1.02c Simultaneous equations: two variables by elimination and substitution

284 questions

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Edexcel C3 2016 June Q1
5 marks Standard +0.3
  1. The functions \(f\) and \(g\) are defined by
$$\begin{aligned} & \mathrm { f } : x \rightarrow 7 x - 1 , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \rightarrow \frac { 4 } { x - 2 } , \quad x \neq 2 , x \in \mathbb { R } \end{aligned}$$
  1. Solve the equation \(\operatorname { fg } ( x ) = x\)
  2. Hence, or otherwise, find the largest value of \(a\) such that \(\mathrm { g } ( a ) = \mathrm { f } ^ { - 1 } ( a )\)
Edexcel P4 2023 January Q2
6 marks Standard +0.3
  1. A set of points \(P ( x , y )\) is defined by the parametric equations
$$x = \frac { t - 1 } { 2 t + 1 } \quad y = \frac { 6 } { 2 t + 1 } \quad t \neq - \frac { 1 } { 2 }$$
  1. Show that all points \(P ( x , y )\) lie on a straight line.
  2. Hence or otherwise, find the \(x\) coordinate of the point of intersection of this line and the line with equation \(y = x + 12\)
Edexcel F1 2017 June Q4
7 marks Standard +0.3
4. The rectangular hyperbola \(H\) has parametric equations $$x = 4 t , \quad y = \frac { 4 } { t }$$ The straight line with equation \(3 y - 2 x = 10\) intersects \(H\) at the points \(A\) and \(B\). Given that the point \(A\) is above the \(x\)-axis,
  1. find the coordinates of the point \(A\) and the coordinates of the point \(B\).
  2. Find the coordinates of the midpoint of \(A B\).
Edexcel F1 2018 Specimen Q6
10 marks Standard +0.3
  1. The rectangular hyperbola \(H\) has equation \(x y = 25\)
    1. Verify that, for \(t \neq 0\), the point \(P \left( 5 t , \frac { 5 } { t } \right)\) is a general point on \(H\).
    The point \(A\) on \(H\) has parameter \(t = \frac { 1 } { 2 }\)
  2. Show that the normal to \(H\) at the point \(A\) has equation $$8 y - 2 x - 75 = 0$$ This normal at \(A\) meets \(H\) again at the point \(B\).
  3. Find the coordinates of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-13_2261_50_312_36}
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Edexcel FP1 2012 January Q9
9 marks Standard +0.8
9. The rectangular hyperbola \(H\) has cartesian equation \(x y = 9\) The points \(P \left( 3 p , \frac { 3 } { p } \right)\) and \(Q \left( 3 q , \frac { 3 } { q } \right)\) lie on \(H\), where \(p \neq \pm q\).
  1. Show that the equation of the tangent at \(P\) is \(x + p ^ { 2 } y = 6 p\).
  2. Write down the equation of the tangent at \(Q\). The tangent at the point \(P\) and the tangent at the point \(Q\) intersect at \(R\).
  3. Find, as single fractions in their simplest form, the coordinates of \(R\) in terms of \(p\) and \(q\).
Edexcel F2 2023 June Q3
7 marks Challenging +1.2
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.} Given that $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { k ( x - 1 ) }$$ where \(k\) is a positive constant,
  1. show that $$( x + 4 ) ( x - 1 ) \left( p x ^ { 2 } + q x + r \right) \leqslant 0$$ where \(p , q\) and \(r\) are expressions in terms of \(k\) to be determined.
  2. Hence, or otherwise, determine the values for \(x\) for which $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { 3 ( x - 1 ) }$$
Edexcel F3 2015 June Q5
9 marks Challenging +1.2
  1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)
The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),
  1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
  2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).
Edexcel C1 2015 June Q2
7 marks Moderate -0.3
Solve the simultaneous equations $$\begin{gathered} y - 2 x - 4 = 0 \\ 4 x ^ { 2 } + y ^ { 2 } + 20 x = 0 \end{gathered}$$
Edexcel F3 2022 June Q9
10 marks Standard +0.8
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$ The line \(l\) has equation \(y = k x - 3\), where \(k\) is a constant.
Given that \(E\) and \(l\) meet at 2 distinct points \(P\) and \(Q\)
  1. show that the \(x\) coordinates of \(P\) and \(Q\) are solutions of the equation $$\left( 9 k ^ { 2 } + 4 \right) x ^ { 2 } - 54 k x + 45 = 0$$ The point \(M\) is the midpoint of \(P Q\)
  2. Determine, in simplest form in terms of \(k\), the coordinates of \(M\)
  3. Hence show that, as \(k\) varies, \(M\) lies on the curve with equation $$x ^ { 2 } + p y ^ { 2 } = q y$$ where \(p\) and \(q\) are constants to be determined.
OCR C1 Q5
6 marks Moderate -0.3
5
  1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7 .$$
  2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
  3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 2005 January Q4
5 marks Moderate -0.5
4 Solve the simultaneous equations $$x ^ { 2 } - 3 y + 11 = 0 , \quad 2 x - y + 1 = 0$$
OCR C1 2006 January Q8
11 marks Moderate -0.3
8
  1. Given that \(y = x ^ { 2 } - 5 x + 15\) and \(5 x - y = 10\), show that \(x ^ { 2 } - 10 x + 25 = 0\).
  2. Find the discriminant of \(x ^ { 2 } - 10 x + 25\).
  3. What can you deduce from the answer to part (ii) about the line \(5 x - y = 10\) and the curve \(y = x ^ { 2 } - 5 x + 15\) ?
  4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
  5. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 5 x + 15\) at the point \(( 5,15 )\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 2008 January Q7
8 marks Moderate -0.8
7
  1. Find the gradient of the line \(l\) which has equation \(x + 2 y = 4\).
  2. Find the equation of the line parallel to \(l\) which passes through the point ( 6,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Solve the simultaneous equations $$y = x ^ { 2 } + x + 1 \quad \text { and } \quad x + 2 y = 4$$
OCR C1 2005 June Q8
8 marks Moderate -0.8
8
  1. Describe completely the curve \(x ^ { 2 } + y ^ { 2 } = 25\).
  2. Find the coordinates of the points of intersection of the curve \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(2 x + y - 5 = 0\).
OCR C1 2008 June Q10
14 marks Moderate -0.8
10
  1. Express \(2 x ^ { 2 } - 6 x + 11\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 6 x + 11\).
  3. Calculate the discriminant of \(2 x ^ { 2 } - 6 x + 11\).
  4. State the number of real roots of the equation \(2 x ^ { 2 } - 6 x + 11 = 0\).
  5. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 6 x + 11\) and the line \(7 x + y = 14\).
OCR C1 Specimen Q5
11 marks Moderate -0.3
5
  1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7$$
  2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
  3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR MEI C1 2008 January Q4
4 marks Easy -1.8
4 Find, algebraically, the coordinates of the point of intersection of the lines \(y = 2 x - 5\) and \(6 x + 2 y = 7\).
OCR MEI C1 2009 January Q6
3 marks Easy -1.8
6 Solve the equation \(\frac { 3 x + 1 } { 2 x } = 4\).
OCR MEI C1 2009 January Q9
4 marks Moderate -0.8
9 Rearrange \(y + 5 = x ( y + 2 )\) to make \(y\) the subject of the formula.
OCR MEI C1 2009 January Q12
11 marks Moderate -0.8
12
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
  2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
  3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.
OCR MEI C1 2007 June Q7
3 marks Easy -1.2
7 Solve the equation \(\frac { 4 x + 5 } { 2 x } = - 3\).
OCR MEI C1 2008 June Q10
12 marks Moderate -0.8
10
  1. Express \(x ^ { 2 } - 6 x + 2\) in the form \(( x - a ) ^ { 2 } - b\).
  2. State the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 2\).
  3. Sketch the graph of \(y = x ^ { 2 } - 6 x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis.
  4. Solve the simultaneous equations \(y = x ^ { 2 } - 6 x + 2\) and \(y = 2 x - 14\). Hence show that the line \(y = 2 x - 14\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 2\).
OCR MEI C1 2015 June Q5
4 marks Easy -1.2
5 Find the coordinates of the point of intersection of the lines \(y = 5 x - 2\) and \(x + 3 y = 8\).
OCR MEI C1 2015 June Q12
12 marks Moderate -0.8
12
  1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
  2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
  3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).
OCR MEI C1 Q8
5 marks Moderate -0.8
8 Find the points where the line \(y = 2 x - 3\) cuts the curve \(y = x ^ { 2 } - 4 x + 5\).