6.The solution of the equations x-y=2 and x+y=4 is:
(a) 3 and 1
(b) 4 and 3
(c) 5 and 1
(d) -1 and -3
Answer: (a) 3 and 1
Explanation: x-y =2
x=2+y
Substituting the value of x in the second equation we get;
2+y+y=4
2+2y=4
2y = 2
y=1
Now putting the value of y, we get;
x=2+1 = 3
Hence, the solutions are x=3 and y=1.
7. A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is:
(a) 3/12
(b) 4/12
(c) 5/12
(d) 7/12
Answer: (c) 5/12
Explanation: Let the fraction be x/y
So, as per the question given,
(x -1)/y = 1/3 => 3x – y = 3…………………(1)
x/(y + 8) = 1/4 => 4x –y =8 ………………..(2)
Subtracting equation (1) from (2), we get
x = 5 ………………………………………….(3)
Using this value in equation (2), we get,
4×5 – y = 8
y= 12
Therefore, the fraction is 5/12.
8. The solution of 4/x+3y=14 and 3/x-4y=23 is:
(a) ⅕ and -2
(b) ⅓ and ½
(c) 3 and ½
(d) 2 and ⅓
Answer: (a) ⅕ and -2
Explanation: Let 1/x = m
4m + 3y = 14
3m – 4y = 23
By cross multiplication we get;
m/(-69-56) = y/(-42-(-92)) = 1/(-16-9)
m/-125=y/50=-1/25
m/-125 = -1/25 and y/50=-1/25
m=5 and y=-2
m=1/x or x=1/m = ⅕
9. Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Her speed of rowing in still water and the speed of the current is:
(a) 6km/hr and 3km/hr
(b) 7km/hr and 4km/hr
(c) 6km/hr and 4km/hr
(d) 10km/hr and 6km/hr
Answer: (c) 6km/hr and 4km/hr
Explanation: Let, Speed of Ritu in still water = x km/hr
Speed of Stream = y km/hr
Now, speed of Ritu, during,
Downstream = x + y km/h
Upstream = x – y km/h
As per the question given,
2(x+y) = 20
Or x + y = 10……………………….(1)
And, 2(x-y) = 4
Or x – y = 2………………………(2)
Adding both the equations, we get,
2x=12
x = 6
Putting the value of x in eq.1, we get,
y = 4
Therefore,
Speed of Ritu is still water = 6 km/hr
Speed of Stream = 4 km/hr
10. The angles of cyclic quadrilaterals ABCD are: A = (6x+10), B=(5x)°, C = (x+y)° and D=(3y-10)°. The value of x and y is:
(a) x=20° and y = 10°
(b) x=20° and y = 30°
(c) x=44° and y=15°
(d) x=15° and y=15°
Answer: (b) x=20° and y = 30°
Explanation: We know, in cyclic quadrilaterals, the sum of the opposite angles are 180°.
Hence,
A + C = 180°
6x+10+x+y=180 =>7x+y=170°
And B+D=180°
5x+3y-10=180 =>5x+3y=190°
By solving the above two equations we get;
x=20° and y = 30°