Zadanie 1g
Under the double-root assumption, state the b-values for which f(x)=0 has (i) at least one real root; (ii) at least one purely imaginary root.
Complex coordinates and conjugates
Represent complex numbers in Cartesian form and on an Argand diagram; calculate conjugates, magnitudes and arguments.
Tę umiejętność sprawdza 46 zadań wzorowanych na arkuszach IB z lat 2022–2026.
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Zadanie 2b
Sketch the locus r=6 on an Argand diagram, labelling every axis intercept.
Zadanie 2f
Sketch , −π/2 ≤ θ ≤ π/2, on an Argand diagram. Label all axis intercepts.
Zadanie 2b
(i) Express in terms of a,b,c,d. (ii) Hence express w in these integers.
Zadanie 2c
Take z1 = 4 + 8i, z2 = c + 7i. (i) Show that w=16c−56. (ii) Show that the only possible positive integer c are 4 and 5. (iii) For c=5, find u,v,w,t.
Zadanie 2d
Use 39 = 12 + 22 + 32 + 52 and the complex construction to find one valid route with t=39.
Zadanie 2e
For a route define z = (v + wi)/(t − u). Show that the route (48,54,16;74) gives z = (27 + 8i)/13.
Zadanie 2f
For the route (48,54,16;74), the generating numbers also satisfy . (i) Use t−u to prove c²+d²=13. (ii) Given c>d, find z₁ and z₂.
Zadanie 8d
The three solutions in (c) are the roots of (−iz)3 = c. The three solutions in (c) form a triangle in the complex plane. Find its area.
Zadanie 12a
Solve Q(z)=0, giving the roots (i) in Cartesian form; (ii) in modulus–argument form.
Zadanie 12c
Plot these six roots on an Argand diagram. Label them z₁,…,z₆ in counterclockwise order, starting with z₁ on the positive real axis.
Zadanie 12e
The already-specified quadratic factor is z2 − 9. P is the product of three quadratic polynomials with real coefficients. Find the other two quadratic factors.
Zadanie 12f
Label the six roots counterclockwise, with z1 = 3 on the positive real axis. Find the total length of the five line segments joining z₁ to the other roots.
Zadanie 12b
In Cartesian form, (i) find z1, z2; (ii) deduce z3, z4.
Zadanie 12c
(i) Plot the four roots on an Argand diagram. (ii) Find the area of their convex quadrilateral.
Zadanie 12d
These four roots obey . Their reciprocals obey , where p, q, r are integers with no common factor and p > 0. Find p, q, r.
Zadanie 12e
(i) Find 1/z1 in Cartesian form. (ii) Hence deduce the area of the convex quadrilateral whose vertices represent 1/z1, 1/z2, 1/z3, 1/z4.
Zadanie 12a
Prove that points on this locus satisfy x2 + y2 + 2x − 4y − 20 = 0.
Zadanie 12b
For real p, suppose , where the quotient is defined and nonzero. Show that x2 + y2 + (p − 2)x − (p + 2)y − 2p = 0.
Zadanie 12c
Two distinct roots z1, z2 of a monic quartic z4 + az3 + bz2 + cz + d = 0 with real coefficients satisfy the circle condition and . Use (a) and (b) to find these roots.
Zadanie 11a
Show that w=2+2i.
Zadanie 11d
(i) Show that . (ii) Find b².
Zadanie 12a
In terms of a,b, find (i) the real part of φ; (ii) its imaginary part.
Zadanie 2b
Show that the complex roots of x2 − x + 1 = 0 form a reciprocal pair.
Zadanie 2g
Hence find all roots of F(x) = 0 exactly, stating multiplicities.
Zadanie 8a
Prove for every complex w that ww∗ = ∣w∣2.
Zadanie 8b
Find z in Cartesian form.
Zadanie 12c
Plot A,B,C on an Argand diagram, labelling the axes and points.
Zadanie 7
For m>0, let P(z) = 5m − mz + (20/m)z2 − z3. Given that z−2i is a factor, find every root of P(z)=0.
Zadanie 11d
Express z=a+bi to solve z²=3iz*, with z≠0.
Zadanie 7
The complex number z=3+qi satisfies z2 + 2iz = −p + 30i, where p,q are real. Find p and q.
Zadanie 7
The complex number z=3+qi satisfies z2 + 2iz = −p + 30i, where p,q are real. Find p and q.
Zadanie 12a
Expand in the form A+iB, with A and B expressed using c,s.
Zadanie 12a
Expand in the form A+iB, with A and B expressed using c,s.
Zadanie 2a
(i) Given the nonreal root 2+2i of f(z)=0, state the other nonreal root. (ii) Verify that the mean of these two roots is 2.
Zadanie 2e
For a≠r, use g′(a) = 2(a − r)b2 to deduce that the two nonreal roots are .
Zadanie 2f
The figure concerns g(x) = (x + 1)2[(x − a)2 + 4]. The curve touches the x-axis at R=(−1,0). Its tangent at A=(a,64) crosses the x-axis at T=(1,0). Points C₁,C₂ represent respectively the nonreal roots with positive and negative imaginary parts of g(z)=0, drawn on…
- Materiał graficzny
Zadanie 2c
Using (u − v)2 + (v − w)2 + (w − u)2 = 2(q2 − 3pr)/r2, prove that q²<3pr forces at least one original root α,β,γ to be nonreal.
Zadanie 2d
Consider Fc(x) = x3 + cx2 − 6x + 1, where c is real. You may use the sufficient condition: a real cubic x³+px²+qx+r, r≠0, has a nonreal root if q²<3pr. Show that F₁₃(x)=0 has a nonreal root.
Zadanie 2e
Consider Fc(x) = x3 + cx2 − 6x + 1, where c is real. You may use the sufficient condition: a real cubic x³+px²+qx+r, r≠0, has a nonreal root if q²<3pr. A student claims that q²≥3pr guarantees all roots are real. (i) By varying the positive integer c, find the…
Zadanie 2f
A real quartic x4 + px3 + qx2 + rx + s = 0, s≠0, has nonzero roots α,β,γ,δ. Their reciprocals u,v,w,z satisfy u+v+w+z=−r/s and uv+uw+uz+vw+vz+wz=q/s. (i) Express u²+v²+w²+z² in terms of q,r,s. (ii) Hence state a sufficient inequality in q,r,s for the original…
Zadanie 2g
A real quartic with coefficients q,r,s as in x⁴+px³+qx²+rx+s has a nonreal root whenever r²<2qs. Use this to show that x4 + 2x3 + 4x2 − 3x + 2 = 0 has a nonreal root.
Zadanie 2h
Consider x4 + 8x3 − 15x2 − 9x − 2 = 0, which has exactly one integer root. You may use two sufficient criteria: a quartic x⁴+px³+qx²+rx+s has a nonreal root if r²<2qs; a cubic x³+px²+qx+r, r≠0, has one if q²<3pr. (i) State what the quartic criterion tells you here.…
Zadanie 9a
Express z₁z₂ in Cartesian form.
Zadanie 12d
For this 120° configuration, suppose z₁,z₂ are the distinct roots of z2 + az + b = 0 with real a,b. Use z12 + z22 = −z1z2 to show that a²−b=0.
Zadanie 5a
Use the product of the four roots to find a.