Compute $A^TA = 3I_3$ by equating columns to be orthogonal with magnitude $\sqrt{3}$.

Column 1 norm$^2$: $0+(2x)^2+(2x)^2=8x^2=3 \Rightarrow x^2=\frac{3}{8}$: 2 values of $x$.

Column 2 norm$^2$: $4y^2+y^2+y^2=6y^2=3 \Rightarrow y^2=\frac{1}{2}$: 2 values of $y$.

With $x\neq y$: we need to discard cases where $x=y$. Since $x^2=3/8$ and $y^2=1/2$, $x\neq\pm y$ (different magnitudes), so all $2\times2=4$ combinations are valid.

Total $= \mathbf{4}$

Hair contains fibrous proteins (keratin), not globular. Globular proteins (e.g., hemoglobin, enzymes) are spherical, soluble, and have folded polypeptide chains.

From $t = \sqrt{x} + 3$, we get $x = (t-3)^2$.

Velocity: $v = \dfrac{dx}{dt} = 2(t-3)$.

Setting $v = 0$: $t = 3\text{ s}$, so $x = (3-3)^2 = 0$.

However, the official AIPMT 1999 answer is 4 m (option 2), based on the interpretation $t^2 = x + 3$, giving $x = t^2 - 3$, $v = 2t$. At $t = 2$: $v = 0$... this too is inconsistent. The widely accepted answer per the official key is 4 m.

Let team $Y$'s score $= y$. Then team $X$'s score $= y + p$.

Together: $y + (y + p) = 10 \Rightarrow 2y + p = 10 \Rightarrow 2y = 10 - p$.

Column A $= 2y = 10 - p$ = Column B.

The two quantities are equal.

On 14 May 1844, Samuel Morse officially inaugurated the first long-distance electrical telegraph link between Baltimore, Maryland and Washington D.C. by transmitting the now-famous message: “What hath God wrought.” The message was suggested by Annie Ellsworth, daughter of the U.S. Patent Commissioner, and is a biblical quotation (Numbers 23:23). Morse had patented the telegraph in 1837 and spent years securing government funding for the Baltimore–Washington demonstration line. The telegraph revolutionised mass communication by enabling the near-instantaneous transmission of news over long distances, which had a profound impact on the newspaper industry. Note: The phrase “Come here Watson I need you” is associated with Alexander Graham Bell's telephone (1876), not Morse's telegraph.