Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set G

Question. The value of ‘a’ for which one root of the quadratic equation (a² −5a + 3) x² + (3a −1)x + 2 = 0 is twice as large as the other is:
a. 2/3
b. – 2/3
c. 1/3
d. – 1/3

Answer: a

Question. The roots of the equation a(x² +1) − (a² +1)x = 0 are:
a. a,1/a
b. a, 2a
c. a,1/2a
d. None of these

Answer: a

Question. The roots of the equation ix² − 4x − 4i = 0 are:
a. – 2i
b. 2i
c. –2i, –2i
d. 2i, 2i

Answer: c

Question. If the sum of the roots of the quadratic equation 0 ax² + bx + c = is equal to the sum of the squares of their reciprocals, then a / c,b / a, c / b are in:
a. A.P.
b. G.P.
c. H.P.
d. None of these

Answer: c

Question. If x+1/x = √3, then x =?
a. cos Π/3 + isin Π/3
b. cos Π/2 + isin Π/2
c. sin Π/6 + icos Π/6
d. cos Π/6 + isin Π/6

Answer: d

Question. If α is a complex constant such that 2 α z + z +α = 0 has a real root, then:
a. α +α = 1
b. α +α = 0
c. α +α = −1
d. the absolute value of the real root is 1.

Answer: a,c,d

Question. The value of x = √2+√2+√2+........is:
a. –1
b. 1
c. 2
d. 3

Answer: c

Question. If the difference between the corresponding roots of x²+ ax + b = 0 and x²+ bx + a = 0 is same and a ≠ b, then:
a. a + b + 4 = 0
b. a + b − 4 = 0
c. a − b − 4 = 0
d. a − b + 4 = 0

Answer: a

Question. If the product of the roots of the equation (a +1)x² + (2a + 3)x + (3a + 4) = 0 be 2, then the sum of roots is:
a. 1
b. –1
c. 2
d. –2

Answer: b

Question. If one root of a quadratic equation is 1/2+√5 ,then the equation is:
a. x2 + 4x + 1 = 0
b. x2 + 4x − 1 = 0
c. x2 − 4x + 1 = 0
d. None of these

Answer: b

Question. If one of the roots of the equation x² + ax + b = 0 and x² + bx + a = 0 is coincident. Then the numerical value of (a + b) is:
a. 0
b. – 1
c. 2
d. 5

Answer: d

Question. Both the roots of given equation (x − a)(x − b) + (x − b)(x − c) + (x − c)(x − a) = 0 are always: 
a. Positive
b. Negative
c. Real
d. Imaginary

Answer: c

Question. The solution of the equation x+1/x = 2
a. 2, –1
b. 0, –1, -1/5
c. -1 , 1/5
d. None of these

Answer: d

Question. The number of roots of the quadratic equation 2 8sec θ − 6secθ +1 = 0 is:
a. Infinite
b. 1
c. 2
d. 0

Answer: d

Question. If Xr = cos (π/2r) + sin (π/2r) then x₁ . x₂ .x₃ ...∞ is: C
a. –3
b. –2
c. –1
d. 0

Answer: c

Question. If α and β are the roots of the equation 2x² − 3x + 4 = 0 , then the equation whose roots are α² and β² is:
a. 4x² + 7x +16 = 0
b. 4x² + 7x + 6 = 0
c. 4x² + 7x +1 = 0
d. 4x² − 7x +16 = 0

Answer: a

Question. The number which exceeds its positive square root by 12 is:
a. 9
b. 16
c. 25
d. None of these

Answer: b

Question. If one root of the equation 2x + px + q = 0 is the square of the other, then:
a. p3 + q2 − q(3p +1) = 0
b. p3 + q2 + q(1+ 3p) = 0
c. p3 + q2 + q(3p −1) = 0
d. p3 + q2 + q(1− 3p) = 0

Answer: d

Question. If x₁ , x₂ , x₃ are distinct roots of the equation ax² + bx + c = 0 then:
a. a = b = 0,c∈R
b. a = c = 0,b∈R
c. 2 b − 4ac ≥ 0
d. a = b = c = 0

Answer: d

Question. If 3 is a root of x² + kx – 24 = 0, it is also a root of :
a. 2 x + 5x + k = 0
b. 2 x − 5x + k = 0
c. 2 x − kx + 6 = 0
d. 2 x + kx + 24 = 0

Answer: c

Question. If one root of 5x² +13x + k = 0 is reciprocal of the other, then k = ?
a. 0
b. 5
c. 1/6
d. 6

Answer: c

Question. Let α and β be the roots of the equation 2x + x +1 = 0 , the equation whose roots are α19, β7 is:
a. 2x − x − 1 = 0
b. 2x − x +1 = 0
c. 2x + x −1 = 0
d. 2x + x + 1 = 0

Answer: d

Question. If z = 1+i√3 /√3+i then (Z̅)100 lies in: C
a. I quadrant
b. II quadrant
c. III quadrant
d. IV quadrant

Answer: c

Question. If a > 0,b > 0,c > 0 then both the roots of the equation ax² + bx + c = 0 ?
a. Are real and negative
b. Have negative real parts
c. Are rational numbers
d. None of these

Answer: b

Question. For what values of k will the equation 2x − 2(1+ 3k)x + 7 (3 + 2k) = 0 have equal roots?
a. 1, –10/9
b. 2, –10/9
c. 3, –10/9
d. 4, –10/9

Answer: b

Question. If x be real, then the minimum value of x² − 8x +17 is:
a. 0– 1
b. 0
c. 1
d. 2

Answer: c

Question. If α ,β are the roots of the equation ax2 + bx + c = 0 then the equation whose roots are α+1/β and β+1/α is:
a. acx² + (a + c)bx + (a + c)² = 0
b. abx² + (a + c)bx + (a + c)² = 0
c. acx² + (a + b)cx + (a + c)² = 0
d. None of these

Answer: a

Question. The equation e^x − x −1 = 0 has:
a. Only one real root x = 0
b. At least two real roots
c. Exactly two real roots
d. Infinitely many real roots

Answer: a

Question. If α and β be the roots of the equation 2x² + 2(a+b)x + a²+b² = 0,then the equation whose roots are (α + β)² and (α −β)² is:
a. x² - 2abx - (a²-b²)² = 0
b. x² - 4abx + (a²-b²)² = 0
c. x² - 4abx - (a²-b²)² = 0
d. None of these

Answer: b

Question. Ifα,β are roots of x² − 3x +1 = 0, then the equation whose roots are 1/α − 2 , 1/β − 2 is
a. x2 + x −1 = 0
b. x2 + x +1 = 0
c. x2 − x −1 = 0
d. None of these

Answer: c

Question. A real root of the equation log4 {log2(√x+8 -√x )} = 0 is:
a. 1
b. 2
c. 3
d. 4

Answer: a

Question. If the roots of the equations px² + 2qx + r = 0 and qx² - 2√prx+q = 0 be real, then:
a. p = q
b. q² = pr
c. p² = qr
d. r² = pq

Answer: b

Question. The solution set of the equation xlog(1-x)² = 9 is
a. {– 2, 4}
b. {4}
c. {0, – 2, 4}
d. None of these

Answer: a

Question. If α ,β be the roots of x² − px + q = 0 and α ′,β ′ be the roots of x² − p'x + q' = 0 then the value of (α −α ')² + (β −α ′)² + (a −β ′)² + (β −β ′)² is:
a. 2{p² − 2q + p′ − 2q′ − pp′}
b. 2{p² − 2q + p′ − 2q′ − qq′}
c. 2{p² − 2q − p′ − 2q′ − pp′}
d. 2{p² − 2q − p′ − 2q′ − qq′}

Answer: a

Question. If the roots of the equation x² - 5x + 16 = 0 are α, β and the roots of equation x² + px + q = 0 are α² +β² , αβ / 2 , then:
a. p = 1, q = −56
b. p = −1, q = −56
c. p = 1, q = 56
d. p = −1, q = 56

Answer: b

Question. If (1+i√3 )⁹ , + i = a + ib then b is equal to:
a. 1
b. 256
c. 0
d. 9³

Answer: c

Question. The common roots of the equations z³+(1+i)z² +(1+i) z +i = 0 (where i = √-1) and z¹⁹⁹³ + z¹⁹⁹⁴ + 1= 0 are:
a. 1
b. ω
c. ω²
d. ω⁹⁸¹

Answer: b,c

Question. If the roots of the equation 1/x+p + 1/x+q = 1/r are equal in magnitude but opposite in sign, then the product of the roots will be:
a. p²+q²/2
b. (p²+q²)/2
c. (p²-q²)/2
d. (p²-q²)/2

Answer: b

Question. If α ,β be the roots of the equation x² − 2x + 3= 0, then the equation whose roots are 1/α² and 1/β² is:
a. x2 + 2x + 1 = 0
b. 9x2 + 2x + 1 = 0
c. 9x2 - 2x + 1 = 0
d. 9x2 + 2x −1= 0

Answer: b

Question. If ω is the cube root of unity, then (3+5ω +3ω²)² + (3+5ω +3ω²)² ?
a. 4
b. 0
c. – 4
d. 5

Answer: c

Question. i log (x-i/x+i) is equal to:
a. Π+2 tan^-1x
b. Π–2 tan^-1x
c. –Π+2 tan^-1x
d. –Π–2 tan^-1x

Answer: b

Question. If eiθ = cosiθ +i sinθ = + , then in %ABC value of e^iA .e^iB .e^iC is:
a. – i
b. 1
c. –1
d. None of these

Answer: c

Question. If x²/3 − 7x +10 = 0, then x = ?
a. {125}
b. {8}
c. φ
d. {125, 8}

Answer: d

Question. If the sum of the roots of the equation λx² - 2x + 3λ = 0 be equal to their product, then λ = ?
a. 4
b. −4
c. 6
d. None of these

Answer: d

Question. If the ratio of the roots of the equation ax² + bx + c = 0 be p : q , then:
a. pqb² + ( p + q) ac = 0
b. pqb² − ( p + q) ac = 0
c. pqa² − ( p + q) bc = 0
d. None of these

Answer: b

Question. If -1+√3 = reiθ then θ is equal to
a.π/3
b. -π/3
c. 2π/3
d. -2π/3

Answer: c

Question. The number of real roots of the equation esin e^sinx − e^-sinx -4 = 0 are:
a. 1
b. 2
c. Infinite
d. None

Answer: d

Question. If α ,β are the roots of (x − a)(x − b) = c, c ≠ 0, then the roots of (x −α )(x −β ) + c = 0 shall be:
a. a, c
b. b,c
c. a,b
d. a + c,b + c

Answer: c

Question. min |1 –3i –z | is equal to:
a. 2–√3/2
b. 2+√3/2
c. 3–√3/2
d. 3+√3/2

Answer: c

Question. If 1/x + x = 2cosθ , then xⁿ + 1/xⁿ is equal to
a. 2cos nθ
b. 2 sin nθ
c. cos nθ
d. sin nθ

Answer: a

Question. The roots of the equation x⁴ – 1 = 0, are:
a. 1, 1, i, – i
b. 1, –1, i, – i
c. 1, –1, ω, ω²
d. None of these

Answer: b

Question. If z is any complex number satisfying | z − 3 − 2i |≤ 2, then the maximum value of |2z – 6 + 5i|is:

Answer: 3

Question. If | z₁ |= 2,| z₂ |= 3,| z₃ |= 4 and | 2z₁ + 3z₂ + 4z₃ |= 9, then absolute value 8z₂z₃ +27z₃z₁ + 64z₁z₂ must be equal to:

Answer: 216