IB Mathematics Analysis and Approaches Topic AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

Consider the quadratic function $f (x) = x^{2} - 2 k x + k^{2} - 3$ , where $k \in R$ , with roots $α$ and $β$ .

Express the sum and product of the roots of $f (x) = 0$ in terms of $k$ .

[2]

Show that $α^{2} + β^{2} = 2 (k^{2} + 3)$ .

[2]

Given that $α^{2} + β^{2} = \frac{1}{α ^{2} + β ^{2}}$ , find the possible values of $k$ .

[5]

For the larger positive value of $k$ found previously, determine the range of values of $m$ such that $f (x) = m$ has exactly two distinct real roots.

[3]

Question 2

HLPaper 1

Let $f (x) = 3 x^{4} + a x^{3} + b x^{2} + c x + d$ , where $a, b, c, d \in Z$ , and suppose $x^{2} - x - 6$ is a factor of $f (x)$ . Additionally, it is given that $f (2) = - 72$ and the sum of the roots of $f (x) = 0$ is $- 2$ .

Find the values of $a$ , $b$ , $c$ , and $d$ .

[5]

Factorize $f (x)$ completely into linear factors over the real numbers.

[4]

Question 3

HLPaper 1

Consider the polynomial $P (t) = t^{4} - 5 t^{3} + a t^{2} + b t + c$ , where $a, b, c \in R$ .

If $(t + 1)$ is a factor of $P (t)$ , determine an equation involving $a$ , $b$ , and $c$ .

[2]

When $(t + 1)^{2}$ is a factor of $P (t)$ , state the value of $P^{'} (- 1)$ .

[1]

Given that $(t + 1)^{2}$ is a factor of $P (t)$ and that $a = 3$ , calculate the values of $b$ and $c$ .

[3]

Question 4

HLPaper 2

Consider $f(x)=\frac{x^3+4x^2+x+\lambda}{x^2 - x - 6},\qquad x\in\mathbb{R}\setminus\{-2,3\},\ \lambda\in\mathbb{R}\setminus\{-6, -66\}.$

Show the vertical asymptotes are independent of $λ$ and find them.

[2]

Write $f (x) = q (x) + \frac{r ( x )}{x ^{2} - x - 6}$ with $q$ and $r$ are linear expressions.

[4]

Deduce the oblique asymptote.

[2]

Find the intersection point(s) of the graph with its oblique asymptote.

[4]

Solve $f (x) \geq x + 5$ for $λ = - 30$ .

[6]

Question 5

HLPaper 1

The cubic equation $2 x^{3} - 5 x^{2} + k x - 3 = 0$ has roots $α$ , $β$ , and $γ$ .

Given that $α^{2} + β^{2} + γ^{2} = \frac{13}{4}$ , find the value of $k$ .

[3]

IB Mathematics Analysis and Approaches AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

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IB Mathematics Analysis and Approaches AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

Unlock all 22 questions with a free account