VACATION ASSIGNMENT

1. Find the conjugate of (3-2i)(2+3i)/(1+2i)(2-i)2. Find the multiplicative inverse of (1-√2i)/(5+√2i) 3. Express -16/(1+i√3) in polar form4. Find the real numbers x and y if (x-iy)(3+5i) is the conjugate of -6-24i.5. If (x+iy)3 = u+iv, show that (u/x)+(v/y) = 4(x2 +y2)6. If (1+i/1-i)m = 1 then find the least positive integral value of m.7. If x-iy = √(a-ib/c-id), Prove that (x2 +y2)2 = a2 +b2/c2+d2

Linear inequalities

1. Rani obtained 70 and 75 marks in first two unit tests. Find the minimum marks she should obtain in the third test to have an average of at least 60 marks.

2. Solve 5−2 x3 ≤

x6−5.

3. Find all pairs of consecutive even positive integers, both of which are larger that 5 such that their sum is less than 23.

4. Solve the following system of linear inequalities graphicallyi) 3x+2y≤150, x+4y ≤ 80, x≤15, y ≥0, x ≥ 0.ii) 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0.

PMI

1. Prove the following by using the principle of mathematical induction for all n∈N .

i. 1+ 1(1+2 )

+ 1(1+2+3)

+…+ 1(1+2+3+..n )

= 2n(n+1 )

.

ii.12.5

+ 15.8

+ 18.11

+…+ 1(3n−1 ) (3n+2 )

= n(6n+4 )

.

iii. n (n+1 ) (n+5 ) is amultiple of 3 .

iv. 102n−1+1is divisible by11 .

v. x2n− y2nis divisible by x+ y .