DP Mathematics: Analysis and Approaches Questionbank

AHL 2.12—Factor and remainder theorems, sum and product of roots
Description
[N/A]Directly related questions
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17N.1.AHL.TZ0.H_3a:
Given that q(x)q(x) has a factor (x−4)(x−4), find the value of kk.
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21M.3.AHL.TZ1.1f:
Consider the function g(x)=x3-3cx+dg(x)=x3−3cx+d for x∈ℝ and where c , d∈ℝ.
Find all conditions on c and d such that the graph of y=g(x) has exactly one x-axis intercept, explaining your reasoning.
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21M.1.AHL.TZ2.7:
The cubic equation x3-kx2+3k=0 where k>0 has roots α, β and α+β.
Given that αβ=-k24, find the value of k.
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21N.2.AHL.TZ0.6b:
The equation 2x2-5x+1=0 has two real roots, α and β.
Consider the equation x2+mx+n=0, where m, n∈ℤ and which has roots 1α3 and 1β3.
Without solving 2x2-5x+1=0, determine the values of m and n. -
21N.2.AHL.TZ0.6a:
Prove the identity (p+q)3-3pq(p+q)≡p3+q3.
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22M.3.AHL.TZ2.2f.i:
Find an expression for α2+β2+γ2+δ2 in terms of p and q.
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22M.3.AHL.TZ2.2h.ii:
Write down the integer root of this equation.
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22M.2.AHL.TZ1.8a:
Write down an expression for the product of the roots, in terms of k.
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22M.2.AHL.TZ1.8b:
Hence or otherwise, determine the values of k such that the equation has one positive and one negative real root.
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22M.1.AHL.TZ2.12d:
Use the result from part (c)(ii) to show that a2-3b=0.
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22M.1.AHL.TZ2.12e:
Consider the equation z2+az+12=0, where z∈ℂ and a∈ℝ.
Given that 0<α-θ<π, deduce that only one equilateral triangle Z1OZ2 can be formed from the point O and the roots of this equation.
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SPM.1.AHL.TZ0.11c:
Find the area of triangle UVW.
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SPM.1.AHL.TZ0.11b:
Find u, v and w expressing your answers in the form reiθ, where r>0 and −π<θ⩽π.
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SPM.1.AHL.TZ0.11d:
By considering the sum of the roots u, v and w, show that
cos5π18+cos7π18+cos17π18=0.
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SPM.1.AHL.TZ0.11a:
Express −3+√3i in the form reiθ, where r>0 and −π<θ⩽π.
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17N.1.AHL.TZ0.H_3b:
Hence or otherwise, factorize q(x) as a product of linear factors.
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17M.2.AHL.TZ2.H_11a:
Given that x2−1 is a factor of f(x) find the value of a and the value of b.
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17M.2.AHL.TZ2.H_11b:
Factorize f(x) into a product of linear factors.
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17M.2.AHL.TZ2.H_11d:
Using your graph state the range of values of c for which f(x)=c has exactly two distinct real roots.
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18M.1.AHL.TZ1.H_1:
Let f(x) = x4 + px3 + qx + 5 where p, q are constants.
The remainder when f(x) is divided by (x + 1) is 7, and the remainder when f(x) is divided by (x − 2) is 1. Find the value of p and the value of q.
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18M.2.AHL.TZ2.H_2:
The polynomial x4+px3+qx2+rx+6 is exactly divisible by each of (x−1), (x−2) and (x−3).
Find the values of p, q and r.
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18N.2.AHL.TZ0.H_6a:
Given that (x−5) is a factor of P(x), find a relationship between a, b and c.
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18N.2.AHL.TZ0.H_6b:
Given that (x−5)2 is a factor of P(x), write down the value of P′(5).
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18N.2.AHL.TZ0.H_6c:
Given that (x−5)2 is a factor of P(x), and that a=2, find the values of b and c.
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18N.1.AHL.TZ0.H_8:
Consider the equation z4+az3+bz2+cz+d=0, where a, b, c, d∈R and z∈C.
Two of the roots of the equation are log26 and i√3 and the sum of all the roots is 3 + log23.
Show that 6a + d + 12 = 0.
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16N.1.AHL.TZ0.H_5:
The quadratic equation x2−2kx+(k−1)=0 has roots α and β such that α2+β2=4. Without solving the equation, find the possible values of the real number k.
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19M.2.AHL.TZ1.H_11a:
Show that abc=8.
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19M.2.AHL.TZ1.H_11b:
Show that one of the real roots is equal to 1.
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19M.2.AHL.TZ1.H_11c:
Given that q=8d2, find the other two real roots.