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3. trignometrical ratios,functions and identities — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS3. trignometrical ratios,functions and identities1 Mark
MCQ
Answer the following by appropriately matching the lists based on the information given in the paragraph

Let $f( x )=\sin (\pi \cos x )$ and $g ( x )=\cos (2 \pi \sin x )$ be two functions defined for $x >0$. Define the following sets whose elements are written in the increasing order :

$X =\{ x : f( x )=0\},  Y =\left\{ x : f^{\prime}( x )=0\right\}$

$Z =\{ x : g ( x )=0\},  W =\left\{ x : g ^{\prime}( x )=0\right\}.$

$List-I$ contains the sets $X , Y , Z$ and $W$. $List -II$ contains some information regarding these sets.

$List-I$ $List-II$
$(I)$ $X$ $(P)$ $\supseteq\left\{\frac{\pi}{2}, \frac{3 \pi}{2}, 4 \pi, 7 \pi\right\}$
$(II)$ $Y$ $(Q)$ an arithmetic progression
$(III)$ $Z$ $(R)$ $NOT$ an arithmetic progression
$(IV)$ $W$ $(S)$ $\supseteq\left\{\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{13 \pi}{6}\right\}$
  $(T)$ $\supseteq\left\{\frac{\pi}{3}, \frac{2 \pi}{3}, \pi\right\}$
  $( U )$ $\supseteq\left\{\frac{\pi}{6}, \frac{3 \pi}{4}\right\}$

($1$) Which of the following is the only $CORRECT$ combination?

$(1) (II), (R), (S)$ $(2) (I), (P), (R)$ $(3) (II), (Q), (T)$ $(4) (I), (Q), (U)$

($2$) Which of the following is the only $CORRECT$ combinations?

$(1) (IV), (Q), (T)$  $(2) (IV), (P), (R), (S)$  $(3) (III), (R), (U)$ $(4) (III), (P), (Q), (U)$

Give the answer the quetion ($1$) and ($2$)

  • A
    $1,2$
  • ✓
    $3,2$
  • C
    $1,4$
  • D
    $1,3$
✓

Answer

Correct option: B.
$3,2$
b
($2$) $f(x)=\sin (\pi \cos x)$

$X :\{ x : f ( x )=0\}$

$f(x)=0 \Rightarrow \sin (x \cos x)=0 \Rightarrow \cos x=n \Rightarrow \cos x=1,-1,0 \Rightarrow x=\frac{n \pi}{2}$

$x =\left\{\frac{ n \pi }{2}: \pi \in N \right\}-\left\{\frac{\pi}{2}, \pi \cdot \frac{3 \pi}{2}, 2 \pi,\right\}$

$g(x)=\cos (2 \pi \sin x)$

$Z=\{x: g(x)=0\}$

$\cos (2 \pi \sin x)=0 \Rightarrow 2 \pi \sin x=(2 n+1) \frac{\pi}{2} \Rightarrow \sin x-\frac{(2 n+1)}{4}$

$\sin x=-\frac{1}{4} \cdot \frac{1}{4} \cdot \frac{-3}{4} \cdot \frac{3}{4}$

$Z=\left\{n \pi \pm \sin ^{-1}\left(\frac{1}{4}\right), m \pi \pm \sin ^{-1}\left(\frac{3}{4}\right), n \in I\right\}$

$Y =\{ x : f ( x )=0\}$

$f(x)=\sin (\pi \cos x) \Rightarrow f^{\prime}(x)=\cos (\pi \cos x) \cdot(-\pi \sin x)=0$

$\sin x=0 \Rightarrow x=m \pi \text {. }$

$\cos (\pi \cos x)=0 \Rightarrow \pi \cos x=(2 n+1) \frac{\pi}{2} \Rightarrow \cos x-\frac{(2 n+1)}{2} \Rightarrow \cos x=-\frac{1}{2} \cdot \frac{1}{2}$

$Y=\left\{2 \pi, n \pi \pm \frac{\pi}{3}\right\}-\left\{\frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}, 2 \pi, \ldots . .\right\}$

$W=\left\{x: g^{\prime}(x)=0\right\}$

$g(x)=\cos (2 \pi \sin x) \Rightarrow g^{\prime}(x)=-\sin (2 \pi \sin x) \cdot(2 \pi \cos x)=0$

$\cos x =0 \Rightarrow x =(2 n +1) \frac{\pi}{2}$

$\sin (2 \pi \sin x)=0 \Rightarrow 2 \pi \sin x=n \pi \Rightarrow \sin x=\frac{\pi}{2}=-1-\frac{1}{2} \cdot 0 \cdot \frac{1}{2} \cdot 1$

$W=\left\{\frac{n \pi}{2}, n \pi \pm \frac{\pi}{6}, \pi \in I\right\}-\left\{\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \pi, \frac{7 \pi}{6}, \frac{3 \pi}{2}, \ldots\right\}$

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