2 Marks Questions

Question 12 Marks

Evaluate $: \frac{\cos 45^{\circ}+\sin 60^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

Answer

$\frac{\cos 45^{\circ}+\sin 60^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$
$=\frac{\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}+2}$
$=\frac{2 \sqrt{3}+3 \sqrt{2}}{4 \sqrt{2}(1+\sqrt{3})}$

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Question 22 Marks

In the given figure, $\frac{ EA }{ EC }=\frac{ EB }{ ED }$, prove that $\triangle EAB \sim \triangle ECD$

Answer

In $\triangle EAB$ and $\triangle ECD$
$\frac{EA}{EC}=\frac{EB}{ED}$
$\angle AEB=\angle CED$
$\triangle EAB \sim \triangle ECD$

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Question 32 Marks

Points $A (-1, y)$ and $B (5,7)$ lie on a circle with centre $O (2,-3 y)$ such that $AB$ is a diameter of the circle. Find the value of $y$. Also, find the radius of the circle.

Answer

Centre $O(2,-3 y)$ is the mid point of $A B$
$\therefore \frac{y+7}{2}=-3 y$
$\Rightarrow y=-1$
$\text { Radius }=OB=\sqrt{(5-2)^2+(7-3)^2}=5$

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Question 42 Marks

Find a relation between $x$ and $y$ such that the point $P (x, y )$ is equidistant from the points $A (7,1)$ and $B (3,5)$.

Answer

$ PA = PB$
$\Rightarrow PA ^2= PB ^2$
$( x -7)^2+( y -1)^2=( x -3)^2+( y -5)^2$
$\Rightarrow-8 x +8 y +16=0 \text { or } x - y -2=0$

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Question 52 Marks

Sum of two numbers is $105$ and their difference is $45$ . Find the numbers.

Answer

Let the numbers be $x , y ( x > y )$
$x+y=105 ......(i)$
$x-y=45......(ii)$
on solving $(i)$ and $(ii)$
$\Rightarrow x=75 \ y=30$
$\therefore$ Numbers are $75,30$

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Question 62 Marks

If $2 x+y=13$ and $4 x-y=17$, find the value of $(x-y)$.

Answer

$2 x+y=13 .....\text{(i)}$
$4 x-y=\text{17}.....\text { (ii) }$
$\text { Solving (i) and (ii) }$
$x=5 \ y=3$
$x-y=2$

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Question 72 Marks

One card is drawn at random from a well shuffled deck of 52 cards. Find the probability that the card drawn
(i) is queen of hearts;
(ii) is not a jack.

Answer

Total outcomes $=52$
(i) $\quad P$ (card is queen of hearts) $=\frac{1}{52}$
(ii) $P($ not a jack $)=\frac{48}{52}$ or $\frac{12}{13}$

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