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Pair of Linear Equations in Two Variables

Short Answer Type

201. Places A and B are 100 km. apart on a highway. On car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars ?

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202. The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each saves Rs. 200 per month find their monthly incomes.

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203. A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But, if he travels 130 km by train and the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.

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204. The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?

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205.

Solve the following system of equations by elimination method

6(ax + by) = 3a + 2b
6(bx - ay) =3b - 2a.

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206. A man has only 20 paisc coins and 25 paise coins in his purse. If he has 50 coins in all, totalling Rs. 11.25. How many coins of each type does he share ? (Use Elimination Method).

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207. A person starts his job with a monthly income and earns a fixed increment every year. If his salary was Rs. 4500 after 4 years of service and Rs. 5400 after 10 years of service, find the initial salary and the annual increment by using elimination method.

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208.

A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32km upstream than to return downstream to the same spot. Find the speed of the stream.

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Long Answer Type

209. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

Let the average speed of train for the first 54 km be x km/h

⇒ Average speed for the next 63 km = ( x + 6) km/h

We know

$Time space equals fraction numerator space Distance over denominator Speed end fraction therefore space Time space taken space by space the space train space to space cover space 54 space km space equals space 54 over straight x space straight h Also comma space time space taken space by space the space train space to space cover space the space next space 63 space km space equals space fraction numerator 63 over denominator straight x space plus space 6 end fraction space straight h According space to space the space question comma 54 over straight x space plus space fraction numerator 63 over denominator straight x space plus space 6 end fraction space equals space 63 rightwards double arrow space fraction numerator 54 space left parenthesis straight x space plus space 6 space right parenthesis space plus 63 straight x over denominator straight x space left parenthesis straight x space plus space 6 right parenthesis end fraction space space equals space 3$

⇒ 117x + 324 = 3 ( x² + 6x)

⇒ 117x + 324 = 3x² + 18x

⇒ 3x² - 99x -324 = 0 ..(i)

Taking common from the above equation (i)

we have

x² - 33x - 108 = 0

⇒ x² - 36x + 3x -108 = 0

⇒ x (x - 36) + 3 (x - 36) = 0

⇒ (x - 36) (x +3 )=0

⇒ x -36 = 0 Or x + 3 =0

x = 36 or x = -3

The speed of the train cannot be negative. Thus, x = 36

Hence, the speed of the train to cover 54 km or its first speed is 36 km/h.

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210.

If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k – 1, 5k) are collinear, then find the value of k.