match the column :
Column I Column II
1. Thermal power plant. (a)Methane
2. Hydro power plant. (b) Nuclear fission
3. Major component of biogas to produce
(c) Use coal, petroleum and natural gas thermal electricity
4. Device which converts (d) Solar cell solar energy into electricity
5. Nuclear power plant generation
(e) Dams are constructed for power
Answers
Answer:
Appropriate Question :-
If f(x) = sin[π²]x + sin[-π²]x where [] denotes the greatest integer less than or equal to x then
[A] none of these
[B] f(π/2) = 1
[C] f(π) = 2
[D] f(π/4) = -1
\large\underline{\sf{Solution-}}
Solution−
Given that
\begin{gathered}\rm \: f(x) = sin[ {\pi}^{2}]x + sin[ - {\pi}^{2}]x \\ \end{gathered}
f(x)=sin[π
2
]x+sin[−π
2
]x
We know
Greatest Integer function reduces any real number to its nearest lowest Integer.
\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y = [x]\\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf [0,1) & \sf 0 \\ \\ \sf [1,2) & \sf 1 \\ \\ \sf [2,3) & \sf 2\\ \\ \sf [ - 1,0) & \sf - 1\\ \\ \sf [ - 2, - 1) & \sf - 2 \end{array}} \\ \end{gathered} \\ \end{gathered}
x
[0,1)
[1,2)
[2,3)
[−1,0)
[−2,−1)
y=[x]
0
1
2
−1
−2
Now, We know
\begin{gathered}\rm \: \pi = 3.14 \\ \end{gathered}
π=3.14
\begin{gathered}\rm\implies \: {\pi}^{2} = 9.8596 \\ \end{gathered}
⟹π
2
=9.8596
So,
\begin{gathered}\rm\implies \:[{\pi}^{2}] = [9.8596] = 9 \\ \end{gathered}
⟹[π
2
]=[9.8596]=9
and
\begin{gathered}\rm\implies \:[ - {\pi}^{2}] = [ - 9.8596] = - 10\\ \end{gathered}
⟹[−π
2
]=[−9.8596]=−10
Thus, given function
\begin{gathered}\rm \: f(x) = sin[ {\pi}^{2}]x + sin[ - {\pi}^{2}]x \\ \end{gathered}
f(x)=sin[π
2
]x+sin[−π
2
]x
can be rewritten as
\begin{gathered}\rm \: f(x) = sin9x + sin( - 10x) \\ \end{gathered}
f(x)=sin9x+sin(−10x)
\begin{gathered}\rm \: f(x) = sin9x - sin10x \\ \end{gathered}
f(x)=sin9x−sin10x
So,
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{2} \bigg) = sin\bigg( \dfrac{9\pi}{2} \bigg) - sin\bigg( \dfrac{10\pi}{2} \bigg) \\ \end{gathered}
f(
2
π
)=sin(
2
9π
)−sin(
2
10π
)
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{2} \bigg) = sin\bigg( \pi + \dfrac{\pi}{2} \bigg) - sin5\pi \\ \end{gathered}
f(
2
π
)=sin(π+
2
π
)−sin5π
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{2} \bigg) = sin\bigg( \dfrac{\pi}{2} \bigg) - 0 \\ \end{gathered}
f(
2
π
)=sin(
2
π
)−0
\begin{gathered}\rm\implies \:\rm \: f\bigg( \dfrac{\pi}{2} \bigg) = 1 \\ \end{gathered}
⟹f(
2
π
)=1
So, option [B] is correct.
Now, Consider
\begin{gathered}\rm \: f(\pi) = sin9\pi - sin10\pi \\ \end{gathered}
f(π)=sin9π−sin10π
\begin{gathered}\rm \: f(\pi) = 0 - 0 \\ \end{gathered}
f(π)=0−0
\begin{gathered}\rm\implies \:\rm \: f(\pi) = 0 \\ \end{gathered}
⟹f(π)=0
It implies, option [C] is not correct.
Now, Consider
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{4} \bigg) = sin\bigg( \dfrac{9\pi}{4} \bigg) - sin\bigg( \dfrac{10\pi}{4} \bigg) \\ \end{gathered}
f(
4
π
)=sin(
4
9π
)−sin(
4
10π
)
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{4} \bigg) = sin\bigg(2\pi + \dfrac{\pi}{4} \bigg) - sin\bigg( \dfrac{5\pi}{2} \bigg) \\ \end{gathered}
f(
4
π
)=sin(2π+
4
π
)−sin(
2
5π
)
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{4} \bigg) = sin\bigg(\dfrac{\pi}{4} \bigg) - sin\bigg(2\pi + \dfrac{\pi}{2} \bigg) \\ \end{gathered}
f(
4
π
)=sin(
4
π
)−sin(2π+
2
π
)
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{4} \bigg) = \dfrac{1}{ \sqrt{2} } - sin\bigg( \dfrac{\pi}{2} \bigg) \\ \end{gathered}
f(
4
π
)=
2
1
−sin(
2
π
)
\begin{gathered}\rm \: f\bigg( \dfrac{\pi}{4} \bigg) = \dfrac{1}{ \sqrt{2} } - 1\\ \end{gathered}
f(
4
π
)=
2
1
−1
It implies, option [D] is not correct.
Hence, from above we concluded that
\begin{gathered}\rm\implies \: \:\boxed{\tt{ \: \rm \: f\bigg( \dfrac{\pi}{2} \bigg) = 1 \: \: }} \\ \end{gathered}
⟹
f(
2
π
)=1
It implies, option [B] is correct.