Algebra

State the quotient and the remainder after dividing $2x^3 + 3x^2 + 10$ by $(x + 2)$.

Feb/March 2016

Solve the inequality $|x - 5| < |2x + 3|$ for $x$.

Feb/March 2016

The polynomial $p(x)$ is given by $p(x)=ax^3+bx^2-17x-a$, where $a$ and $b$ are constants. It is also known that $(x+2)$ is a factor of $p(x)$, and that the remainder on dividing $p(x)$ by $(x-2)$ is $28$.

Feb/March 2017

Solve the inequality $|5x + 2| > |4x + 3|$.

Feb/March 2018

Where $x$ satisfies $|2x + 3| = |2x - 1|$, determine the value of

Feb/March 2019

Determine the quotient when $4x^3 + 8x^2 + 11x + 9$ is divided by $(2x + 1)$, and prove that the remainder equals $5$.

Feb/March 2019

Determine the quotient when $4x^3 + 17x^2 + 9x$ is divided by $x^2 + 5x + 6$, and show that the remainder is $18$.

Feb/March 2020

Sketch, on the same diagram, the graphs of $y = |x + 2k|$ and $y = |2x - 3k|$, where $k$ is a positive constant. Give, in terms of $k$, the coordinates of the points where each graph crosses the axes.

Feb/March 2020

Sketch the graphs of $y = |3x - 5|$ and $y = x + 2$ on the same set of axes.

Feb/March 2021

Solve for $x$ in $|5x - 2| = |4x + 9|$.

Feb/March 2022

The polynomial $p(x)$ has the definition $p(x) = 4x^3 + 16x^2 + 9x - 15$.

Feb/March 2022

The polynomial $p(x)$ is given by $p(x) = ax^3 - ax^2 + ax + b$, with $a$ and $b$ as constants. You are told that $(x + 2)$ is a factor of $p(x)$, and that the remainder is $35$ when $p(x)$ is divided by $(x - 3)$.

Feb/March 2023

Sketch the graph of $y = \lvert 3x - 7 \rvert$, giving the coordinates of the points at which the graph crosses the axes.

Feb/March 2024

Solve the inequality $|2x - 3| > 5$.

May/June 2010

The polynomial $x^3 + 3x^2 + 4x + 2$ is represented by $f(x)$.

May/June 2010

Solve the inequality $|2x - 1| < |x + 4|$.

May/June 2010

The polynomial $2x^3 + ax^2 + bx + 6$, with $a$ and $b$ as constants, is written as $p(x)$. It is stated that dividing $p(x)$ by $(x - 3)$ leaves a remainder of $30$, and dividing $p(x)$ by $(x + 1)$ leaves a remainder of $18$.

May/June 2010

Solve $|2x - 1| < |x + 4|$.

May/June 2010

Let $p(x)$ denote the polynomial $2x^3 + ax^2 + bx + 6$, where $a$ and $b$ are constants. It is stated that dividing $p(x)$ by $(x - 3)$ leaves a remainder of $30$, and dividing $p(x)$ by $(x + 1)$ leaves a remainder of $18$.

May/June 2010

Solve the equation $|3x + 4| = |2x + 5|$.

May/June 2011

The polynomial $f(x)$ is given by $f(x) = 3x^3 + ax^2 + ax + a$, with $a$ as a constant.

May/June 2011

The cubic polynomial $p(x)$ is given by $p(x) = 6x^3 + ax^2 + bx + 10$, with $a$ and $b$ as constants. It is also stated that $(x + 2)$ is a factor of $p(x)$ and that, on division of $p(x)$ by $(x + 1)$, the remainder equals 24.

May/June 2011

The cubic polynomial $p(x)$ is specified by $p(x) = 6x^3 + ax^2 + bx + 10$, where $a$ and $b$ are constants. It is given that $(x + 2)$ is a factor of $p(x)$ and that, when $p(x)$ is divided by $(x + 1)$, the remainder is $24$.

May/June 2011

Find the solution of $|x^3 - 14| = 13$, and include every step in your working.

May/June 2012

The polynomial $p(x)$ is given by $p(x) = ax^3 - 3x^2 - 5x + a + 4$, with $a$ a constant.

May/June 2012

Determine the values of $x$ for which the inequality $|x + 3| < |2x + 1|$ holds.

May/June 2012

Determine the quotient when $8x^3 - 4x^2 - 18x + 13$ is divided by $4x^2 + 4x - 3$, and show that the remainder is $4$.

May/June 2012

Solve $|x^3 - 14| = 13$, showing all of your working.

May/June 2012

The polynomial $p(x)$ is specified as $p(x) = ax^3 - 3x^2 - 5x + a + 4$, where $a$ is constant.

May/June 2012

The polynomial $ax^3 - 5x^2 + bx + 9$, with $a$ and $b$ as constants, is written as $p(x)$. You are told that $(2x + 3)$ is a factor of $p(x)$, and that the remainder when $p(x)$ is divided by $(x + 1)$ is $8$.

May/June 2013

Determine the solution of the inequality $|x - 8| > |2x - 4|$.

May/June 2013

The polynomial $2x^3 + ax^2 - ax - 12$, with $a$ as a constant, is represented by $p(x)$.

May/June 2013

The polynomial $ax^3 - 5x^2 + bx + 9$, where $a$ and $b$ are constants, is represented by $p(x)$. It is stated that $(2x + 3)$ is a factor of $p(x)$, and that dividing $p(x)$ by $(x + 1)$ leaves a remainder of $8$.

May/June 2013

Solve $|3x - 2| \ge |x + 4|$.

May/June 2014

Determine the quotient when $6x^4 - x^3 - 26x^2 + 4x + 15$ is divided by $(x^2 - 4)$, and check that the remainder is $7$.

May/June 2014

The polynomial $p(x)$ is given by $p(x)=x^3+2x+a$, with $a$ a constant.

May/June 2014

The polynomial $p(x)$ is given by $p(x) = x^3 + 2x + a$, where $a$ is a constant.

May/June 2014

Since $(x + 2)$ is a factor of $4x^3 + ax^2 - (a + 1)x - 18$, determine the value of the constant $a$.

May/June 2015

Find the value of the constant $a$ if $(x + 2)$ is a factor of $4x^3 + ax^2 - (a + 1)x - 18$.

May/June 2015

The polynomial $p(x)$ is given by $p(x) = 8x^3 + 30x^2 + 13x - 25$.

May/June 2016

Determine the quotient and remainder when $2x^3 - 7x^2 - 9x + 3$ is divided by $x^2 - 2x + 5$.

May/June 2016

Find the quotient together with the remainder when $2x^3 - 7x^2 - 9x + 3$ is divided by $x^2 - 2x + 5$.

May/June 2016

Solve for the values of $x$ in $|4 - x| \leq |3 - 2x|$.

May/June 2017

Find the solutions of $|x + a| = |2x - 5a|$, expressing $x$ in terms of the positive constant $a$.

May/June 2017

Apply the factor theorem to demonstrate that $(x + 2)$ is a factor of $6x^3 + 13x^2 - 33x - 70$, then factorise the expression fully.

May/June 2017

Find the solutions to $|x + a| = |2x - 5a|$, giving $x$ in terms of the positive constant $a$.

May/June 2017

Use the factor theorem to prove that $(x + 2)$ is a factor of $6x^3 + 13x^2 - 33x - 70$, and then factorise the expression fully.

May/June 2017

The variables $x$ and $y$ are linked by $y = A \times B^{\ln x}$, with $A$ and $B$ as constants. The plot of $\ln y$ against $\ln x$ is a straight line that goes through the points $(2.2,\,4.908)$ and $(5.9,\,11.008)$, as illustrated in the diagram.

May/June 2018

The cubic polynomial $f(x)$ is written as $f(x) = x^3 + ax^2 + 14x + a + 1$, with $a$ as a constant. It is stated that $(x + 2)$ is a factor of $f(x)$.

May/June 2018

Solve $|3x - 2| < |x + 5|$.

May/June 2018

Determine the quotient when $x^4 - 2x^3 + 8x^2 - 12x + 13$ is divided by $x^2 + 6$ and confirm that the remainder is $1$.

May/June 2018

Solve for $x$ in the inequality $|3x - 2| < |x + 5|$.

May/June 2018

Determine the quotient when $x^4 - 2x^3 + 8x^2 - 12x + 13$ is divided by $x^2 + 6$, and show that the remainder equals $1$.

May/June 2018

The polynomial $p(x)$ has the form $p(x) = 5x^3 + ax^2 + bx - 16$, with $a$ and $b$ as constants. It is known that $(x - 2)$ divides $p(x)$ exactly and that the remainder is $27$ when $p(x)$ is divided by $(x + 1)$.

May/June 2019

The polynomial $p(x)$ is given by $p(x) = 4x^3 + (k + 1)x^2 - mx + 3k$, where $k$ and $m$ are constants.

May/June 2019

The polynomial $p(x)$ is given by $p(x) = 4x^3 + (k + 1)x^2 - mx + 3k$, with $k$ and $m$ as constants.

May/June 2019

Let the polynomial $p(x)$ be $p(x) = 6x^3 + ax^2 + 9x + b$, with $a$ and $b$ as constants. You are told that $(x - 2)$ and $(2x + 1)$ are factors of $p(x)$.

May/June 2020

On one diagram, sketch the graphs of $y = |3x + 2a|$ and $y = |3x - 4a|$, where $a$ is a positive constant. State the coordinates of the points at which each graph meets the axes.

May/June 2020

The variables $x$ and $y$ obey $y = Ax^{-2p}$, where $A$ and $p$ are constants. The graph of $\ln y$ plotted against $\ln x$ is a straight line passing through the points $(-0.68, 3.02)$ and $(1.07, -1.53)$, as illustrated in the diagram.

May/June 2020

On one set of axes, sketch the graphs of $y = |2x - 3|$ and $y = 3x + 5$.

May/June 2020

On one set of axes, sketch the graphs of $y = |2x - 3|$ and $y = 3x + 5$.

May/June 2020

Solve for $x$ the inequality $|3x - 7| < |4x + 5|$.

May/June 2021

The roots of the equation $5|x| = 5 - 2x$ are $x = a$ and $x = b$, with $a < b$.

May/June 2021

For the equation $5|x| = 5 - 2x$, the two solutions are $x = a$ and $x = b$, with $a < b$.

May/June 2021

On one set of axes, sketch the graphs of $y = |2x - 9|$ and $y = 5x - 3$.

May/June 2022

Sketch the graphs of $y = |2x - 9|$ and $y = 5x - 3$ on the same diagram.

May/June 2022

The polynomial $p(x)$ is given by $p(x) = 2x^3 + 3x^2 + kx - 30$, with $k$ as a constant. You are told that $(x - 3)$ is a factor of $p(x)$.

May/June 2023

The variables $x$ and $y$ are linked by the equation $y = Ae^{(A-B)x}$, where $A$ and $B$ are constants. The plot of $\ln y$ against $x$ is a straight line and passes through the points $(0.4, 3.6)$ and $(2.9, 14.1)$, as shown in the diagram.

May/June 2023

On a single diagram, sketch the graphs of $y = |3x - 8|$ and $y = 5 - x$.

May/June 2024

Solve for the values of $x$ in $|5x + 7| > |2x - 3|$.

May/June 2024

Solve the inequality given by $|5x + 7| > |2x - 3|$.

May/June 2024

On one diagram, sketch the graphs of $y = |2x - 9|$ and $y = 4x - 5$.

May/June 2025

On one set of axes, sketch the graphs of $y = |2x - 9|$ and $y = 4x - 5$.

May/June 2025

On one set of axes, sketch the graphs of $y = |2x - 9|$ and $y = 4x - 5$.

May/June 2025

Solve $|x + 1| > |x - 4|$.

Oct/Nov 2010

The polynomial $3x^3 + 2x^2 + ax + b$, in which $a$ and $b$ are constants, is written as $p(x)$. It is stated that $(x - 1)$ divides $p(x)$ exactly, and that the remainder when $p(x)$ is divided by $(x - 2)$ is $10$.

Oct/Nov 2010

Solve for $x$ in the inequality $|x + 1| > |x - 4|$.

Oct/Nov 2010

Let $p(x)$ represent the polynomial $3x^3 + 2x^2 + ax + b$, with $a$ and $b$ as constants. It is stated that $(x - 1)$ is a factor of $p(x)$, and that the remainder on dividing $p(x)$ by $(x - 2)$ is $10$.

Oct/Nov 2010

Solve for $x$ in the inequality $|3x + 1| > 8$.

Oct/Nov 2010

The polynomial $x^3 + 4x^2 + ax + 2$, with $a$ a constant, is written as $p(x)$. You are told that the remainder on dividing $p(x)$ by $(x + 1)$ is the same as the remainder on dividing $p(x)$ by $(x - 2)$.

Oct/Nov 2010

Solve $|4 - 5x| < 3$.

Oct/Nov 2011

Let $p(x)$ represent the polynomial $4x^3 + ax^2 + 9x + 9$, where $a$ is a constant. When $p(x)$ is divided by $(2x - 1)$, the remainder is $10$.

Oct/Nov 2011

Solve the inequality $|x + 2| > \left|\tfrac{1}{2}x - 2\right|$ to find the allowed values of $x$.

Oct/Nov 2011

The polynomial $ax^3 - 3x^2 - 11x + b$, with $a$ and $b$ as constants, is written as $p(x)$. It is stated that $(x + 2)$ divides $p(x)$ exactly, and that the remainder when $p(x)$ is divided by $(x + 1)$ is $12$.

Oct/Nov 2011

Solve for x in the inequality $|2x - 3| \leq |3x|$.

Oct/Nov 2011

We write $p(x)$ for the polynomial $x^4 + ax^3 - x^2 + bx + 2$, with $a$ and $b$ fixed constants.

Oct/Nov 2011

Solve the inequality $|x - 2| \ge |x + 5|$.

Oct/Nov 2012

Let $p(x)$ be the polynomial $2x^3 - 4x^2 + ax + b$, with $a$ and $b$ as constants. It is known that the remainder is $4$ when $p(x)$ is divided by $(x + 1)$, and that the remainder is $12$ when $p(x)$ is divided by $(x - 3)$.

Oct/Nov 2012

Solve the inequality $|2x + 1| < |2x - 5|$ for $x$.

Oct/Nov 2012

$p(x)$ is the notation used for the polynomial $x^4 - 4x^3 + 3x^2 + 4x - 4$.

Oct/Nov 2012

Solve the inequality $|x - 2| \ge |x + 5|$.

Oct/Nov 2012

Let $p(x)$ represent the polynomial $2x^3 - 4x^2 + ax + b$, with $a$ and $b$ as constants. It is stated that the remainder is $4$ when $p(x)$ is divided by $(x + 1)$, and that the remainder is $12$ when $p(x)$ is divided by $(x - 3)$.

Oct/Nov 2012

Solve the inequality $|x + 1| < |3x + 5|$ for $x$.

Oct/Nov 2013

The polynomial $x^3 + ax^2 + bx + 8$, with $a$ and $b$ as constants, is called $p(x)$. It is stated that the remainder on dividing $p(x)$ by $(x - 3)$ is $14$, and that the remainder on dividing $p(x)$ by $(x + 2)$ is $24$. Determine the values of $a$ and $b$.

Oct/Nov 2013

Define p(x) to be the polynomial ax^3 + bx^2 - 25x - 6, where a and b are constants.

Oct/Nov 2013

Solve $|x + 1| < |3x + 5|$.

Oct/Nov 2013

The polynomial $x^3 + ax^2 + bx + 8$, in which $a$ and $b$ are constants, is represented by $p(x)$. It is given that when $p(x)$ is divided by $(x - 3)$ the remainder is $14$, and that when $p(x)$ is divided by $(x + 2)$ the remainder is $24$. Find the values of $a$ and $b$.

Oct/Nov 2013

Solve $|3x - 1| = |2x + 5|$ for $x$.

Oct/Nov 2014

Determine the quotient and the remainder when $x^4 + x^3 + 3x^2 + 12x + 6$ is divided by $(x^2 - x + 4)$.

Oct/Nov 2015

Find the quotient obtained when $3x^3 + 5x^2 - 2x - 1$ is divided by $(x-2)$, and show that the remainder comes to $39$.

Oct/Nov 2015