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Maths True False[1 Marks ]

Trigonometric Identities · Jharkhand Board (JAC) STD 10 (Jharkhand - English Medium). 8 questions.

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True False[1 Marks ]

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8 questions · self-marked practice — reveal the answer and mark yourself.

Question 11 Mark
Write 'True' or 'False' and justify your answer in the following:
The value of $\sin\theta+\cos\theta$ is always greater than 1.
Answer
False.
The value of $(\sin\theta+\cos\theta)\text{ for }\theta=0^\circ$ is 1.
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Question 21 Mark
Write 'True' or 'False' and justify your answer in the following:
The value of the expression $\sin80^\circ-\cos80^\circ$ is negative.
Answer
False.
$\sin80^\circ-\cos80^\circ$
$=\sin80^\circ-\sin(90^\circ-80^\circ)$
$=\sin80^\circ-\sin10^\circ\text{is }+\text{ve} $
$\because\ \sin\theta$ increase as $\theta$ increase.
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Question 31 Mark
Write 'True' or 'False' and justify your answer in the following:
$\cos\theta=\frac{\text{a}^2+\text{b}^2}{2\text{ab}}$, where a and b are two lab distinct numbers such that ab > 0.
Answer
False.
Given, a and b are two distinct numbers such that ab > 0.
Using, AM > GM
[since, AM and GM of two number a and b are $\frac{\text{a}+\text{b}}{2}$ and $\sqrt{\text{ab}},$ respectively]
$\Rightarrow\ \frac{\text{a}^2+\text{b}^2}{2}>\sqrt{\text{a}^2\times\text{b}^2}$
$\Rightarrow\ \text{a}^2+\text{b}^2>2\text{ab}$
$\Rightarrow\ \frac{\text{a}^2+\text{b}^2}{2\text{ab}}>1\ \Big[\because\ \cos\theta=\frac{\text{a}^2+\text{b}^2}{2\text{ab}}\Big]$
$\Rightarrow\ \cos\theta>1\ \big[\because\ -1\leq\cos\theta\leq1\big]$
Which is not possible.
Hence, $\cos\theta\neq\frac{\text{a}^2+\text{b}^2}{2\text{ab}}$
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Question 41 Mark
Write 'True' or 'False' and justify your answer in the following:
The value of $\sin\theta$ is $\text{x}+\frac{1}{\text{x}},$ where 'x' is a positive real number.
Answer
False. We know that $\Big(\sqrt{\text{x}}-\frac{1}{\sqrt{\text{x}}}\Big)^2\geq0$ or $\Big(\text{x}+\frac{1}{\text{x}}\Big)\geq2,$ but $\sin\theta$ is not greater than 1. Alternatively, there exists the following three posibilities: Case 1: If $\text{x}<1,\text{then}\Big(\text{x}+\frac{1}{\text{x}}\Big)<1$Case 2:
If $\text{x}=1,\text{then}\Big(\text{x}+\frac{1}{\text{x}}\Big)=1$Case 3:
If $\text{x}>1,\text{then}\Big(\text{x}+\frac{1}{\text{x}}\Big)>1$ However, $\sin\theta$ cannot be greater than 1.
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Question 51 Mark
Write 'True' or 'False' and justify your answer in the following:
The value of $\cos^2{23}-\sin^2{67}$ is positive.
Answer
False.
$\cos^2{23}-\sin^2{67}$
$=\sin^2(90-23)-\sin^2{67}$
$=\sin^2{67}-\sin^2{67}=0$
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Question 71 Mark
$\cos \theta=\frac{a^2+b^2}{2 a b}$, where $a$ and $b$ are two distinct numbers such that $a b>0$.
Answer
False
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