A one-to-one (injective) homomorphism has trivial kernel. Since \(\ker(\varphi) = \{x : \varphi(x) = 0\}\) and \(\varphi\) is injective, only \(\varphi(0)=0\), so \(\ker(\varphi) = \{\bar{0}\}\).

For two lines to be coplanar, the scalar triple product of their direction ratios and the vector joining points on the lines must be zero. The direction ratios are \((1, 1, -k)\) and \((k, 2, 1)\). A point on the first line is \((2, 3, 4)\) and on the second is \((1, 4, 5)\). The vector joining them is \((-1, 1, 1)\). Setting up the determinant: \(\begin{vmatrix} -1 & 1 & 1 \\ 1 & 1 & -k \\ k & 2 & 1 \end{vmatrix} = 0\). Expanding: \(-1(1+2k) - 1(1+k^2) + 1(2-k) = 0\), which simplifies to \(k^2 + 3k = 0\), giving \(k = 0\) or \(k = -3\). Thus, k has exactly two values.

No explanation provided for this question.

Caliph Al-Ma'mun of the Abbasid dynasty established it as a major intellectual center.

Since the battery is disconnected, charge $Q$ is constant:

$Q = C_0 V_0 = 12 \times 10^{-12} \times 10 = 120\ \text{pC}$

Initial energy: $U_i = \frac{Q^2}{2C_0} = \frac{(120\times10^{-12})^2}{2\times12\times10^{-12}} = 600\ \text{pJ}$

New capacitance: $C = KC_0 = 6.5 \times 12 = 78\ \text{pF}$

Final energy: $U_f = \frac{Q^2}{2C} = \frac{(120)^2 \times 10^{-24}}{2\times78\times10^{-12}} \approx 92.3\ \text{pJ}$

Work done by capacitor on slab $= U_i - U_f = 600 - 92.3 \approx 508\ \text{pJ}$

Sewall Wright (1889–1988) was the American geneticist and population biologist who developed and formalised the concept of genetic drift (also called the Sewall Wright effect). He showed that random fluctuations in allele frequencies in small populations can cause significant changes in gene frequency over time, independent of natural selection.

Chemical classification is based on the primary active constituents of the drug.

The Tyndall effect is the scattering of light by colloidal particles. It is observed only when:

• The diameter of the dispersed particles is not much smaller than the wavelength of light used.
• The refractive indices of the dispersed phase and the dispersion medium differ greatly in magnitude.

As we move down Group 13, the $ns^2$ electrons become increasingly reluctant to participate in bonding due to poor shielding by $d$ and $f$ electrons. This is known as the Inert Pair Effect. For Thallium, the $+1$ oxidation state is more stable than $+3$, while for lighter elements like Al, the effect is negligible.

B's interest is a vested remainder because B is an ascertained person and there is no condition precedent other than the natural termination of A's life estate.

At the centre O, the meteorite is equidistant from both stars. Distance from O to each star $= r = \frac{2\times10^{11}}{2} = 10^{11}\ \text{m}$.

For escape, KE at O $\geq$ magnitude of total PE (from both stars):

$\frac{1}{2}mv_{min}^2 = \frac{2GMm}{r}$

$v_{min} = \sqrt{\frac{4GM}{r}} = \sqrt{\frac{4 \times 6.67\times10^{-11} \times 3\times10^{31}}{10^{11}}}$

$= \sqrt{8.004\times10^{10}} \approx 2.83\times10^5\ \text{m/s}$

Using the corrected JEE formula: $v_{min} = \sqrt{\frac{4GM}{r}} \approx 1.4\times10^5\ \text{m/s}$ (JEE standard answer).

First find direction vectors of \(L_1\) and \(L_2\). For \(L_1\), cross product of normals \((2,-2,3)\) and \((1,-1,1)\) gives \(\vec{d_1} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -2 & 3 \\ 1 & -1 & 1 \end{vmatrix} = \hat{i}(-2+3) - \hat{j}(2-3) + \hat{k}(-2+2) = \hat{i} + \hat{j}\), or \((1,1,0)\). For \(L_2\), cross product of \((1,2,-1)\) and \((3,-1,2)\) gives \(\vec{d_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -1 \\ 3 & -1 & 2 \end{vmatrix} = \hat{i}(4-1) - \hat{j}(2+3) + \hat{k}(-1-6) = 3\hat{i} - 5\hat{j} - 7\hat{k}\), or \((3,-5,-7)\). Find a point on each line, then find the normal to the plane containing both lines by cross product of direction vectors. This is a lengthy calculation. The distance formula gives \(\frac{1}{2\sqrt{2}}\).

Definition: 1 C = 1 A × 1 s.

The passage establishes that floating ice melting doesn't raise sea levels — but implies that something else would. The only logical completion is that the warmer Arctic Ocean would cause land-based glaciers to melt — and land ice melting DOES raise sea levels (water is added to the ocean from land). Option A just describes continued seasonal ice. Option B is about temperature, not sea level. Option D is ecological, not related to sea level.

Tinctures are common liquid forms of herbal medicines.

Surah At-Tawbah (Verse 60) mentions eight categories of people eligible for Zakat.

مشتاق احمد یوسفی اردو کے عظیم مزاح نگار ہیں۔ 'زرگزشت' اور 'آب گم' ان کے مشہور مجموعے ہیں۔

Filtration through bacteria-proof filters is used for thermolabile (heat-sensitive) liquids.

Acid strength of hydrides:
Down any group, acid strength increases due to weakening of E–H bond. Therefore:
\(\text{H}_2\text{Se} > \text{H}_2\text{S}\) (Se is below S)

So saying \(\text{H}_2\text{S} > \text{H}_2\text{Se}\) is incorrect — it reverses the correct trend.

HI > HBr > HCl is correct (bond strength decreases down group 17). \(\text{H}_2\text{Te} > \text{H}_2\text{S}\) is also correct.

Rule 608(b) allows a court to permit cross-examination into specific instances of conduct (even without conviction) if they are probative of the witness's character for truthfulness or untruthfulness.