📝 Exam Questions — Pure Mathematics 1

A biologist is studying a population growth model that can be approximated by the quadratic equation 5x^2 - 7x + 1 = 0, where x represents a specific time parameter. The biologist needs to find the exact values of x when the population satisfies this condition. (a) Solve the equation 5x^2 - 7x + 1 = 0 using the quadratic formula, giving your answers correct to 3 significant figures. [5]

A mathematical function is defined as y = (x^2 - x)^2 - 8(x^2 - x) + 12. Understanding its roots and graphical behaviour is important in certain optimisation problems. (a) Find the real roots of the equation (x^2 - x)^2 - 8(x^2 - x) + 12 = 0. [5] (b) Sketch a possible graph of y = (x^2 - x)^2 - 8(x^2 - x) + 12, indicating the x-intercepts. [3]

A student is investigating the number of real roots for different quadratic equations. (a) Sketch the graph of a quadratic function that has no real roots. [3] (b) Explain how the discriminant value relates to your sketch in part (a). [3] (c) Calculate the discriminant for the equation x^2 + 2x + 5 = 0. [3]

Completing the square can be used both to rewrite quadratic expressions and to solve quadratic equations by isolating the variable. (a) Use the method of completing the square to express x^2 + 6x - 2 in the form (x + p)^2 + q. [4] (b) Hence, solve the equation x^2 + 6x - 2 = 0, leaving your answers in surd form. [3]

Fig 1.1 shows the graph of a quadratic function given by y = ax^2 + bx + c. The curve represents the trajectory of a ball thrown into the air. (a) Analyse Fig 1.1. State the coordinates of the vertex of the parabola. [4] (b) Compare the completed square form y = a(x - h)^2 + k with the graph, and evaluate the values of a, h, and k for the function shown in Fig 1.1. [8]

A civil engineer is designing a parabolic arch for a bridge. The shape of the arch can be modelled by a quadratic equation. (a) Find the values of x that satisfy the equation (x - 1)^2 = 2x + 5, giving your answers correct to 2 decimal places. [6] (b) Deduce the nature of the roots of the equation in part (a) based on your solutions. [4]

Fig 1.1 shows a graph of the function y = 2x - 7√x + 3. (a) Solve the equation 2x - 7√x + 3 = 0, giving your answers in exact form. [7] (b) Show that one of the solutions is extraneous if we do not consider the domain of √x to be restricted to the principal (positive) square root. [2]

The method of completing the square is a powerful technique for rewriting quadratic expressions into a standard form that reveals important properties of the function. (a) Express x^2 - 8x + 19 in the form (x + a)^2 + b, where a and b are constants to be found. [5]

The number of real roots of a quadratic equation can be determined by evaluating its discriminant. Consider the quadratic equation 3x^2 - 7x + 2 = 0. (a) Determine the number of distinct real roots for this equation. [4]

A quadratic function is given in the completed square form y = (x - h)^2 + k. Fig 1.2 shows the graph of such a function. (a) Identify the values of h and k from the graph in Fig 1.2. [3] (b) Explain how the vertex of a parabola relates to its completed square form. [3]

A line and a curve are defined by the equations y = 2x + 3 and y = x^2 + x - 1 respectively. (a) Calculate the coordinates where the line y = 2x + 3 intersects the curve y = x^2 + x - 1. [6]

Consider the equation x^4 - 5x^2 + 4 = 0. (a) Solve the equation x^4 - 5x^2 + 4 = 0. [5]

The height of a projectile, h metres, above the ground after t seconds is given by the expression h(t) = t^2 - 6t + 1. (a) Express t^2 - 6t + 1 in the form (t + a)^2 + b. [3] (b) State the minimum value of the expression and the value of t at which it occurs. [2]

A metal plate has an area that can be modelled by a quadratic expression. Understanding its factors is important for its design. (a) Find two linear factors of the expression 3x^2 - 10x - 8. [4] (b) Hence, solve the equation 3x^2 - 10x - 8 = 0. [3]

Fig 1.1 shows the graph of a straight line and a parabola. The equation of the line is y = x + 1 and the equation of the parabola is y = x^2 - 3x + 4. (a) Find the coordinates of the points of intersection of the line and the curve shown in Fig 1.1. [5] (b) Show that the x-coordinates of these points of intersection satisfy the equation x^2 - 4x + 3 = 0. [2]

A straight line with equation y = kx + 2 is tangent to the curve y = x^2 - 3x + 6. (a) Determine the values of the constant k for which the line is tangent to the curve. [6] (b) For the smaller value of k found in part (a), find the coordinates of the point of tangency. [4]

A parabolic arch is modelled by the curve y = 3x^2 + 18x - 10. (a) Find the coordinates of the turning point of the curve y = 3x^2 + 18x - 10. [3] (b) Sketch the curve, clearly showing the turning point and y-intercept. [3]

Fig 1.1 shows the graph of a quadratic function y = f(x). (a) State the number of real roots for the quadratic function y = f(x) shown in Fig 1.1. [3] (b) Identify the sign of the discriminant for the quadratic function shown in Fig 1.1. [2]

A student is analysing the trajectory of a projectile and models its height using a quadratic equation. To understand the vertex of the trajectory, the student needs to express the equation in completed square form. (a) Show that the equation 2x^2 - 10x + 11 = 0 can be written in the form 2(x - p)^2 + q = 0, stating the values of p and q. [6] (b) Determine the exact solutions to the equation 2x^2 - 10x + 11 = 0 using the completed square form. [4]

A scientist is modelling the population growth of a certain bacteria culture, where the number of bacteria at time x (in hours) is given by an equation involving powers of 2. (a) Solve the equation 2^(2x) - 9(2^x) + 8 = 0. [7] (b) Explain why there are fewer solutions for x than expected for a standard quadratic equation. [3]

Engineers often use graphical methods to find approximate solutions to equations before carrying out precise calculations. Fig 1.1 shows the graph of the function y = x^2 - 2x - 4. (a) Use the graph in Fig 1.1 to estimate the roots of the quadratic equation x^2 - 2x - 4 = 0. [4] (b) Verify your estimates from part (a) by solving the equation x^2 - 2x - 4 = 0 using the quadratic formula, giving your answers correct to 1 decimal place. [5]

A line and a circle are represented by the equations 3x + y = 7 and x^2 + y^2 = 13 respectively. (a) Find the coordinates of the points of intersection of the line and the circle. [6] (b) Plot the line and the circle on a coordinate grid, indicating the points of intersection. [3]

In a study of radioactive decay, the remaining amount of a substance at time x (in years) can be modelled by an equation that involves powers of 3. (a) Solve the equation 3^(2x+1) - 28(3^x) + 9 = 0. [8] (b) Analyse how the structure of the equation in part (a) allows it to be transformed into a quadratic form. [4]

A straight line with equation y = 3x + c is considered in relation to a quadratic curve with equation y = x^2 - 5x + 10. (a) Determine the range of values for 'c' such that the line y = 3x + c does not intersect the curve y = x^2 - 5x + 10. [7] (b) Discuss the implications of a negative discriminant in this context. [4]

An engineer is designing a parabolic antenna dish, and the cross-section can be modelled by a quadratic equation. To understand the properties of the curve, it is important to know if it intersects the x-axis and how many times. (a) Calculate the discriminant of the quadratic equation 3x^2 - 5x - 2 = 0. [4] (b) Determine the number of real roots for the equation 3x^2 - 5x - 2 = 0, justifying your answer. [3]

Completing the square is a versatile algebraic technique that allows us to find the vertex and minimum/maximum values of quadratic functions. (a) Express 3x^2 + 18x + 10 in the form a(x + b)^2 + c, where a, b and c are constants to be found. [5] (b) Find the minimum value of the quadratic function 3x^2 + 18x + 10. [3]

A mathematical model for a physical phenomenon involves fractional powers of a variable x. To understand this model, it is necessary to solve the following equation. (a) Determine the values of x that satisfy the equation x^(2/3) - 7x^(1/3) + 12 = 0. [6]

When solving quadratic equations, it is crucial to ensure all possible solutions are found. Consider the equation 6x^2 + 5x = 4. (a) Find the values of x that satisfy the equation 6x^2 + 5x = 4. [6] (b) Discuss the common misconception of dividing by x when solving equations of this type and explain why it can lead to a loss of solutions. [4]

A parabolic archway is designed such that its shape can be modelled by the equation y = x^2 - 4x + 1. The architect needs to visualise this curve and identify its key features. (a) Sketch the graph of y = x^2 - 4x + 1, clearly labelling the vertex and the y-intercept. [5] (b) State the equation of the line of symmetry for this graph. [3]

Fig 1.1 shows the graphs of a straight line and a quadratic curve. The straight line has the equation y = 3x - 1 and the curve has the equation y = 2x^2 - x + 2. (a) Find the values of x and y that satisfy the simultaneous equations. [6] (b) Explain the geometric meaning of your solutions. [2]

The path of a ball thrown into the air can be modelled by the quadratic function f(x) = -2x^2 - 8x + 5, where f(x) is the height of the ball and x is the horizontal distance from the thrower. The graph of this function is shown in Fig 1.1. (a) Find the maximum or minimum value of the function f(x) = -2x^2 - 8x + 5 and the x-coordinate where it occurs. [5] (b) Identify whether it is a maximum or minimum value. [2]

The general form of a quadratic function is y = ax^2 + bx + c. (a) Derive the formula for the x-coordinate of the vertex of a parabola y = ax^2 + bx + c using the method of completing the square. [6] (b) Evaluate the maximum value of the function h(x) = -3x^2 + 12x - 5 using the derived formula. [3] (c) Compare the efficiency of finding the vertex using the derived formula versus directly completing the square for a given quadratic. [3]

For a quadratic equation, the nature of its roots depends on the value of its discriminant. Consider the equation kx^2 + 6x + 2 = 0, where k is a constant. (a) Find the set of values of k for which the equation has two distinct real roots. [6]

Fig 1.4 shows the graph of a quadratic function y = ax^2 + bx + c. (a) Interpret the meaning of the discriminant for the parabola shown in Fig 1.4. [3] (b) Explain how changing the constant term of the quadratic function in Fig 1.4 might affect the number of real roots. [3]

An algebraic expression is given by (x + 1)^2 + 3(x + 1) - 10 = 0. (a) Find all real solutions to the equation (x + 1)^2 + 3(x + 1) - 10 = 0. [5] (b) Verify one of your solutions by substituting it back into the original equation. [2]

A student is asked to solve a quadratic equation as part of a larger problem involving projectile motion. (a) Solve the equation 2x^2 + 4x - 3 = 0, giving your answers in exact surd form. [5] (b) Explain why factorisation might not be the preferred method for solving this equation. [3]

The cost of producing a certain item, g(x) (in hundreds of dollars), depends on the number of items produced, x (in thousands), according to the function g(x) = x^2 - 4x + 7. The production is limited to a domain of -1 ≤ x ≤ 5. The graph of this function is shown in Fig 1.2. (a) Determine the range of the function g(x) = x^2 - 4x + 7 for the domain -1 ≤ x ≤ 5. [6] (b) Discuss how the restricted domain affects finding the maximum and minimum values. [4]

The profit, P, from selling x units of a product is modelled by a quadratic expression. Consider a related problem involving the inequality (x - 1)(x + 3) ≥ x + 3. (a) Determine the set of values of x for which (x - 1)(x + 3) ≥ x + 3. [6] (b) Explain how you would use a sign diagram to verify your answer for part (a). [4]

A projectile is launched from a platform, and its height h (in metres) at time t (in seconds) is modelled by a quadratic expression. Consider a related mathematical problem involving an inequality. (a) Solve the inequality (x + 2)^2 > 4x + 7. [6] (b) Discuss common student misconceptions when solving quadratic inequalities, particularly regarding division by variables or negative numbers. [4]

A manufacturer is designing a square plate for a new product. The area of the plate must be less than 9 units squared. Let x be the side length of the square plate. Find the range of values of x for which x^2 - 9 < 0.

The height of a projectile, h metres, above the ground at time t seconds is modelled by a quadratic function. Consider a similar quadratic relationship given by y = 3x^2 + 10x - 8. (a) Find the range of values of x for which 3x^2 + 10x ≤ 8. [4] (b) Sketch the graph of y = 3x^2 + 10x - 8, clearly indicating the x-intercepts. [3]

Fig 1.5 shows the graph of a quadratic function y = f(x). (a) Use the graph in Fig 1.5 to determine the range of x for which the function y = f(x) is positive (y > 0). [4] (b) Determine the equation of the line of symmetry for the parabola shown. [3] (c) State whether the vertex is a maximum or minimum point. [2]

A farmer wants to build a rectangular enclosure for his animals. He has 100 metres of fencing available. (a) Calculate the dimensions of the rectangular enclosure that maximises its area. [6] (b) Interpret the meaning of the maximum area in the context of the problem. [2]

An engineer is analysing the displacement of a component, which is described by the quadratic expression 4x^2 - 12x. (a) Determine the values of x for which 4x^2 - 12x = 0. [5] (b) Show that x = 0 is a valid solution to the equation. [3]

Consider the quadratic function y = x^2 - x - 6. (a) Draw the graph of y = x^2 - x - 6 for x values from -3 to 4, labelling the x-intercepts and y-intercept. [4] (b) From your graph, solve the equation x^2 - x - 6 = 0. [3]

A scientist is analysing experimental data which produces a parabolic trend. Fig 1.2 shows the graph of a quadratic function f(x). (a) Fig 1.2 shows the graph of a quadratic function. Identify the roots of the equation f(x) = 0 from the graph. [3] (b) Explain how you would factorise the quadratic expression corresponding to this graph, given its roots. [3]

A student is solving a quadratic equation as part of a larger problem. (a) Solve the quadratic equation 2x^2 + 7x - 15 = 0 by factorisation. [5]

A gardener is designing a rectangular flower bed. The length is (x + 3) metres and the width is (x - 5) metres. The area of the flower bed is 9 square metres. (a) Solve the equation (x + 3)(x - 5) = 9 by first expanding and then factorising. [6]

A gardener is designing a rectangular flower bed. The area of the flower bed must satisfy certain conditions related to its length, x metres. (a) Solve the quadratic inequality 2x^2 - 5x - 3 > 0. [3] (b) Express your answer in set notation. [2]

The trajectory of a ball thrown into the air can be modelled by a quadratic function. Fig 1.1 shows the graph of a quadratic function y = x^2 - 2x - 3. (a) Use the graph in Fig 1.1 to find the values of x for which x^2 - 2x - 3 < 0. [4] (b) State the coordinates of the vertex of the parabola shown. [2] (c) Deduce the range of values of x for which x^2 - 2x - 3 > 5. [2]

Fig 1.1 shows the graph of a line and a curve, representing the path of a moving object and a boundary, respectively. (a) Use Fig 1.1 to estimate the coordinates of the points of intersection of the line y = -x + 4 and the curve y = x^2 - 2x + 1. [4] (b) Verify your estimates by solving the equations algebraically. [5] (c) Estimate the range of x values for which the curve is below the line. [3]

The nature of the roots of a quadratic equation is determined by its discriminant. Consider the equation (p + 1)x^2 + 4px + 9 = 0, where p is a constant. (a) Show that the equation has no real roots if p satisfies 9p^2 - 9p + 9 < 0. [5] (b) Find the range of values of p for which the equation has two equal real roots. [5]

Consider the line y = x - 3 and the curve y = 2x^2 + 3x - 1. (a) Determine the number of points of intersection between the line and the curve. [5] (b) Sketch the line and the curve on the same axes, showing their relative positions. [3]

A quadratic equation models the height of a ball thrown vertically upwards. Consider the equation 2x^2 - 5x + c = 0, where x represents time and c is a constant related to the initial height. (a) Determine the condition for the equation 2x^2 - 5x + c = 0 to have no real roots. [4] (b) Illustrate this condition graphically by sketching a parabola that satisfies it. [3]

The number of real roots of a quadratic equation relates to how its graph intersects the x-axis. Consider the equation x^2 + (m - 2)x + 4 = 0. (a) Determine the value of m for which the equation has exactly one real root. [4] (b) Explain the geometric significance of having exactly one real root for a quadratic equation. [3]

A designer is modelling a parabolic arch for a bridge and a horizontal support beam. The arch is represented by the curve y = (x - 1)^2 + 2, and the support beam is represented by the line y = 4. (a) Find the coordinates of the points where the line y = 4 intersects the curve y = (x - 1)^2 + 2. [5] (b) Calculate the distance between these two points of intersection. [3]

A line and a curve are defined by the equations y = 3x + c and y = x^2 - x + 4, where c is a constant. (a) Find the values of c for which the line y = 3x + c is a tangent to the curve y = x^2 - x + 4. [5] (b) Show that for these values of c, there is only one point of intersection. [3] (c) Deduce the coordinates of the point of tangency for one of the values of c found in part (a). [3]

The trajectory of a projectile can be modelled by a quadratic curve, and a laser beam follows a straight line path. Consider a scenario where a laser beam given by the equation y = kx - 2 interacts with a parabolic trajectory given by the equation y = x^2 + 2x + 3. (a) Find the range of values of k for which the line y = kx - 2 does not intersect the curve y = x^2 + 2x + 3. [6] (b) Interpret your answer geometrically in terms of the positions of the line and the curve. [3]

A straight line and a quadratic curve are given by the equations y = 2x + 1 and y = x^2 - 3x + 5 respectively. (a) Find the coordinates of the points of intersection of the line y = 2x + 1 and the curve y = x^2 - 3x + 5. [6]

Fig 1.2 shows a parabola y = x^2 - 4 and a straight line y = 2x - 1. (a) State the y-intercept of the straight line. [2] (b) Determine the coordinates of the vertex of the parabola by completing the square for y = x^2 - 4. [3] (c) Calculate the area of the triangle formed by the two intersection points and the y-intercept of the line. [4]

Fig 1.4 shows six graphs of quadratic functions, labelled A to F. (a) Identify which graph represents a quadratic equation with two equal real roots. [2] (b) Deduce the sign of the discriminant for graph C and explain your reasoning. [3] (c) Determine the values of m for which the equation 2x^2 + 4x + m = 0 has no real roots, referring to similar graphs in the figure. [4]

Fig 1.2 shows two parabolas on the same axes. (a) Read the coordinates of the vertex for the inverted U-shaped parabola. [2] (b) State whether the U-shaped parabola has a maximum or minimum value, and give this value. [2]

Fig 1.3 shows six graphs of quadratic functions, labelled A to F. (a) Classify the nature of the roots for graph A and graph F, referring to their intersections with the x-axis. [3] (b) Explain why the discriminant is zero for graph B, given that its equation is y = (x - 3)^2. [4] (c) Derive the condition for the line y = mx + c to be tangent to the parabola y = x^2. [4]

Fig 1.1 shows six graphs of quadratic functions, labelled A to F. (a) State the number of real roots for: (i) Graph A (ii) Graph C (b) Interpret the graphical representation of two distinct real roots, referring to specific examples from Fig 1.1. (c) The quadratic equation x^2 - 6x + k = 0 represents a parabola similar to Graph A. Find the range of values for k such that this equation has two distinct real roots.

Fig 1.1 shows a parabola and a straight line on the same axes. (a) Derive the equation of the parabola in the form y = ax^2 + bx + c, given its vertex and one x-intercept. [4] (b) Solve the simultaneous equations y = x^2 - 4 and y = 2x - 1 algebraically. [3] (c) Interpret the significance of the solutions found in part (b) with reference to the diagram. [3]

Fig 1.1 shows the graph of the quadratic function y = x^2 - 5x - 14. (a) Identify the values of x where the parabola intersects the x-axis. [2] (b) Solve the equation x^2 - 5x - 14 = 0 by factorisation, and state the roots. [3]

Fig 1.3 shows a U-shaped parabola representing the function y = x^2 - 5x - 14. (a) Identify the x-intercepts of the parabola shown in the figure. (b) Write down the equation of the parabola in factored form, given that its y-intercept is -14. (c) Solve the inequality x^2 - 5x - 14 > 0 using the information from the graph.

Fig 1.4 shows a U-shaped parabola with equation y = x^2 - 5x - 14. (a) Read the x-intercepts of the parabola y = x^2 - 5x - 14 from the graph. [2] (b) Determine the values of x for which the function y = x^2 - 5x - 14 is positive. [3] (c) Sketch a graph of y = (x+2)(x-7) and indicate the region where y < 0. [3]

Fig 1.3 shows the graph of the quadratic function y = x^2 - 5x - 14. (a) Read the x-intercepts of the parabola from the graph. [2] (b) Factorise the quadratic expression x^2 - 5x - 14 using the x-intercepts found in part (a). [2]

Fig 1.4 shows the graph of the quadratic function y = x^2 - 5x - 14. (a) Identify the x-coordinates where the parabola intersects the x-axis. [2] (b) Determine the range of x for which the quadratic function is negative (y < 0). [3] (c) Deduce the solution to the inequality (x+2)(x-7) ≥ 0, referencing the graph. [4]

Fig 1.3 shows the graph of the quadratic function y = x^2 - 5x - 14. (a) Identify the roots of the quadratic equation x^2 - 5x - 14 = 0 from the graph. [2] (b) Determine the range of x-values for which the curve y = x^2 - 5x - 14 is below the x-axis. [3] (c) Sketch a similar parabola for y = -(x + 2)(x - 7) and state the range of x for which y > 0. [3]

Fig 1.4 shows a U-shaped parabola with the equation y = x^2 - 5x - 14. (a) Read the y-intercept of the parabola from the diagram. [2] (b) Determine the coordinates of the minimum point of the parabola. [3] (c) Find the range of x for which x^2 - 5x - 14 < 0. [3]

Fig 1.1 shows six graphs of quadratic functions, labelled A to F. (a) Identify which graphs represent quadratic equations with no real roots, and state the characteristic of these graphs that indicates this. [3] (b) Explain the significance of the point of tangency on the x-axis for graphs B and E in relation to their discriminants. [4] (c) Find the value of k for which the equation x^2 + kx + 9 = 0 has two equal real roots, similar to graph B (but shifted). [4]

Fig 1.2 shows two parabolas on the same axes. (a) Identify the coordinates of the vertex of the inverted U-shaped parabola. (b) Write the equation of the inverted U-shaped parabola in the form y = a(x + b)^2 + c, given its vertex is (1, 3) and it crosses the y-axis at (0, 2).

Fig 1.4 shows six graphs of quadratic functions, labelled A to F. (a) Analyse the relationship between the discriminant (b^2 - 4ac) and the position of the parabolas relative to the x-axis for graphs A, B, and C. [4] (b) Deduce the sign of the coefficient of x^2 for graph D and explain your reasoning. [3] (c) Find the possible values of k for the quadratic equation x^2 - 4x + k = 0, such that its graph would resemble graph B. [5]

Fig 1.1 shows a parabola with equation y = x^2 - 4 and a straight line with equation y = 2x - 1 on the same set of axes. (a) Identify the y-intercept of the parabola. [2] (b) Determine the gradient of the straight line using the two intersection points of the line and the parabola. [3] (c) Calculate the value of y for the parabola when x = 1, and verify it matches the line's y-value at x=1. [3]

Fig 1.1 shows a parabola with equation y = x^2 - 4 and a straight line. (a) State the equation of the straight line shown in the figure. [2] (b) Calculate the y-coordinate of the point on the parabola where x = 0. [3] (c) Find the x-coordinates of the intersection points by solving the equations simultaneously, and compare your findings with the diagram. [4]

Fig 1.2 shows two parabolas on the same axes. (a) Identify the coordinates of the turning point for the U-shaped parabola. [2] (b) Determine the equation of the U-shaped parabola in the form y = x^2 + bx + c, given its vertex and y-intercept. [3] (c) The inverted U-shaped parabola has the equation y = -(x - 1)^2 + 3. Calculate the y-coordinate of this parabola when x = 0, and verify it matches the diagram. [3]

Fig 1.3 shows two parabolas on the same axes. (a) Determine the equation of the line of symmetry for the U-shaped parabola. [3] (b) Formulate the equation of the inverted U-shaped parabola in the form y = a(x + b)^2 + c, given its vertex and y-intercept. [4] (c) Explain how the sign of the coefficient of x^2 determines whether a parabola has a maximum or minimum point. [3]

Fig 1.4 shows six graphs of quadratic functions (A to F). (a) Match each graph from Row 1 (A, B, C) to the correct condition for its discriminant (Δ > 0, Δ = 0, Δ < 0). (b) Explain how the number of x-intercepts relates to the nature of the roots for graph C. (c) Determine the value of c for graph B, given its equation is y = (x - 3)^2 + c and it touches the x-axis at x = 3.

Fig 1.2 shows two parabolas on the same axes. (a) Estimate the minimum value of the U-shaped parabola from the graph. [2] (b) Express the equation of the U-shaped parabola in the form (x+q)^2+r, given its minimum point is (-2, -4). [3]

Fig 1.4 shows two parabolas on the same axes. (a) Deduce the equation of the line of symmetry for the U-shaped parabola. (b) Formulate the equation of the inverted U-shaped parabola, given its vertex (1, 3) and y-intercept (0, 2), in the form y = ax^2 + bx + c. (c) The equation of the U-shaped parabola is y = x^2 + 4x. Solve the equation formed by setting this quadratic function equal to the equation you found in part (b), to find their points of intersection, if any.

Fig 1.1 shows a parabola with equation y = x^2 - 4 and a straight line with equation y = 2x - 1. (a) Read the coordinates of the two points of intersection from the graph. (b) Verify algebraically that one of the points read in part (a) satisfies both equations y = x^2 - 4 and y = 2x - 1.

Fig 1.2 shows a parabola with equation y = x^2 - 4 and a straight line with equation y = 2x - 1. (a) Determine the equation of the line passing through the two intersection points identified in Fig 1.2. [3] (b) Formulate a single quadratic equation by equating the expressions for y, and show that its discriminant is positive. [4] (c) Explain the geometric interpretation of a positive discriminant in the context of a line intersecting a parabola. [3]

Fig 1.3 shows two parabolas on the same axes. (a) Read the coordinates of the minimum point of the U-shaped parabola and the maximum point of the inverted U-shaped parabola. [2] (b) Express the equation of the U-shaped parabola in the form y = p(x+q)^2+r, given its vertex and y-intercept. [4] (c) Compare the y-intercepts of both parabolas as shown in the diagram. [3]

Fig 1.2 shows two parabolas on the same set of axes. (a) Identify the coordinates of the minimum point of the U-shaped parabola. (b) Determine the equation of the line of symmetry for the inverted U-shaped parabola. (c) State the y-intercept for the U-shaped parabola.