Practicing Success
If $A=\{θ:2 \cos^2 θ+ \sin θ ≤2\}$ and $B=\{θ:\frac{π}{2}≤θ ≤\frac{3π}{2}\}$ then AB is equal to |
$\{θ:π/2≤θ≤5π/6\}$ $\{θ:π≤θ≤3π/2\}$ $\{θ:π/2≤θ≤5π/6\}∪\{θ:π≤θ≤3π/2\}$ none of these |
$\{θ:π/2≤θ≤5π/6\}∪\{θ:π≤θ≤3π/2\}$ |
We have, $2 \cos^2θ+ \sin θ≤2$ $⇒2(1- \sin^2θ) + \sin θ ≤2$ $⇒2\sin^2θ-\sin θ≥0$ $⇒\sin θ (2 \sin θ-1) ≥0$ $⇒\sin θ≥0$ and $2\sin θ-1≥0$ or, $⇒\sin θ≤0$ and $2\sin θ-1≤0$ CASE I When $\sin θ>0$ and $2\sin θ-1 >0$ In this case, we have $\sin θ≥0$ and $2\sin θ-1≥0$ $⇒\sin θ≥0$ and $2\sin θ≥\frac{1}{2}$ $⇒\frac{π}{6}≤θ≤\frac{5π}{6}$ $∴A∩B=\{θ:π/2≤θ≤5π/6\}$ $[∵B=\{θ:π/2≤θ≤3π/2\}]$ CASE II When $\sin θ≤0$ and $2 \sin θ-1≤0$ In this case, we have $⇒\sin θ≤0$ and $2\sin θ-1≤0$ $⇒\sin θ≤0$ and $2\sin θ≤\frac{1}{2}$ $⇒\sin θ≤0$ $⇒π≤θ≤2π$ $∴A∩B=\{θ:π≤θ≤3π/2\}$ $[∵B=\{θ:π/2≤θ≤3π/2\}]$ Thus, $A∩B=\{θ:π/2≤θ≤5π/6\}∪\{θ:π≤θ≤3π/2\}$ |