Question 1
According to De Moivre’s theorem what is the value of z1/n ?
A. r1/n [cos(2kπ + θ) + i sin(2kπ + θ)]
B. r1/n [cos(2kπ + θ)/n – i sin(2kπ + θ)/n]
C. r1/n [cos(2kπ + θ)/n + i sin(2kπ + θ)/n]
D. r1/n [cos(2kπ + θ) – i sin(2kπ + θ)]
View Answer
Answer: Option C
Explanation:
If n is any integer, then (cosθ + isinθ)n = cos(nθ) + i sin(nθ).Writing the binomial expansion of (cosθ + isinθ)n and equating real parts of cos(nθ) and the imaginary part to sin(nθ), we get,cos(nθ) = cosnθ – nC2 cosn-2θ sin2θ + nC4 cosn-4θ sin4θ + ……….sin(nθ) = nC1 cosn-1θ sinθ – nC3 cosn-3θ sin3θ + ……….If, n is a rational number, then one of the value of (cosθ + isinθ)n = cos(nθ) + i sin(nθ).If, n = p/q, where, p and q are integers (q>θ) and p, q have no common factor, then (cosθ + isinθ)n has q distinct values one of which is cos(nθ) + i sin(nθ)If, z1/n = r1/n [cos(2kπ + θ)/n + i sin(2kπ + θ)/n], where k = 0, 1, 2, ……….., n – 1.
Question 2
If |z1| = 4, |z2| = 3, then what is the value of |z1 + z2 + 3 + 4i|?
A. Less than 2
B. Less than 5
C. Less than 7
D. Less than 12
View Answer
Answer: Option D
Explanation:
As, we know | z1 + z2 + …….. +zn| ≤ |z1| + |z2| + ………. + |zn|So, |z1 + z2 + 3 + 4i| ≤ |z1| + |z2| + |3 + 4i|Now, putting the given values in the equation, we get,=> |z1 + z2 + 3 + 4i| ≤ 4 + 3 + √(9 + 16)=> |z1 + z2 + 3 + 4i| ≤ 4 + 3 + 5=> |z1 + z2 + 3 + 4i| ≤ 12.
Question 3
If acosθ + bsinθ = c have roots α and β. Then, what will be the value of sinα * sinβ ?
A. 2bc/(a2 + b2)
B. 0
C. 1
D. (c2 + a2)/(a2 + b2)
View Answer
Answer: Option D
Explanation:
Given, acosθ + bsinθ = cSo, this implies acosθ = c – bsinθNow squaring both the sides we get,(acosθ)2 = (c – bsinθ)2a2 cos2 θ = c2 + b2 sin2 θ – 2b c sinθa2 (1- sin2 θ) = c2 + b2 sin2 θ – 2b c sinθa2 – a2 sin2 θ = c2 + b2 sin2 θ – 2b c sinθNow rearranging the elements,(a2 + b2) sin2 θ – 2b c sinθ +( c2 – a2) = θSo, as sum of the roots are in the formc/a if there is a quadratic equation ax2 + bx + c = 0Now , we can conclude thatsinα + sinβ = (c2 + a2)/(a2 + b2).
Question 4
If acosθ + bsinθ = c have roots α and β. Then, what will be the value of sinα + sinβ?
A. 2bc/(a2 + b2)
B. 0
C. 1
D. (c2 + a2)/(a2 + b2)
View Answer
Answer: Option A
Explanation:
Given, acosθ + bsinθ = cSo, this implies acosθ = c – bsinθNow squaring both the sides we get,(acosθ)2 = (c – bsinθ)2a2 cos2 θ = c2 + b2 sin2 θ – 2b c sinθa2 (1- sin2 θ) = c2 + b2 sin2 θ – 2b c sinθa2 – a2 sin2 θ = c2 + b2 sin2 θ – 2b c sinθNow rearranging the elements,(a2 + b2) sin2 θ – 2b c sinθ +( c2 – a2) = θSo, as sum of the roots are in the form –b/a if there is a quadratic equation ax2 + bx + c = 0Now, we can conclude thatsinα + sinβ = 2bc/(a2 + b2).
Question 5
If p and q are the roots of the equation x2 + px + q =0 then, what are the values of p and q?
A. p = 1, q = -2
B. p = 0, q = 1
C. p = -2, q = 0
D. p = -2, q = 1
View Answer
Answer: Option A
Explanation:
Since, p and q are the roots of the equation x2 + px + q =0So, p + q = -p and pq = qSo, pq = qAnd, q = 0 or p = 1If, q = 0 then, p = 0 and if p = 1 then q = -2.
Question 6
If x2 + ax + b = 0 and x2 + bx + a = 0 have exactly 1 common root then what is the value of (a + b)?
A. 1
B. 0
C. -1
D. 3
View Answer
Answer: Option C
Explanation:
Subtracting the equation x2 + ax + b = 0 to x2 + bx + a = 0 by solving the equation simultaneously, we get,(a – b)x + (b – a) = 0So, (a – b)x = (a – b)Therefore, x = 1Now, putting the value of x = 3 in any one of the equation, we get,1 + a + b = 0Therefore, a + b = -1.
Question 7
If x2 + px + 1 = 0 and (a – b)x2 + (b – c)x + (c – a) = 0 have both roots common, then what is the form of a, b, c?
A. a, b, c are in A.P
B. b, a, c are in A.P
C. b, a, c are in G.P
D. b, a, c are in H.P
View Answer
Answer: Option B
Explanation:
Given, (a – b)x2 + (b – c)x + (c – a) = 0 and x2 + px + 1 =0So, 1 / (a – b) = p / (b – c) = 1 / (c – a)Equating the above equation, we get,(b – c) = p(a – b) and(b – c) = p(c – a)So, p(a – b) = p(c – a)=> a – b = c – aSo, 2a = b + c which means that b, a, c are in A.P.
Question 8
Roots of a quadratic equation are imaginary when discriminant is ______________
A. zero
B. greater than zero
C. less than zero
D. greater than or equal to zero
View Answer
Question 9
Roots of a quadratic equation are real when discriminant is ______________
A. zero
B. greater than zero
C. less than zero
D. greater than or equal to zero
View Answer
Question 10
Solve x2+1 = 0.
A. x=1, -1
B. x=i, -i
C. x=-1
D. x=i
View Answer
Question 11
What will be the product of b * c if the equations x2 + bx + c = 0 and x2 + 3x + 3 = 0 have one common root?
A. 3
B. 4
C. 6
D. 9
View Answer
Answer: Option D
Explanation:
Comparing the coefficients of the above equation we get,1/1 = b/3 = c/3This means b = 3 and c = 3Therefore, b * c = 9.
Question 12
What will be the sum of b + c if the equations x2 + bx + c = 0 and x2 + 3x + 3 = 0 have one common root?
A. 2
B. 4
C. 6
D. 8
View Answer
Answer: Option C
Explanation:
Comparing the coefficients of the above equation we get,1/1 = b/3 = c/3This means b = 3 and c = 3Therefore, b + c = 6.