📝 Exam Questions — Pure Mathematics 2 & 3

Consider the polynomial P(x) = 2x^3 + 3x^2 - 3x + 1/2. (a) Prove that (2x - 1) is a factor of P(x). [4] (b) Determine the other factors of P(x). [4] (c) Evaluate P(1/2) and explain its significance in relation to the factor theorem. [3]

Consider the modulus inequality |2x - 4| > x + 1. (a) Sketch the graph of y = |2x - 4| on a coordinate plane, clearly showing its vertex and intercepts. [4] (b) Solve the inequality |2x - 4| > x + 1 algebraically. [4]

An engineer is designing a control system where the tolerance for two variables, x and y, must satisfy certain conditions expressed as modulus inequalities. (a) Solve the inequality |x - 3| < |2x + 1|. [6] (b) Verify if x = -5 satisfies the inequality. [2]

Consider the graph of y = |2x - 4|. (a) Analyse the transformation from the graph of y = x to the graph of y = |2x - 4|. Describe the sequence of transformations. [6] (b) Determine the area of the region bounded by the graph of y = |2x - 4|, the x-axis, and the lines x = 0 and x = 4. [3] (c) Sketch the graph of y = -|2x - 4|, showing the coordinates of the vertex and any intercepts with the axes. [3]

Fig 1.1 shows the graph of y = |x^2 - 4|. This graph illustrates the behaviour of a modulus function applied to a quadratic expression. (a) Identify the coordinates of the vertex and the x-intercepts of the graph y = |x^2 - 4| shown in Fig 1.1. [4] (b) Solve the equation |x^2 - 4| = 5 using the graph. [3] (c) Describe how the graph of y = |x^2 - 4| relates to the graph of y = x^2 - 4. [3]

Graphs of modulus functions are characterised by their 'V' shape, and their domain can be restricted. (a) Draw the graph of y = |2x - 4| for -1 ≤ x ≤ 5, clearly labelling the x and y intercepts. [4] (b) State the range of y for the given domain. [2]

The graph of a modulus function has a distinctive 'V' shape. (a) Sketch the graph of y = |x + 2|, showing the coordinates of the vertex and the y-intercept. [3] (b) Identify the gradient of each linear part of the graph. [2]

Modulus functions can be represented graphically, and their intersections can indicate solutions to equations. (a) Sketch the graphs of y = |x - 2| and y = |2x - 1| on the same set of axes, labelling any intercepts and vertices. [5] (b) Determine the x-coordinates of the intersection points of these two graphs from your sketch. [4]

A student is learning about modulus functions and needs to solve inequalities involving them. (a) State the property for |x| < a. [2] (b) Solve the inequality |3x - 2| < 7. [3]

Modulus inequalities can be represented on a number line. (a) Interpret the meaning of |x - a| < b graphically, using Fig 1.2. [3] (b) Solve the inequality |2x + 3| ≤ 5 algebraically. [4] (c) Illustrate the solution from part (b) on a number line. [2]

Fig 1.1 shows the graphs of y = |x + 1| and y = 3x - 1 on the same set of axes. These graphs can be used to visually identify solutions to modulus equations. (a) Read the x-coordinate of the intersection point of the graphs y = |x + 1| and y = 3x - 1 from Fig 1.1. [4] (b) Confirm your reading by solving the equation |x + 1| = 3x - 1 algebraically. [3]

A student is exploring graphical methods to solve inequalities involving modulus functions. (a) Sketch the graphs of y = |x - 2| and y = 4 on the same axes. Clearly label any intercepts with the axes and the vertex of the modulus function graph. [4] (b) Use your graph to determine the solution to the inequality |x - 2| < 4. [6]

A student is investigating the properties of modulus functions. (a) Sketch the graph of y = |3x - 6|, showing the coordinates of the vertex and any intercepts with the axes. [4] (b) Determine the values of x for which |3x - 6| = 3. [4]

The graph of a polynomial P(x) is shown in Fig 1.2. (a) Identify the x-intercepts of the graph of P(x) in Fig 1.2. [2] (b) Explain how the factor theorem relates to these x-intercepts. [2] (c) Check if x = 0 is a factor of P(x). [2]

Polynomial division is a fundamental technique for simplifying rational expressions and finding factors of polynomials. Consider the division of a cubic polynomial by a linear factor. (a) Find the quotient and remainder when 2x^3 - 3x^2 + x - 5 is divided by x - 2. [5] (b) Explain why it is important to include zero coefficients for missing terms in polynomial long division. [2]

When a polynomial P(x) is divided by a linear factor (x - c), the remainder is given by P(c). (a) Identify the value of 'c' for which P(c) gives the remainder when P(x) is divided by (3x - 6). [2] (b) Calculate the remainder for P(x) = x^2 + 5x - 3 when divided by (3x - 6). [2]

Fig 1.1 shows the graph of a cubic polynomial P(x). (a) Use the graph in Fig 1.1 to identify two linear factors of the polynomial P(x) whose roots are shown. [4] (b) Deduce a possible expression for P(x) in factorised form. [3] (c) Sketch the graph of y = |P(x)| based on the identified factors. [3]

The remainder theorem is a useful tool for quickly finding the remainder of polynomial division without performing the full division. (a) Give the general form of the remainder when a polynomial P(x) is divided by (ax + b). [2] (b) Find the remainder when P(x) = x^3 + x^2 - 4x + 6 is divided by (2x - 2). [3]

A mathematician is working with polynomials and needs to find remainders quickly without performing lengthy division. The Remainder Theorem provides an elegant solution for this. (a) Determine the remainder when the polynomial P(x) = 2x^3 - x^2 + 4x - 1 is divided by (x - 3). [4] (b) Explain how the remainder theorem simplifies finding the remainder compared to long division in this case. [2]

The factor theorem is a fundamental concept in polynomial algebra. (a) State the factor theorem. [2] (b) Show that (x - 1) is a factor of P(x) = x^3 - 2x^2 + x. [2]

Polynomial division is a fundamental operation in algebra, allowing us to break down complex polynomials into simpler parts. (a) State the division algorithm for polynomials. [2] (b) Perform the division of x^2 + 5x + 6 by x + 2, stating the quotient. [3]

Consider a polynomial P(x) that is to be divided by a linear expression. The Remainder Theorem can be used to efficiently determine the remainder. (a) Identify the value of 'c' if a polynomial P(x) is divided by (x + 2). [2] (b) Calculate the remainder when P(x) = x^3 - 2x + 7 is divided by (x + 2). [3]

Fig 1.1 shows the graph of a polynomial P(x). (a) Read the value of P(2) from the graph in Fig 1.1. [3] (b) State the remainder when P(x) is divided by (x - 2). [2]

Fig 1.4 shows the graph of a polynomial P(x). (a) Read the y-intercept from Fig. 1.4. [2] (b) Interpret the remainder when P(x) is divided by x from the graph. [2] (c) Draw a line representing a divisor (x - c) on the graph such that the remainder is positive. [2]

A student is learning about polynomial division and the Remainder Theorem. They are given a polynomial P(x) and asked to find the remainder when it is divided by a linear factor. (a) State the Remainder Theorem for a polynomial P(x) divided by (x - c). [2] (b) Find the remainder when P(x) = x^2 + 3x + 5 is divided by (x - 1). [2]

Polynomials are fundamental in algebra, and understanding their properties, such as factors and remainders, is crucial. The Remainder Theorem provides a direct method for finding remainders, which is closely related to the Factor Theorem. (a) Find the remainder when P(x) = 3x^3 + 2x^2 - 5x + 1 is divided by (x + 1). [4] (b) Show that if P(c) = 0, then (x - c) is a factor of P(x). [3]

Polynomial division can be extended to cases where the divisor is a quadratic polynomial. This process requires careful algebraic manipulation. (a) Determine the quotient and remainder when x^4 - 3x^2 + 2x - 1 is divided by x^2 + x - 1. [6] (b) Show that the degree of the remainder is less than the degree of the divisor. [2]

A student is analysing the behaviour of a polynomial P(x) using its graph. Fig 1.2 shows the graph of a polynomial P(x) for -4 ≤ x ≤ 4. (a) Determine the remainder when the polynomial shown in Fig. 1.2 is divided by (x + 1). [4] (b) Sketch a possible graph of the polynomial y = P(x) if the remainder when divided by (x - 3) was 0. [3]

Polynomial division can be performed using various methods, including synthetic division. Fig 1.1 shows a synthetic division setup. (a) Analyse the provided synthetic division setup in Fig 1.1 to identify the dividend and the divisor. [4] (b) Determine the quotient and remainder from the synthetic division shown in Fig 1.1. [3] (c) Sketch a graph of a linear divisor (x-c) and a quadratic dividend, showing a possible intersection point that relates to the remainder. [2]

Consider the polynomial P(x) = x^4 + 2x^3 - 5x^2 - 6x + 8 and the divisor D(x) = x^2 - x - 2. (a) Perform the polynomial division of P(x) by D(x), finding the quotient Q(x) and remainder R(x). [6] (b) Analyse the relationship P(x) = D(x)Q(x) + R(x) for the given polynomials. [3] (c) Explain why the remainder must have a degree less than the divisor. [3]

A student is analysing the intersection points of a modulus function and a linear function. (a) Sketch the graphs of y = |2x + 1| and y = x + 2 on the same axes, showing the coordinates of any intercepts with the axes and the vertices. [6] (b) Solve the equation |2x + 1| = x + 2 using your sketch. [4]

Modulus inequalities are frequently encountered in various fields of mathematics and engineering to define ranges or bounds. Consider the inequality involving two modulus functions. (a) Solve the inequality |3x + 4| ≥ |x - 2|. [8] (b) Discuss a common misconception when squaring both sides of a modulus inequality. [3]

A polynomial P(x) is given by P(x) = x^3 + ax^2 + bx - 6. It is known that P(x) is exactly divisible by (x - 1) and (x + 2). (a) Find the values of a and b. [6] (b) Deduce the quotient when the polynomial with these values of a and b is divided by (x - 1)(x + 2). [4]

Fig 1.1 shows a number line with the number 2 marked. This number line can be used to interpret and represent solutions to inequalities involving the modulus function. (a) Interpret the inequality |x - 2| < 3 on the number line shown in Fig 1.1. [4] (b) Shade the region on the number line that represents the solution to |x - 2| < 3. [3]

A mathematician is analysing the conditions under which certain functions are defined, leading to modulus inequalities. (a) Solve the inequality |2x + 1| ≥ 5. [5] (b) Express the solution in interval notation. [2]

Fig 1.1 shows the graph of y = |x - 3|. (a) Use the graph in Fig 1.1 to estimate the solutions to |x - 3| = 2. [5] (b) Find the exact solutions to |x - 3| = 2 using algebraic methods. [4]

Consider the polynomial P(x) = x^3 + 5x^2 + 7x + 2. (a) Determine if (x + 2) is a factor of P(x). [3] (b) Factorise P(x) completely, given that (x + 1) is a factor. [3]

The polynomial P(x) is defined as P(x) = x^3 - 4x^2 + kx + 6. (a) Find the value of k if (x - 3) is a factor of P(x). [4] (b) Hence factorise P(x) completely for the value of k found in part (a). [4]

The polynomial P(x) = x^3 - 7x + 6 is given. The graph of P(x) is shown in Fig 1.3. (a) Use the given polynomial P(x) = x^3 - 7x + 6 and the divisor (x - 1) to set up and perform synthetic division. [4] (b) Deduce the quotient and remainder from your division. [3] (c) Interpret the meaning of the remainder in the context of the polynomial graph shown in Fig 1.3, specifically at x=1. [2]

A student is working with a polynomial P(x) = 2x^3 + 5x^2 - 7x - 1. (a) Calculate the remainder when P(x) is divided by (x - 1/2). [4] (b) Verify your result using the factor theorem if the remainder was 0. [2]

The modulus function is a fundamental concept in mathematics, representing the absolute value of a number. Understanding its definition is crucial for solving equations and inequalities involving absolute values. (a) Define the modulus function |x| using a piecewise definition. [2] (b) State the value of |-7.3|. [2]

The modulus function has several key properties that simplify expressions and are crucial for solving more complex problems. Understanding these properties allows for efficient manipulation of absolute value expressions. (a) Explain why |x| = |-x| for any real number x. [3] (b) Simplify the expression |x - 2| + |2 - x|. [4]

The properties of the modulus function are essential for solving equations and understanding its algebraic behaviour. These properties allow us to manipulate expressions involving absolute values effectively. (a) Determine the range of values for k such that |2k - 3| = 5. [4] (b) Show that |ab| = |a||b| for any real numbers a and b. [4]

The modulus function is used to represent the magnitude of a number, regardless of its sign. This concept is important for understanding distances and ranges of values. (a) Evaluate the expression |5 - 9| + |-3|. [2] (b) Identify all integers x such that |x| < 4. [3]

The modulus function has several important properties that are fundamental in mathematics. One such property relates to the sum of two numbers, x and y. (a) Discuss the conditions under which |x + y| = |x| + |y| is true. [5] (b) Prove the triangle inequality: |x + y| ≤ |x| + |y|. [5]

Modulus equations can sometimes lead to extraneous roots, particularly when squaring both sides of an equation. (a) Determine all real values of x for which |x^2 - 4| = 2x + 1. [7] (b) Explain why checking solutions is crucial when solving modulus equations of the form |f(x)| = g(x). [4]

Equations involving two modulus expressions on either side of the equality can be solved by squaring both sides or by considering different cases based on the critical points. (a) Find the solutions to the equation |3x - 4| = |x + 2|. [5] (b) Find the solutions to the equation |2x - 3| = |x - 1|, giving your answers as exact fractions. [4]

When solving modulus equations where one side is an algebraic expression, it is important to check for extraneous solutions. This often occurs when squaring both sides. (a) Solve the equation |2x + 1| = x + 4. [4] (b) Solve the equation |x - 5| = 2x - 1. [4]

Modulus equations often arise in various mathematical contexts, requiring careful consideration of both positive and negative cases. Consider the following equations. (a) Solve the equation |3x - 2| = 7. [3] (b) Solve the equation |x/2 + 1| = 4. [3]

Solving modulus equations sometimes requires careful consideration of different cases or conditions. (a) Solve the equation |x^2 - 3x - 10| = x + 2. [8] (b) Verify your solutions by substituting them back into the original equation. [4]

A polynomial P(x) is defined as P(x) = x^3 - kx^2 + 2x - 4. (a) Determine the value of 'k' such that when P(x) is divided by (x - 2), the remainder is 0. [6] (b) Discuss what it means for (x - 2) to be a factor of P(x). [3]

Fig 1.3 shows the graph of a polynomial P(x) for -2 ≤ x ≤ 2. A scientist uses this polynomial to model a natural phenomenon. (a) Analyse the graph in Fig. 1.3 to determine the remainders when P(x) is divided by (x - 0.5) and (x + 1.5). [6] (b) Predict the remainder when P(x) is divided by (2x - 1) and explain your reasoning. [4]

A polynomial P(x) is defined as P(x) = ax^3 + 4x^2 + bx - 2. This polynomial has specific remainder properties when divided by linear factors. (a) Evaluate the constants 'a' and 'b' such that when P(x) is divided by (x - 1), the remainder is 3, and when divided by (x + 2), the remainder is -12. [6] (b) Analyse the efficiency of using the remainder theorem to solve this problem compared to setting up a system of equations from long division. [4]

A student is investigating polynomial division. They are given the polynomial P(x) = 4x^4 - 2x^3 + x - 5. (a) Calculate the remainder when P(x) is divided by (2x - 1). [5] (b) Deduce whether (2x - 1) is a factor of P(x). [3]

Consider the polynomial P(x) = x^4 - px^3 + qx^2 - 3x + 5. This polynomial exhibits specific remainder properties when divided by linear factors. (a) Determine the values of 'p' and 'q' if P(x) leaves a remainder of 3 when divided by (x - 1) and a remainder of 21 when divided by (x + 2). [7] (b) Discuss the implications if one of the remainders was zero. [4]

The polynomial P(x) is defined as P(x) = x^3 + ax^2 + bx + c. (a) Derive an expression for the remainder when the polynomial P(x) is divided by (x - k). [7] (b) Evaluate the remainder when P(x) = x^3 - 2x^2 + 3x - 1 is divided by (x - (1 + i)), where 'i' is the imaginary unit. [4]

A polynomial P(x) is defined as P(x) = Ax^3 + Bx^2 - 5x + 1. (a) Given that P(x) leaves a remainder of 7 when divided by (x - 1) and a remainder of -1 when divided by (x + 1), solve for the constants 'A' and 'B'. [8] (b) Interpret the meaning of these constants in the context of the polynomial's behavior at x = 1 and x = -1. [4]

Consider the polynomial P(x) = 6x^3 - 11x^2 + 7x - 1. (a) Find the remainder when P(x) is divided by (3x - 1). [5] (b) Compare the value of P(1/3) with your answer from part (a). [3]

Fig 1.5 shows the graph of a polynomial P(x). (a) Determine the numerical values of the remainders when the polynomial P(x) shown in Fig. 1.5 is divided by (x + 2) and (x - 2). [4] (b) Sketch the graph of a linear divisor (x - c) on Fig. 1.5 such that the remainder P(c) is -3. [4]

A student is asked to divide the polynomial P(x) = 2x^4 - 3x^3 + 5x - 1 by (x + 3). (a) Find the remainder when P(x) = 2x^4 - 3x^3 + 5x - 1 is divided by (x + 3). [4] (b) Explain why P(-3) directly gives the remainder in this case. [3]

Fig 1.4 shows a polynomial long division of x^3 + 4x^2 - 11x + 6 by x - 1. (a) Identify the divisor that results in a zero remainder in Fig 1.4. (b) State the value of c such that (x - c) is a factor of x^3 + 4x^2 - 11x + 6, based on Fig 1.4. (c) Show that P(c) = 0 for the value of c found in part (b), where P(x) = x^3 + 4x^2 - 11x + 6.

Fig 1.5 shows the graph of y = |x - 4| and the line y = 2x + 1. (a) Determine the gradient of the line y = 2x + 1 shown in Fig 1.5. [2] (b) Calculate the gradient of the right branch of y = |x - 4| for x > 4. [3] (c) Compare the gradients found in parts (a) and (b) and explain why there is only one intersection point shown in Fig 1.5. [2] (d) Justify why the line y = 2x + 1 does not intersect the left branch of y = |x - 4| (for x < 4) based on their gradients and y-intercepts. [3]

Fig 1.7 shows the definition of the modulus function. (a) Apply the definition from Fig 1.7 to write |2 - 5| without the modulus sign. [2] (b) Calculate the value of |(-3)^2 - 10|. [2] (c) Explain why |x| = |-x| for any real number x, using the definition in Fig 1.7. [2] (d) Determine the values of x for which |x - 3| = x - 3, based on the definition in Fig 1.7. [2]

Fig 1.1 shows the graph of y = |x - 1/2|. (a) State the equation of the line that is reflected to form the graph y = |x - 1/2| for x < 1/2. [1] (b) Determine the y-coordinate when x = 0 on the graph shown in Fig 1.1. [2] (c) Sketch the graph of y = x - 1/2 on the same axes as Fig 1.1, showing how it relates to y = |x - 1/2|. [3] (d) Calculate the total length of the graph segment shown in Fig 1.1 from x = 0 to x = 1. [3]

Fig 1.6 shows the graphs of y = |2x - 1| and y = |x + 2|. (a) Identify the x-coordinates of the intersection points of y = |2x - 1| and y = |x + 2| from Fig 1.6. (b) Write down the inequality represented by the region where the graph of y = |2x - 1| is below the graph of y = |x + 2|. (c) Solve the equation 2x - 1 = x + 2 algebraically to confirm one of the intersection points identified in part (a). (d) Explain how the solution to |2x - 1| < |x + 2| can be deduced from the intersection points and the relative positions of the graphs in Fig 1.6.

Fig 1.7 shows the definition of the modulus function. (a) Interpret the definition of |x| for x ≥ 0 from Fig 1.7. [1] (b) Evaluate the value of |5| using the definition. [1] (c) State the value of |-7| and explain which part of the definition from Fig 1.7 applies. [2]

Fig 1.3 shows the graph of y = |2x - 5| and the horizontal line y = 3. a) Identify the coordinates of the vertex of the graph y = |2x - 5| from Fig 1.3. b) Calculate the area of the triangle formed by the graph y = |2x - 5| and the x-axis between x = 1 and x = 4. c) Determine the y-intercept of the graph y = |2x - 5|. d) State the gradient of the line segment of y = |2x - 5| for x < 2.5.

Fig 1.4 shows a polynomial long division setup. (a) Identify the divisor used in the polynomial long division shown in Fig 1.4. [1] (b) State the remainder when x^3 + 4x^2 - 11x + 6 is divided by the identified divisor, as shown in Fig 1.4. [2] (c) Use the Remainder Theorem to show that P(1) equals the remainder found in part (b), where P(x) = x^3 + 4x^2 - 11x + 6. [3] (d) Find the value of P(0) and relate it to the constant term of the polynomial. [2]

Fig 1.5 shows the graph of y = |x - 4| and the line y = 2x + 1. (a) Read the x-coordinate of the intersection point of y = |x - 4| and y = 2x + 1 from Fig 1.5. (b) Determine the exact value of the y-coordinate at the intersection point by substituting the x-coordinate found in part (a) into y = 2x + 1. (c) Show algebraically that solving x - 4 = -(2x + 1) leads to an extraneous root that is not shown in Fig 1.5.

Fig 1.4 shows a polynomial long division setup. (a) Identify the divisor polynomial shown in the long division setup in Fig 1.4. (b) State the quotient polynomial from the long division shown in Fig 1.4. (c) Perform the first step of the polynomial long division by subtracting x^2(x - 1) from the dividend shown in Fig 1.4, and show the resulting polynomial. (d) Write down the remainder from the long division in Fig 1.4.

Fig 1.1 shows the graph of y = |x - 1/2|. a) Identify the x-intercept of the graph in Fig 1.1. b) State the coordinates of the point where the graph meets the y-axis. c) Calculate the exact value of y when x = -1, using the equation of the reflected part of the graph.

Fig 1.3 shows the graph of y = |2x - 5| and a horizontal line. (a) Determine the vertex of the graph y = |2x - 5| from Fig 1.3. [2] (b) Sketch the graph of y = 2x - 5 without the modulus function, clearly indicating the part that is reflected to form y = |2x - 5|. [3] (c) Show algebraically how one of the intersection points in Fig 1.3 is found by solving 2x - 5 = 3. [2] (d) State the interval of x for which |2x - 5| ≥ 3, using the information from Fig 1.3. [2]

Fig 1.6 shows the graphs of y = |2x - 1| and y = |x + 2|. (a) Determine the y-coordinate of the intersection point at x = 3 by substituting x = 3 into either equation shown in Fig 1.6. [3] (b) Solve the equation |2x - 1| = |x + 2| algebraically by squaring both sides to find the exact x-coordinates of both intersection points. [3] (c) Compare the graphical solution for |2x - 1| > |x + 2| from Fig 1.6 with the algebraic solution obtained in part (b). [3] (d) Deduce the range of x for which |2x - 1| < |x + 2| and justify your answer using the graphical information from Fig 1.6. [3]

Fig 1.2 shows the graph of y = |2x - 1| and a straight line intersecting it. (a) Identify the equation of the straight line intersecting y = |2x - 1| in Fig 1.2. [1] (b) Read the y-coordinate of the intersection points from Fig 1.2. [2] (c) Verify that the x-coordinate x = -1 satisfies the equation |2x - 1| = 3. [2] (d) Discuss how the symmetry of the graph y = |2x - 1| explains the two intersection points with y = 3 shown in Fig 1.2. [3]

Fig 1.4 shows a polynomial long division setup. a) Identify the quotient polynomial from the long division shown in Fig 1.4. b) Calculate P(-2) for P(x) = x^3 + 4x^2 - 11x + 6. c) Deduce the remainder when P(x) is divided by (x + 2) using the Remainder Theorem and compare it with the calculation in part (b). d) Formulate the expression P(x) = D(x)Q(x) + R(x) using the polynomials shown in Fig 1.4, where D(x) is the divisor, Q(x) is the quotient, and R(x) is the remainder.

Fig 1.2 shows the graph of y = |2x - 1| and the horizontal line y = 3. (a) Read the x-coordinates of the intersection points of y = |2x - 1| and y = 3 from Fig 1.2. (b) State the equation of the line that forms the right branch of y = |2x - 1|. (c) Verify that one of the x-coordinates found in part (a) satisfies the equation 2x - 1 = 3.

Fig 1.2 shows the graph of y = |2x - 1| and the horizontal line y = 3. (a) Identify the coordinates of the vertex of the graph y = |2x - 1| from Fig 1.2. [2] (b) Determine the exact values of x where y = |2x - 1| intersects y = 3, by solving the equation 2x - 1 = 3 and -(2x - 1) = 3. [3] (c) Solve the equation |2x - 1| = x + 1 graphically by adding the line y = x + 1 to Fig 1.2 and estimating the intersection points. [3] (d) Explain why a horizontal line y = k, where k < 0, would have no intersection points with the graph y = |2x - 1| shown in Fig 1.2. [4]

Fig 1.5 shows the graph of y = |x - 4| and the line y = 2x + 1. a) Read the y-intercept of the line y = 2x + 1 from Fig 1.5. b) Determine the y-intercept of the graph y = |x - 4|. c) Explain why the graph of y = |x - 4| has a 'V' shape with its vertex on the x-axis, referring to the definition of the modulus function.

Fig 1.3 shows the graph of y = |2x - 5| and the horizontal line y = 3. (a) Identify the x-coordinates of the intersection points of y = |2x - 5| and y = 3 from Fig 1.3. (b) Write down the range of x values for which |2x - 5| < 3, based on Fig 1.3. (c) Solve the inequality 2x - 5 < 3 algebraically to confirm one boundary of the solution.

Fig 1.4 shows a polynomial long division setup. (a) Identify the dividend polynomial from the long division shown in Fig 1.4. [1] (b) Calculate the value of the dividend polynomial at x = 1. [3] (c) Explain the relationship between the remainder found in Fig 1.4 and the value calculated in part (b), referencing the Remainder Theorem. [3] (d) Predict what the remainder would be if the dividend was x^3 + 4x^2 - 11x + 7 and the divisor remained x - 1. [3]