2 Marks Questions

Question 12 Marks

Show that the number $5 \times 11 \times 17+3 \times 11$ is a composite number.

Answer

$
\begin{aligned}
5 \times 11 \times 17+3 \times 11 & =11 \times(5 \times 17+3) \\
& =11 \times 88 \text { or } 11 \times 11 \times 2^3
\end{aligned}
$
It means the number can be expressed as a product of two factors other than
1 , therefore the given number is a composite number.

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Question 22 Marks

Prove that $5-2 \sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.

Answer

Assuming $5-2 \sqrt{3}$ to be a rational number.
Let $5-2 \sqrt{3}=\frac{a}{b}$ where $a$ and $b$ are integers & $b \neq 0$
$\Rightarrow \sqrt{3}=\frac{5 b-a}{2 b}$
Here RHS is rational but LHS is irrational.
Therefore our assumption is wrong.
Hence, $5-2 \sqrt{3}$ is an irrational number.

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Question 32 Marks

In the given figure, $\text{ABCD}$ is a quadrilateral. Diagonal $BD$ bisects $\angle B$ and $\angle D$ both. Prove that :
$(i) \triangle \text{ABD} \sim \triangle CBD$
$(ii) \text{AB = BC}$

Answer

$\text { (i) } \operatorname{In} \triangle ABD \ \triangle CBD$
$ \angle 3=\angle 4$
$\angle 1=\angle 2$
$\therefore \triangle ABD \sim \triangle CBD$
$\text { (ii) } \triangle ABD \cong \triangle CBD$
$\therefore AB = BC $

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Question 42 Marks

If $A=60^{\circ}$ and $B=30^{\circ}$, verify that :
$\sin (A+B)=\sin A \cos B+\cos A \sin B$

Answer

$ \text { LHS }=\sin \left(60^{\circ}+30^{\circ}\right)=\sin 90^{\circ}=1$
$ \text { RHS } =\sin 60^{\circ} \cos 30^{\circ}+\cos 60^{\circ} \sin 30^{\circ}$
$ =\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}+\frac{1}{2} \times \frac{1}{2}=1$
$\therefore \text { LHS }=\text { RHS }$

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Question 52 Marks

Evaluate : $2 \sqrt{2} \cos 45^{\circ} \sin 30^{\circ}+2 \sqrt{3} \cos 30^{\circ}$

Answer

$2 \sqrt{2} \times \frac{1}{\sqrt{2}} \times \frac{1}{2}+2 \sqrt{3} \times \frac{\sqrt{3}}{2}$
$=4$

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Question 62 Marks

In a pack of 52 playing cards one card is lost. From the remaining cards, a card is drawn at random. Find the probability that the drawn card is queen of heart, if the lost card is a black card.

Answer

Total number of remaining cards $=51$
$
P(\text { getting queen of heart })=\frac{1}{51}
$

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Question 72 Marks

Solve the following system of linear equations $7 x-2 y=5$ and $8 x+7 y=15$ and verify your answer.

Answer

$7 x-2 y=5 .....(i)$
$8 x+7 y=15....(ii)$
Solving equation $(i)$ and $(ii)$, we get
$x=1, y=1$
Verification of answer

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