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Complex Number And Binomial Theorem

Solution for Class 11, Mathematics, Complex Number And Binomial Theorem

Question 1

समांतर श्रेणी क्रमागत 3 पदों का योगफल 24 है उनकी उनकी मदद की संख्या का मान ज्ञात कीजिए

Topic:

Complex Number and Binomial Theorem

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Question 2

Express the following expression in the form of a + IB 3 + iota root 5 into 3 minus iota √5 / root under 3 + root 2 iota - √3 - √2

Topic:

Complex Number and Binomial Theorem

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Question 3

सिग्मा न३ का मान कितना होगा

Topic:

Complex Number and Binomial Theorem

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Question 4

सिग्मा n स्क्वायर का मान क्या होगा

Topic:

Complex Number and Binomial Theorem

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Question 5

1. री

\overset{z}{ˉ} = - 3 + 5 i

है, धढ़

x

(1 + - 5)

बiि भालड़ सकीजिए = (d) 2 (c)

- 2

(de

3 + 5 i 5 + 3 i

(4)

5 - 31

2. री

x = - 2 - - 5

हह,

ृ 64 \overset{z}{ˉ} =

Q - 2 + - 5

(4)

2 - - 5

(4)

4 (cos \frac{π}{4} - i sin \frac{π}{4})

(2)

- 4 (cos \frac{2}{4} + i \frac{t}{4})

2 + - 5

(4)

- 5 + 21

(9)

3 (cos \frac{π}{4} - i sin \frac{π}{4}) (3 (cos \frac{π}{4} + lin \frac{5}{4})

1 . 64नา ris (8)

2 + 3 i > 1 + 4 i

(4)

3 + 3 i > 6 + 2 i

5 + 9 i > 5 + 6 i

(2)

\frac{π}{2}

Topic:

Complex Number and Binomial Theorem

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Question 6

(5)

1^{3} + 3^{3} + 5^{3} + \dots

n

terms

= n^{2} (2 n^{2} - 1)

(6)

1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots . + n (n + 1) = \frac{n}{3} (n + 1) (n + 2)

(7)

1 \cdot 3 + 3 \cdot 5 + 5 \cdot 7 + \dots .

. ton terms

= \frac{n}{3} (4 n^{2} + 6 n - 1)

Topic:

Complex Number and Binomial Theorem

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Question 7

(i)

\frac{1 + - 3}{2}

এর বর্গমূল नির্ণয় করো। (ii) 2310-এর মযেেে কত বিভিন্ন রকম উৎ্পাদক আছে? (iii)

(1 + x)^{n}

-এর বিন্ত্তির

p

-তম,

(p + 1)

তম এবং

(p + 2)

তম পদের সহগ তিনটি সমান্তর প্রগতিতে থাকলে দেখাও যে,

n^{2} - n (4 p + 1) + 4 p^{2} - 2 = 0

(iv)

9, 9^{\frac{1}{2}}, 9^{\frac{1}{4}}, 9^{\frac{1}{8}} \dots \infty

শ্রেণিটির সমষ্টি नির্ণয় করো। যে-कেाনেा একটি প্নেत উিষ্ত দা(3 : (i)

3 x + y + 2 = 0

সরলরেখার অভিলম্ব আকার

x cos θ + y sin θ = P

হলে,

θ

এবং P-এর মাन नির্ণয় করো।

Topic:

Complex Number and Binomial Theorem

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Question 8

5). Express

\frac{3 + 4 i}{4 - 5 i}

a + ib

form. if

\frac{3 + 2 i s i n ^{4 - 5 i}}{1 - 2 i s i n θ}

is purely real

t

Topic:

Complex Number and Binomial Theorem

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Question 9

Find the value of

x^{3} + 7 x^{2} - x + 16

When x = 1 + 2 i

Topic:

Complex Number and Binomial Theorem

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Question 10

LONG ANSWER TYPE QUESTIONS 8. Find the square root of the following complex numbers (i)

5 + 12 i

(iii)

4 i

(ii)

- 7 - 24 i

(iv)

- i

9. Prove that

\frac{1 - c o s 2 θ + i s i n 2 θ}{1 + c o s 2 θ - i s i n 2 θ} = i tan θ

10. Find the complex number which when multiplied by

5 +

. 11. Change the following complex numbers to cartesian form ANSWERS

Topic:

Complex Number and Binomial Theorem

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Question 11

A Text Book of Algebrai (i)

7 C_{1} +^{8} C_{3} +^{8} C_{5} + \dots + 8 C_{7} = 128

(iii)

(1 + n C_{1} +^{n} C_{2} + n C_{3} + \dots + n C_{n})^{2}

= 1 + 2 n C_{1} + 2^{2 n C_{2}} + 2 n C_{3} + \dots +^{in} C_{in}

Topic:

Complex Number and Binomial Theorem

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Question 12

Reasl

(a^{m})^{n} = a^{m \times n}

JeE Maln 29in June, 2021 Shift-2 (a) Let the coefficients of

x^{- 1}

and

x^{- 1}

in the expansion of

(2 x x^{\frac{1}{3}} - \frac{1}{x ^{3}})^{18}, x > 0

, be

m

and

n

respoctivoly. If

r

is a positivo integor such

m n^{2} =^{15} C_{1}, 2^{1}

, then the value of

r

is equal to

(2 x^{1/5} - \frac{3}{x ^{1/5}})^{15} 3 - 2 k /5 = - 1

Topic:

Complex Number and Binomial Theorem

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Question 13

Maths-Section B Read More If

z_{1}

and

z_{2}

are two complex numbers such that

∣ z_{1} ∣ = ∣ z_{2} ∣ = 1

, then minimum value of

∣ z_{1} + 1 ∣ + ∣ z_{2} + 1 ∣ + ∣ z_{1} z_{2} + 1 ∣

is Answer: Enter your answer here. Back Space

7/8/9

Topic:

Complex Number and Binomial Theorem

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Question 14

Q:- Find the fifth term in expansion of

(1) - 2 x^{3}) \frac{11}{2} (1 - 2 x^{3})^{\frac{1}{2}}

Topic:

Complex Number and Binomial Theorem

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Question 15

Q:- Find the fifth term in expansion of

(1 - 2 x^{3})^{11/2} \frac{15}{2} 5.51

Topic:

Complex Number and Binomial Theorem

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Question 16

Find the moduly - amplitude borm of the conplse number

1 + i

Topic:

Complex Number and Binomial Theorem

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Question 17

Find remainder when

7^{103}

is divided by 25 (18)

7 \times (49)^{51} 7 \times (50 - 1)^{5} 7 \times (\frac{( 99}{025 k} + (- 1)) θ cms^{2}

Topic:

Complex Number and Binomial Theorem

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Question 18

>) Euler's Ideutity and Important Relation (1)

(cos θ + i sin θ) = e^{i (θ)}

(2)

(cos θ - i sin θ) = cos (- θ) + i sin (- θ) = e^{i (- θ)}

(3)

(sin θ + i cos θ) = i (cos θ + \frac{1}{i} sin θ) = i (cos θ - i sin θ) = e^{i (π /2)} i e^{i (- θ)} = e^{i (π /2 - θ)}

(2)

L sin θ - i cos θ) = - i (cos θ + i sin θ) = e^{i (- \frac{π}{2})} \cdot e^{i (θ)} = e^{i (θ - π /2})

Topic:

Complex Number and Binomial Theorem

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Question 19

(b) Find numerically (i.e. in magnitude) greatest term in expansion of

(3 - 5 x)^{15}

x = \frac{1}{5}

\frac{15 - K + 1}{K} \times \frac{1}{3} \times \frac{5 ( x ) ^{e} /5}{K = 1} = 1

Vedantum MASTERTEAST Shy:

T_{y} = TF

Topic:

Complex Number and Binomial Theorem

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Question 20

4 At Gl JEE Main 26th June, 2022 Shift-2 m If (°C%) + ('C1) + (±C2) + ... + (°°C20) = M m and n are coprime, then m + n is equal to. ६ + म पटटा पउ + 60%20 6 60C 20, ७८

Topic:

Complex Number and Binomial Theorem

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Solution for Class 11, Mathematics, Complex Number And Binomial Theorem

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