Question

The statements in Column-I are labelled A, B, C and D, while the statements in Column-II are labelled p, q, r, s and t. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column-II.

Answer 1

(A) $|\sin 2x|+|\cos 2x|\in [1,\sqrt{2}]and|\sin y|\in [0,1]$
so, equation is valid only
for $|\sin y|=1and|\sin 2x|+|\cos 2x|=1$
No. of values of $y\in [-2\pi ,2\pi ]=4$
No. of values of $x\in [-2\pi ,2\pi ]=17$
$hencenumberoforderedpair=68$
(B) $f\left(x\right)>0\Rightarrow x^2-18+a>0f\left(x\right)\in R$
$\Rightarrow (18{)}^2-4a<0$
$a>81$
$henceleastintergralvalueofa=82$
(C) $\therefore {\cos }^{-1}|\cos x|isperiodicwithperiod\pi$
$\Rightarrow \stackrel{80\pi }{\underset{0}{\int }}\left[{\cos }^{-1}|\cos x|\right]dx=80\stackrel{\pi }{\underset{0}{\int }}{\cos }^{-1}|\cos x|]dx$
$=80[\stackrel{1}{\underset{0}{\int }}0.dx+\stackrel{\pi /2}{\underset{1}{\int }}1.dx+\stackrel{\pi -1}{\underset{\pi -2}{\int }}1.dx+\stackrel{\pi }{\underset{\pi -1}{\int }}0.dx]$
$=80(\pi -2)$
(D) $TherequiredNumberofNumber=6^n-{2.5}^n+4^n$
$\therefore thegivenexpressionis=(A^n+2B^n+C^n)$
hence $A=6,B=5,C=4$
$\Rightarrow (10A+2B+C)=10\times 6+2\times 5+4=74$