Exams › JEE Main › Maths › Complex Numbers and Quadratic Equations
453 questions with worked solutions.
Q1. Given z1 = √3 + i√3 and z2 = √3 + i, determine the quadrant in which the complex number z1/z2 lies.
Answer: I
z1/z2 = (sqrt3 + i sqrt3)/(sqrt3 + i). Multiplying by the conjugate gives a positive real part (~1.18) and positive imaginary part (~0.32), so the quotient lies in Quadrant I.
Answer: (3−5i)/2
Solving 2(1+i)x^2 -4(2-i)x -(5+3i)=0 gives roots x = (3-5i)/2 (modulus sqrt(34)/2 ~ 2.92) and x = (-1-i)/2 (modulus ~0.71). The root of larger absolute value is (3-5i)/2.
Q3. Find the value of ((cosθ+isinθ)⁴)/((cosθ-isinθ)³).
Answer: cos 7θ+isin 7θ
(cosT+isinT)^4 = e^(i4T) and (cosT-isinT)^3 = e^(-i3T), so the ratio is e^(i4T)/e^(-i3T) = e^(i7T) = cos7T + isin7T.
Q4. How many complex numbers z satisfy the equation z³ + (3|z|²)/(z) = 0, given that |z| = √(3)?
Answer: 4
With |z|=sqrt(3), |z|^2=3, so z^3 + 3*3/z = z^3 + 9/z = 0, giving z^4 = -9. This quartic has exactly 4 roots, each of modulus 9^(1/4)=sqrt(3), so 4 complex numbers satisfy it.
Answer: x² + y² − 2y = 1
Setting z = x+iy and equating the argument of (z-1)/(z+1) to pi/4 gives an arc of a circle through +1 and -1; algebra (or direct testing of points on the locus) yields x^2 + y^2 - 2y = 1.
Answer: Both roots have negative real parts
The roots of the quadratic equation can be determined using the quadratic formula, and since the coefficients a, b, and c are all positive, the sum of the roots (given by -b/a) is negative, indicating that both roots must have negative real parts.
Answer: None of these
The area of the triangle formed by the points z, 0, and z+ω can be calculated using the formula for the area of a triangle in the complex plane. Given that the area is 4√3 square units, and knowing that the area of a triangle is also related to the radius of the circumcircle, we can deduce that the radius must be greater than the options provided, leading to the conclusion that 'None of these' is the correct answer.
Q8. For a complex number z, what is the least possible value of |z + i| + |z - 2|?
Answer: None of these
|z+i|+|z-2| is the sum of distances from z to (0,-1) and to (2,0). The minimum is the straight-line distance between these points = sqrt((2-0)^2+(0+1)^2) = sqrt5 ≈ 2.236, which is none of 1, 2 or 3, so the answer is None of these.
Answer: 2i
Let x=alpha/a, y=beta/b, z=gamma/c. Then x+y+z=1+i and 1/x+1/y+1/z=0 gives xy+yz+zx=0. So x^2+y^2+z^2 = (x+y+z)^2 - 2(xy+yz+zx) = (1+i)^2 = 2i.
Answer: 2, 2 ± √2
With k = x^2-4x+3, putting a = (7+4sqrt3)^k gives a + 1/a = 14, so a = 7 +/- 4sqrt3, i.e. k = 1 or k = -1. k=-1 gives x^2-4x+4=0 -> x=2; k=1 gives x^2-4x+2=0 -> x = 2 +/- sqrt2. Solutions: 2, 2 +/- sqrt2.
Answer: a² l² x² - abmx + b² l n + acm² - 4acln = 0
For roots (alpha*gamma+beta*delta) and (alpha*delta+beta*gamma): their sum = (alpha+beta)(gamma+delta) = (b/a)(m/l) and product reduces to b^2 l n + a c m^2 - 4 a c l n over a^2 l^2. Forming the monic equation and clearing denominators gives a^2 l^2 x^2 - a b m x + b^2 l n + a c m^2 - 4 a c l n = 0.
Q12. Evaluate the value of [(-1 + √-3)/2]¹⁰⁰ + [(-1 - √-3)/2]¹⁰⁰.
Answer: -1
(-1 +/- sqrt(-3))/2 are the complex cube roots of unity omega and omega^2. Since 100 ≡ 1 and 200 ≡ 2 (mod 3), the sum is omega^100 + (omega^2)^100 = omega + omega^2 = -1.
Answer: 41
Let t = 3x^2+9x, whose minimum is -27/4. The expression = 1 + 10/(t+7); t+7 is smallest at t = -27/4, giving t+7 = 1/4. Then the maximum value = 1 + 10/(1/4) = 1 + 40 = 41.
Q14. Evaluate the complex expression ((1)/(1-2i)+(3)/(1+i)) ((3+4i)/(2-4i)):
Answer: (1)/(4)+(9)/(4)i
1/(1-2i) = (1+2i)/5 and 3/(1+i) = 3(1-i)/2; their sum is (2/10+4i/10)+(15/10-15i/10) = (17 -11i)/10. Multiplying by (3+4i)/(2-4i) = (3+4i)(2+4i)/20 = (-10+20i)/20 = (-1+2i)/2 gives (17-11i)(-1+2i)/20 = (5+45i)/20 = 1/4 + (9/4)i.
Answer: have no common root
The two quadratic equations can be analyzed for common roots by examining their discriminants and coefficients. If the conditions for having common roots are not satisfied, as indicated in the problem, then the equations will not share any roots, confirming that option A is correct.
Answer: 1 + i(1 ± √3)
Vector from centre i to vertex (2+i) is 2. Adjacent vertices: 2*e^(+-i*pi/3)=2(1/2 +- i*sqrt(3)/2)=1 +- i*sqrt(3). Add centre i: 1 + i(1 +- sqrt(3)).
Answer: γ = α
The correct option is that b3 = b1 because b1 is one of the roots of the quadratic equation formed by b1 and b2. Since b1 is a root, it must satisfy the equation, making it a valid solution.
Answer: ab + bc + ca
Expanding gives 3x^2-2(a+b+c)x+(ab+bc+ca)=0. Equal roots need discriminant 0: 4(a+b+c)^2-12(ab+bc+ca)=0, so (a+b+c)^2=3(ab+bc+ca). Expanding, a^2+b^2+c^2+2(ab+bc+ca)=3(ab+bc+ca), hence a^2+b^2+c^2=ab+bc+ca.
Answer: Statement-1 is true, Statement-2 is false
Statement-1 is true because it reflects a relationship between the coefficients and roots of the quadratics, indicating a certain condition on the discriminants. However, Statement-2 is false as it incorrectly suggests a necessary condition that does not hold true given the established relationships between the variables.
Answer: ax² − |b|x + c = 0
The correct option is right because the roots |α| and |β| are both positive, and the quadratic formed by these roots must maintain the leading coefficient 'a' while adjusting the linear coefficient to account for the absolute value of 'b', which reflects the change in sign of the roots.
Answer: −2
Sum=-3, product=b/2. alpha/beta+beta/alpha=((alpha+beta)^2-2*product)/product=(9-b)/(b/2)=18/b-2. Since b<0, 18/b<0, so the value is < -2.
Answer: sin ucos v [cos(x+y-u-v)+isin(x+y-u-v)]
Numerator = e^(i(x+y)). cot u + i = e^(iu)/sin u and 1 + i tan v = e^(iv)/cos v, so denominator = e^(i(u+v))/(sin u cos v). Dividing gives sin u cos v * e^(i(x+y-u-v)) = sin u cos v[cos(x+y-u-v) + i sin(x+y-u-v)].
Answer: a hyperbola
The expression for the complex number z can be separated into its real and imaginary parts, where the real part is linear in t and the imaginary part is quadratic in t. This combination of a linear term and a quadratic term in the complex plane typically represents a hyperbolic path.
Answer: e^(iα/2)
Product z_1...z_n = exp(i*(a/n^2)*sum r) = exp(i*(a/n^2)*n(n+1)/2) = exp(i*a(n+1)/(2n)) -> exp(ia/2) as n->infinity, i.e. e^(ia/2).
Answer: Both tan⁻¹ θ and tan⁻¹ ϕ are real.
The roots θ and ϕ of the quadratic equation are real numbers, and the inverse tangent function (tan⁻¹) is defined for all real numbers, making both tan⁻¹ θ and tan⁻¹ ϕ real.
Answer: less than α
The polynomial p(x) has no constant term, so p(0)=0, and it also has p(alpha)=0. The second equation is p'(x)=0. By Rolle's theorem applied on [0, alpha], p' has a root strictly between 0 and alpha, i.e. a positive root less than alpha.
Answer: |tan nθ|
With z + 1/z = 2cosθ we have z = e^(iθ). Then (z^(2n) - 1)/(z^(2n) + 1) = (e^(inθ)·2i·sin nθ)/(e^(inθ)·2·cos nθ) = i·tan nθ, whose magnitude is |tan nθ|. Since a modulus cannot be negative, the answer is |tan nθ|.
Answer: None of these
Since 1/w = w^2 and 1/w^2 = w, we have 1/w + 1/w^2 = w + w^2 = -1. Each pair (k+1/w)+(k+1/w^2) = 2k - 1, so the sum over k=1..n is sum(2k-1) = n^2, which is not among the first three forms, hence None of these.
Answer: x² - 2x + 16 = 0
With p+q=2, p^2+q^2=4-2pq and p^4+q^4=(p^2+q^2)^2-2(pq)^2=(4-2pq)^2-2(pq)^2=272 -> 2(pq)^2-16pq-256=0 -> (pq)^2-8pq-128=0 -> pq=16 (taking the positive root). So p,q are roots of x^2-(p+q)x+pq=0 = x^2-2x+16=0.
Q30. How many real values of x satisfy the equation x² - 3|x| + 2 = 0?
Answer: 4
The equation can be analyzed by considering the cases for |x|, leading to two separate quadratic equations: one for x ≥ 0 and another for x < 0. Each case yields two distinct real solutions, resulting in a total of four real solutions.
Q31. Let z = x - i y and z³ = p + i q. Then the value of (x/p + y/q)/(p² + q²) is
Answer: -2
Writing z=(p+iq)^3 gives x=p(p^2-3q^2) and -y=q(3p^2-q^2), so x/p=p^2-3q^2 and y/q=q^2-3p^2. Their sum is -2(p^2+q^2), hence (x/p + y/q)/(p^2+q^2) = -2.
Answer: 0, -1
Put x = (1-p) into x^2 + px + (1-p) = 0: (1-p)[(1-p)+p+1] = 2(1-p) = 0, so p = 1. The equation becomes x^2 + x = 0 -> x(x+1) = 0, giving roots 0 and -1.
Answer: 49/4
Since 4 is a root of x^2+px+12=0: 16+4p+12=0 -> p=-7. For x^2-7x+q=0 to have repeated roots, discriminant 49-4q=0 -> q=49/4.
Answer: 1
The sum of the squares of the roots of a quadratic equation can be expressed in terms of its coefficients. By analyzing the expression derived from the equation, we find that the minimum occurs when a equals 1, leading to the least value for the sum of the squares.
Answer: -1, 1 - 2ω, 1 - 2ω²
The equation can be rewritten as (x - 1)³ = -8, which simplifies to (x - 1) = -2, leading to x = -1. The other roots are derived from the cube roots of -8, which correspond to the expressions 1 - 2ω and 1 - 2ω², making option C the correct choice.
Answer: 0
The triangle-inequality becomes an equality only when z1 and z2 point in the same direction, i.e. they have equal arguments. Therefore arg z1 - arg z2 = 0.
Q37. If ω = (z)/(z - (i)/(3)) and |ω| = 1, then the point z must lie on
Answer: a straight line
With w=z/(z-i/3) and |w|=1, we get |z|=|z-i/3|, i.e. z is equidistant from 0 and i/3. That locus is the perpendicular bisector of the segment joining them, a straight line. Answer: option 2.
Answer: (-∞, 4)
The roots of the quadratic equation must be less than 5, which can be determined by analyzing the vertex and the behavior of the parabola. For the roots to be smaller than 5, the value of k must be such that the vertex of the parabola, which occurs at x = k, is less than 5, leading to the conclusion that k must be in the interval (-∞, 4).
Q39. If the complex number z satisfies |z + 4| ≤ 3, what is the greatest possible value of |z + 1|?
Answer: 6
The inequality |z + 4| ≤ 3 describes a circle in the complex plane centered at -4 with a radius of 3. The farthest point from -1 within this circle occurs at the point on the boundary of the circle that is directly opposite -1, which results in a maximum distance of 6.
Q40. If the conjugate of a complex number is given by 1/(i−1), then the complex number is:
Answer: −1/(i+1)
The conjugate of a complex number is obtained by changing the sign of its imaginary part. Since the given conjugate is 1/(i−1), the original complex number must be the negative of the conjugate of that expression, which simplifies to −1/(i+1).
Answer: greater than −4ab
The quadratic equation has non-real roots when its discriminant is negative, which implies that the parabola opens upwards and does not intersect the x-axis. Therefore, the expression 3b²x² + 6bcx + 2c², being a quadratic in x with positive leading coefficient, will always yield values greater than a certain minimum, which in this case is greater than -4ab.
Q42. If the complex number z satisfies |z - 4/z| = 2, what is the greatest possible value of |z|?
Answer: √5 + 1
The equation |z - 4/z| = 2 represents a geometric condition in the complex plane, where the distance from the point z to the point 4/z is constant. By analyzing the relationship and maximizing the modulus |z| under this constraint, we find that the greatest possible value occurs when z is positioned optimally, leading to the result of √5 + 1.
Q43. How many complex numbers z satisfy the condition |z - 1| = |z + 1| = |z - i|?
Answer: 1
The condition |z - 1| = |z + 1| describes the perpendicular bisector of the segment connecting the points 1 and -1 on the complex plane, which is the vertical line x = 0. The condition |z - i| = |z + 1| describes the perpendicular bisector of the segment connecting -1 and i, which intersects the first line at a single point, resulting in exactly one complex number that satisfies all three conditions.
Answer: 1
The roots α and β of the equation x² - x + 1 = 0 can be expressed in terms of complex numbers, specifically as α = e^(iπ/3) and β = e^(-iπ/3). Using De Moivre's theorem, we find that αⁿ + βⁿ can be computed using the periodicity of complex exponentials, leading to α²⁰⁰⁹ + β²⁰⁰⁹ = 1.
Answer: β is greater than 1
For the quadratic equation to have two distinct roots on the vertical line Re(z) = 1, the roots must be complex conjugates of the form 1 + yi and 1 - yi. This requires that the discriminant is positive, which leads to the condition that β must be greater than 1 to ensure the roots are real and distinct.
Answer: 6, 1
Sachin got the x-coefficient right, so the true sum of roots = 4+3 = 7. Rahul got the constant right, so the true product = 3*2 = 6. The correct equation is x^2-7x+6=0, whose roots are 6 and 1.
Answer: 18
The polynomial p(x) is a quadratic function that has a double root at x = -1, meaning it can be expressed as p(x) = k(x + 1)² for some constant k. Given that p(-2) = 2, we can find k and subsequently calculate p(2), which results in p(2) = 18.
Q48. If z ≠ 1 and z²/(z − 1) is real, then the point represented by the complex number z lies on:
Answer: either on the real axis or on a circle passing through the origin.
The expression z²/(z − 1) being real implies that the complex number z must satisfy certain conditions related to its argument and magnitude, which geometrically corresponds to points either lying on the real axis or on a circle that intersects the origin.
Answer: 1: 2: 3
The equations share a common root, which implies that the coefficients of the second equation can be expressed in terms of the coefficients of the first equation. By substituting the common root into both equations and simplifying, we find that the ratio of the coefficients a, b, and c must be 1:2:3 to maintain the equality.
Q50. If z is a complex number of unit modulus and argument θ, then arg((1 + z)/(1 + z̄)) equals:
Answer: θ
The argument of the quotient of two complex numbers is the difference of their arguments. Since z has unit modulus, its conjugate z̄ has the same modulus, and the argument of (1 + z) and (1 + z̄) can be shown to differ by θ, leading to the conclusion that arg((1 + z)/(1 + z̄)) equals θ.