Mathematics Test 217

Since the line passes through origin the equation of line is Y=mX,it also passes through the center (3,-1),Y= -1/3(X)

X+3Y=0

Point of intersection of 3x − 4y − 7 = 0 and 2x − 3y − 5 = 0 is (1 , − 1), which is the centre of the circle and radius = 7.

∴ Equation is  (x−1)²+(y+1)²=49

⇒x²+y²−2x+2y−47=0.

Let z₁=a+ib , z₂=c+id 

|z₁+z₂|=|z₁|+|z₂| 

|a+ib+c+ib|=|a+ib|+|c+id| 

(a+c+i(b+d)=sqrt(a²+b²)+sqrt(c²+d²) 

sqrt((a+c)²+(b+d)² = sqrt(a²+b²)+sqrt(c²+d²)......on simplifying this we get....

a=c and b=d 

argz₁=tan^-1|(b/a)|...... argz₂=tan^-1|(d/c)| = tan^-1|b/a| 

argz₁ - argz₂ = 0

Presuming the question actually means

[ ( 1 + √3i ) / [( 1 - √3i ) ⁿ]

i.e. the whole thing raised to the power of n, not just the denominator.

Let w = ( 1 + √3i ) / ( 1 - √3i )

Multiplying numerator and denominator by ( 1 + √3i ) gives

w = ( 1 + √3i )² / [ ( 1 - √3i ) ( 1 + √3i ) ]

  = ( 1 - 3 + 2√3i ) / ( 1 + 3 )

  = ( -2 + 2√3i ) / 4

  = ( -1 + √3i) / 2.

So w is a primitive cubed root of unity.     (*) ... see note below

So wⁿ is an integer

<=> wⁿ = 1

<=> n is a multiple of 3.

Toolbox:

If f(g(x))=x then g is inverse of x and f is inverse of g

Given f is inverse of itself.

⇒f(f(x)) = x

⇒1+α(1+αx) = x

⇒1+α+(α)²x = x

Equating the like terms on either side, we get

1+α=0and(α)²=1

⇒ α = −1.

Y= mx + c; x²= 4ay

Subst y in the equation of parabola

x²= 4a(mx + c) 

x²-4amx-4ac = 0

Since the line touches the parabola at one point The above quadratic will have one root 

b²-4ac = 0

b²= 4ac

16a²m² = 4(1)(-4ac) 

16a²m²= -16ac

c=-am²

We know 7^k, k∈ N, has 1, 3, 9, 7 at the unit's place for

k = 4p, 4p - 1, 4p - 2, 4p - 3 respectively, where p = 1, 2, 3.....

7^m + 7ⁿ will be divisible by 5 if 7^m has 3 or 7 in the unit's place

and 7ⁿ has 7 or 3 in the unit's place or 7^m has 1 or 9 is the unit's place and 7ⁿ has

9 or 1 in the unit's place

∴ For any choice of m, n the digit in the unit's place of 7^m + 7ⁿ is 2, 4, 6, 0 or 8

It is divisible by 5 only when this digit is 0

∴ the required probability = 1/5