Aman Kumar Gupta

Let Rehman's present age be x

So before 3 years age=(x-3) years

His age after 5 years= (x+5) years

So a/c to given data

x(x-7)+3(x-7)=0

(x-7)(x+3)=0

So, x=7 or -3

Age can not be negative soRehman's present age is 7 years.

1) x^2-1=3x

x^2-3x-1=0

Apply quadratic formula i.e. {-b±√(b^2-4ac)}/2a

So, roots= {3±√(9+4)}/2 = {3±√13}/2  

So roots are {3+√13}/2 and {3-√13}/2  i.e. and

2)  (x-7-x-4)×30 =11×(x+4)(x-7)

-330=11×(x^2-3x-28)

x^2-3x-28+30=0

X^2-3x+2=0

X^2-2x-x+2=0

x(x-2)-1(x-2)=0

(x-2)(x-1)=0

x=1,2 i.e. ans

Length of semicircle=circumference/2 = 2Πr/2=Πr

There are 13 semicircles with radius

0.5,1,1.5,2,2.5,3,3.5,4,4.5,5,5.5,6,6.5

I.e. in AP whose 1St term=0.5 &

common difference=0.5

Now, total length of spiral=Π(0.5+1+1.5+2+2.5+3+3.5+4+4.5+5+5.5+6+6.5)

As we know sum of an AP is=

no. of terms(1St term+last term)/2= 13(0.5+6.5)/2=13×7/2=91/2

Now put value

Total length of spiral =(22/7)×(91/2)=11×13=143cm i.e. ans

To verify the solution of the question.just put either value of x or y then you all get another's value i.e. y or x or put value of both x and y in the equation you will get the LHS=RHS.

Put x=1

2×1-3y=8

2-3y=8

2-8=3y

y=(-6/3)

y=-2 i.e. given so verified.

Similarly when you put y=-2 you all get x=1.

Now, put both x=1 & y=-2

2×1-3×(-2)=8

2+6=8

8=8

LHS=RHS so verified.