Question 11

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

Solution

x=−72 or 8 (speed of the stream cannot be negative)

Therefore, speed of stream is 8 km/hr.

Question 12

The diagonal of a rectangular field is 60  metres  more than the shorter side.If the shorter side is 30 metres less than the longer side, find the sides of the field .  Also find the length of diagonal.

Solution

Let ABCD be a rectangular field .The shorter  Side be BC of rectangle = x metres,AB=(x+30)m Diagonal AC  =(x +60 )metres

(x-90)(x+30) = 0  => x -90  = 0 or x+30 = 0

X  =90 or x = -30 (side can not be negative)

So shorter side of rectangle = 90m ,longer side = 120m

Length of diagonal =90m +60m =150 m

Question 13

A plane is flying along the line 3x – y = 18. A volcanic eruption has happened and the edge of volcanic cloud is defined by y = x² -8x +12. For safety reasons the pilot should not enter this air space. If the plane continues along this line, would you advise the pilot to change direction?

Solution

Question 14

A farmer prepares a rectangular vegetable garden of area 180 m². With 39 m of barbed wire, he can fence the three sides of the garden, leaving one of the longer sides unfenced. Find the dimensions of the garden.

Solution

Question 15

A plane left 30 mints later than the schedule time and in order to reach its destination 1500 Km away in time it has to increase its speed by 250km/hr from its usual speed. Find its usual speed.

Solution

Let usual speed = x km/hr

New speed = ( x + 250) km/hr Total distance = 1500 km Time taken by usual speed =1500/x  Time taken by new speed =1500/(x+250) 

ATQ  1500/x  – 1500/(x+250)  =1/2

x² + 250x – 750000 = 0x=750  or x=-1000(not possible)

Question 16

A train travels at a certain average speed for a distance of 63 km and then travels a distance
of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to
complete the total journey, what is its original average speed?

Solution

x2 – 39x – 126 = 0 i.e., (x + 3) (x – 42) = 0 i.e., x = – 3 or x = 42 Since x is the
average speed of the train, x cannot be negative. Therefore, x = 42. So, the speed of
the train is 42 km/h.

Question 17

Ritu is x years old ,while her mother is x² year old. 5 years hence her mother
will be 3 times as old as Ritu. Find present age of RITU and her mother.

Solution

RITU`s present age is 5 years and mother~s age is 25 years

Question 18

In a group of children,each child will gives a gift to every other child. If the
number of gifts is2450, what will be the number of children ?

Solution

Let the number of children be x. The no. of gifts each child gives =x – 1
Total no. of gifts = 2450.
ATQ,
x.( x – 1) = 2450 ,or x² –x =2450 ,or x² –x -2450 =0 ,
or x² -50x +49x – 2450 = 0 ,or ( x -50) (x +49) =0
So x = 50 0r x = – 49(not possible). Therefore number of children is 50

Question 19

Find the root of the equation:
1/x + 4 – 1/ x-7 =11/30, x is not equal to-4 and 7

Solution

1 and 2 are roots of the given quadratic equation

Question 20

The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller
number, then find the two numbers.

Solution

Smaller number =9
⇒ Larger number =2x−5=2×9−5=18−5=13
⇒ Product of two numbers =9×13=117

SELF ASSESSMENT

Q1. 1/(𝑝 + 𝑞 + 𝑥) = 1/𝑝 + 1/𝑞 + 1/𝑥 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑏𝑦 𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑡ℎ𝑜𝑑.

Q2. The difference of the ages of a boy and his brother is 3 and the product of their ages in years
is 504. Find their ages

Q3. The sum of areas of two squares is 468m² If the difference of their perimeters is 24cm, find the sides
of the two squares

Q4. Solve by factorization 4x² – 4a²x + (a⁴ – b⁴) = 0

Q5. A passenger, while boarding the plane, slipped from the stairs and got hurt. The pilot took the passenger in the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passengers could catch the connecting flight, the speed of the plane was increased by 250 km/hour than the usual speed. Find the usual speed of the plane. What value is depicted in this question?