Justify elementary numerical or algebraic claims through a logically connected sequence of equivalent statements.
Tę umiejętność sprawdza 31 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Rectangle k has width 1, spans k − 1 ≤ x ≤ k, and has height (2k + 1)2. A quadratic curve y=f(x) passes through their top-right corners. (i) State f(x). (ii) Compare areas to show . (iii) Use a similar area comparison to show…
For un = (2n + 1)3 − (2n − 1)3, (i) suggest an expression for ; (ii) prove it using the definition of u.
The two bounds to verify are and . Again un = (2n + 1)3 − (2n − 1)3. (i) Express u_n as a quadratic in v = 2n + 1. (ii) Hence express in terms of T2(n), T1(n), T0(n). (iii) Hence show that…
For general 0<p<1 with p≠q, replace Rₙ by (p+q)Rₙ in the recurrence and prove Rn+1 − Rn = (q/p)(Rn − Rn−1).
(i) Use R₀=0 to express R₂−R₁ in terms of p,q,R₁. (ii) Show that R3 − R2 = (q/p)2R1. (iii) Deduce the general expression for Rₙ−Rₙ₋₁.
(i) By adding successive differences, or otherwise, prove . (ii) Given R1 = [1 − q/p]/[1 − (q/p)w], deduce Rn = [1 − (q/p)n]/[1 − (q/p)w].
Use the coordinate identities to show that every point of , −π/2 ≤ θ ≤ π/2, satisfies (x − 3)2 + y2 = 9.
The comparison area from 11d is bounded by y = −7/4 + (17/16)x − (37/64)x2, the x-axis and the lines x = 0, x = 1. Given ln(486/125) ≈ 1.36 to three significant figures, decide whether the area from 11d overestimates or underestimates the exact area.…
Now instead suppose f has a vertical asymptote, oblique asymptote y=x−5, and at least one point of zero gradient. Show that a=−3.
(i) Prove the identity . (ii) Hence give the equation of its oblique asymptote. (iii) Give one point on the curve at which the tangent is parallel to that asymptote.
The solution relation is , with A > 0, x ≠ 0, y ≠ 3x/2 and y ≠ −x/2. Restrict the relation in (f) to . Show algebraically that it produces y = FA(x).
Show that V = 9x3 − 3(a + b)x2 + abx.
For the remaining parts, set a = s2 + 2st and b = t2 + 2st, where s,t are positive integers and a<b. Show that t > s.
For the remaining parts, set a = s2 + 2st and b = t2 + 2st, where s,t are positive integers and a<b. You are given . Substitute into to show that xm = st/3.
Let xm denote the interior height maximizing V on [0, a/3]. For the remaining parts, set a = s2 + 2st and b = t2 + 2st, where s,t are positive integers and a<b. Let Vm = xm(a − 3xm)(b − 3xm). (i) Explain why s divisible by 3 guarantees that both xm and…
A separate function is h(x) = 9x2/(3x + 2), x ≠ −2/3. Region R is bounded by its curve, the line y = 1, and x = 1, x = 2. (i) Establish . (ii) Hence or otherwise, calculate the area of R in the form p + qln r, with positive…
Show that .
Prove that (5n + 3)2 − (5n − 3)2 is divisible by 30 for every positive integer n.
(i) Show that the arithmetic and geometric means of z_j=x_j² are equal when all parameters x_j are equal. (ii) Prove Mn(S) = (S/n − 1)n/2 for every admissible n.
(i) Show that r=−c/m. (ii) For admissible L, show that L(x) = mx + 2m2/(m − 1). (iii) State the further excluded value of m.
Exactly three admissible L have integer r,m,c. One is L(x) = 3x + 9. Use the general expression to find the other two. For an admissible linear function, c = 2m2/(m − 1) and r = −c/m, with m ≠ 0, 1.
Prove that a<0 and 0<b<a²/4 imply exactly four distinct x-axis intercepts.
Solve E_a and H_b simultaneously to show that their four intersections have coordinates , where the two signs are independent.
(i) Verify algebraically that C1(n) = n(n − 1) + 1. (ii) The count 421 occurs in this r=1 scheme. Determine its index n.
(i) Show that C1(n) + C1(n + 1) = 2n2 + 2. (ii) Describe this result in words as a rearrangement of tokens into two equal square arrays and isolated tokens. (iii) For n=3 sketch such a rearrangement, distinguishing tokens from the two counts.
Show that 4C1(n) − 3 is the square of an odd integer for every positive integer n.
Prove that their sum is divisible by 3.
Prove that the sum of their squares is never divisible by 3.
You may use u+v+w=−q/r and uv+vw+wu=p/r. (i) Show that u2 + v2 + w2 = (q2 − 2pr)/r2. (ii) Hence show that (u − v)2 + (v − w)2 + (w − u)2 = 2(q2 − 3pr)/r2.
Prove that A contains more than B whenever 7<t≤8.
Show that (4t)2 + (t2 − 4)2 = (t2 + 4)2.
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