Question

Consider the following statements :
1. $n({\sin }^{2}67\frac{1^o}{2}-{\sin }^{2}22\frac{1^o}{2})>1$ for all positive integers $n\ge 2$.
2. If x is any positive real number, then $nx>1$ for all positive integers $n\ge 2$.
Which of the above statements is/are correct ?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is A 1 only

${\sin }^{2}67{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }={\cos }^2({90}^{\circ }-67{\frac{1}{2}}^{\circ })-{\sin }^{2}22{\frac{1}{2}}^{\circ }={\cos }^{2}22{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }={\cos }^2{45}^{\circ }=\frac{1}{2}$

Now, $n({\sin }^{2}67{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ })=\frac{n}{2}$

For $n\ge 2,n({\sin }^{2}67{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ })\ge 2$ and not $>2$

Statement 2 is false as all positive real numbers do not give $nx>1$ eg: $x=\frac{1}{4},n=3⟹nx=\frac{3}{4}\le 1$