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All questions of Complex Numbers and Quadratic Equation for JEE Exam
Imaginary part of −i(3i + 2) isa)−2b)2c)3d)−3Correct answer is option 'A'. Can you explain this answer?
|
Geetika Shah answered |
(-i)(3i) +2(-i) =-3(i^2)-2i =-3(-1)-2i =3-2i since i=√-1 =3+(-2)i comparing with a+bi,we get b=(-2)
Write in the simplest form: (i)-997- a)-i
- b)1
- c)-1
- d)i
Correct answer is 'A'. Can you explain this answer?
Write in the simplest form: (i)-997
a)
-i
b)
1
c)
-1
d)
i
|
Muskaan Mishra answered |
I^-997= 1/i^997, 1/(i^4)^249 × i, since i^4 = 1, i^4/i= i^3= -i
If z1 and z2 are non real complex numbers such that z1+z2 and z1z2 are real numbers, then- a)
- b)
- c)
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
If z1 and z2 are non real complex numbers such that z1+z2 and z1z2 are real numbers, then
a)
b)
c)
d)
none of these
|
|
Siddharth answered |
since
Z1 + Z2 & Z1Z2 are the real numbers
therefore
Z1 = conjugate of Z2
Z2 = conjugate of Z1
so
option D is correct
according to me
apke according kon sa option sahi h
Find the reciprocal (or multiplicative inverse) of -2 + 5i - a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
Find the reciprocal (or multiplicative inverse) of -2 + 5i
a)
b)
c)
d)
|
|
Gaurav Kumar answered |
-2 + 5i
multiplicative inverse of -2 + 5i is
1/(-2+5i)
= 1/(-2+5i) * ((-2-5i)/(-2-5i))
= -2-5i/(-2)^2 -(5i)^2
= -2-5i/4-(-25)
= -2-5i/4+25
= -2-5i/29
= -2/29 -5i/29
multiplicative inverse of -2 + 5i is
1/(-2+5i)
= 1/(-2+5i) * ((-2-5i)/(-2-5i))
= -2-5i/(-2)^2 -(5i)^2
= -2-5i/4-(-25)
= -2-5i/4+25
= -2-5i/29
= -2/29 -5i/29
Express the following in standard form : 
- a)3+3i
- b)2 + 2i
- c)1 + 2i
- d)0 + 2i
Correct answer is option 'D'. Can you explain this answer?
Express the following in standard form :
a)
3+3i
b)
2 + 2i
c)
1 + 2i
d)
0 + 2i
|
|
Gaurav Kumar answered |
(8 - 4i) - (-2 - 3i) + (-10 + 3i)
=> 8 - 4i + 2 + 3i-10 + 3i
=> 8 + 2 - 10 - 4i + 3i + 3i =>0 + 2i
=> 8 - 4i + 2 + 3i-10 + 3i
=> 8 + 2 - 10 - 4i + 3i + 3i =>0 + 2i
The argument of the complex number -1 – √3- a)π/3
- b)4π/3
- c)-π/3
- d)-2π/3
Correct answer is option 'D'. Can you explain this answer?
The argument of the complex number -1 – √3
a)
π/3
b)
4π/3
c)
-π/3
d)
-2π/3
|
Om Desai answered |
z = a + ib
a = -1, b = -√3
(-1, -√3) lies in third quadrant.
Arg(z) = -π + tan-1(b/a)
= - π + tan-1(√3)
= - π + tan-1(tan π/3)
= - π + π/3
= -2π/3
a = -1, b = -√3
(-1, -√3) lies in third quadrant.
Arg(z) = -π + tan-1(b/a)
= - π + tan-1(√3)
= - π + tan-1(tan π/3)
= - π + π/3
= -2π/3
Find the real numbers x and y such that : (x + iy)(3 + 2i) = 1 + i- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
Find the real numbers x and y such that : (x + iy)(3 + 2i) = 1 + i
a)
b)
c)
d)
|
Suresh Iyer answered |
(x + iy)(3 + 2i) = (1 + i)
x + iy = (1 + i)/(3 + 2i)
x + iy = [(1 + i) * (3 - 2i)] / [(3 + 2i)*(3 - 2i)]
x + iy = (3 + 3i - 2i + 2) / [(3)2 + (2)2]
x + iy = (5 + i)/[ 9 + 4]
= (5 + i) / 13
=> 13x + 13iy = 5+i
13x = 5 13y = 1
x = 5/13 y = 1/13
x + iy = (1 + i)/(3 + 2i)
x + iy = [(1 + i) * (3 - 2i)] / [(3 + 2i)*(3 - 2i)]
x + iy = (3 + 3i - 2i + 2) / [(3)2 + (2)2]
x + iy = (5 + i)/[ 9 + 4]
= (5 + i) / 13
=> 13x + 13iy = 5+i
13x = 5 13y = 1
x = 5/13 y = 1/13
Find the real numbers x and y such that :a)
b)
c)
d)
Correct answer is option 'c'. Can you explain this answer?
|
Hansa Sharma answered |
(x + iy) (3 + 2i)
= 3x + 2xi + 3iy + 3i*y = 1+i
= 3x-2y + i(2x+3y) = 1+i
= 3x-2y-1 = 0 ; 2x + 3y -1 = 0
on equating real and imaginary parts on both sides
on solving two equations
x= 5/13 ; y = 1/13
= 3x + 2xi + 3iy + 3i*y = 1+i
= 3x-2y + i(2x+3y) = 1+i
= 3x-2y-1 = 0 ; 2x + 3y -1 = 0
on equating real and imaginary parts on both sides
on solving two equations
x= 5/13 ; y = 1/13
Express the following in standard form :
- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
Express the following in standard form :
a)
b)
c)
d)
|
Pooja Shah answered |
first write above equation in complex number format , ie using iota
(3-4i) / (2-3i)*(2+3i) / (2+3i) = (6+9i-8i+12) / 13=(18/13)+(i/13)
(3-4i) / (2-3i)*(2+3i) / (2+3i) = (6+9i-8i+12) / 13=(18/13)+(i/13)
For a complex number x + iy, if x > 0 and y < 0 then the number lies in the.- a)Second quadrant
- b)Third quadrant
- c)First quadrant
- d)Fourth quadrant
Correct answer is option 'D'. Can you explain this answer?
For a complex number x + iy, if x > 0 and y < 0 then the number lies in the.
a)
Second quadrant
b)
Third quadrant
c)
First quadrant
d)
Fourth quadrant
|
Naina Sharma answered |
When x > 0 & y < 0, the quadrant contains positive X-axis and negative Y-axis
Therefore, the point lies in the fourth quadrant.
Therefore, the point lies in the fourth quadrant.
If If ω is a non real cube root of unity and (1+ω)9 = a+bω;a,b ∈ R, then a and b are respectively the numbers :- a)1, 0
- b)-1, 0
- c)0, 1
- d)0, -1
Correct answer is option 'B'. Can you explain this answer?
If If ω is a non real cube root of unity and (1+ω)9 = a+bω;a,b ∈ R, then a and b are respectively the numbers :
a)
1, 0
b)
-1, 0
c)
0, 1
d)
0, -1
|
Aryan Khanna answered |
Since w is the cube root of unity.
(1+w)9 =A+Bw
⇒(−w2)9 =A+Bw
⇒−(w3)6 = A+Bw
⇒−1=A+Bw
∴ A= -1 & B=0
(1+w)9 =A+Bw
⇒(−w2)9 =A+Bw
⇒−(w3)6 = A+Bw
⇒−1=A+Bw
∴ A= -1 & B=0
Multiplicative inverse of the non zero complex number x + iy (x,y ∈ R,)- a)
- b)
- c)
- 
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
Multiplicative inverse of the non zero complex number x + iy (x,y ∈ R,)
a)
b)
c)
- d)
none of these
|
Impact Learning answered |
Let u be multiplicative inverse
zu = 1
u = 1/z
u = 1/(x+iy)
Rationalise it
[1/(x+iy)]*[(x-iy)/(x-iy)]
= (x-iy)/(x2+y2)
u = x/(x2+y2) +i(-y)/(x2+y2)
zu = 1
u = 1/z
u = 1/(x+iy)
Rationalise it
[1/(x+iy)]*[(x-iy)/(x-iy)]
= (x-iy)/(x2+y2)
u = x/(x2+y2) +i(-y)/(x2+y2)
- a)- i
- b)1
- c)i
- d)-1
Correct answer is option 'C'. Can you explain this answer?
a)
- i
b)
1
c)
i
d)
-1
|
Raghav Bansal answered |
x = (√3+i)/2
x3 = 1/8(√3+i)3
Apply formula (a+b)3 = a3 + b3 + 3a2b + 3ab2
= (3√3 + i3 + 3*3*i + 3*√3*i2)/8
= (3√3 - i + 9i - 3√3)/8
= 8i/8
= i
x3 = 1/8(√3+i)3
Apply formula (a+b)3 = a3 + b3 + 3a2b + 3ab2
= (3√3 + i3 + 3*3*i + 3*√3*i2)/8
= (3√3 - i + 9i - 3√3)/8
= 8i/8
= i
If points corresponding to the complex numbers z1, z2, z3 and z4 are the vertices of a rhombus, taken in order, then for a non-zero real number k- a)z1 – z3 = i k( z2 –z4)
- b)z1 – z2 = i k( z3 –z4)
- c)z1 + z3 = k( z2 +z4)
- d)z1 + z2 = k( z3 +z4)
Correct answer is option 'A'. Can you explain this answer?
If points corresponding to the complex numbers z1, z2, z3 and z4 are the vertices of a rhombus, taken in order, then for a non-zero real number k
a)
z1 – z3 = i k( z2 –z4)
b)
z1 – z2 = i k( z3 –z4)
c)
z1 + z3 = k( z2 +z4)
d)
z1 + z2 = k( z3 +z4)
|
|
Riya Banerjee answered |
AC = z3 = z1 eiπ
= z1 (cosπ + i sinπ)
= z3 = z1(-1 + i(0))
= z3 = -z1
AC = z1 - z3
BC = z2 - z4
(z1 - z3)/(z2 - z4) = k
(z1 - z3) = eiπ/2(z2 - z4)
(z1 - z3) k(cosπ/2 + sinπ/2) (z2 - z4)
z1 - z3 = ki(z2 - z4)
z1 - z3 = ik(z2 - z4)
= z1 (cosπ + i sinπ)
= z3 = z1(-1 + i(0))
= z3 = -z1
AC = z1 - z3
BC = z2 - z4
(z1 - z3)/(z2 - z4) = k
(z1 - z3) = eiπ/2(z2 - z4)
(z1 - z3) k(cosπ/2 + sinπ/2) (z2 - z4)
z1 - z3 = ki(z2 - z4)
z1 - z3 = ik(z2 - z4)
If P and Q are two sets such that n(P) = 120, n(Q) = 50 and n(P ∪ Q) = 140 then, n(P ∩ Q) is:- a)90
- b)30
- c)20
- d)70
Correct answer is option 'B'. Can you explain this answer?
If P and Q are two sets such that n(P) = 120, n(Q) = 50 and n(P ∪ Q) = 140 then, n(P ∩ Q) is:
a)
90
b)
30
c)
20
d)
70
|
Seelam Yogitha answered |
N(AUB)=N(A)+N(B)-N(A^B)=120 +50-140=30
A point z = P(x,y) lies on the negative direction of y axis, so amp(z) =- a)π
- b)-π
- c)3(π/2)
- d)π/2
Correct answer is option 'C'. Can you explain this answer?
A point z = P(x,y) lies on the negative direction of y axis, so amp(z) =
a)
π
b)
-π
c)
3(π/2)
d)
π/2
|
|
Geetika Shah answered |
P(-x,-y) lies on the 3rd or 4th quadrant
So, according to the question, points lies in 3rd quadrant
i.e. 3(π/2)
So, according to the question, points lies in 3rd quadrant
i.e. 3(π/2)
then a and b are respectively :- a)64 and - 64√3
- b)128 and 128√3
- c)512 and - 512√3
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
then a and b are respectively :a)
64 and - 64√3
b)
128 and 128√3
c)
512 and - 512√3
d)
none of these
|
|
Om Desai answered |
(√3 + i)10 = a + ib
Z = √3 + i = rcosθ + i rsinθ
⇒ √3 = rcosθ i = rsinθ
⇒ (√3)2 + (1)2 = r2cos2θ + r2sin2θ
⇒ 4 = r2
⇒ r = 2
tan = 1/√3
⇒ tan π/6
Therefore, Z = √3 + i = 2(cos π/6 + i sin π/6)
(Z)10 = √3 + i = (2cos π/6 + 2i sin π/6)10
= 210 (cos π/6 + i sin π/6)10
210 (cos 10π/6 + i sin 10π/6)
= 210 (cos(2π - π/3) + i sin(2π - π/3))
= 210 (cos π/3 - i sin π/3)
= 210 (1/2 - i√3/2)
29(1 - i√3)
a = 29 = 512
b = - 29(√3) = -512√3
Z = √3 + i = rcosθ + i rsinθ
⇒ √3 = rcosθ i = rsinθ
⇒ (√3)2 + (1)2 = r2cos2θ + r2sin2θ
⇒ 4 = r2
⇒ r = 2
tan = 1/√3
⇒ tan π/6
Therefore, Z = √3 + i = 2(cos π/6 + i sin π/6)
(Z)10 = √3 + i = (2cos π/6 + 2i sin π/6)10
= 210 (cos π/6 + i sin π/6)10
210 (cos 10π/6 + i sin 10π/6)
= 210 (cos(2π - π/3) + i sin(2π - π/3))
= 210 (cos π/3 - i sin π/3)
= 210 (1/2 - i√3/2)
29(1 - i√3)
a = 29 = 512
b = - 29(√3) = -512√3
If z1 = 2 + i, z2 = 1 + 3i, then Re ( z1 - z2) =- a)ι
- b)1
- c)2 ι
- d)2
Correct answer is option 'B'. Can you explain this answer?
If z1 = 2 + i, z2 = 1 + 3i, then Re ( z1 - z2) =
a)
ι
b)
1
c)
2 ι
d)
2
|
Aditi Basu answered |
1
To find Re(z1 - z2), we first need to subtract z2 from z1:
z1 - z2 = (2i) - (1 + 3i) = 1 - i
Now, to find the real part of this complex number, we simply take the real component, which is 1. Therefore, Re(z1 - z2) = 1.
To find Re(z1 - z2), we first need to subtract z2 from z1:
z1 - z2 = (2i) - (1 + 3i) = 1 - i
Now, to find the real part of this complex number, we simply take the real component, which is 1. Therefore, Re(z1 - z2) = 1.
The amplitude of a complex number is called the principal value amplitude if it lies between.- a)-2π/3 , 2π/3
- b)π , 2π/3
- c)-π , π
- d)-π/2 , π/2
Correct answer is option 'C'. Can you explain this answer?
The amplitude of a complex number is called the principal value amplitude if it lies between.
a)
-2π/3 , 2π/3
b)
π , 2π/3
c)
-π , π
d)
-π/2 , π/2
|
Akshay Kumar Gupta answered |
We know complex number is a form of a straight line so maximum angle of the line is range of Π.
The modulas of the complex number -1 + √3i is
- a)2
- b)4
- c)1
- d)3
Correct answer is option 'A'. Can you explain this answer?
The modulas of the complex number -1 + √3i is
a)
2
b)
4
c)
1
d)
3
|
Shruti Mehta answered |
Method to Solve :
Given the complex number is z=2+3i.
Then,
|z|=√22+32
=√4+9
=√13
.
Then,
|z|=√22+32
=√4+9
=√13
.
- a)none of these
- b)a circle
- c)a straight line
- d)an ellipse
Correct answer is option 'B'. Can you explain this answer?
a)
none of these
b)
a circle
c)
a straight line
d)
an ellipse
|
|
Geetika Shah answered |
This is Equation of a Circle.If S is the set of all real x such that 
is positive, then S contains- a)(– ∞, –3/2)
- b)(–3/2, 1/4)
- c)(–1/4, 1/2)
- d)(–1/2, 3)
Correct answer is option 'A'. Can you explain this answer?
If S is the set of all real x such that is positive, then S contains
is positive, then S containsa)
(– ∞, –3/2)
b)
(–3/2, 1/4)
c)
(–1/4, 1/2)
d)
(–1/2, 3)
|
|
Rishi Raj answered |
Here we go:We get the roots (1/2), 0, -1, and (-1/2).
Let x,y ∈ R, hen x + iy is a non real complex number if- a)y = 0
- b)x ≠ 0
- c)x = 0
- d)y ≠ 0
Correct answer is option 'D'. Can you explain this answer?
Let x,y ∈ R, hen x + iy is a non real complex number if
a)
y = 0
b)
x ≠ 0
c)
x = 0
d)
y ≠ 0
|
Muskaan Chakraborty answered |
I'm sorry, but your question is incomplete. Please provide more information or specify what you would like me to do.
If the roots of the equation kx2 – 3x -1= 0 are the reciprocal of the roots of the equation x2 + 3x – 4 = 0 then K =- a)4
- b)-4
- c)3
- d)-3
Correct answer is option 'A'. Can you explain this answer?
If the roots of the equation kx2 – 3x -1= 0 are the reciprocal of the roots of the equation x2 + 3x – 4 = 0 then K =
a)
4
b)
-4
c)
3
d)
-3
|
KP Classes answered |
∵ x2 + 3x – 4 = 0
or; x2 – 4x + x – 4 = 0
or; x(x – 4) + 1(x – 4) = 0
or; (x – 4)(x + 1) = 0
x = 4; -1
Eqn. having roots 1/2 & 1/−1 = 1/4 & – 1 is.
or x2 – (1/4 – 1) + 1/4(-1) = 0
or x2 + 3/4x – 1/4 = 0
Multiplying by 4 ; we get
4x2 + 3x -1 = 0
Comparing it with kx2 + 3x -1 = 0
We get K = 4
Tricks : Eqn. having roots the reciprocal of the roots of ax2 + bx + c = 0 is cx2 + bx +a = 0 i.e. 1st and last term interchanges.
or; x2 – 4x + x – 4 = 0
or; x(x – 4) + 1(x – 4) = 0
or; (x – 4)(x + 1) = 0
x = 4; -1
Eqn. having roots 1/2 & 1/−1 = 1/4 & – 1 is.
or x2 – (1/4 – 1) + 1/4(-1) = 0
or x2 + 3/4x – 1/4 = 0
Multiplying by 4 ; we get
4x2 + 3x -1 = 0
Comparing it with kx2 + 3x -1 = 0
We get K = 4
Tricks : Eqn. having roots the reciprocal of the roots of ax2 + bx + c = 0 is cx2 + bx +a = 0 i.e. 1st and last term interchanges.
Find the condition that one root is double the of ax2 + bx + c = 0- a)2b2 = 3ac
- b)b2 = 3ac
- c)2b2 = 9ac
- d)None
Correct answer is option 'C'. Can you explain this answer?
Find the condition that one root is double the of ax2 + bx + c = 0
a)
2b2 = 3ac
b)
b2 = 3ac
c)
2b2 = 9ac
d)
None
|
Dishani Kulkarni answered |
Understanding the Condition for Roots
In the quadratic equation ax² + bx + c = 0, we seek the condition such that one root is double the other. Let's denote the roots as α and β, where α = 2β (one root is double the other).
Roots Relationship
Using Vieta's formulas, we know:
- Sum of the roots (α + β) = -b/a
- Product of the roots (αβ) = c/a
Given α = 2β, we can express the roots in terms of β:
- α + β = 2β + β = 3β
- αβ = 2β * β = 2β²
Substituting in Vieta's Formulas
Now substituting into Vieta's formulas:
1. Sum of roots:
- 3β = -b/a
2. Product of roots:
- 2β² = c/a
Finding Relationships
From the first equation:
- β = -b/(3a)
Now substituting this value of β into the second equation:
- 2(-b/(3a))² = c/a
This simplifies to:
- 2(b²/(9a²)) = c/a
Cross-multiplying gives us:
- 2b² = 9ac
Final Condition
Thus, the derived condition is:
- 2b² = 9ac
This means that the correct answer is option 'C'.
Conclusion
To summarize, the condition that ensures one root of the quadratic equation ax² + bx + c = 0 is double the other is represented mathematically as 2b² = 9ac. This establishes a specific relationship between the coefficients of the quadratic equation.
In the quadratic equation ax² + bx + c = 0, we seek the condition such that one root is double the other. Let's denote the roots as α and β, where α = 2β (one root is double the other).
Roots Relationship
Using Vieta's formulas, we know:
- Sum of the roots (α + β) = -b/a
- Product of the roots (αβ) = c/a
Given α = 2β, we can express the roots in terms of β:
- α + β = 2β + β = 3β
- αβ = 2β * β = 2β²
Substituting in Vieta's Formulas
Now substituting into Vieta's formulas:
1. Sum of roots:
- 3β = -b/a
2. Product of roots:
- 2β² = c/a
Finding Relationships
From the first equation:
- β = -b/(3a)
Now substituting this value of β into the second equation:
- 2(-b/(3a))² = c/a
This simplifies to:
- 2(b²/(9a²)) = c/a
Cross-multiplying gives us:
- 2b² = 9ac
Final Condition
Thus, the derived condition is:
- 2b² = 9ac
This means that the correct answer is option 'C'.
Conclusion
To summarize, the condition that ensures one root of the quadratic equation ax² + bx + c = 0 is double the other is represented mathematically as 2b² = 9ac. This establishes a specific relationship between the coefficients of the quadratic equation.
If the roots of the equation x2 – 15x2 + kx – 45 = 0 are in A.P., find value of k:- a)56
- b)59
- c)-56
- d)-59
Correct answer is option 'B'. Can you explain this answer?
If the roots of the equation x2 – 15x2 + kx – 45 = 0 are in A.P., find value of k:
a)
56
b)
59
c)
-56
d)
-59
|
|
KP Classes answered |
∵ Roots are in A.R
Let roots are a – d; a; a + d
So, (a – d)+a + (a + d) = 15
or; 3a = 15
or; a = 5
And Product of roots
(a – d ). a . (a + d ) = 45
or (5 – d);5. (5 + d) = 45
or 25 – d2 = 9
or; d2 = 25 – 9 = 16
or; d = √16 = 4
Hence; roots are
a – d, a, a + d = 5 – 4; 5; 5 + 4
= 1; 5 ; 9.
The value of K
= Sum of product of two roots in a order
= (1 × 5) + (5 × 9) + (9 × 1)
= 5 + 45 + 9 = 59
(b) is correct.
Let roots are a – d; a; a + d
So, (a – d)+a + (a + d) = 15
or; 3a = 15
or; a = 5
And Product of roots
(a – d ). a . (a + d ) = 45
or (5 – d);5. (5 + d) = 45
or 25 – d2 = 9
or; d2 = 25 – 9 = 16
or; d = √16 = 4
Hence; roots are
a – d, a, a + d = 5 – 4; 5; 5 + 4
= 1; 5 ; 9.
The value of K
= Sum of product of two roots in a order
= (1 × 5) + (5 × 9) + (9 × 1)
= 5 + 45 + 9 = 59
(b) is correct.
If coefficients of the equation ax2 + bx + c = 0, a < 0 are real and roots of the equation are non–real complex and a + c + b < 0, then- a)4a + c < 2b
- b)4a + c> 2b
- c)4a + c = 2b
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
If coefficients of the equation ax2 + bx + c = 0, a < 0 are real and roots of the equation are non–real complex and a + c + b < 0, then
a)
4a + c < 2b
b)
4a + c> 2b
c)
4a + c = 2b
d)
none of these
|
Arindam Unni answered |
Method to Solve
Puting x=-1
a-b+c
but
a+c<b or a+c-b<0
hence f(x)<0 for all real values of x
therefore
putting x=-2
we get f(X)=4a+c-2b<0
or 4a+c<2b
a-b+c
but
a+c<b or a+c-b<0
hence f(x)<0 for all real values of x
therefore
putting x=-2
we get f(X)=4a+c-2b<0
or 4a+c<2b
Which of the following is a finite set?- a)Set of roots of the equation x2 – 25 = 0
- b)The set of numbers which are multiples of 5
- c)The set of lines parallel to y – axis
- d)Set of natural numbers
Correct answer is option 'A'. Can you explain this answer?
Which of the following is a finite set?
a)
Set of roots of the equation x2 – 25 = 0
b)
The set of numbers which are multiples of 5
c)
The set of lines parallel to y – axis
d)
Set of natural numbers
|
Umesh Kumar answered |
All 3 options have infinite many roots but
x^2-25=0 has only two roots. +5& -5
hense it is a finite set.
x^2-25=0 has only two roots. +5& -5
hense it is a finite set.
- a)48
- b)24
- c)-24
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
a)
48
b)
24
c)
-24
d)
none of these
|
Khushi Mishra answered |
B is right answer because root -16*root-9 Will be root +144 and its value is 12 and 12*2=24
The argument of the complex number -i- a)-π/2
- b)3π/4
- c)-π/3
- d)-π/4
Correct answer is option 'A'. Can you explain this answer?
The argument of the complex number -i
a)
-π/2
b)
3π/4
c)
-π/3
d)
-π/4
|
Rohit Shah answered |
An argument of the complex number z = x + iy, denoted arg(z), is defined in two equivalent ways: Geometrically, in the complex plane, as the angle φ from the positive real axis to the vector representing z. The numeric value is given by the angle in radians and is positive if measured counterclockwise.
£ 4, then least and the highest values of 4x2 are
If i2=−1, then sum i+i2+i3+....... to 1000 terms is equal to
- a)1
- b)0
- c)-1
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
If i2=−1, then sum i+i2+i3+....... to 1000 terms is equal to
a)
1
b)
0
c)
-1
d)
none of these
|
Mahi Pillai answered |
Sum of arithmetic series:
The given sequence is an arithmetic series with a common difference of 1. Let's denote the first term as 'a' and the number of terms as 'n'. The sum of an arithmetic series can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)
Where Sn is the sum of the series, a is the first term, n is the number of terms, and d is the common difference.
Calculating the sum:
In this case, the first term 'a' is 1 and the common difference 'd' is also 1. So, the formula becomes:
Sn = (n/2)(2 + n - 1)
Simplifying further:
Sn = (n/2)(n + 1)
Now, let's substitute the given value of 'n' and calculate the sum:
S4n = (4n/2)(4n + 1)
S4n = 2n(4n + 1)
S4n = 8n^2 + 2n
Determining the sum's value:
To find the value of the sum, we need to substitute the value of 'n' into the equation. However, since the options are not expressed in terms of 'n', we can do some algebraic manipulation to simplify the expression further.
Let's factor out n from the equation:
S4n = n(8n + 2)
We can see that 'n' is a common factor in both terms. Since 'n' represents the number of terms, it is always a positive value. Therefore, the sum can only be zero when '8n + 2' equals zero.
Setting '8n + 2' equal to zero and solving for 'n':
8n + 2 = 0
8n = -2
n = -2/8
Since 'n' represents the number of terms, it cannot be a negative value. Therefore, there is no value of 'n' for which the sum is zero. Hence, the correct answer is option 'B' - 0.
Conclusion:
The sum of the given arithmetic series is equal to zero.
The given sequence is an arithmetic series with a common difference of 1. Let's denote the first term as 'a' and the number of terms as 'n'. The sum of an arithmetic series can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)
Where Sn is the sum of the series, a is the first term, n is the number of terms, and d is the common difference.
Calculating the sum:
In this case, the first term 'a' is 1 and the common difference 'd' is also 1. So, the formula becomes:
Sn = (n/2)(2 + n - 1)
Simplifying further:
Sn = (n/2)(n + 1)
Now, let's substitute the given value of 'n' and calculate the sum:
S4n = (4n/2)(4n + 1)
S4n = 2n(4n + 1)
S4n = 8n^2 + 2n
Determining the sum's value:
To find the value of the sum, we need to substitute the value of 'n' into the equation. However, since the options are not expressed in terms of 'n', we can do some algebraic manipulation to simplify the expression further.
Let's factor out n from the equation:
S4n = n(8n + 2)
We can see that 'n' is a common factor in both terms. Since 'n' represents the number of terms, it is always a positive value. Therefore, the sum can only be zero when '8n + 2' equals zero.
Setting '8n + 2' equal to zero and solving for 'n':
8n + 2 = 0
8n = -2
n = -2/8
Since 'n' represents the number of terms, it cannot be a negative value. Therefore, there is no value of 'n' for which the sum is zero. Hence, the correct answer is option 'B' - 0.
Conclusion:
The sum of the given arithmetic series is equal to zero.
In z = a + bι, if i is replaced by −ι, then another complex number obtained is said to b- a)additive inverse of z
- b)prime factor of z
- c)Complex conjugate of z
- d)multiplicative inverse of z
Correct answer is option 'C'. Can you explain this answer?
In z = a + bι, if i is replaced by −ι, then another complex number obtained is said to b
a)
additive inverse of z
b)
prime factor of z
c)
Complex conjugate of z
d)
multiplicative inverse of z
|
Kirti Choudhary answered |
The equation given, z = a b, is incomplete and does not specify the relationship between a and b. Without further information, it is not possible to determine the meaning or solve for any variables.
The set of all solutions of the inequality 
< 1/4 contains the set- a) (–¥, 0)
- b)(–¥, 1)
- c) (1, ¥)
- d) (3, ¥)
Correct answer is option 'D'. Can you explain this answer?
The set of all solutions of the inequality < 1/4 contains the set
< 1/4 contains the seta)
(–¥, 0)
b)
(–¥, 1)
c)
(1, ¥)
d)
(3, ¥)
|
|
Hansa Sharma answered |
(1/2)(x2 - 2x) < (1/4)
(1/2)(x2 - 2x) < (1/2)2
x2 − 2x > 2......(as after multiplicative inverse sign of inequality changes)
x2 − 2x − 2 > 0
x2 - 2x + 1 - 3 >
(x-1)2 - 3 > 0
(x-1)2 > 3
So for the above to hold good both the expression must be positive or both must be negative. After finding the solution the range of the solution will be
either x > 3
(3,¥)
(1/2)(x2 - 2x) < (1/2)2
x2 − 2x > 2......(as after multiplicative inverse sign of inequality changes)
x2 − 2x − 2 > 0
x2 - 2x + 1 - 3 >
(x-1)2 - 3 > 0
(x-1)2 > 3
So for the above to hold good both the expression must be positive or both must be negative. After finding the solution the range of the solution will be
either x > 3
(3,¥)
If (1 – p) is root of quadratic equation x2 + px + (1 – p) = 0, then its roots are- a)0, 1
- b)–1, 1
- c)0, –1
- d)–1, 2
Correct answer is option 'C'. Can you explain this answer?
If (1 – p) is root of quadratic equation x2 + px + (1 – p) = 0, then its roots are
a)
0, 1
b)
–1, 1
c)
0, –1
d)
–1, 2
|
|
Anand Kumar answered |
Put (1-p) In the given equation and solve for p.
after solving it u will have two values of p
now, for the solution put values p one by one in (1-p)
after solving it u will have two values of p
now, for the solution put values p one by one in (1-p)
- a)24i
- b)24
- c)-24
- d)-24i
Correct answer is option 'D'. Can you explain this answer?
a)
24i
b)
24
c)
-24
d)
-24i
|
Preeti Iyer answered |
√-8 * √-18 * √-4
= 2√2(√-1) × 3√2(√-1) × 2(√-1)
= 2√2(i) × 3√2(i) × 2(i)
= (2√2 * 3√2 * 2) *(i * i * i)
= (24)(-i2 * i)
= 24(-i)
= -24i
= 2√2(√-1) × 3√2(√-1) × 2(√-1)
= 2√2(i) × 3√2(i) × 2(i)
= (2√2 * 3√2 * 2) *(i * i * i)
= (24)(-i2 * i)
= 24(-i)
= -24i
Let x,y ∈ R, then x + iy is a purely imaginary number if- a)x ≠ 0 , y = 0
- b)x = 0 , y ≠ 0
- c)x = 0 , y = 0
- d)x ≠ 0 , y ≠ 0
Correct answer is option 'B'. Can you explain this answer?
Let x,y ∈ R, then x + iy is a purely imaginary number if
a)
x ≠ 0 , y = 0
b)
x = 0 , y ≠ 0
c)
x = 0 , y = 0
d)
x ≠ 0 , y ≠ 0
|
|
Sujith Reddy answered |
A complex number is purely imaginary if the real term is zero, and imaginary term is non-zero. So, x being the real term is zero and y being the coefficient of the imaginary term is non-zero. So, (B) x=0, y≠0
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