5 Marks Questions

Question 15 Marks

(B)Places A and B are 180 km apart on a highway. One car starts from A and another from B at the same time. If the car travels in the same direction at different speeds, they meet in 9 hours. If they travel towards each other with the same speeds as before, they meet in an hour. What are the speeds of the two cars?

Answer

(B)Let car I starts from A with speed x km/hr and car Il Starts from B with speed y km/hr (x>y)

Case I- when cars are moving in the same direction.
Distance covered by car I in 9 hours = 9x.
Distance covered by car II in 9 hours = 9y
Therefore 9 (x-y) = 180
=> x-y= 20 ……………. (i)

case II- when cars are moving in opposite directions.

Distance covered by Car I in 1 hour = x
Distance covered by Car II in 1 hour = y

Therefore x + y=180 ………….. (ii)
Solving (i) and (ii) we get, x=100 km/hr, y=80 km/hr.

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Question 25 Marks

The monthly expenditure on milk in $200$ families of a Housing Society is given below

Monthly Expenditure $($in$)$	$1000- 1500$	$1500- 2000$	$2000- 2500$	$2500- 3000$	$3000- 3500$	$3500- 4000$	$4000- 4500$	$4500- 5000$
Number of families	$24$	$40$	$33$	$\times$	$30$	$22$	$16$	$7$

Find the value of $x$ and also find the mean expenditure

Answer

$\text{Monthly Expenditure}$	$f_i$	$x_i$	$f_ix_i$
$1000-1500$	$24$	$1250$	$30,000$
$1000-1500$	$40$	$1750$	$70,000$
$2000-2500$	$33$	$2250$	$74,250$
$2500-3000$	$X=28$	$2750$	$77,000$
$3000-3500$	$30$	$3250$	$97,500$
$3500-4000$	$22$	$3750$	$82,500$
$4000-4500$	$16$	$4250$	$68,000$
$4500-5000$	$7$	$4750$	$33,250$

$172+x=200$
$ X=28$
$\text { Mean }=\frac{532500}{200}$
$=2662.5$

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Question 35 Marks

Find the mean and median of the following dat

Class	85-90	90-95	95-100	100-105	105-110	110-115
frequency	15	22	20	18	15	25

Answer

Class	x	f	$u =\frac{x-102.5}{5}$	fu	cf
85-90	87.5	15	-3	-45	15
90-95	92.5	22	-2	-44	37
95-100	97.5	20	-1	-20	57
100-105	102.5	18	0	0	75
105-110	107.5	20	1	20	95
110-115	112.5	25	2	50	120
		$\Sigma f=120$		$\Sigma fu =-39$

$\begin{aligned}
\text { Mean }=\bar{x} & =102.5-5 \times \frac{39}{120} \\
& =100.875
\end{aligned}$
Median class is $100-105$
$\text { Median }=100+\frac{5}{18}(60-57)=100.83$

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Question 45 Marks

A boy whose eye level is $1.35 m$ from the ground, spots a balloon moving with the wind in a horizontal line at some height from the ground. The angle of elevation of the balloon from the eyes of the boy at an instant is $60^{\circ}$. After $12$ seconds, the angle of elevation reduces to $30^{\circ}$. If the speed of the wind is $3 m / s$ then find the height of the balloon from the ground. $($Use $\sqrt{3}=1.73)$

Answer

Let $A$ be the eye level $B, C$ are positions of balloon
Distance covered by balloon in $12 \sec = 3 \times 12 = 36 m$
$BC = GF = 36 m$
$\tan 60^{\circ}=\sqrt{3}=\frac{h}{x}$
$=>h=x \sqrt{3} \ldots (i)$
$\tan 30^{\circ}=\frac{1}{\sqrt{3}}=\frac{h}{x+36}$
$=h=\frac{x+36}{\sqrt{3}} \ldots \ldots . . (ii)$
Solving $(i)$ and $(ii)\ h=18 \sqrt{3}=31.14 m$
Height of balloon from ground $=1.35+31.14=32.49 m$

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Question 55 Marks

Prove that the lengths of tangents drawn from an external point to a circle are equal.
Using above result, find the length $B C$ of $\triangle A B C$. Given that, a circle is inscribed in $\triangle A B C$ touching the sides $A B, B C$ and $C A$ at $R, P$ and $Q$ respectively and $A B=10 cm$, $A Q=7 cm, C Q=5 cm$.

Answer

Correct given, to prove, construction, figure
Correct proof
AR = AQ = 7cm
BP = BR = AB-AR = 3cm
CP =CQ = 5cm
BC = BP+PC = 3+5 = 8 cm

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Question 65 Marks

(A)Solve the following system of linear equations graphically:
$x+2 y=3,2 x-3 y+8=0$

Answer

x+2y=3, 2x-3y+8=0
Correct graph of each equation
Solution x=-1 and y=2

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Maths 5 Marks Questions

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