Question

1. in a class test the sum of shefali's marks in mathematics and English is 30 . Had she got 2 marks more in mathematics and 3 marks less in English, the product of their marks would have been 210. find her marks in the two subject.

2. the diagonal of a rectangular field is 60 metres more than the shorter side. if the longer side is 13 more than the shorter side ,find the sides of the field.

3. the difference of squares of two number is 180. the square of the smaller number is 8 times the larger number. Find two numbers.

Answer 1

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(1)

Let take Shefali’s marks in Mathematics = x

Given that

the sum of Shefali’s marks in Mathematics and English is 30

so that marks of Shefali’s in English = 30 – x
given that .

she got 2 marks more in Mathematics and 3 marks less in English,

so that marks in mathematics = x +2
marks in English = 30 – x -3 = 27 – x

the product of their marks would have been 210.

So that

(x+2)(27-x) = 210
27 x - x2 + 54 – 2x = 210
25 x - x2 + 54 – 210 = 0
- x2 + 25 x – 156 =0

Change the sign we get
x2-25x +156 =0

x2-25x +156 =0
x2 – 12 x – 13 x + 156 = 0
x(x-12) -13(x-12) =0
(x-12)(x-13)=0

x-12= 0 then x = 12
x-13 = 0 then x = 13

if Shefali marks in mathematics = 12
then marks in English = 30 – 12 = 18

if marks of Shefali in mathematics is 13
then marks in English = 30 – 13 = 17

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(2)
Let the shorter side of the rectangular field be 'x' meters.

Therefore the longer side will be (x + 30) meters.

And the length of the diagonal will be (x + 60) meters.

Now, according to the question

The diagonal divides the rectangular into two right angled triangles and the diagonal is the common side of the two triangles and it is also the longest side of the triangles i.e. the hypotenuse.

So, by Pythagoras Theorem,

(Diagonal)² = (Smaller Side)² + (Longer Side)²

(x + 60)² = (x)² + (x + 30)²
x² + 120x + 3600 = x² + x² + 60x + 900
x² + 60x - 120x + 900 - 3600 = 0
x² - 60x - 2700 = 0
x² - 90x + 30x - 2700 = 0
x(x - 90) + 30(x - 90) = 0
(x - 90) (x + 30) = 0

x = 90 because x = - 30 as length cannot be possible.

So the length of the shorter side is 90 meters and the length of the longer side is 90 + 30 = 120 meters.

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(3)

Let the larger number = x

Then the square of the smaller number = 8

times the larger number = 8x

and the square of the larger numbe r = x

According to the question,

x - 8x = 180
=> x - 8x - 180 = 0
=> x - 18x + 10x - 180 = 0
=> x(x - 18) + 10(x - 18) = 0
=> (x - 18) (x + 10) = 0
=> x - 18 = 0 or x + 10 = 0
=> x = 18 or x = -10

Thus, the larger number = 18 or -10

Then, the square of the smaller number = 8(18) or 8(-10)
= 144 or -80

The square of a number can't be negative, so, the square of smaller number = 144

Hence,

the smaller number = sqrt(144) = 12
The numbers are 12 and 18

hope it will help you

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