Ncert – Class10 – Linear Equations in Two Variables- Hots Questions
Question- 1
A boat takes 12 hours to cover a distance of 24 km upstream and it takes only 3 hours to come back the same distance. Find the speed of the boat in still water and that of the stream.
Solution
Time taken for upstream journey = distance / time = 24 / ( x-y )
Time taken for downstream journey= 24 / ( x+y )
According to the given condition, 24 / (x-y) = 12 ⇒ x-y = 2
Also 24 / (x+y) = 3 ⇒ x+y = 8
Adding we get x-y+x+y + 10 ⇒ 2x = 10 ;
⇒ x=5 .putting in the equation we get, y = 3
Hence speed of the motor boat in still water = 5 km/hour
and of the speed of the stream is 3 km/hr
Question -2
A man can row a boat downstream 20km in 2 hours and upstream 4km in 2 hours. Find his speed of rowing in still water and also find the speed of the stream
Solution
Let speed of rowing is x km/h and speed of current is y km/h so upstream speed is x-y and downstream speed is x+y km/h. So 20/x+y =2 and 4/x-y =2 . so x+y=10 and x-y=2 solving we get x=6 and y=4.
Question -3
Points A and B are 80km apart on a highway. A car starts from point A and another from B simultaneously. If they travel in the same direction they meet in 8 hours , but if they travel towards each other , they meet in 1 hour. Find the speed of each car.
Solution
Let the speed of cars be x and y . if they move in same direction the relative speed is x-y and in opposite direction relative speed is x+y. Then x-y=10 and x+y=80 so x=45 km/h and y=35 km/h
Question -4
Determine graphically the coordinates of the vertices of a triangle , the equations of whose sides are given by 2y-x=8 5y-x=14 and y-2x=1`
Solution
Correct graph plotted. Vertices are (-4,2),(1,3),(2,5)
Question -5
From Bengaluru bus stand, if Riddhima buys 2 tickets to Malleswaram and
3 tickets to Yeswanthpur, then total cost is Rs 46; but if she buys 3 tickets
to Malleswaram and 5 tickets to Yeswanthpur, then total cost is Rs 74.
Consider the fares from Bengaluru to Malleswaram and that to
Yeswanthpur as Rs x and Rs y respectively and answer the following
questions.
(i) 1st situation can be represented algebraically as
(a) 3x-5y=74 (b) 2x+5y=74 (c) 2x-3y=46 (d) 2x+3y=46
(ii) 2nd situation can be represented algebraically as
(a) 5x + 3y = 74 (b) 5x- 3y= 74 (c) 3x + 5y = 74 (d) 3x-5y=74
(iii), Fare from Ben~aluru to Malleswaram is
(a) Rs 6 (b) Rs 8 (c) Rs 10 (d) Rs 2
Solution
(i) (d): 1st situation can be represented algebraically as 2x + 3y = 46
(ii) (c): 2nd situation can be represented algebraically as 3x + 5y = 74
(iii) (b): We have, 2x + 3y = 46 ………(i)
3x+5y=74……….. (ii)
Multiplying (i) by 5 and (ii) by 3 and then subtracting,
we get 10x – 9x = 230 – 222 ⇒⇒ x = 8
∴∴ Fare from Bengaluru to Malleswaram is Rs 8
Question -6
Raman usually go to a dry fruit shop with his mother. He observes the
following two situations.
On 1st day: The cost of 2 kg of almonds and 1 kg of cashew was Rs 1600.
On 2nd day: The cost of 4 kg of almonds and 2 kg of cashew was Rs 3000.
Denoting the cost of 1 kg almonds by Rs x and cost of 1 kg cashew by Rs
y, answer the following questions.
(i) Represent algebraically the situation of day-I.
(a) x + 2y = 1000 (b) 2x + y = 1600 (c) x – 2y = 1000 (d) 2x – y = 1000
(ii) Represent algebraically the situation of day- II.
(a) 2x + y= 1500 (b) 2x- y= 1500 (c) x + 2y=1500 (d) 2x + y = 750
(iii) The linear equation represented by day-I, intersect the x axis at
(a) (0,800) (b) (0,-800) (c) (800,0) (d) (-800,0)
(iv) The linear equation represented by day-II, intersect the y-axis at
(a) (1500,0) (b) (0, -1500) (c) (-1500,0) (d) (0,1500)
(v) Linear equations represented by day-I and day -II situations, are
(a) non parallel (b) parallel
(c) intersect at one point (d) overlapping each other.
Solution
(i) (b) 2x + y = 1600
(ii) (a) 2x + y= 1500
(iii) (c) (800,0)
(iv) (d) (0,1500)
(v) (b) parallel
Question -7
A train covered a certain distance at a uniform speed. If the train could have been 10 km/hr faster, it would have been taken 2 hours less than the scheduled time and if the train were slower by 10 km/hr it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.
Solution
Let the speed of the train be x km/h and the time has taken be y h.
The total distance is then xy km.
Case1:
Speed increases by 10 km/h and the time taken reduces by 2 hours.
The distance traveled remains xy km.
(x+10)(y−2)=xy
Therefore, xy−2x+10y−20=xy
⇒−2x+10y−20=0
⇒−2x+10y=20 ———- (1)
Case 2:
Speed decreases by 10 km/h, then the time taken increases by 3 hours.
However, the distance remains,xy km.
(x−10)(y+3)=xy
Therefore, xy+3x−10y−30=xy
⇒3x−10y−30=0
⇒3x−10y=30 ———- (2)
Adding (1) and (2) we solve x=50 km/h
Using this in (2), we get
150−10y=30
⇒y=12
Therefore, the distance is 50×12=600 km
Question -8
A and B are two points 150 km apart on a highway. Two cars start from A and B at the same time. If they move in the same direction they meet in 15 hours.But if they move in opposite directions, they meet in 1 hour. Find their speed
Solution
Let the speed of Car which starts from A be x km per hour
And the speed of the Car which starts from B be y km per hour
Then according to the question
When cars move in the same direction till 15 Hour . then,
Distance covered by the car at A− distance covered by the car at B =150km
i.e.,15x−15y=150 …………………….1
Also according to the question when the car moves in opposite direction till 1
Hour. Then,
Distance covered by the car at A + distance covered by the car at B =150km
i.e.,1x+1y=150 ……………….2
Solving eq. 1 and 2 we get,
x=80 km per hour
and ,y=70 km per hour
Question -9
Use elimination method to find all possible solution of the following pair of linear equations
3x-5y=20 6x-10y =40
Solution
3x- 5y =20 ———–(1)
6x+10y=40———–(2)
From equation (1)
⟹3x=20+5y
⟹x=(20+5𝑦)/3
⟹putting the value of x in equation (2)
⟹6( (20+5𝑦)/3) + 10y=40
⟹20y=0
⟹y=0
⟹putting the value of y=0 in equation (1)
⟹3x=20
⟹x=20/3
Question -10
The students of a class are made to stand in rows. If 4 students are extra in each row , there would be 2 rows less .If 4students are less in each row , there would be 4 rows more . Find the number of students in the class.
Solution
Answer 96
