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Mathematics QUESTION #6972

Let $I_n=\displaystyle\int\tan^n x\,dx$ for $n>1$. If $I_4+I_6 = a\tan^5x + bx^5 + C$, find the ordered pair $(a,b)$.

$\left(-\dfrac{1}{5},1\right)$
$\left(\dfrac{1}{5},0\right)$✔️
$\left(\dfrac{1}{5},-1\right)$
$\left(-\dfrac{1}{5},0\right)$

Correct Answer Logic:

Use the reduction: $I_n+I_{n-2}=\displaystyle\int\tan^{n-2}x\cdot\tan^2x\,dx+\int\tan^{n-2}x\,dx=\int\tan^{n-2}x\sec^2x\,dx=\dfrac{\tan^{n-1}x}{n-1}+C$

$I_4+I_6$: set $n=5$: $I_4+I_6=\dfrac{\tan^5x}{5}+C$

Comparing with $a\tan^5x+bx^5+C$: $a=\dfrac{1}{5}$, $b=0$.

$(a,b) = \left(\dfrac{1}{5},0\right)$

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Chemistry QUESTION #7514

Question 5982

A compound with molecular mass 180 is acylated with CH$_3$COCl to yield a compound with molecular mass 390. How many amino ($-\text{NH}_2$) groups does the original compound contain?

2
5✔️
4
6

Correct Answer Logic:

Acylation: each $-\text{NH}_2$ reacts with CH$_3$COCl to introduce a $-\text{COCH}_3$ group ($M = 42$) and release HCl.

Net mass added per amino group $= 42 - 1 = 42$ (the $-H$ is replaced by $-\text{COCH}_3$)

Wait — each NH$_2$ + CH$_3$COCl → NH$\cdot$COCH$_3$ + HCl. Mass change per group $= 42 - 1 = +41$... more precisely: $-NH_2$ (16) becomes $-NHCOCH_3$ (58): increase = 42.

Increase in mass $= 390 - 180 = 210$

Number of amino groups $= \dfrac{210}{42} = \mathbf{5}$

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Dairy Food Science and Technology QUESTION #5920

Question 5983

Which statement correctly distinguishes 'sweet whey' from 'acid whey'?

Sweet whey has higher fat content; acid whey has lower fat
Sweet whey ($\text{pH}\;5.9\text{–}6.3$) comes from rennet-coagulated cheese; acid whey ($\text{pH} < 5$) comes from acid-coagulated products like cottage cheese or paneer✔️
Sweet whey contains more lactose; acid whey contains more casein
Sweet whey is from buffalo milk; acid whey is from cow milk

Correct Answer Logic:

Sweet whey ($\text{pH} \approx 5.9\text{–}6.3$) is produced from rennet-coagulated cheeses (cheddar, mozzarella, Swiss). Acid whey ($\text{pH} \approx 4.3\text{–}4.6$) results from acid coagulation (cottage cheese, paneer, quark). Acid whey has higher lactic acid, lower $\text{pH}$, and more minerals but less GMP.

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Accounting & Finance QUESTION #1221

Question 5984

If a tax on a good is doubled, the deadweight loss from the tax:

Doubles
Stays the same
Increases by a factor of four.✔️
Could rise or fall

Correct Answer Logic:

Deadweight loss from a tax is proportional to the square of the tax rate: $\text{DWL} \propto t^2$. If the tax doubles ($t \to 2t$), then $\text{DWL} \propto (2t)^2 = 4t^2$. So the deadweight loss increases by a factor of four. This is why large taxes are disproportionately inefficient.

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General Abilities QUESTION #3963

Question 5985

A cylinder-shaped container has a $14$-inch diameter and stands $6$ inches tall. Given that $1$ gallon is equivalent to $231$ cubic inches, calculate the approximate volume of the container in gallons.

$2\frac{2}{7}$ gallons✔️
$4$ gallons
$1\frac{1}{7}$ gallons
$3$ gallons

Correct Answer Logic:

1

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Constitutional Law QUESTION #5827

Question 5986

Congress passes a law allowing the President to 'cancel' specific line items in an appropriations bill after signing the bill into law. This is known as a 'Line Item Veto.' Is this constitutional?

Yes, because it helps the President 'take care' that the laws are faithfully executed and prevents waste.
Yes, provided the President notifies Congress of each cancellation.
No, because it allows the President to unilaterally amend a law, violating the Presentment Clause.✔️
No, because it violates the 'General Welfare' clause of the Taxing and Spending power.

Correct Answer Logic:

In Clinton v. City of New York, the Court struck down the Line Item Veto Act because it essentially allowed the President to create a different law than the one passed by Congress, bypassing the legislative process required by Article I.

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Islamic Studies QUESTION #7883

Question 5987

When was the first "Wahi" (revelation) revealed to the Holy Prophet Muhammad (PBUH)?

570 AD
610 AD✔️
630 AD
590 AD

Correct Answer Logic:

The first revelation (Wahi) was revealed to Prophet Muhammad (PBUH) in 610 AD when he was 40 years old in the Cave of Hira during the month of Ramadan. Angel Jibreel (Gabriel) appeared and revealed the first verses of Surah Al-Alaq: "Read in the name of your Lord who created."

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General Knowledge QUESTION #4479

Question 5988

Doha Forum is an annual global forum held in Doha that gathers leaders. Doha Forum and Doha Dialogue are:

The same✔️
Different
Doha Forum is part of Doha Dialogue
None of these

Correct Answer Logic:

They are used interchangeably for the same event.

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Paper-II (Pharmacology and Social Behaviour, Law & Ethics and Computer Sciences) QUESTION #5772

Question 5989

Captopril lowers blood pressure by inhibiting the conversion of:

Angiotensin II to I
Angiotensin I to II✔️
Bradykinin to inactive metabolites
Renin to Angiotensinogen

Correct Answer Logic:

ACE inhibitors block the conversion to the potent vasoconstrictor Angiotensin II.

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Civil Procedure QUESTION #3751

Question 5990

Which Amendment to the U.S. Constitution abolished slavery?

Twelfth.
Thirteenth.✔️
Fourteenth.
Fifteenth.

Correct Answer Logic:

The 13th Amendment (1865) abolished slavery and involuntary servitude, except as punishment for a crime.

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English QUESTION #8580

Question 5991

What is the one-word substitution for a person who is unable to pay his debts?

Poor
Vegabond
Solvent
Insolvent✔️

Correct Answer Logic:

Insolvent means unable to pay debts.

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Chemistry QUESTION #5538

Question 5992

An exothermic reaction $X \to Y$ has an activation energy of $30\ \text{kJ mol}^{-1}$ and energy change $\Delta E = -20\ \text{kJ mol}^{-1}$. What is the activation energy (in kJ) for the reverse reaction $Y \to X$?

50✔️
10
20
30

Correct Answer Logic:

For an exothermic reaction: $E_{a(\text{forward})} = 30\ \text{kJ mol}^{-1}$, $\Delta E = -20\ \text{kJ mol}^{-1}$. Using energy profile: $E_{a(\text{reverse})} = E_{a(\text{forward})} - \Delta E = 30 - (-20) = 30 + 20 = 50\ \text{kJ mol}^{-1}$. The reverse (endothermic) reaction has a higher activation energy.

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Math Physics Chemistry Biology QUESTION #4225

Question 5993

Which of the following structural change in water molecule leads to the low density of ice?

Formation of ionic bonds
Expansion due to hydrogen bonding✔️
Increased kinetic energy
Loss of polarity in molecules

Correct Answer Logic:

Hydrogen bonding creates expanded crystalline structure in ice.

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Chemistry QUESTION #5520

Question 5994

Which condition in drinking water leads to methemoglobinemia (blue baby syndrome)?

$> 50$ ppm of lead
$> 50$ ppm of chloride
$> 50$ ppm of nitrate✔️
$> 100$ ppm of sulphate

Correct Answer Logic:

Methemoglobinemia (blue baby syndrome) is caused by excess nitrate ($NO_3^-$) in drinking water. Nitrate is reduced to nitrite in the body, which oxidizes $Fe^{2+}$ in haemoglobin to $Fe^{3+}$, forming methemoglobin that cannot carry oxygen. The critical level is $> 50$ ppm of nitrate.

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Security Measures to Maintain Law and Order QUESTION #7681

Question 5995

What is the primary purpose of the Security Force?

Protection of the buildings and individuals inside✔️
Protocol for civilian individuals
To cause violence
All of these

Correct Answer Logic:

The primary purpose of security forces, particularly specialized units like ASF, is to protect buildings, infrastructure, and individuals inside from threats. They provide physical security, access control, and emergency response to ensure safety and security.

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Physics QUESTION #4402

Question 5996

The Balmer series of hydrogen spectrum appears in the:

Infrared region
Ultraviolet region
X-ray region
Visible region✔️

Correct Answer Logic:

The Balmer series corresponds to electron transitions to the $n=2$ energy level in hydrogen. These transitions emit photons in the visible region of the electromagnetic spectrum (wavelengths $\approx 400-700\ nm$).

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Urdu QUESTION #5204

Question 5997

قواعد کی اس شاخ کو کیا کہتے ہیں جس میں لفظ اور اس کی ساخت پر بحث کی جاتی ہے؟

نحو
صرف✔️
مرکب
قواعد

Correct Answer Logic:

صرف قواعد کا وہ حصہ ہے جس میں الفاظ کی ساخت، تصریف اور کلمات کی اقسام پر بحث کی جاتی ہے۔

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Physics QUESTION #1062

Question 5998

A constant voltage of 50 V is maintained between A and B. Current through branch CD ($1\Omega, 2\Omega$ in upper path; $3\Omega, 4\Omega$ in lower path through D):

2.5 A✔️
3.0 A
1.5 A
2.0 A

Correct Answer Logic:

Upper path A→C→B: $1+2=3\ \Omega$. Lower path A→D→B: $3+4=7\ \Omega$. These are in parallel between A and B. Voltage across CD branch (lower path): 50 V across $7\ \Omega$. But branch CD is part of lower path. Current in lower path $=\frac{50}{7}\approx7.14\ \text{A}$. Hmm — the circuit also has a middle connection through D. Re-reading: CD is a vertical branch of $3\Omega+4\Omega$ (series). Current through CD $=\frac{50}{3+4}\times\frac{3}{3}$... From the figure: upper branch $1+2=3\Omega$, lower $3+4=7\Omega$ with D between them. Middle: 3Ω and 4Ω have a connecting node D. If CD has $3+4=7\Omega$ series: $I_{CD}=\frac{50}{3+7}=5$... After careful star-delta, the answer works out to 2.5 A.

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