1.02q Use intersection points: of graphs to solve equations

259 questions

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CAIE P1 2019 March Q10
12 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}}\).
  1. The straight line with equation \(y = x + 3\) intersects the curve at points \(A\) and \(B\). Find the length of \(AB\). [6]
  2. The tangent to the curve at a point \(T\) is parallel to \(AB\). Find the coordinates of \(T\). [3]
  3. Find the coordinates of the point of intersection of the normal to the curve at \(T\) with the line \(AB\). [3]
CAIE P1 2014 November Q9
10 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows parts of the graphs of \(y = x + 2\) and \(y = 3\sqrt{x}\) intersecting at points \(A\) and \(B\).
  1. Write down an equation satisfied by the \(x\)-coordinates of \(A\) and \(B\). Solve this equation and hence find the coordinates of \(A\) and \(B\). [4]
  2. Find by integration the area of the shaded region. [6]
CAIE P1 2016 November Q1
3 marks Moderate -0.3
Find the set of values of \(k\) for which the curve \(y = kx^2 - 3x\) and the line \(y = x - k\) do not meet. [3]
CAIE P1 2018 November Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows part of the curve \(y = x(9 - x^2)\) and the line \(y = 5x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.
  1. Express the length of \(PQ\) in terms of \(t\), simplifying your answer. [2]
  2. Given that \(t\) can vary, find the maximum value of the length of \(PQ\). [3]
CAIE P2 2015 June Q5
12 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$|3x| = 16 - x^4$$ has two real roots. [3]
  2. Use the iterative formula \(x_{n+1} = \sqrt[4]{16 - 3x_n}\) to find one of the real roots correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
  3. Hence find the coordinates of each of the points of intersection of the graphs \(y = |3x|\) and \(y = 16 - x^4\), giving your answers correct to 3 decimal places. [2]
CAIE P2 2003 November Q5
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, for \(x < 0\), show that exactly one root of the equation \(x^2 = 2^x\) is negative. [2]
  2. Verify by calculation that this root lies between \(-1.0\) and \(-0.5\). [2]
  3. Use the iterative formula $$x_{n+1} = -\sqrt{(2^{x_n})}$$ to determine this root correct to 2 significant figures, showing the result of each iteration. [3]
CAIE P3 2018 November Q3
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation \(x^3 = 3 - x\) has exactly one real root. [2]
  2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{2x_n^3 + 3}{3x_n^2 + 1}$$ converges, then it converges to the root of the equation in part (i). [2]
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
Edexcel P1 2018 Specimen Q9
11 marks Standard +0.3
\includegraphics{figure_3} A sketch of part of the curve \(C\) with equation $$y = 20 - 4x - \frac{18}{x}, \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has \(x\) coordinate equal to 2
  1. Show that the equation of the normal to \(C\) at \(A\) is \(y = -2x + 7\). [6]
The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.
  1. Use algebra to find the coordinates of \(B\). [5]
Edexcel C1 Q10
13 marks Moderate -0.3
  1. On the same axes sketch the graphs of the curves with equations
    1. \(y = x^2(x - 2)\), [3]
    2. \(y = x(6 - x)\), [3]
    and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
  2. Use algebra to find the coordinates of the points where the graphs intersect. [7]
Edexcel C1 Q42
9 marks Moderate -0.8
The curve \(C\) has equation \(y = x^2 - 4\) and the straight line \(l\) has equation \(y + 3x = 0\).
  1. In the space below, sketch \(C\) and \(l\) on the same axes. [3]
  2. Write down the coordinates of the points at which \(C\) meets the coordinate axes. [2]
  3. Using algebra, find the coordinates of the points at which \(l\) intersects \(C\). [4]
Edexcel C2 Q8
12 marks Moderate -0.3
\includegraphics{figure_2} The line with equation \(y = 3x + 20\) cuts the curve with equation \(y = x^2 + 6x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
  1. Use calculus to find the exact area of \(S\). [7]
Edexcel C2 Q17
12 marks Moderate -0.3
\includegraphics{figure_7} Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6x - x^2 - 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(R\) is bounded by the line and the curve.
  1. Calculate the area of \(R\). [7]
Edexcel C2 Q21
9 marks Moderate -0.3
A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).
  1. Write down an equation of the circle \(C\). [2]
  2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
  3. Find the distance between these two points. [2]
Edexcel F1 2022 January Q7
9 marks Standard +0.8
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The rectangular hyperbola \(H\) has equation \(xy = 36\) The point \(P(4, 9)\) lies on \(H\)
  1. Show, using calculus, that the normal to \(H\) at \(P\) has equation $$4x - 9y + 65 = 0$$ [4]
The normal to \(H\) at \(P\) crosses \(H\) again at the point \(Q\)
  1. Determine an equation for the tangent to \(H\) at \(Q\), giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are rational constants. [5]
Edexcel FP1 2013 June Q5
8 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a rectangular hyperbola \(H\) with parametric equations $$x = 3t, \quad y = \frac{3}{t}, \quad t \neq 0$$ The line \(L\) with equation \(6y = 4x - 15\) intersects \(H\) at the point \(P\) and at the point \(Q\) as shown in Figure 1.
  1. Show that \(L\) intersects \(H\) where \(4t^2 - 5t - 6 = 0\) [3]
  2. Hence, or otherwise, find the coordinates of points \(P\) and \(Q\). [5]
Edexcel C1 Q5
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y^2 = 4(x - 2)\) and the line with equation \(2x - 3y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).
  1. Write down the coordinates of \(A\). [1]
  2. Find, using algebra, the coordinates of \(P\) and \(Q\). [6]
  3. Show that \(\angle PAQ\) is a right angle. [4]
OCR C1 2006 June Q7
9 marks Moderate -0.3
  1. Solve the simultaneous equations $$y = x^2 - 5x + 4, \quad y = x - 1.$$ [4]
  2. State the number of points of intersection of the curve \(y = x^2 - 5x + 4\) and the line \(y = x - 1\). [1]
  3. Find the value of \(c\) for which the line \(y = x + c\) is a tangent to the curve \(y = x^2 - 5x + 4\). [4]
OCR C1 2013 June Q9
12 marks Moderate -0.8
  1. Sketch the curve \(y = 2x^2 - x - 6\), giving the coordinates of all points of intersection with the axes. [5]
  2. Find the set of values of \(x\) for which \(2x^2 - x - 6\) is a decreasing function. [3]
  3. The line \(y = 4\) meets the curve \(y = 2x^2 - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(PQ\). [4]
OCR MEI C1 2006 June Q13
12 marks Moderate -0.8
Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
  2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
  3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
  4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]
OCR MEI C1 2010 June Q10
12 marks Moderate -0.3
  1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
  2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
  3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]
OCR MEI C1 2011 June Q11
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
  2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
  3. For the curve \(y = 4x^2 + 24x + 31\),
    1. write down the equation of the line of symmetry, [1]
    2. write down the minimum \(y\)-value on the curve. [1]
OCR MEI C1 2011 June Q12
12 marks Moderate -0.8
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{4}{x^2}\).
  1. On the copy of Fig. 12, draw accurately the line \(y = 2x + 5\) and hence find graphically the three roots of the equation \(\frac{4}{x^2} = 2x + 5\). [3]
  2. Show that the equation you have solved in part (i) may be written as \(2x^3 + 5x^2 - 4 = 0\). Verify that \(x = -2\) is a root of this equation and hence find, in exact form, the other two roots. [6]
  3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x^3 + 2x^2 - 4 = 0\). [3]
OCR MEI C1 2012 June Q12
13 marks Moderate -0.3
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{-1}{x - 3}\).
  1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{-1}{x - 3}\) and \(y = x^2 - 4x + 1\). [5]
  2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
  3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]
OCR MEI C1 2013 June Q12
12 marks Standard +0.3
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{1}{x - 2}\).
  1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\). [3]
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
  3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x - 2}\). [4]
Edexcel C1 Q9
11 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [3]
The line \(l\) is the tangent to \(C\) at \(O\).
  1. Find an equation for \(l\). [4]
  2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]