Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set 04

Question: If z₁ and z₂ are two non-zero com plex num bers such that |z₁+ z₂ | =| z₁| +|z₂| ,then arg ( z₁ ) -arg ( z₂ ) equal to 
(a)- π/ 2
(b) 0
(c) -π
(d) π/2 
Answer: b

Question: If ω=z/z-i/3 and |ω|= 1, then z lies on
(a) a parabola
(b) a straight line
(c) a circle
(d) an ellipse
Answer: b

Question: If(1+i/1+i)^x=1, then
(a) x = 4n, where n is any positive integer
(b) x = 2n, where n is any positive integer
(c) x = 4n + 1, where n is any positive integer
(d) x = 2n + 1, where n is any positive integer
Answer: a

Question:  If |z - iRe ( z)|=|z - Im ( z)| (where i = √-1), then z lies on
(a) Re (z) = 2
(b) Im (z) = 2
(c) Re (z) + Im (z) = 2
(d) None of the above 
Answer: a

Question: If z₁ and z₂ be complex numbers such that z1≠ z2 and| z₁ | |z₂ |. If z₁ has positive real part and z₂ has negative imaginary part, then [( z₁+ z₂ )/( z₁ -z₂)] may be
(a) purely imaginary
(b) real and positive
(c) real and negative
(d) None of the above 
Answer: a

Question: If z₁ = 2+5i , z₂ = 3–i then projection of z₂ on z₂ is
a. 1/10
b. 1 / 10
c. −7 /10
d. None of these
Answer: b

Question: If complex numbers z₁,z₂ and z₃ represent the vertices A, B and C respectively of an isosceles triangle ABC of which ∠C is right angle, then correct statement is
a. z₁² + z₂² + z₃² = z₁z₂z₃
b. (z₃ – z₁)² = z₃ – z₂
c. (z₁ – z₂)² = (z₁ – z₃)(z₃ – z₂)
d. (z₁ – z₂)² = 2(z₁ – z₃)(z₃ – z₂)
Answer: d

Question: If |z₁|= |z₂| =1 and amp z₁+ ampz₂= = 0, then
(a) z₁z₂ = 1
(b) z₁ + z₂ = 0
(c) z₁ =¯z2 
(d) None of these
Answer: a,c

Question: If |z₁|< √2 - 1, then |z²+2z cos α| is
(a) less than 1
(b) 2 + 1
(c) 2 - 1
(d) None of these 
Answer: a

Question: If A(z₁), B(z₂) and C(z₃) are the vertices of the ΔABC such that ( z₁- z₂ )/( z₃ -z₂ ) = (1/√2 - (i/1/ √2 , then Δ ABC is
(a) equilateral
(b) right angled
(c) isosceles
(d) obtuse angled 
Answer: c

Question: Complex number z₁¯z₂ is
(a) purely real
(b) purely imaginary
(c) zero
(d) None of the above 
Answer: b

Question: One of the possible argument of complex number
i(z₁/z₂) 
(a) π/2
(b) - π/2
(c) 0
(d) None of these 
Answer: c

Question: If z is a complex number of unit modulus and argument q, then arg(1+z/1+z) is equal to
(a) -θ
(b)π/2-θ
(c) q
(d) θ - q
Answer: c

Question: The point of intersection of the curves arg ( z - 3 i) =3π/4 and arg (2z+1- 2i )= π/4,(where i = √-1) is 
(a)1/4(3+9i)
(b)1/4(3-9i)
(c)1/2(3+2i
(d) No point 
Answer: d

Question: The number of complex numbers z such that
|z - 1|=|z + 1|=|z - i| is equal to 
(a) 0
(b) 1
(c) 2
(d) ∞
Answer: b

Question: If z≠1 and z²/z-1 is real, then the point represented by the complex number z lies 
(a) either on the real axis or on a circle passing through the origin
(b) on a circle with centre at the origin
(c) either on the real axis or on a circle not passing through the origin
(d) on the imaginary axis 
Answer: a

Question: If |z + 4|≤ 3, then the max i mum value of |z + 1| is
(a) 4
(b) 10
(c) 6
(d) 0 
Answer: c

Question: The conjugate of a complex number is 1/i - 1 . Then,that complex number is 
(a)1/i - 1
(b) -1/-1
(c)1/i + 1
(d) -1/i+ 1 
Answer: d

Question: If the cube roots of unity are 1,ω and ω² , then the roots of the equation (x - 1)³ + 8 = 0, are 
(a) -1,1 + 2ω,1 + 2ω²
(b) -1, 1 - 2ω,1- 2 ω²
(c) -1, -1, -1
(d) -1 -1 + 2ω-1-2ω² 
Answer: b

Question: If |z²-1|= |z|²+1,then z lies on
(a) the real axis
(b) the imaginary axis
(c) a circle
(d) an ellipse 
Answer: b

Question: Let z₁ and z₂ be two roots of the equation z²+ az+ b= 0,z being com plex. Further, assumethat the origin z₁ and z₂ form an equilateral triangle. 
Then,
(a) a² =b
(b) a² = 2b
(c) a² = 3b
(d) a² = 4b 
Answer: c

Question: Let z and w be com plex num bers such that ¯z + i¯w = 0 and arg ( zω) = π. Then, arg (z) is equal to 
(a)π /4
(b) π /2
(c) 3π /4
(d) 5π /2 
Answer: c

Question: If z1,z2,z3,z4 are roots of the equation a₀z₄ + a₁z₃ + a₂z₂ + a₃z + a₄ = 0
where a0, a1, a2, a3, and a4 are real, then:
a. z̅1,z̅2,z̅3,z̅4, are also roots of the equation
b. z̅1 is equal to at least one of z̅1,z̅2,z̅3,z̅4
c. –z̅1,–z̅2,–z̅3,–z̅4 are also roots of the equation d. none of the above.
d. none of the above
Answer: a,b

Question: If z=7-i/3-4i then z¹⁴ = ?
a. 2⁷
b. 2⁷ i
c. 2¹⁴ i
d. – 2⁷i
Answer: d

Question: Inverse of a point a with respect to the circle |z – c| = R (a and c are complex numbers, centre C and radius R) is the point c+R2/a̅ –c̅
a. c + R²/a̅ –c̅
b. c–R²/a̅ –c̅
c. c +R/c̅ -a̅
d. None of these
Answer: a

Question: If zi = a + ib and z₂ = c + ib are complex numbers such that |z₁|=| z₂|=1 and 1 2 Re(z₁z₂) = 0, then the pair of complex numbers w₁ = a + ic and w₂ = b + id satisfies?
a. |w1| = 1
b. |w2| = 1
c. 1 2 Re | w w |= 0
d. None of these
Answer: a,b,c

Question: Let z₁ and z₂ be complex numbers such that z₁ ≠ z₂ and |z₁|=|z₂|, If z₁ has positive real part and z2 has negative imaginary part, then z₁+z₂/z₁–z₂ may be:
a. zero
b. real and positive
c. 0
d. 0
Answer: a,d

Question: The imaginary part of tan-1 (5i/3) is:
a. 0
b. ∞
c. log 2
d. log 4
Answer: c

Question: If ω is an imaginary cube root of unity, then
(1 ) + ω - ω²⁾⁷ is equal to 
(a) 128 w
(b) -128 w
(c) 128 w2
(d) -128 w2 
Answer: d

Question: If w(≠ 1) is a cube root of unity and (1+ω )⁷ = A + Bω .
Then, (A, B) is equal to
(a) (1, 1)
(b) (1, 0)
(c) (-1, 1)
(d) (0, 1) 
Answer: a

Question: Let ω be a complex cube root of unity with ω ≠1 and P = [p_ij] be a n×n matrix with . Pij=ω^i+j Then, P2 ≠ 0 when n is equal to:
a. 57
b. 55
c. 58
d. 56
Answer: c,d

Question: If z₁,z₂, z₃, z₄ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z₁–z₄=z₂ –z3and amp (z₄–z₁/z₂–z₁) = π/2 , then the quadrilateral is:
a. rhombus
b. square
c. rectangle
d. cyclic quadrilateral
Answer: c,d

Question: If |z₁|= 15 and |z₂-3-4|5,then
(a) | z₁ - z₂ ^|_min = 5
(b) | z₁ - z₂ ^|_min = 10
(c) | z₁ - z₂ ^|_max = 20
(d) | z₁ - z₂ ^|_max = 25 
Answer: a,d

Question: If |z₁| =|z₂ |, arg ( z₁ / z₂ ) = π, then z₁+ z₂ is equal to
(a) 0
(b) purely imaginary
(c) purely real
(d) None of these
Answer: a

Question: If the roots of the equation bx² + cx + α = 0 be imaginary, then for all real values of x, the expression 3b²x² + 6bcx + 2c² is :
(a) less than 4αb
(b) greater than – 4αb
(c) 1ess than – 4αb
(d) greater than 4αb
Answer: b

Question: If α, β are roots of the equation x² – 5x + 6 = 0, then the equation whose roots are α + 3 and β + 3 is
(a) 2x² – 11x 30 = 0
(b) –x² +11x = 0
(c) x² –11x + 30 = 0
(d) 2x² – 5x + 30 = 0 
Answer: c

Question: (1 + i)⁸ + (1 – i)⁸ equal to
(a) 2⁸
(b) 2⁵
(c) 2⁴ cos π/4
(d) 2⁸ cos π/8 
Answer: b

Question: If (1 + i)³/(1 – i)³ – (1 – i)³/(1 + i )³ = x + iy
(a) x = 0, y = –2
(b) x = –2, y = 0
(c) x = 1, y = 1
(d) x = –1, y = 1 
Answer: a

Question: If (x + iy) (2 – 3i) = 4 + i, then
(a) x = c 14/13, y = 5/13
(b) x = 5/13, y = 14/13
(c) x = 14/13, y = 5/13
(d) x = 5/13, y = – 14/13
Answer: b

Question: If α, β ∈ C are the distinct roots, of the equation x² – x +1 = 0 , then α¹⁰¹ + β¹⁰⁷ is equal to :
(a) 0
(b) 1
(c) 2
(d) – 1
Answer: b

Question: If A.M. of the roots of a quadratic equation is 8/5 and A.M. of their reciprocals is 8/7 ,then the equation is
(a) 5x² – 16x + 7 = 0
(b) 7x² – 16x + 5 = 0
(c) 7x² – 16x + 8 = 0
(d) 3x² – 12x + 7 = 0 
Answer: a

Question: Which of the following options defined ‘imaginary number’?
(a) Square root of any number
(b) Square root of positive number
(c) Square root of negative number
(d) Cube root of number 
Answer: c

Question: All the values of m for which both roots of the equation x² – 2mx + m² – 1 = 0 are greater than – 2 but less than 4, lie in the interval
(a) -2 < m < 0
(b) m > 3
(c) -1 < m < 3
(d) 1 < m < 4
Answer: c

Question: Product of real roots of the equation t² x² + | x | + 9 = 0
(a) is always positive
(b) is always negative
(c) does not exist
(d) none of these
Answer: a