2 Marks Questions

Question 12 Marks

In a library $136$ copies of a certain book require a shelf-length of $3.4m. $ How many copies of the same book would occupy a shelf-length of $5.1m?$

Answer

Number of copies	$136$	$x$
Length the Shelf (in m)	$3.4$	$5.1$

Let $x$ be the number of copies that would occupy a shelf-length of $5.1m.$
Since the number of copies and the length of the shelf are in direct variation, we have:
$\frac{136}{\text{x}}=\frac{3.4}{\text{5.1}}$
$\Rightarrow 136\times5.1 =\text{x}\times3.4$
$\Rightarrow\text{x}=\frac{136\times5.1}{3.4}$
$= 204$
Thus, $204$ copies will occupy a shelf of length $5.1m.$

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Question 22 Marks

The second class railway fare for $240\ km$ of Journey is $Rs.15.00$. What would be the fare for a journey of $139.2\ km?$

Answer

Let $Rs.x$ be the fare for a journey of $139.2km.$

Distance (in km)	$240$	$139.2$
Fare (in Rs.)	$15$	$x$

Since the distance travelled and the fare are in direct variation, we have: $\frac{240}{139.2}=\frac{15}{\text{x}}$
$\Rightarrow 240\times\text{x} =15\times139.2$
$\Rightarrow\text{x}=\frac{15\times139.2}{240}$
$=\frac{2088}{240}$
$= 8.7$ Thus, the fare for a journey of $139.2\ km$ will be $Rs. 8.70.$

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Question 32 Marks

Anupama takes $125$ minutes in walking a distance of $100$ metre. What distance would she cover in $315$ minutes$?$

Answer

Let the distance travelled in $315$ minutes be $x\ km.$

Time (in minute):	$125$	$315$
Distance (in metre):	$100$	$x$

If the distance travelled is more, the time needed to cover it will also be more.
Therefore, it is a direct variation.
We get: 1$25 : 315 = 100 : x$
$\Rightarrow\frac{125}{315}=\frac{100}{\text{x}}$
Applying cross multiplication, we get: $\text{x}=\frac{100\times315}{125} =252$
Thus, Anupama would cover $252$ metre in $315$ minutes.

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Question 42 Marks

In which of the following tables $x$ and $y$ vary inversely:

$x$	$9$	$24$	$15$	$3$
$y$	$8$	$3$	$4$	$25$

Answer

If $x$ and $y$ vary inversely, the product $xy$ should be constant. Here, product is different for all cases. Thus, in this case, $x$ and $y$ do not vary inversely.

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Question 52 Marks

If $x$ and $y$ vary inversely as: $x = 30,$ find y when constant of variation $= 900.$

Answer

Given: $\text{x}=30\text{ and k}=900$
$\therefore\text{xy}=\text{k}$
$\Rightarrow30\text{y}=900$
$\Rightarrow\text{y}=\frac{900}{30}$
$=30$
$\therefore\text{y}=30$

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Question 62 Marks

If $x$ and $y$ vary inversely as: $y = 35,$ find $x$ when constant of variation $= 7.$

Answer

Given: $y = 35$ and $k = 7$ Now, $xy = k$
$\Rightarrow35\text{x}=7$
$\Rightarrow\text{x}=\frac{7}{35}$
$=\frac{1}{5}$
$\therefore\text{x}=\frac{1}{5}$

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Question 72 Marks

In which of the following tables $x$ and $y$ vary inversely:

$x$	$4$	$3$	$6$	$1$
$y$	$9$	$12$	$8$	$36$

Answer

If $x$ and $y$ vary inversely, the product $xy$ should be constant. Here, in one case, product $= 6 × 8 = 48$ and in the rest, product $= 36$ Thus, in this case, $x$ and $y$ do not vary inversely.

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Question 82 Marks

Rohit bought $12$ registers for $Rs.156,$ find the cost of $7$ such registers.

Answer

Let the cost of $7$ registers be $Rs.x.$

Register:	$12$	$7$
Cost (in Rs.):	$156$	$x$

If he buys less number of registers, the cost will also be less. Therefore, it is a direct variation. We get: $12 : 7 = 156 : x$ $\Rightarrow\frac{12}{7}=\frac{156}{\text{x}}$ Applying cross multiplication, we get: $\text{x}=\frac{156\times7}{12} =91$ Thus, the cost of such registers will be $Rs.91.$

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Question 92 Marks

Suneeta types $1080$ words in one hour. What is her $GWAM ($gross words a minute rate$)?$

Answer

Number of words	$1080$	$x$
Time (in minute)	$60$	$1$

Let $x$ be her $GWAM.$
If the time taken is less, $GWAM$ will also be less.
Therefore, it is a direct variation.
$1080 : x = 60 : 1$
$\Rightarrow\frac{1080}{\text{x}}=\frac{60}{\text{1}}$
Applying cross multiplication, we get:
$\text{x}=\frac{1080\times1}{60}$
$=18$
Thus, her $GWAM$ will be $18.$

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Question 102 Marks

A car is travelling at the average speed of $50\ km/ hr.$ How much distance would it travel in $12$ minutes$?$

Answer

Distance (in km)	$50$	$x$
Time (in minute)	$60$	$12$

Let the distance be $x\ km.$
If the time taken is less, the distance covered will also be less.
Therefore, it is a direct variation.
$50 : x = 60 : 12$
$\Rightarrow\frac{50}{\text{x}}=\frac{60}{\text{12}}$
Applying cross multiplication, we get:
$\text{x}=\frac{50\times12}{60}$
$=10$
Thus, the required distance will be $10\ km.$

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Question 112 Marks

In which of the following tables $x$ and $y$ vary inversely:

$x$	$5$	$20$	$10$	$4$
$y$	$20$	$5$	$10$	$25$

Answer

In all cases, the product $xy$ is constant for any two pairs of values for $x$ and $y.$
Here, $xy = 100$ for all cases Thus, in this case, $x$ and $y$ vary inversely.

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Question 122 Marks

Find the constant of variation from the table given below:

$x$	$3$	$5$	$7$	$9$
$y$	$12$	$20$	$28$	$36$

Set up a table and solve the following problems. Use unitary method to verify the answer.

Answer

Since it is a direct variation, $\frac{\text{x}}{\text{y}}=\text{k}.$ For $x = 3$ and $y = 12,$
we have: $\text{k}=\frac{3}{12}=\frac{1}{4}$ Thus, in all cases, $\text{k}=\frac{1}{4}$

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Question 132 Marks

$68$ boxes of a certain commodity require a shelf-length of $13.6m.$ How many boxes of the same commodity would occupy a shelf length of $20.4m?$

Answer

Number of Box	$68$	$x$
Shelf-length (in m)	$13.6$	$20.4$

Let $x$ be the number of boxes that occupy a shelf-length of $20.4m.$
If the length of the shelf increases, the number of boxes will also increase.
Therefore, it is a case of direct variation.
$\frac{68}{\text{x}}=\frac{13.6}{\text{20.4}}$
$\Rightarrow 68 \times 20.4 = \text{x}\times13.6$
$\Rightarrow\text{x}=\frac{68\times20.4}{13.6}$
$=\frac{1387.2}{13.6}$
$= 102$
Thus, 102 boxes will occupy a shelf-length of $24.4m.$

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Question 142 Marks

In which of the following tables $x$ and $y$ vary inversely:

$x$	$4$	$3$	$12$	$1$
$y$	$6$	$8$	$2$	$24$

Answer

Since $x$ and $y$ vary inversely, we have: $\text{y}=\frac{\text{k}}{\text{x}}$
$\Rightarrow\text{xy}=\text{k}$
$\therefore$ The product of $x$ and $y$ is consant In all cases, the product $xy$ is consant $(i.e., 24)$ Thus, in this case, $x$ and $y$ vary inversely.

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