Zadanie 2
A differentiable function has derivative f′(x) = 2x2 + x − 4 for all real x. Find where f is decreasing.
Quadratic solutions and discriminants
Solve quadratic equations and inequalities by factorization, completing the square or the quadratic formula; use the discriminant to classify roots and constrain parameters.
Tę umiejętność sprawdzają 53 zadania wzorowane na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 53 Otwórz w wyszukiwarce z filtrami
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 5b
A and B are independent. Find all possible x.
Zadanie 10b
The graph has horizontal tangents at x=3 and at a distinct x=k. (i) Show that c=9. (ii) Find k. (iii) Classify the stationary point at x=k, with justification.
Zadanie 11b
Hence solve for 0 < θ < π, with θ ≠ π/2.
Zadanie 12d
Now instead suppose f has a vertical asymptote, oblique asymptote y=x−5, and at least one point of zero gradient. Show that b ≥ −9.
Zadanie 2c
Show that the only solutions of dV/dx = 0 are .
Zadanie 4a
Show that their intersection coordinates satisfy 2x2 + (3 − m)x + 3 = 0.
Zadanie 4b
Hence determine every slope m for which the line is tangent to the parabola.
Zadanie 5a
Find the allowable values of k.
Zadanie 5b
Find the allowable values of k if the two solutions also have the same sign.
Zadanie 8
A calibration network contains n sensors. Each unordered pair is scheduled to exchange three test messages. One sensor is removed after participating in exactly eight messages; all pairs among the remaining sensors complete their three messages. In total 701…
Zadanie 10d
Rods are supplied in boxes of 50. Find the maximum number of tiers that can be built using 12 boxes.
Zadanie 10c
The already-investigated geometric case is k = −6. A second value of k also makes the sequence geometric. (i) Show that k2 + 7k + 6 = 0. (ii) Find the first three terms for the second value. (iii) State the sum S2m of its first 2m terms.
Zadanie 12d
A line y = mx, where m is an integer, is tangent to C. Determine every possible m.
Zadanie 12e
The tangent y = −5x touches C at B. Find yB.
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 3a
(i) State c. (ii) Find a.
- Materiał graficzny
Zadanie 4a
Show that p=−4.
Zadanie 4b
Find q.
Zadanie 9a
Show that the greatest possible k is 6.
Zadanie 9b
Take k=6. Find (i) the equation of the axis of symmetry; (ii) the coordinates of the minimum.
Zadanie 2
Solve 2 ⋅ 16x + 7 ⋅ 4x − 4 = 0 over the reals.
Zadanie 2b
Verify that the allowed choice a=4,b=8,c=4 gives a graph with exactly one x-intercept.
Zadanie 2c
Show that the probability of exactly one x-intercept is 3/64.
Zadanie 2d
For generated quadratics with two distinct x-intercepts, determine the possible values of ac.
Zadanie 2e
For quadratics with two distinct x-intercepts: (i) show that ac=1 gives six possible quadratics; (ii) show that ac=2 gives twelve.
Zadanie 2f
Find the probability p that a generated quadratic has two distinct x-intercepts. Show how you count the possibilities.
Zadanie 2g
Now use the continuous model q(x) = x2 + 2Zx + 4, where Z is normal with mean 0.5 and standard deviation 1.5. Find the probability of two distinct x-intercepts.
Zadanie 2h
In the continuous model q(x) = x2 + 2Zx + 4, with Z normal of mean 0.5 and standard deviation 1.5, suppose the graph has two distinct x-intercepts and . Find the conditional probability that both intercepts are greater…
Zadanie 4b
(i) Show that 16k2 − 40k + 9 = 0. (ii) The roots of this equation are 1/4 and 9/4. State the valid value of k and justify your choice.
Zadanie 5b
(i) For m > 0, state the number of real solutions of f(x) = g(x). (ii) Find m for which there is exactly one real solution. (iii) Find all m for which there are two distinct solutions and both are nonnegative.
Zadanie 10c
Show that p=8/3.
Zadanie 10d
Find b and c.
Zadanie 10e
Find the y-coordinate of the vertex of g.
Zadanie 2a
(i) Solve x2 − 6x + 1 = 0 exactly. (ii) Verify that the two roots are reciprocals.
Zadanie 5
The expression h(x) = log3(3x2 − rx + r − 2) must define a real-valued function for every real x. Find all possible real r.
Zadanie 6a
Solve Q(x)<0.
Zadanie 6b
Find the greatest k for which f is real-valued throughout x≤k and has an inverse on that domain. Justify your answer.
Zadanie 12b
Find all k>0 for which the curve has a local maximum or minimum.
Zadanie 4
Find all real k for which e2x + ln k = 4ex has at least one real solution x.
Zadanie 8c
Find sink exactly.
Zadanie 9c
Find all θ for which there are two distinct possible positions of M on AB.
Zadanie 11b
Using the quadratic for experiment A, show that . The relation for experiment A is y2 − (4r + 1)y + r2 − r = 0.
Zadanie 11e
One yellow counter is added to the container in experiment B. The probability of exactly one red and two yellow counters becomes 3/2 times its previous value. Find the original y.
Zadanie 12b
Find t.
Zadanie 10b
(i) Show that intersection abscissas satisfy x2 − 3dx + 6d = 0. (ii) Hence show that 9d2 − 24d > 0. (iii) Find the possible values of d.
Zadanie 10c
For d=4, find q−p in simplest surd form. Intersection abscissas satisfy x2 − 3dx + 6d = 0.
Zadanie 10b
(i) Show that intersection abscissas satisfy x2 − 3dx + 6d = 0. (ii) Hence show that 9d2 − 24d > 0. (iii) Find the possible values of d.
Zadanie 10c
For d=4, find q−p in simplest surd form. Intersection abscissas satisfy x2 − 3dx + 6d = 0.
Zadanie 12c
(i) Using , show that θ=π/10 and θ=3π/10 solve . (ii) Hence show that .
Zadanie 12c
(i) Using , show that θ=π/10 and θ=3π/10 solve . (ii) Hence show that .
Zadanie 5b
The third term is the arithmetic mean of the second and fourth terms. Find all possible x.