IIT-JEE 2006Question 1
Match the following:
Column I
(i) Two rays in the first quadrant $x+y=|a|$ and $ax-y=1$ intersect each other in the interval $a\in(a_0,\infty)$, the value of $a_0$ is
(ii) Point $(\alpha,\beta,\gamma)$ lies on the plane $x+y+z=2$. Let $\vec a=\alpha\hat i+\beta\hat j+\gamma\hat k$, $\hat k\times(\hat k\times\vec a)=0$, then $\gamma=$
(iii) $\left|\int_0^1(1-y^2)dy\right|+\left|\int_1^0(y^2-1)dy\right|$
(iv) If $\sin A\sin B\sin C+\cos A\cos B=1$, then the value of $\sin C=$
Column II
(A) $2$
(B) $4/3$
(C) $\left|\int_0^1\sqrt{1-x}\,dx\right|+\left|\int_{-1}^0\sqrt{1+x}\,dx\right|$
(D) $1$
Points and Straight Lines
Solution
(i) For $a>0$: $x=1,\ y=a-1>0$ needs $a>1$; $a<0$ gives no first-quadrant point, so $a_0=1$. (ii) $\vec a\parallel\hat k$ gives $\alpha=\beta=0$, $\gamma=2$. (iii) $\frac23+\frac23=\frac43$, equal to (B) and (C). (iv) $\cos(A-B)\ge\sin A\sin B\sin C+\cos A\cos B=1$ forces $\sin C=1$.