5 Marks Questions

Question 15 Marks

Ravi starts for his school at $8:20a.m$. on his bicycle. If he travels at a speed of $10\ km/h$, then he reaches his school late by $8$ minutes but on travelling at $16\ km/h$ he reaches the school $10$ minutes early. At what time does the school start?

Answer

Let the total distance $= x\ km$
Let the time taken by Ravi to reach the school at sharp time $= t\ min$
If speed of the bicycle is $10\ km/h$, then he reach his school late by $8$ $min$
$\frac{\text{x}}{10}=\text{t}+\frac{8}{60}$
$\frac{\text{x}}{10}=\text{t}+\frac{2}{15}$
If speed of the bicycle is $16\ km/h$, then he reach his school $10\ min.$
early $\frac{\text{x}}{16}=\text{t}-\frac{10}{60}$
$\frac{\text{x}}{16}=\text{t}-\frac{1}{6}$
On solving eqs. $(i)$ and $(ii)$ we get $\frac{\text{x}}{10}-\frac{\text{x}}{16}=\frac{2}{15}+\frac{1}{6}$
$\frac{\text{8x}-\text{5x}}{80}=\frac{4+5}{30}$
$\frac{3\text{x}}{80}=\frac{9}{30}$
$\text{x}=\frac{9\times80}{30\times3}=8\text{km}$
Now, put $x = 8$ in eq. $(i),$ we get $\frac{8}{10}=\text{t}+\frac{2}{15}$
$\text{t}=\frac{8}{10}-\frac{2}{15}=\frac{24-4}{30}$
$\text{t}=\frac{20}{30}=\frac{2}{3}\text{h}$
$=\frac{2}{3}\times60=40\text{min}$
Hence, starting time of school is $8:20 + 40$ min i.e, $9:00\ am$

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Question 25 Marks

The table shows the time four elevators take to travel various distances. Find which elevator is fastest and which is slowest.

	Distance (m)	Time (sec.)
Elevator $- A$	$435$	$29$
Elevator $- B$	$448$	$28$
Elevator $- C$	$130$	$10$
Elevator $- D$	$85$	$5$

How much distance will be travelled by elevators $B$ and $C$ seperately in $140$sec? Who travelled more and by how much?

Answer

Elevator $A$ takes $29$ sec to cover $435m$ Distance covered by elevator $A$ in $=\frac{435}{29}=15\text{ m}$
Elevator $B$ takes $28\ s$ to cover $448m$ Distance covered by elevator $B$ in $1\ s$ $=\frac{448}{28}=16\text{m}$
Elevator $C$ takes $10\ s$ to cover $130m$ Distance covered by elevator $C$ in $1\ s$ $ =\frac{130}{1} =13\text{m}$
Elevator $D$ takes $5\ s$ to cover $85m$ Distance covered by elevator $D$ in $1\ s$ $=\frac{95}{5}=17\text{m}$
Elevator $D$ is fastest, while $C$ covers least.
Hence, elevator $C$ is slowest.
Elevator $B$ covers distance is $140\ s = 140 \times 16 = 2240\ m$
Elevator $C$ covers distance in $140\ s = 140 \times 13 = 1820\ m$
Elevator $B$ covers more distance than $C = 2240 − 1820 = 420\ m$

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Question 35 Marks

If $a$ and $b$ vary inversely to each other, then find the values of $p, q, r ; x, y, z$ and $l, m, n.$

$a$	$6$	$8$	$q$	$50$
$b$	$18$	$p$	$39$	$r$

$a$	$2$	$y$	$6$	$10$
$b$	$x$	$12.5$	$15$	$z$

$a$	$l$	$9$	$n$	$6$
$b$	$5$	$m$	$25$	$10$

Answer

It a and b are vary inversely to each other i.e., $ab = k$ (constant)
For table $(a)$
If $a = 6$ and $b = 18$
Then, $a \times b = 6 \times 18 = 108$
$k = 108$
When $a = 8$ and $b = p$, then
$ab = k$
$8 \times p = 108$
$\text{p}=\frac{27}{2}$
When $a = q$ and $p = 39$, then
$ab = k$
$q \times 39 = 10$
$\text{q}=\frac{108}{39}=\frac{36}{13}$
When $a = 25$ and $b = r,$ then
$ab = k$
$25 \times r = 108$
$\text{r}=\frac{108}{25}$
For table $(b)$
If $a = 6$ and $b = 15$, thena
$\times b = 6 \times 5 = 90$
$k = 90$
When $a = 2$ and $b = x$, then
$ab = k$
$2 \times x = 90$
$x = 45$
When, $a = y$ and $b = 12.5$ then,
$ab = k$
$y \times 12.5 = 90$
$\text{y}=\text{y}=\frac{90}{12.5}=7.2$
When $a = 0$ and $y = z$ then
$ab = k$
$10 \times z = 90$
$z = 9$
For table $(c)$
If $a = 6$ and $b = 10$ then
$ab = 6 \times 10 = 60$
$k = 60$
When $a = l$ and $b = 5$ then
$ab = k$
$9 \times m = 60$
$\text{m}=\frac{20}{3}$
When, $a = n$ and $b = 25$, then
$ab = k$
$n \times 25 = 60$
$n =$ $\frac{60}{25}$
$n =$ $\frac{12}{5}$

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Question 45 Marks

Match each of the entries in Column $I$ with the appropriate entry in Column $II$

S.No	Column $I$	S.No	Column II
$1.$	x and y vary inversely to each other	$A.$	$\frac{\text{x}}{\text{y}}=\text{constant}$
$2.$	Mathematical representation of inverse variation of quantities $p$ and $q$	$B.$	$y$ will increase in proportion
$3.$	Mathematical representation of direct variation of quantities $m$ and $n$	$C.$	$xy$ = Constant
$4.$	When $x = 5, y = 2.5$ and when $y = 5, x = 10$	$D.$	$\text{p} \propto\frac{1}{\text{q}}$
$5.$	When $x = 10 , y = 5$ and when $x = 20, y = 2.5$	$E.$	$y$ will decrease in proportion
$6.$	x and y vary directly with each other	$F.$	$x$ and $y$ are directly proportional
$7.$	If x and y vary inversely then on decreasing x	$G.$	$\text{m }\alpha \text{ n}$
$8.$	If x and y vary directly then on decreasing	$H.$	$x$ and $y$ vary inversely
		$I.$	$\text{p } \alpha \text{ q}$
		$J.$	$\text{m }\alpha \frac{1}{\text{n}}$

Answer

S.No	Column I	S.No	Column II
$1.$	$x$ and $y$ vary inversely to each other	$6.$	$\frac{\text{x}}{\text{y}}=\text{constant}$
$2.$	Mathematical representation of inverse variation of quantities $p$ and $q$	$7.$	$y$ will increase in proportion
$3.$	Mathematical representation of direct variation of quantities $m$ and $n$	$1.$	$xy =$ Constant
$4.$	When $x = 5, y = 2.5$ and when $y = 5, x = 10$	$2.$	$\text{p} \propto\frac{1}{\text{q}}$
$5.$	When $x = 10 , y = 5$ and when $x = 20, y = 2.5$	$8.$	$y$ will decrease in proportion
$6.$	$x$ and $y$ vary directly with each other	$4.$	$x$ and $y$ are directly proportional
$7.$	If $x$ and $y$ vary inversely then on decreasing $x$	$3.$	$\text{m }\alpha \text{ n}$
$8.$	If $x$ and $y$ vary directly then on decreasing	$5.$	$x$ and $y$ vary inversely
		$I.$	$\text{p } \alpha \text{ q}$
		$J$	$\text{m }\alpha \frac{1}{\text{n}}$

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