This document provides 25 math problems across various topics for students to complete as part of their class assignment. The problems cover topics like sets, trigonometry, complex numbers, and polar forms. Students are asked to perform tasks like writing sets in set-builder form, finding intersections and unions of sets, solving trigonometric equations, working with complex numbers, and evaluating expressions in polar form.

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