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A regular octahedron can be viewed as two square pyramids base-to-base. The distance between opposite vertices (the apexes) equals $a\sqrt{2}$. This can be derived from the fact that the octahedron's vertices can be placed at $(\pm a/\sqrt{2}, 0, 0)$, $(0, \pm a/\sqrt{2}, 0)$, $(0, 0, \pm a/\sqrt{2})$, giving opposite vertex distance $a\sqrt{2}$.

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Mathematics QUESTION #7470

Question 1964

If $0 \leq x < \frac{\pi}{2}$, then the number of values of x for which $\sin x - \sin 2x + \sin 3x = 0$, is:

3
1
4
2✔️

Correct Answer Logic:

Using sum-to-product: $(\sin x + \sin 3x) - \sin 2x = 0$, which gives $2\sin 2x \cos x - \sin 2x = 0$, or $\sin 2x(2\cos x - 1) = 0$. Either $\sin 2x = 0$ or $\cos x = \frac{1}{2}$. For $\sin 2x = 0$: $2x = 0, \pi, 2\pi, \ldots$, giving $x = 0$ (boundary, included) or $x = \frac{\pi}{2}$ (boundary, excluded). So one solution $x=0$. For $\cos x = \frac{1}{2}$: $x = \frac{\pi}{3}$ in the given range. Total solutions = 2.

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MBBS QUESTION #10245

Question 1965

The dicrotic notch seen on an arterial pressure waveform tracing corresponds to which physiological event?

Opening of the aortic valve at the start of systolic ejection
Closure of the aortic valve, marking the onset of diastole✔️
Opening of the mitral valve during rapid ventricular filling
The peak of systolic ejection

Correct Answer Logic:

The dicrotic notch represents a brief backflow of blood toward the closing aortic valve, producing a small pressure rebound just after peak systolic pressure — marking the transition from systole to diastole.

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Disaster Management QUESTION #6518

Question 1966

Which statement best describes the relationship between climate change adaptation and disaster risk reduction in Pakistan's strategy?

They are separate domains managed by different institutions
DRR is a subset of climate adaptation
Climate adaptation is integrated as a cross-cutting theme within DRR frameworks✔️
Climate change only affects long-term planning, not immediate disaster response

Correct Answer Logic:

Section 13.2 establishes climate adaptation as a cross-cutting theme integrated throughout the DRR strategy. The document emphasizes conducting climate vulnerability assessments and incorporating climate risk into all DRR plans, recognizing the interconnection between climate-driven hazards and disaster risks.

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Mathematics QUESTION #4154

Question 1967

For a uniform lamina in the shape of a triangle with vertices $(0,0)$, $(a,0)$, $(0,b)$, what is the $y$-coordinate of the center of mass?

$\frac{b}{2}$
$\frac{b}{3}$✔️
$\frac{2b}{3}$
$\frac{b}{4}$

Correct Answer Logic:

For a triangular lamina, center of mass is at the centroid: $(\frac{a}{3}, \frac{b}{3})$. So $\bar{y} = \frac{b}{3}$.

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Accounting & Finance QUESTION #6222

Question 1968

When an auditor performs a 'Walk-through' procedure, they are:

Counting the physical inventory at the warehouse
Calculating the Zakat deductible at source
Tracing a transaction through the information system✔️
Reading the minutes of the board of directors

Correct Answer Logic:

Tracing transactions through the information system relevant to financial reporting is specifically defined as a 'walk-through'.

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Quantitative Reasoning QUESTION #8145

Question 1969

There is nothing quite like this film, and I am not entirely certain there is much more to it than its lovely (i)_______. At a moment when so many films try to be as (ii)_______ as possible, it is refreshing to find one that is so subtle and puzzling.

peculiarity .. assertive✔️
pellucidity .. enigmatic
conventionality .. indirect
peculiarity .. enigmatic

Correct Answer Logic:

The film is described as 'subtle and puzzling' — this implies the reviewer values its peculiarity (distinctive strangeness/uniqueness). For blank (ii), many films try to be as assertive as possible (forceful, direct, in-your-face) — the contrast with 'subtle and puzzling' makes this fit. 'Pellucidity' means clarity/transparency, which contradicts 'puzzling.' 'Enigmatic' for blank (ii) would contradict the contrast being set up.

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Dairy Food Science and Technology QUESTION #5888

Question 1970

Which milk protein fraction is most responsible for heat stability of milk?

$\beta$-casein due to its hydrophobic character
$\kappa$-casein due to its surface-active glycopeptide moiety stabilizing micelles✔️
$\alpha$-lactalbumin due to its calcium-binding domain
$\beta$-lactoglobulin due to disulfide bonding

Correct Answer Logic:

$\kappa$-casein, located on the outer surface of casein micelles, provides steric and electrostatic stabilization through its negatively charged hydrophilic glycomacropeptide (GMP) tail. Its removal by chymosin or denaturation during heating ($>70^{\circ}\text{C}$) destabilizes micelles.

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Boilers Mechanical Engineering QUESTION #4044

Question 1971

'Firebricks' are categorized as:

Insulators
Refractories✔️
Conductors
Fuels

Correct Answer Logic:

They line the furnace to protect the outer steel structure.

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Psychology QUESTION #9904

Question 1972

The prototype of the modern intelligence test for children was developed by:

Terman
Wechsler
Binet✔️
Stanford

Correct Answer Logic:

Alfred Binet and Théodore Simon created the first practical intelligence scale (1905) for the French Ministry of Education. This Binet-Simon Scale is the direct prototype of modern tests. Terman adapted it into the Stanford-Binet.

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CISA QUESTION #6845

Question 1973

In evaluating cryptographic controls, an auditor finds that SSL 3.0 is still enabled on web servers despite known vulnerabilities and availability of TLS 1.2 and 1.3. Management states they need SSL 3.0 for backward compatibility. What should the auditor recommend?

Accept management's reasoning
Disable SSL 3.0 and conduct impact analysis to identify legitimate compatibility requirements✔️
Enable all protocol versions
Document as acceptable risk

Correct Answer Logic:

Known vulnerabilities in deprecated protocols like SSL 3.0 create unacceptable risk. Keeping vulnerable protocols enabled 'just in case' without identifying specific requirements is poor security practice. The auditor should recommend disabling the vulnerable protocol and conducting proper analysis to identify any legitimate compatibility needs that can be addressed through secure means.

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Pakistan Affairs QUESTION #161

Question 1974

What was the result of the authority granted to the Governor General by the 1935 Act?

It limited the role to ceremonial duties
It established the Governor General as the epicentre of authority✔️
It decentralised power to provinces
It abolished the office

Correct Answer Logic:

The Act granted 'Special Responsibilities' and discretionary powers to the Governor General.

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Math Physics Chemistry Biology QUESTION #11255

Question 1975

Which vitamin is synthesized by bacteria in the human large intestine?

Vitamin C
Vitamin D
Vitamin K✔️
Vitamin A

Correct Answer Logic:

Vitamin K (specifically K$_2$ or menaquinone) is synthesized by beneficial bacteria in the large intestine. It is essential for blood clotting and bone metabolism.

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FIA Act QUESTION #9999

Question 1976

If a Provincial Police officer wants to know the source of the FIA's arrest powers analogous to ordinary police, the correct answer is that FIA members have powers, duties, privileges and liabilities of:

Federal Ministers
Provincial Police officers investigating under the Code✔️
Judicial Magistrates
Customs officials

Correct Answer Logic:

Section 5(1) equates FIA members' powers/duties with those of Provincial Police under the Code.

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Statistics QUESTION #1990

Question 1977

For the Poisson distribution with parameter $\lambda$, which of the following statements is INCORRECT?

Mean = Variance = $\lambda$
The distribution is always positively skewed for finite $\lambda$
The mgf is $M(t) = e^{\lambda(e^t - 1)}$
As $\lambda \to \infty$, the Poisson distribution approaches the t-distribution✔️

Correct Answer Logic:

The mean, variance, and mgf statements are all correct properties of the Poisson distribution. As $\lambda \to \infty$, the Poisson distribution approaches the NORMAL distribution (not the t-distribution), by the Central Limit Theorem. This is what makes statement (D) incorrect.

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Paper-II (Pharmacology and Social Behaviour, Law & Ethics and Computer Sciences) QUESTION #5789

Question 1978

According to the material, the 'Trial' stage in the adoption process involves:

Interest seeking
Evaluating benefits
Applying the idea✔️
Awareness

Correct Answer Logic:

Trial involves the application of an idea on a small scale[cite: 7].

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Islamic Studies QUESTION #7921

Question 1979

What is the backbone of economic system of Islam?

Zakat✔️
Tax
Jihad
Ushr

Correct Answer Logic:

Zakat is considered the backbone of the Islamic economic system. It is a mandatory 2.5% annual charity on savings, ensuring wealth circulation from rich to poor. Zakat purifies wealth, reduces inequality, and creates social welfare. Along with Ushr (agricultural tax), it forms the foundation of Islamic economic justice.

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Planning QUESTION #4933

Question 1980

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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