1.02q Use intersection points: of graphs to solve equations

259 questions

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Edexcel C1 Q8
10 marks Moderate -0.3
  1. Describe fully the single transformation that maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = \text{f}(x - 1)\). [2]
  2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
  3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]
Edexcel C1 Q5
7 marks Moderate -0.8
  1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
  3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]
Edexcel C1 Q4
6 marks Moderate -0.3
  1. Sketch on the same diagram the curves \(y = x^2 - 4x\) and \(y = -\frac{1}{x}\). [4]
  2. State, with a reason, the number of real solutions to the equation $$x^2 - 4x + \frac{1}{x} = 0.$$ [2]
OCR C1 Q7
10 marks Standard +0.3
  1. Describe fully the single transformation that maps the graph of \(y = f(x)\) onto the graph of \(y = f(x - 1)\). [2]
  2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
  3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]
OCR C1 Q8
11 marks Moderate -0.3
\includegraphics{figure_8} The diagram 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]
OCR C1 Q9
10 marks Standard +0.3
A curve has the equation \(y = x + \frac{3}{x}\), \(x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate \(1\).
  1. Show that the gradient of the curve at \(P\) is \(-2\). [3]
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = mx + c\). [3]
  3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]
OCR C1 Q7
11 marks Moderate -0.8
The curve \(C\) has the equation \(y = x^2 + 2x + 4\).
  1. Express \(x^2 + 2x + 4\) in the form \((x + p)^2 + q\) and hence state the coordinates of the minimum point of \(C\). [4]
The straight line \(l\) has the equation \(x + y = 8\).
  1. Sketch \(l\) and \(C\) on the same set of axes. [3]
  2. Find the coordinates of the points where \(l\) and \(C\) intersect. [4]
OCR C1 Q4
7 marks Moderate -0.3
  1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
  3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]
OCR MEI C1 Q1
12 marks Moderate -0.8
\includegraphics{figure_1} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \((-5, 0)\), \((-2, 0)\), \((1.5, 0)\) and \((0, -30)\).
  1. Use the intersections with both axes to express the equation of the curve in a factorised form. [2]
  2. Hence show that the equation of the curve may be written as \(y = 2x^3 + 11x^2 - x - 30\). [2]
  3. Draw the line \(y = 5x + 10\) accurately on the graph. The curve and this line intersect at \((-2, 0)\); find graphically the \(x\)-coordinates of the other points of intersection. [3]
  4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2x^2 + 7x - 20 = 0.$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. [5]
OCR MEI C1 Q2
12 marks Standard +0.3
\includegraphics{figure_2} 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]
OCR MEI C1 Q3
13 marks Moderate -0.3
\includegraphics{figure_3} 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 Q4
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 Q6
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 Q6
11 marks Moderate -0.3
  1. \includegraphics{figure_1} Fig. 10 shows a sketch of the graph of \(y = \frac{1}{x}\). Sketch the graph of \(y = \frac{1}{x-2}\), showing clearly the coordinates of any points where it crosses the axes. [3]
  2. Find the value of \(x\) for which \(\frac{1}{x-2} = 5\). [2]
  3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac{1}{x-2}\). Give your answers in the form \(a \pm \sqrt{b}\). Show the position of these points on your graph in part (i). [6]
OCR MEI C1 Q2
11 marks Moderate -0.3
Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\).
  1. On the insert, on the same axes, plot the graph of \(y = x^2 - 5x + 5\) for \(0 \leq x \leq 5\). [4]
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac{1}{x}\) and \(y = x^2 - 5x + 5\) satisfy the equation \(x^3 - 5x^2 + 5x - 1 = 0\). [2]
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x^3 - 5x^2 + 5x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x^3 - 5x^2 + 5x - 1 = 0\) is rational. [5]
OCR MEI C1 Q4
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]
Edexcel C2 Q9
12 marks Standard +0.3
\includegraphics{figure_2} 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 Q4
9 marks Moderate -0.3
\includegraphics{figure_1} Fig. 1 shows the curve with equation \(y = 5 + 2x - x^2\) and the line with equation \(y = 2\). The curve and the line intersect at the points \(A\) and \(B\).
  1. Find the \(x\)-coordinates of \(A\) and \(B\). [3]
The shaded region \(R\) is bounded by the curve and the line.
  1. Find the area of \(R\). [6]
Edexcel C2 Q7
12 marks Standard +0.3
\includegraphics{figure_2} Fig. 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]
OCR C3 2013 January Q6
11 marks Standard +0.3
  1. By sketching the curves \(y = \ln x\) and \(y = 8 - 2x^2\) on a single diagram, show that the equation $$\ln x = 8 - 2x^2$$ has exactly one real root. [3]
  2. Explain how your diagram shows that the root is between 1 and 2. [1]
  3. Use the iterative formula $$x_{n+1} = \sqrt{4 - \frac{1}{2}\ln x_n},$$ with a suitable starting value, to find the root. Show all your working and give the root correct to 3 decimal places. [4]
  4. The curves \(y = \ln x\) and \(y = 8 - 2x^2\) are each translated by 2 units in the positive \(x\)-direction and then stretched by scale factor 4 in the \(y\)-direction. Find the coordinates of the point where the new curves intersect, giving each coordinate correct to 2 decimal places. [3]
Edexcel AEA 2004 June Q6
17 marks Challenging +1.8
$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).
  1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
  2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]
Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),
  1. find the value of \(k\). [2]
  2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]
The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.
  1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
  2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]
AQA AS Paper 1 2020 June Q7
6 marks Standard +0.3
Curve C has equation \(y = x^2\) C is translated by vector \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give curve \(C_1\) Line L has equation \(y = x\) L is stretched by scale factor 2 parallel to the \(x\)-axis to give line \(L_1\) Find the exact distance between the two intersection points of \(C_1\) and \(L_1\) [6 marks]
AQA AS Paper 1 2020 June Q8
8 marks Standard +0.3
  1. Find the equation of the tangent to the curve \(y = e^{4x}\) at the point \((a, e^{4a})\). [3 marks]
  2. Find the value of \(a\) for which this tangent passes through the origin. [2 marks]
  3. Hence, find the set of values of \(m\) for which the equation $$e^{4x} = mx$$ has no real solutions. [3 marks]
AQA Paper 3 2024 June Q7
5 marks Moderate -0.8
The graphs with equations $$y = 2 + 3x - 2x^2 \text{ and } x + y = 1$$ are shown in the diagram below. \includegraphics{figure_7} The graphs intersect at the points \(A\) and \(B\) \begin{enumerate}[label=(\alph*)] \item On the diagram above, shade and label the region, \(R\), that is satisfied by the inequalities $$0 \leq y \leq 2 + 3x - 2x^2$$ and $$x + y \geq 1$$ [2 marks] \item Find the exact coordinates of \(A\) [3 marks]
OCR MEI Paper 2 Specimen Q1
5 marks Moderate -0.8
In this question you must show detailed reasoning. Find the coordinates of the points of intersection of the curve \(y = x^2 + x\) and the line \(2x + y = 4\). [5]