SSC CPO SI and ASI Model Paper 7

(1)
$\mathit{c}\mathit{o}{\mathit{s}}^{2}1°+\mathit{c}\mathit{o}{\mathit{s}}^{2}3°+\mathit{c}\mathit{o}{\mathit{s}}^{3}5°+\mathit{c}\mathit{o}{\mathit{s}}^{2}7°+\dots +\mathit{c}\mathit{o}{\mathit{s}}^{2}85°+\mathit{c}\mathit{o}{\mathit{s}}^{2}87°+\mathit{c}\mathit{o}{\mathit{s}}^{2}89°$
In the given expression, we can seethat the angles are 1°, 3°, 5°, 7°,9°, …, 81°, 83°, 85°, 87°, 89°
Let the number of terms in theexpression be n, d be the commondifference, and tn be the nth angle.
$\therefore \mathit{a}+\left(\mathit{n}\mathit{–}1\right)\mathit{d}={\mathit{t}}_{\mathit{n}}$$\Rightarrow 1+\left(\mathit{n}\mathit{–}1\right)2=89$⇒n = 45$∴cos^2 1°+ cos^2 3°+ cos^3 5°
+ cos^2 7°+ … + cos^2 85°+ cos^2 87°+ cos^2 89°$
$=\mathit{c}\mathit{o}{\mathit{s}}^{2}1°+\mathit{c}\mathit{o}{\mathit{s}}^{2}89°+\mathit{c}\mathit{o}{\mathit{s}}^{2}3°+\mathit{c}\mathit{o}{\mathit{s}}^{2}87°+\mathit{c}\mathit{o}{\mathit{s}}^{2}5°+\mathit{c}\mathit{o}{\mathit{s}}^{2}85°+\dots +\mathit{c}\mathit{o}{\mathit{s}}^{2}41°+\mathit{c}\mathit{o}{\mathit{s}}^{2}49°+\mathit{c}\mathit{o}{\mathit{s}}^{2}43°+\mathit{c}\mathit{o}{\mathit{s}}^{2}47°+\mathit{c}\mathit{o}{\mathit{s}}^{2}45°$
$= cos^2 1°+ sin^2 1°+ cos^2 3°+ sin^2 3°
+ … + cos^2 43°+ sin^2 43°+ cos^2 45°$
$={\left[1+1+1+\dots +1\right]}_{\left(22\mathit{t}\mathit{i}\mathit{m}\mathit{e}\mathit{s}\right)}$$+{\left[\frac{1}{\sqrt{2}}\right]}^2$
= 45/2