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GSEB std 10 science solution for Gujarati check Subject Chapters Wise::

Evalute the following : (5+√2i)/(1-√2i)

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જવાબ : 1 + 2√2i

Evalute the following : (11-4i-21+i)(1-4i5+i)

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જવાબ : 307/442 + 599/442 i

Evalute the following : 3-4i/(4-2i)(1+i)

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જવાબ : ¼ - ¾ i

Evalute the following : (1+2i)^-3

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જવાબ : −11/125 + 2i/125

Evalute the following : (1-i)³/1-i³

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જવાબ : -2+0i

Find the value of the following : i⁴⁵⁷

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જવાબ : i

Find the value of the following : i⁵²⁸

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જવાબ : 1

Find the value of the following : 1/i⁵⁸

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જવાબ : -1

Find the value of the following : i³⁷+ 1/i⁶⁷

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જવાબ : 2i

Find the value of the following : (i⁴¹ + 1/i²⁵⁷)⁹

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જવાબ : 0

Find the value of the following : (i⁷⁷ + i⁷⁰+ i⁸⁷+ i⁴¹⁴)³

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જવાબ : -8

Find the value of the following : ³⁰+ i⁴⁰+ i⁶⁰

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જવાબ : 1

Find the value of the following : i⁴⁹+i⁶⁸+i⁸⁹+i¹¹⁰

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જવાબ : 2i

Evalute the following : (1+i)(1+2i)

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જવાબ : −1+3i

Evalute the following : (3+2i)/(-2+i)

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જવાબ : −4/5 – 7/5 i

Evalute the following : 1/(2+i)²

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જવાબ : 3/25 – 4/25 i

Evalute the following : (1-i)/(1+i)

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જવાબ : -i

Evalute the following : (2+i)³/2+3i

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જવાબ : 37/13 + 16/13 i

Evalute the following : (1+i)(1+√3i)/(1-i)

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જવાબ : −√3+i

Evalute the following : (2+3i)/(4+5i)

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જવાબ : 23/41 + 2/41 i

Find the values of x and y : (1+i)(x+iy)=2-5i

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જવાબ : x=−32 and y=−72

Find the values of x and y : [(1+i)x-2i]/(3+i) + [(2-3i)y+i]/(3-i)

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જવાબ : x=3 and y=−1

Find the values of x and y : (3x-2iy)(2+i)2=10(1+i)

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જવાબ : x=14/15 y=15

Find the values of x and y : (x+iy)(2-3i)=4+i

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જવાબ : x=5/13 and y=14/13

Find the conjugates of complex numbers : (3-2i)(2+3i)/(1+2i)(2-i)

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જવાબ : (63+16i)/25

Find the conjugates of complex numbers : (1+i)(2+i)/(3+i)

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જવાબ : 3−4i/5

Find the conjugates of complex numbers : (3-i)2/2+i

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જવાબ : 2+4i

Find the conjugates of complex numbers : 1/1+i

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જવાબ : (63+16i)/25

Find the conjugates of complex numbers : 1/(3+5i)

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જવાબ : (3+5i)/34

Find the conjugates of complex numbers : 4 − 5 i

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જવાબ : 4+5i

Find the multiplicative inverse of the complex numbers : √5+3i

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જવાબ : √5/14 – 3/14

Find the multiplicative inverse of the complex numbers : 4 − 3i

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જવાબ : 4/25 + 3/25i

Find the multiplicative inverse of the complex numbers : (1+i√3)²

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જવાબ : −1/8 − 3/√8 i

Find the multiplicative inverse of the complex numbers : 1 − i

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જવાબ : ½ + ½ i

True and False : Region represented by |z+7i|≤5 is On or inside the circle having centre (0, -7) and radius 5 units

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જવાબ : True

True and False : Region represented by |z+7|≤5 is On or inside the circle having centre (–7, 0) and radius 5 units

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જવાબ : True

True and False : The cube root of the unit when represented on the Argand plane form the vertices of an
equilateral triangle

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જવાબ : True

True and False : If three complex numbers are in A.P, then they lie on a circle in the complex plane

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જવાબ : False

Solve the following : i⁶

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જવાબ : -1

Solve the following : i⁵

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જવાબ : i

Solve the following : i^-2

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જવાબ : -1

Solve the following : i^-3

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જવાબ : -i

Solve the following : i^-4

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જવાબ : 1

Solve the following : i^-5

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જવાબ : i

Solve the following : i^-6

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જવાબ : -1

Solve the following : i^-7

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જવાબ : -i

Solve the following : i²

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જવાબ : -1

Solve the following : i²

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જવાબ : -1

Solve the following : i³

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જવાબ : -i

Solve the following : i⁴

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જવાબ : 1

Find the values of i⁸+ i¹⁰ + i¹⁵

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જવાબ : i⁸+ i¹⁰+ i¹⁵ =i⁴ + i^4×2+2 + i^4×3+3 =(i⁴)²+ {(i⁴)²×i²} + {(i⁴)³×i³} =1+i²+i³

[∵i⁴=1] =1−1−i

[∵i²=−1, i³=−i ] =−i

Find the values of i⁶ + i³ + i⁴+ i⁵

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જવાબ : i⁶ + i³+ i⁴+ i⁵= −1 –I +1 + i [∵i2=−1, i3=−i and i4=1]

Find the values of i⁴⁴ + i⁸⁰ + i¹²⁰

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જવાબ : i⁴⁴+ i⁸⁰ +i¹²⁰ = i^4×11 + i^4×20 + i^4×30 = {(i⁴)¹¹} + {(i⁴)²⁰} + {(i⁴)³⁰} = 1+1+1

[∵i⁴=1]=3

Find the values of i⁴⁸ + i⁶⁸ + i⁸⁹ + i¹¹⁴

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જવાબ : i⁴⁸+i⁶⁸+i⁸⁹+i¹¹⁴ =i^4×12 + i^4×17 + i^4×22+1 + i^4×28+2 ={(i⁴)¹²} + {(i⁴)¹⁷} + {(i⁴)²²×i} + {(i⁴)²⁸×i²} =1+1+i+i² [∵i⁴=1] =i+2−1 [∵i²=−1] = i+1

Show that i⁹ + i²⁰ + i³⁴ is a real number.

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જવાબ : I⁹+ i²⁰+ i³⁴ = i^4×2+1+ i^4×5+ i^4×8+2 = [(i⁴)²×i] +(i⁴)⁵ + [(i⁴)⁸×i²] = i+1+i² ( ∵i⁴=1)

= i + 1 −1 (∵ i²=−1)

This isn’t a real number.

Show that i⁹ + i²⁴ is a real number.

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જવાબ : I⁹+ i²⁴ = i^4×2+1+ i^4×6= [(i⁴)²×i] +(i⁴)⁶ = i+1 ( ∵i⁴=1)

= i + 1

This isn’t a real number.

Show that i¹⁴ + i²⁰ is a real number.

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જવાબ : i¹⁴+ i²⁰ = i^4×3+2+ i^4×5= [(i⁴)³×i²] +(i⁴)⁵ = i²+1 ( ∵i⁴=1)

= −1 + 1 (∵ i²=−1)

This is a real number.

Show that i¹⁰ + i²⁴ + i³⁰ is a real number.

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જવાબ : i¹⁰+ i²⁴+ i³⁰ = i^4×2+2+ i^4×6+ i^4×7+2 = [(i⁴)²×i²] +(i⁴)⁶ + [(i⁴)⁷×i²] = i²+1+i² ( ∵i⁴=1)

= −1 + 1 −1 (∵ i²=−1)

=-1

This is a real number.

If [(1-i)/(1+i)]⁹⁸= a+ib, find (a, b).

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જવાબ : (1-i)/(1+i) = (1-i)/(1+i) × (1-i)/(1-i)

=(1-i)²/1²-i²

=1²+i²-2i/(1+1) [∵ i²=-1]

=(1-1-2i)/2

=-2i/2

=-i ....(1)

It is given that,
[(1-i)/(1+i)]¹⁰²=a+ib

⇒(-i)¹⁰²= a+ib [From (1)]

⇒i^4×24+2 = a+ib

⇒-1 + 0i =a+ib [∵ i⁴=1]

⇒a=-1 and b=0

Thus, (a, b) = (-1, 0).

If [(1-i)/(1+i)]⁹⁶= a+ib, find (a, b).

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જવાબ : (1-i)/(1+i) = (1-i)/(1+i) × (1-i)/(1-i)

=(1-i)²/1²-i²

=1²+i²-2i/(1+1) [∵ i²=-1]

=(1-1-2i)/2

=-2i/2

=-i ....(1)

It is given that,
[(1-i)/(1+i)]⁹⁶=a+ib

⇒(-i)⁹⁶= a+ib [From (1)]

⇒i^4×24 = a+ib

⇒1 + 0i =a+ib [∵ i⁴=1]

⇒a=1 and b=0

Thus, (a, b) = (1, 0).

Find the values of i⁴ + i¹⁰ + i¹⁵

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જવાબ : i⁴+ i¹⁰+ i¹⁵ =i⁴ + i^4×2+2 + i^4×3+3 =i⁴+{(i⁴)²×i²} + {(i⁴)³×i³} =1+i²+i³ [∵i4=1] =1−1−i [∵i²=−1, i³=−i ] =−i

Find the values of i² + i³ + i⁴+ i⁵

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જવાબ : i² + i³+ i⁴+ i⁵= −1 –I +1 + i [∵i2=−1, i3=−i and i4=1]

Find the values of i⁴⁰ + i⁸⁰ + i¹²⁰

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જવાબ : i⁴⁰+ i⁸⁰ +i¹²⁰ = i^4×10 + i^4×20 + i^4×30 = {(i⁴)¹⁰} + {(i⁴)²⁰} + {(i⁴)³⁰} = 1+1+1 [∵i⁴=1]=3

Find the values of i⁴⁸ + i⁶⁸ + i⁸⁹ + i¹¹⁰

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જવાબ : i⁴⁸+i⁶⁸+i⁸⁹+i¹¹⁰ =i^4×12 + i^4×17 + i^4×22+1 + i^4×27+2 ={(i⁴)¹²} + {(i⁴)¹⁷} + {(i⁴)²²×i} + {(i⁴)²⁷×i²} =1+1+i+i² [∵i⁴=1] =i+2−1 [∵i²=−1] = i+1

Show that i⁹ + i²⁰ + i³⁰ is a real number.

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જવાબ : I⁹+ i²⁰+ i³⁰ = i^4×2+1+ i^4×5+ i^4×7+2 = [(i⁴)²×i] +(i⁴)⁵ + [(i⁴)⁷×i²] = i+1+i² ( ∵i⁴=1)

= i + 1 −1 (∵ i²=−1)

This isn’t a real number.

Show that i⁹ + i²⁰ is a real number.

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જવાબ : I⁹+ i²⁰ = i^4×2+1+ i^4×5= [(i⁴)²×i] +(i⁴)⁵ = i+1 ( ∵i⁴=1)

= i + 1

This isn’t a real number.

Show that i¹⁰ + i²⁰ is a real number.

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જવાબ : i¹⁰+ i²⁰ = i^4×2+2+ i^4×5= [(i⁴)²×i²] +(i⁴)⁵ = i²+1 ( ∵i⁴=1)

= −1 + 1 (∵ i²=−1)

This is a real number.

Show that i¹⁰ + i²⁰ + i³⁰ is a real number.

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જવાબ : i¹⁰+ i²⁰+ i³⁰ = i^4×2+2+ i^4×5+ i^4×7+2 = [(i⁴)²×i²] +(i⁴)⁵ + [(i⁴)⁷×i²] = i²+1+i² ( ∵i⁴=1)

= −1 + 1 −1 (∵ i²=−1)

=-1

This is a real number.

If [(1-i)/(1+i)]¹⁰²= a+ib, find (a, b).

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જવાબ : (1-i)/(1+i) = (1-i)/(1+i) × (1-i)/(1-i)

=(1-i)²/1²-i²

=1²+i²-2i/(1+1) [∵ i²=-1]

=(1-1-2i)/2

=-2i/2

=-i ....(1)

It is given that,
[(1-i)/(1+i)]¹⁰²=a+ib

⇒(-i)¹⁰²= a+ib [From (1)]

⇒i^4×25+2 = a+ib

⇒-1 + 0i =a+ib [∵ i⁴=1]

⇒a=-1 and b=0

Thus, (a, b) = (-1, 0).

If [(1-i)/(1+i)]¹⁰⁴= a+ib, find (a, b).

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જવાબ : (1-i)/(1+i) = (1-i)/(1+i) × (1-i)/(1-i)

=(1-i)²/1²-i²

=1²+i²-2i/(1+1) [∵ i²=-1]

=(1-1-2i)/2

=-2i/2

=-i ....(1)

It is given that,
[(1-i)/(1+i)]¹⁰⁴=a+ib

⇒(-i)¹⁰⁴= a+ib [From (1)]

⇒i^4×26 = a+ib

⇒1 + 0i =a+ib [∵ i⁴=1]

⇒a=1 and b=0

Thus, (a, b) = (1, 0).

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1	√64	A	-8
2	√-64	B	8
3	-√64	C	8i

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જવાબ :

1-B,2-C,3-A

1	√49	A	-7
2	√-49	B	7i
3	-√49	C	7

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જવાબ :

1-C, 2-B, 3-A

1	√25	A	6
2	√-25	B	6i
3	√-36	C	5i
4	√36	D	5

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જવાબ :

1-D, 2-A, 3-B, 4-C

1	√196	A	-16
2	√-196	B	16i
3	-√196	C	16

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જવાબ :

1-C, 2-A, 3-B

1	√4	A	12
2	√-4	B	12i
3	√-144	C	2i
4	√144	D	2

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જવાબ :

1-D, 2-C, 3-B, 4-A

1	I⁹	A	1
2	I¹⁰	B	-i
3	I¹¹	C	i
4	I¹²	D	-1

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જવાબ :

1-C, 2-D, 3-B, 4-A

1	I⁶	A	-1
2	I⁷	B	-i
3	I⁸	C	1

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જવાબ :

1-C, 2-A, 3-B

1	I^-2	A	i
2	i^-1	B	-1
3	I^-3	C	1
4	i^-4	D	-i

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જવાબ :

1-B, 2-C, 3-D, 4-A

1	i²	A	1
2	I³	B	I
3	I⁴	C	-1
4	I⁵	D	-i

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જવાબ :

1-C, 2-D, 3-A, 4-B

1	i²⁰	A	-i
2	i²¹	B	-1
3	i²²	C	I
4	i²³	D	1

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જવાબ :

1-D, 2-C, 3-B, 4-A

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GSEB std 10 science solution for Gujarati check Subject Chapters Wise::

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