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Complex Number And Binomial Theorem

Complex Number And Binomial Theorem - Mathematics Questions asked by Filo students

Question 1

7. A right triangle has hypotenuse of length

p cm

and one side of length

q cm

. If

p - q = 1

, then the length of the third side of the triangle is :

Topic:

Complex Number and Binomial Theorem

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Question 2

\frac{( x + i ) ^{2}}{x - i} + \frac{( x - - 1 ) ^{2}}{( x + i )}

10.

\frac{( a + i ) ^{3} - ( a - i ) ^{3}}{( a + i ) ^{2} - ( a - i ) ^{2}}

Find the value of

(- i)^{4 n + 3}

; when

n

is a positive integer. Find the square of

9 + 40 i + 9 - 40 i

. Express in the form

a + ib

$\frac{3 + 5 i}{2 - 3 i}$
$\frac{3 - i 2}{2 3 - i 3}$
$\frac{( 1 + i ) ^{2}}{3 - i}$
$\frac{( a + ib ) ^{3}}{a - ib} - \frac{( a - ib ) ^{3}}{a + ib}$
$\frac{1 + i}{1 - i}$

Topic:

Complex Number and Binomial Theorem

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Question 3

1, \infty, ω^{2}

are the three cube roots of unity, prove that (1.

(1 + ω^{2})^{4} = ω

18.

(1 - ω + ω^{2}) (1 + ω - ω^{2}) = 4

(4.

(1 - ω) (1 - ω^{2}) (1 - ω^{4}) (1 - ω^{5}) = 9

12.

(2 + 5 ω + 2 ω^{2})^{6} = (2 + 2 ω + 5 ω^{2})^{6} = 729

21.

(1 - ω + ω^{2}) (1 - ω^{2} + ω^{4}) (1 - ω^{4} + ω^{2}) \dots

2 n

factors

= 2^{2 n}

. 12. Prove that

x^{3} + y^{3} + z^{3} - 3 x yz = (x + y + z) (x + ω y + ω^{2} z) (x + y x

3. If

x = a + b, y = aω + b ω^{2}, z = a ω^{2} + bω

, show that (1)

x yz = a^{3} + b^{3}

(2)

x^{2} + y^{2} + z^{2} = 6 ab

Topic:

Complex Number and Binomial Theorem

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Question 4

1, \infty, ω^{2}

are the three cube roots of unity, prove that (1.

(1 + ω^{2})^{4} = ω

18.

(1 - ω + ω^{2}) (1 + ω - ω^{2}) = 4

(4.

(1 - ω) (1 - ω^{2}) (1 - ω^{4}) (1 - ω^{5}) = 9

12.

(2 + 5 ω + 2 ω^{2})^{6} = (2 + 2 ω + 5 ω^{2})^{6} = 729

21.

(1 - ω + ω^{2}) (1 - ω^{2} + ω^{4}) (1 - ω^{4} + ω^{2}) \dots

2 n

factors

= 2^{2 n}

. 12. Prove that

x^{3} + y^{3} + z^{3} - 3 x yz = (x + y + z) (x + ω y + ω^{2} z) (x + y x

3. If

x = a + b, y = aω + b ω^{2}, z = a ω^{2} + bω

, show that (1)

x yz = a^{3} + b^{3}

(2)

x^{2} + y^{2} + z^{2} = 6 ab

Topic:

Complex Number and Binomial Theorem

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Question 5

If the term independent of

x

in the expansion of

(\frac{3}{2} x^{2} - \frac{1}{3 x})^{9}

k

, then

18 k

is equal to

Topic:

Complex Number and Binomial Theorem

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Question 6

The difference between lenght and breadth of a perimeter is

106 m

, then Hength,

l = 40

length, $l = 45$
breadth, b $= 18$

Topic:

Complex Number and Binomial Theorem

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Question 7

Comprehension Type:

A BC D

is a trapezium in which

A B ∣∣ D C

and

D C = 40 cm

and

A B = 60 cm

. If

X

and

Y

are, the midpoints of

A D

and

BC

, then 7.

X Y =

$40 cm$
$50 cm$
$60 cm$
$30 cm$

Topic:

Complex Number and Binomial Theorem

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Question 8

ar (trap \cdot D C Y X) = k ar (trap.XYBA)

. Then value of

k

$1/2$
2
$1/9$
$9/11$

Topic:

Complex Number and Binomial Theorem

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Question 9

A = {x ∣ x

is a prime number,

x < 10}, B = {x ∣ x

is an even number,

x < 10}

dray Ch

1 - 24

2. If

5 = {x ∣ x \in N, x < 10}

. (v)

n (A^{'}

diagram to find: (ii)

n (A \cap B)

(iii)

n (A \cup B)^{'}

(iv)

n (A \cap B^{'})

information. Hence find

n (A - B)

. 4. The Venn diagram shows:

ξ = {

pupils in class 8

}

A =

\{pupils who play cricket\}

B = {

pupils who play basketball\} (i) are in class

8 ?

(ii) play cricket (iii) play basketball? How many pupils: (iv) play both cricket and basketball (v) play neither cricket nor basketball? 5. In the Venn diagram

ξ = {

people at a function

}

A = {

those who asked for tea

}

B = {

those who watched the ballet

}

Write down the number who: (iii) neither asked for tea nor watched the ballet (iv) attended the function (i) asked for tea 6. In a group of 30 people, 18 play squash and 19 play tennis. How many play both everyone plays at least one game? 7. In a class of 50 students, 22 like History, 25 like Geography and 10 like both a Venn diagram and find the number of students who (i) do not like History (ii) do not like Geography (iii) like neither History nor Geography 8. 2000 candidates appear in a written test in Mathematics and Gerneral Government job. 1800 passed in at least one subject. If 1200 passed in 1 1500 in General Awareness find: (i) how many passed in both the subjects? (ii) how many passed in Mathematics only? (iii) how many failed in General Awareness? 9. In a group of 80 people, 40 like Indian food, 36 like Chinese food and kind of these foods. Draw Venn diagram to find: (i) how many like both kind of food? (ii) how many like only the Indian food? (iii) how many like only the Chinese food? [Hint: Let

A

and

B

be the sets of those people who like Indian and Chine

Topic:

Complex Number and Binomial Theorem

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Question 10

y_{1} + y_{2} + y_{3} + \dots\dots\dots + y_{7} = 10 - 1 = 9, y_{i} \geq 0

for

1 \leq i \leq 7

. So, the required number of solutions,

= C (7 + 9 - 1, 9) = C (15, 9) = 5005 .

Q. 28. Use Binomial theorem to prove that. (i)

3^{n} = \sum_{r = 0}^{n} C (n, r) \cdot 2^{n}

(ii)

C (n, 1) + 2 C (n, 2) + \dots\dots + n C (n, n) = n . 2^{n - 1}

Topic:

Complex Number and Binomial Theorem

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Question 11

z

is a complex number, then the minimum value of

∣ z ∣ + ∣ z - 1∣ is ....?

Topic:

Complex Number and Binomial Theorem

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Question 12

14% Example 10. If z = cos 0 + i sin 0, then find the value of 12 1-Z

Topic:

Complex Number and Binomial Theorem

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Question 13

34 ESSAY TYPE QUESTIONS 1. Find the determinants of the following matrices.

A = (21 7 - 5)

Ans: Det A = a d - b c = 2 (- 5) - 1 = - 10 - 1 = 11

(i 0 0 - i)

Ans:

D

A = - p^{2} - 0 = 1 - 0 = 1

(1 - 3 - 1 1)

Ans:

Det A = 1 (1) - (- 1) (- 3)

= 1 - 3

= - 2

⎝ ⎛ 20 - 3 - 1 - 2 1 453 ⎠ ⎞

Ans : D et A A = 2 (- 2 - 3) + 1 (0 \times 3 + 3 \times 5) - 4 (0 \times 1 - 6) = 2 (- 5) + 1 (15) - 4 (- 6) = - 10 + 15 + 24 = 19

Topic:

Complex Number and Binomial Theorem

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Question 14

There are 100 tickets in a bag numbered 1 through 100 and a ticket is picked at random. Let

A

be the event that the number on the ticket is divisible by 2 and let

B

be the event that number on the ticket is divisible by 5 . Show that

A

and

B

are independent.

8 x - 5

salh

the sample space

S

has

6^{2}

Topic:

Complex Number and Binomial Theorem

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Question 15

\frac{∣ z - z _{1} ∣}{∣ z - z _{2} ∣} = k, k \neq = 0, 1

We have

\frac{A P}{BP} = k

We have two positions of point

P (z)

P_{1} (z^{'})

and

P_{2} (z^{''})

. In the ratio

k : 1, P_{1}

and

P_{2}

divide

A B

internally and externally, respectively. Now, we have another position of

P

in the plane as

P_{3} (z^{'''})

such that

\frac{A P _{3}}{B P _{3}} = k

. So, from the property of angle bisector, in triangle

A P_{3} B

P_{3} P_{1}

is angle bisector of

∠ A P_{3} B

. Also,

P_{3} P_{2}

is external angle bisector of

∠ A P_{3} B

. Now,

P_{3} P_{1} ⊥ P_{3} P_{2}

(as two bisectors are always perpendicular). So, points

P_{1}, P_{2}

and

P_{3}

lie on the circle having

P_{1} P_{2}

as a diameter. Thus,

z

lies on the circle. If

k > 1, z_{2}

lies inside the circle. If

k < 1, z_{1}

lies inside the circle. Diameter of the circle

= P_{1} P_{2} = P_{1} B + B P_{2}

∴ Radius = \frac{∣ z _{1} - z _{2} ∣}{k + 1} + \frac{∣ z _{1} - z _{2} ∣}{k - 1} = \frac{2 k ∣ z _{1} - z _{2} ∣}{k ^{2} - 1}, (k > 1) = \frac{k ∣ z _{1} - z _{2} ∣}{k ^{2} - 1}

Also,

z^{'} = \frac{k z _{2} + z _{1}}{k + 1}

and

z^{''} = \frac{k z _{2} - z _{1}}{k - 1}

. Thus, centre is mid-point of

P_{1} P_{2}

given by

\frac{k ^{2} z _{2} - z _{1}}{k ^{2} - 1}

. ILLUSTRATIDN 3.77

Topic:

Complex Number and Binomial Theorem

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Question 16

Subject Wise syluabus plan

Topic:

Complex Number and Binomial Theorem

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Question 17

Find numcrical greatest and algeibic greated coeff of

x

(3 - 5 x)^{18}

Topic:

Complex Number and Binomial Theorem

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Question 18

ILLUSTRATION 3.6 If

z = 4 + i 7

, then find the value of

z^{3} - 4 z^{2} - 9 z + 91

. Sol.

z = 4 + i 7

z - 4 = i 7

z^{2} - 8 z + 16 = - 7

(Squaring or

z^{2} - 8 z + 23 = 0

Now dividing

z^{3} - 4 z^{2} - 9 z + 91

z^{2} - 8 z + 23

, we get

z^{3} - 4 z^{2} - 9 z + 91 = (z^{2} - 8 z + 23) (z + 4) - 1 = - 1

Topic:

Complex Number and Binomial Theorem

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Question 19

निम्न में कौन रिक्त समुच्चयं का उदाहरण है। (a)

A = {x : x

एक सम अभाज्य संख्या है, जो 2 से बड़ी है।

}

(b)

B = {x : x

एक सम अभाज्य संख्या है।

}

(c)

C = {x : x^{2} - 2 = 0

एवं

x

एक वास्तविक संख्या है।

}

(d)

D = {x : x + 2 = 5

एवं

x

एक प्राकृत संख्या है।

}

2. The given statement

{x : x \in R, - 2 < x < 5}

is also written as दिए गए कथन

{x : x \in R, - 2 < x < 5}

को किस प्रकार से लिख सकते हैं (a)

x \in [- 2, 5]

(b)

x \in [- 2, 5]

(c)

x \in (- 2, 5)

(d)

x \in [- 2, 5)

3. Let

A = {1, 2, {3; 4}, 5, 6

} w hi c h o f t h es t a t e m e n t s a re in correc t

यदि

A = {1, 2, {3, 4}, 5, 6

} होनिम्नमेंकौनगलतविकल्पहै

(a)

{3, 4} \in A

(b)

5 \in A

(c)

{{3, 4}} \in A

(d)

{1, 2, 5, 6} CA

4. If

R

is the set of real numbers and

Q

is the set of rational numbers, then what is

R - Q

? (a)

N

(Set of natural numbers) (b)

Z

(set of integers) (c)

W

(Set of whole numbers) (d)

T

(Set of irrational numbers) यदि

R

वास्तविक संख्याओं का समुच्चय हैं एवं

Q

परिमेय संख्याओं का समुच्चय है तो

R - Q

क्या होगा? (a)

N

(प्राकृत संस्थाओं का समुच्चय) (b) Z (पूर्णाक संख्याओं का समुच्चय (c)

W

(पूर्ण संख्याओं का समुच्चय (d) T (अपरिमेय संख्याओं का समुच्चय 5. If

A \times B = {(- 1, 0), (1, 0), (1, 3), (2, 4), (3, 5)}

, then find

A

. यदि

A \times B = {(- 1, 0), (1, 0), (1, 3), (2, 4), (3, 5)}

, तो

A

ज्ञात करें। (a)

{- 1, 1, 23}

(b)

{0, 3, 4, 5}

(c)

{- 1, 0, 1, 2, 3, 4, 5}

(d)

{1, 2, 3, 4, 5}

6. If

A = {x, y, z}

and

B = {a, b}

How many subsets will

A \times B

have (a) 6 (b) 64 (c) 32 (d) None of these यदि

A = {x, y, z}

और

B = {a, b}

हो तो

A \times B

में उपसम्मूच्वयों की संख्या कितनी होगी? (a) 6 Lb) 64 (c) 32 (d) इनमें से कोई नहीं 7. The domain of real function

f (x) = (x - 3)

is वास्तविक फलन

f (x) = (x - 3)

का प्रांत है- (a)

(- \infty, 3)

(b)

(3, \infty)

(e)

(3, \infty)

(d)

(- \infty, - 3)

8. The value of

Sin 3 0^{\circ} cos 1 5^{\circ} + cos 3 0^{\circ} Sin 1 5^{\circ}

Sin 3 0^{\circ} cos 1 5^{\circ} + cos 3 0^{\circ} Sin 1 5^{\circ}

का मान होगा (a)

\frac{1}{2}

(b) 0 (c) (d)

\frac{1}{2}

9. The value of

tan (\frac{19 π}{3})

tan (\frac{19 π}{3})

कां मान है (a)

3

(b)

- 3

(c)

\frac{1}{3}

(d)

- \frac{1}{3}

10. The Principal value of the equation

cosec x = - 2

are - (a)

\frac{5 π}{6}, \frac{7 π}{6}

(b)

\frac{2 π}{6}, \frac{21 π}{6}

(c)

\frac{π}{6}, \frac{7 π}{6}

(d) None of these

Topic:

Complex Number and Binomial Theorem

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Question 20

29. The smallest positive integer '

n

' for which

(\frac{1 + i}{1 - i})^{n} = 1 is

Topic:

Complex Number and Binomial Theorem

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1 2 3 4 5 6

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