ASSIGNMENT

1. CLASS X1
ASSIGNMENT CHAPTERS – 1,3,4,5
1. Write the set {
𝟏
𝟐
,
𝟐
𝟑
,
𝟑
𝟒
,
𝟒
𝟓
,
𝟓
𝟔
,
𝟔
𝟕
,
𝟕
𝟖
,
𝟖
𝟗
,
𝟗
𝟏𝟎
} in the set – builder form.
2. Writ the set Y = {𝟏,
𝟏
𝟒
,
𝟏
𝟗
,
𝟏
𝟏𝟔
,
𝟏
𝟐𝟓
,… …… …… ..} in the set – builder form.
3. Find sets A, B and C such that A ∩ B, A ∩ C and B ∩ C are non- empty sets and A ∩ B ∩ C = ∅ .
4. Give an examples of three sets A,B and C such that A ∩ B= A ∩ C but B ≠ C
5. If P(A)=P(B) ,then prove that A=B
6. If for any two sets, A ∪ B = A ∩ B , prove that: A= B
7. In a survey of 60 people, it was found that 25 people read newspaper H,26 read newspaper T, 26 read newspaper
I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) The numbers of people who read at least one of the newspapers.
(ii) The numbers of people who read exactly one newspapers.
8. If cos (𝜶 − 𝜷) + 𝒄𝒐𝒔( 𝜷 − 𝜸) + 𝒄𝒐𝒔( 𝜸− 𝜶) = −
𝟑
𝟐
, Prove that
𝒄𝒐𝒔 𝜶+ 𝒄𝒐𝒔 𝜷+ 𝒄𝒐𝒔 𝜸 = 𝒔𝒊𝒏 𝜶+ 𝒔𝒊𝒏 𝜷+ 𝒔𝒊𝒏 𝜸 = 𝟎.
9. If sin 𝜶 + 𝒔𝒊𝒏 𝜷 = 𝒂 𝒄𝒐𝒔 𝜶+ 𝒄𝒐𝒔 𝜷 = 𝒃, show that
(i) cos (𝜶 + 𝜷) =
𝒃 𝟐−𝒂 𝟐
𝒃 𝟐+𝒂 𝟐
(ii) sin (𝜶 + 𝜷) =
𝟐𝒂𝒃
𝒂 𝟐+𝒃 𝟐
10. Prove that : cos 20o
cos 40 o
cos 60 o
cos 80 o
=
𝟏
𝟏𝟔
.
11. Prove that : 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x.
12. Prove that: cosA cos2A cos4A cos8A =
𝐬𝐢𝐧 𝟏𝟔𝑨
𝟏𝟔 𝒔𝒊𝒏𝑨
13. Find the general solution of 4sinxcosx+ 2sinx + 2cosx+1=0
14. Solve for x: cotx + cosecx=√ 𝟑.
15. Prove by the principal of mathematical induction that for all n ∈ N : 1+ 4+ 7 + .......+(3n – 2) =
𝟏
𝟐
𝒏 (3n – 1)
16. If x and y are any two distinct integers, then prove by mathematical induction that (xn
- yn
) is divisible
by (x – y) for all n ∈ N .
17. If a + bi =
𝒄+𝒊
𝒄−𝒊
, where c is real , prove that : a2
+ b2
= 1 and
𝒃
𝒂
=
𝟐𝒄
𝐜 𝟐−𝟏
18. Solve : 2x2
– (3 + 7i) x – (3 – 9i) = 0
19. Convert in polar form :
𝒊−𝟏
𝒄𝒐𝒔
𝝅
𝟑
+ 𝒊 𝒔𝒊𝒏
𝝅
𝟑
20. Express (1 – 2i)-3
in the standard form a + bi.
21. If (x + iy)1/3
= a + ib, x, y, a b ∈ R. Showthat
𝒙
𝒂
+
𝒚
𝒃
= 4(a2
– b2
) .
22. Find the real values ofx and y , if (3x – 7) + 2iy = -5y + (5 + x)i
23. If x + iy = √
𝒂+𝒊𝒃
𝒄+𝒊𝒅
, 𝒑𝒓𝒐𝒗𝒆 𝒕𝒉𝒂𝒕 ∶ ( 𝒙 𝟐 + 𝒚 𝟐)
𝟐
=
𝒂 𝟐+𝒃 𝟐
𝒄 𝟐+𝒅 𝟐
.
24. Find real 𝜽 such that
𝟓+𝟑𝒊𝒔𝒊𝒏𝜽
𝟏−𝟑𝒊𝒔𝒊𝒏𝜽
is purely real.
25. If 𝜶 𝒂𝒏𝒅 𝜷 are different complex numbers with | 𝜷| = 𝟏, 𝒇𝒊𝒏𝒅 |
𝜷−𝜶
𝟏−ᾱ𝜷
| .
26. Find the square root and polar form of i