceae)✔️
Myrtaceae
Lauraceae

Correct Answer Logic:

Plants in the Umbelliferae family typically have umbrella-shaped flower clusters (umbels).

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Boilers Mechanical Engineering QUESTION #2514

Question 9233

For a beam under pure bending, the assumption that plane sections remain plane (Euler-Bernoulli hypothesis) implies that:

Shear strain across the section is constant
Normal strain varies linearly with distance from the neutral axis✔️
Shear stress is zero everywhere
Normal stress is uniform across the cross-section

Correct Answer Logic:

The Euler-Bernoulli hypothesis leads directly to a linear strain distribution: $\varepsilon = -y/\rho$, where $y$ is distance from the neutral axis and $\rho$ is the radius of curvature. This produces the linear bending stress distribution $\sigma = -My/I$.

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Statistics QUESTION #1970

Question 9234

Which of the following actions is most effective in reducing sampling error?

Adopting non-probability sampling
Expanding the population size
Decreasing the sample size
Increasing the sample size✔️

Correct Answer Logic:

Sampling error decreases as sample size increases, because larger samples yield better parameter estimates. The standard error $SE = \dfrac{\sigma}{\sqrt{n}}$ decreases as $n$ increases.

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Library Sciences QUESTION #1561

Question 9235

Resource Description and Access (RDA) replaces which earlier cataloguing code and is structured around which conceptual model?

AACR1; MARC 21
ISBD; Dublin Core
AACR2; FRBR (Functional Requirements for Bibliographic Records)✔️
AACR2; UNIMARC

Correct Answer Logic:

RDA was developed as a successor to AACR2, structured around the FRBR (Functional Requirements for Bibliographic Records) and FRAD (Functional Requirements for Authority Data) conceptual models. It focuses on describing resources and their relationships rather than prescribing card-catalogue-era physical formats.

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Contracts QUESTION #5850

Question 9236

An installment contract states that the Seller will deliver 12 monthly shipments. The first shipment is slightly late. Can the Buyer cancel the whole contract?

Yes, under the 'Perfect Tender Rule'.
No, unless the delay substantially impairs the value of the whole contract.✔️
Yes, but only if the Buyer is a non-merchant.
No, because installment contracts cannot be canceled.

Correct Answer Logic:

The Perfect Tender Rule does not apply to installment contracts. Under UCC §2-612, the buyer can only reject an installment or cancel the contract if the defect substantially impairs the value.

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Paper-I (Anatomy/Physiology and Biochemistry/Microbiology) QUESTION #3249

Question 9237

'Passive Immunity' provides protection that is:

Long-lasting (years)
Immediate but temporary✔️
Slow-acting
Permanent

Correct Answer Logic:

Passive immunity (like breast milk or antivenom) lasts only as long as the antibodies remain in the body.

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Physics QUESTION #393

Question 9238

Cesium coated oxidized silver emits electrons for __________ light.

Infrared✔️
Ultraviolet
Visible
Green

Correct Answer Logic:

This photocathode is sensitive to infrared

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Math Physics Chemistry Biology QUESTION #4278

Question 9239

The neurotransmitters are secreted by the neuron from:

Axon ends✔️
Dendrite end
Nissl's granules
Cell body

Correct Answer Logic:

Neurotransmitters are stored in synaptic vesicles at the axon terminals (synaptic knobs) and released into the synaptic cleft upon stimulation.

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CISA QUESTION #1731

Question 9240

Why is the use of standard terms of reference considered essential in an audit engagement?

To satisfy a legal prerequisite for regulatory compliance.
To establish a clear record of accountability for each party involved.
To facilitate honest, unambiguous, and consistent communication between the auditor and all stakeholders.✔️
To ensure that all regulatory requirements are explicitly identified and documented.

Correct Answer Logic:

Standard terminology creates a shared vocabulary between the auditor and all stakeholders — auditees, clients, management, and regulators. Without consistent definitions, parties may believe they are discussing the same issue when they are not, or may reach different conclusions from the same conversation. This common frame of reference is indispensable for producing a reliable, unbiased, and clearly understood audit report.

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Measure	Formula	Meaning
Sensitivity	\(\frac{TP}{TP + FN}\)	% of true positives correctly identified (rules OUT disease if negative — SnNOut)
Specificity	\(\frac{TN}{TN + FP}\)	% of true negatives correctly identified (rules IN disease if positive — SpPIn)