A casting vote is used to break a tie.

The equinoxes on March 21 and September 21 have equal day and night.

Total cost = (negative cost) + (first 10 prints) + (next 10 prints)

Total $= x + 10 \cdot \frac{3x}{5} + 10 \cdot \frac{x}{5} = 45$

$x + \frac{30x}{5} + \frac{10x}{5} = 45$

$x + 6x + 2x = 45$

$9x = 45$

$x = 5$

Therefore $x = 5$ dollars.

Wait, but the marked answer is A (index 0, value 3). Let me verify: if $x = 3$:

Total $= 3 + 10(\frac{9}{5}) + 10(\frac{3}{5}) = 3 + 18 + 6 = 27 \neq 45$

If $x = 5$: Total $= 5 + 10(3) + 10(1) = 5 + 30 + 10 = 45$ ✓

The mathematically correct answer is $x = 5$ (option D, index 3).

For injectivity, suppose \(f(x_1) = f(x_2)\). Then \(\frac{x_1}{1+x_1^2} = \frac{x_2}{1+x_2^2}\), giving \(x_1(1+x_2^2) = x_2(1+x_1^2)\), which simplifies to \(x_1-x_2 = x_1x_2(x_1-x_2)\). If \(x_1 \neq x_2\), then \(1 = x_1x_2\). But checking the derivative: \(f'(x) = \frac{1-x^2}{(1+x^2)^2}\). This is positive for \(|x|<1\) and negative for \(|x|>1\), meaning f is not strictly monotonic on all of \(\mathbb{R}\). However, for the range \([-\frac{1}{2}, \frac{1}{2}]\), we can verify: maximum of f occurs at \(x=1\) giving \(f(1) = \frac{1}{2}\), and minimum at \(x=-1\) giving \(f(-1) = -\frac{1}{2}\). The function achieves all values in this range, so it's surjective. For injectivity on the full domain \(\mathbb{R}\), note that \(f(2) = \frac{2}{5}\) and \(f(\frac{1}{2}) = \frac{1/2}{1+1/4} = \frac{2}{5}\), so it's not injective. Thus, the answer should be C or D. Based on JEE answer (A), there may be a restriction or different interpretation.

Under the Felony Murder rule, all participants in an inherently dangerous felony (like robbery) are liable for any deaths that occur during the commission of that felony, regardless of who pulled the trigger or whether they intended for a death to occur.

Napoleon is often called the 'Man of Destiny'.

درست جملہ ہے: 'آن پہنچی' کیونکہ گھڑی مونث ہے اور آن پہنچی اس کا درست فعل ہے۔

An ally is a person or organization that cooperates with another. Its antonym is Foe, meaning an enemy or opponent.

Iran was the first country to recognize Pakistan's independence after its creation in August 1947. This recognition came even before Pakistan formally joined the United Nations, demonstrating strong early diplomatic ties between the two Muslim nations.

\(11138 \div 77 = 144\) remainder \(50\).
\(77 - 50 = 27\)... rechecking: \(77 \times 144 = 11088\), \(11138 - 11088 = 50\). So we add \(77 - 50 = 27\). Since 27 is not among the options, re-check: \(77 \times 145 = 11165\), \(11165 - 11138 = 27\). Answer key gives A (9). Students should verify by direct division.

Since \(\vec{u}\) is coplanar with \(\vec{a}\) and \(\vec{b}\), we can write \(\vec{u} = x\vec{a} + y\vec{b} = x(2\hat{i}+3\hat{j}-\hat{k}) + y(\hat{j}+\hat{k}) = 2x\hat{i} + (3x+y)\hat{j} + (-x+y)\hat{k}\). Condition \(\vec{u} \perp \vec{a}\): \(\vec{u} \cdot \vec{a} = 2x(2) + (3x+y)(3) + (-x+y)(-1) = 0\), giving \(4x + 9x + 3y + x - y = 0\), so \(14x + 2y = 0\), thus \(y = -7x\). Condition \(\vec{u} \cdot \vec{b} = 24\): \((3x+y)(1) + (-x+y)(1) = 24\), giving \(3x+y-x+y = 24\), so \(2x+2y = 24\), thus \(x+y = 12\). From \(y=-7x\): \(x-7x = 12\), so \(-6x=12\) and \(x=-2\), giving \(y=14\). Thus \(\vec{u} = -4\hat{i} + 8\hat{j} + 16\hat{k}\), and \(|\vec{u}|^2 = 16 + 64 + 256 = 336\).

No explanation provided for this question.

The ionosphere typically extends from about 60 km to over 1,000 km.+1

Anemia is usually due to low hemoglobin or low RBC count.

Krebs cycle oxidizes acetyl-CoA to CO₂

Advertising revenue in 1980 = $36 million = $36{,}000{,}000

Average daily circulation = 180,000

Ratio $= \frac{36{,}000{,}000}{180{,}000} = \frac{36{,}000{,}000}{180{,}000} = \frac{360}{1.8} = 200$

Therefore, the advertising revenue was 200 times the daily circulation.

However, checking answer index 3 (which is option D with value 20), there may be a unit issue. Let me recalculate considering if the question means something different... If advertising revenue is $36 million and circulation is 180,000, then $36{,}000{,}000 \div 180{,}000 = 200$.

The correct mathematical answer is 200, but if the marked answer is D (index 3, value 20), there may be an interpretation issue with the original problem.

Taking torques about the contact point on the floor. The angle with the wall is \(60°\), so angle with floor is \(30°\). Normal from wall (horizontal) \(\times L\sin30° = mg\times\frac{L}{2}\cos30°\). Wait — using torque about foot: \(N_w\cdot L\sin30° = mg\cdot\frac{L}{2}\cos30°\Rightarrow N_w = \frac{mg\cos30°}{2\sin30°} = \frac{20\times10\times\frac{\sqrt{3}}{2}}{2\times\frac{1}{2}}=100\sqrt{3}\). Friction \(= N_w = 100\sqrt{3}\approx173\ \text{N}\). Answer: \(100\sqrt{3}\ \text{N}\).

Discretionary control is a decentralized method where someone makes decisions based on their own assessment rather than a formal centralized authority. Auditors should investigate who makes decisions and how, as this is often where control failures are found.

Let $x = \dfrac{\pi}{2}-t$ so as $x\to\dfrac{\pi}{2}$, $t\to0$.

$\cot x = \cot\!\left(\tfrac{\pi}{2}-t\right) = \tan t$, $\cos x = \sin t$, $\pi-2x = 2t$

$\lim_{t\to0}\dfrac{\tan t - \sin t}{8t^3} = \lim_{t\to0}\dfrac{\sin t(1-\cos t)}{8t^3\cos t}$

$= \lim_{t\to0}\dfrac{\sin t}{t}\cdot\dfrac{1-\cos t}{t^2}\cdot\dfrac{1}{8\cos t} = 1\cdot\dfrac{1}{2}\cdot\dfrac{1}{8} = \mathbf{\dfrac{1}{16}}$

Official answer is $\dfrac{1}{16}$ (index 1). Rechecking: $\dfrac{1}{2}\cdot\dfrac{1}{8}=\dfrac{1}{16}$.

The recorded value $3.50 \text{ cm}$ has a precision of $0.01 \text{ cm}$. We must find an instrument with a least count (LC) of $0.01 \text{ cm}$.

• Option A: $LC = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm} = 0.001 \text{ cm}$
• Option B: $LC = \frac{1 \text{ mm}}{50} = 0.02 \text{ mm} = 0.002 \text{ cm}$
• Option C: $LC = 0.1 \text{ cm}$
• Option D: $1 \text{ MSD} = \frac{1 \text{ cm}}{10} = 0.1 \text{ cm}$. Since $10 \text{ VSD} = 9 \text{ MSD}$, $1 \text{ VSD} = 0.9 \text{ MSD}$. $LC = 1 \text{ MSD} - 1 \text{ VSD} = 0.1 \text{ MSD} = 0.01 \text{ cm}$.

Thus, Option D matches the required precision.