1.02c Simultaneous equations: two variables by elimination and substitution

The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).

1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
2. Find the \(x\)-coordinates of \(A\) and \(B\).
3. Show that \(A B = 2 \sqrt { 10 }\).
   

8.

1. Given that $$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x ,$$ show that $$y = 2 x + 1 .$$
2. Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x \\ & 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$

3. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0 \\ & \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$

7. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R } ,$$ where \(a\) is a positive constant.

1. Showing the coordinates of any points where each graph meets the axes, sketch on separate diagrams the graphs of
   1. \(\quad y = | \mathrm { f } ( x ) |\),
   2. \(y = \mathrm { f } ( | x | )\). The function g is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
2. Find fg(a) in terms of \(a\).
3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$

1 A particle of mass \(m\) has velocity \(\left[ \begin{array} { l } 4 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides with a particle of mass 3 kg which has velocity \(\left[ \begin{array} { l } - 1 \\ - 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision the particles coalesce and move with velocity \(\left[ \begin{array} { l } 1 \\ V \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that \(m = 2\).
2. Find \(V\).

8 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface.
The particle \(A\) has mass \(m \mathrm {~kg}\) and is moving with velocity \(\left[ \begin{array} { r } 5 \\ - 3 \end{array} \right] \mathrm { ms } ^ { - 1 }\). The particle \(B\) has mass 0.2 kg and is moving with velocity \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find, in terms of \(m\), an expression for the total momentum of the particles.
2. The particles \(A\) and \(B\) collide and form a single particle \(C\), which moves with velocity \(\left[ \begin{array} { c } k \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant.
   1. Show that \(m = 0.1\).
   2. Find the value of \(k\).

4. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and \(O\) is a fixed origin. A pedestrian moves with constant velocity \(\left[ \left( 2 q ^ { 2 } - 3 \right) \mathbf { i } + ( q + 2 ) \mathbf { j } \right] \mathrm { ms } ^ { - 1 }\). Given that the velocity of the pedestrian is parallel to the vector \(( \mathbf { i } - \mathbf { j } )\),

1. Show that one possible value of \(q\) is \({ } ^ { - } 1\) and find the other possible value of \(q\). Given that \(q = { } ^ { - } 1\), and that the pedestrian started walking at the point with position vector \(( 6 \mathbf { i } - \mathbf { j } ) \mathrm { m }\),
2. find the length of time for which the pedestrian is less than 5 m from \(O\).

1. The resultant of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { N }\).

Given that \(\mathbf { F } _ { \mathbf { 1 } } = ( 2 p \mathbf { i } - 3 q \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { \mathbf { 2 } } = ( 5 q \mathbf { i } + 4 p \mathbf { j } ) \mathrm { N }\), calculate the values of \(p\) and \(q\).
(5 marks)

9 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows the curve with equation $$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$ and the straight line with equation $$x + y = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}

1. Write down the exact coordinates of the points where the curve with equation \(\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1\) intersects the coordinate axes.
2. The curve is translated \(k\) units in the positive \(x\) direction, where \(k\) is a constant. Write down, in terms of \(k\), the equation of the curve after this translation.
3. Show that, if the line \(x + y = 2\) intersects the translated curve, the \(x\)-coordinates of the points of intersection must satisfy the equation $$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
4. Hence find the two values of \(k\) for which the line \(x + y = 2\) is a tangent to the translated curve. Give your answer in the form \(p \pm \sqrt { q }\), where \(p\) and \(q\) are integers.
5. On Figure 3, show the translated curves corresponding to these two values of \(k\). \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2 (for use in Question 5)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3 (for use in Question 9)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
   \end{figure}

8 A curve has equation $$y = \frac { x ^ { 2 } } { ( x - 1 ) ( x - 5 ) }$$

1. Write down the equations of the three asymptotes to the curve.
2. Show that the curve has no point of intersection with the line \(y = - 1\).
3. 1. Show that, if the curve intersects the line \(y = k\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$( k - 1 ) x ^ { 2 } - 6 k x + 5 k = 0$$
   2. Show that, if this equation has equal roots, then $$k ( 4 k + 5 ) = 0$$
4. Hence find the coordinates of the two stationary points on the curve.

8 The diagram shows the ellipse \(E\) with equation $$\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$$ and the straight line \(L\) with equation $$y = x + 4$$ \includegraphics[max width=\textwidth, alt={}, center]{9f8cd5ed-f5cf-4cf6-8c92-9fd0819238ca-5_675_1120_708_468}

1. Write down the coordinates of the points where the ellipse \(E\) intersects the coordinate axes.
2. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { c } p \\ 0 \end{array} \right]\), where \(p\) is a constant. Write down the equation of the translated ellipse.
3. Show that, if the translated ellipse intersects the line \(L\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$9 x ^ { 2 } - ( 8 p - 40 ) x + \left( 4 p ^ { 2 } + 60 \right) = 0$$
4. Given that the line \(L\) is a tangent to the translated ellipse, find the coordinates of the two possible points of contact.
   (No credit will be given for solutions based on differentiation.)

1

1. Solve the following simultaneous equations. $$\begin{aligned} & x + \quad y = 1 \\ & x + 0.99 y = 2 \end{aligned}$$
2. The coefficient 0.99 is correct to two decimal places. All other coefficients in the equations are exact. With the aid of suitable calculations, explain why your answer to part (i) is unreliable.

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The line \(l\) has equation \(x - 2 y = c\) The points \(P\) and \(Q\) are the points of intersection of \(H\) and \(l\)

1. Determine, in terms of \(c\), the coordinates of \(P\) and the coordinates of \(Q\) The point \(R\) is the midpoint of \(P Q\)
2. Show that, as \(C\) varies, the coordinates of \(R\) satisfy the equation $$x y = - \frac { c ^ { 2 } } { a }$$ where \(a\) is a constant to be determined.

1. In this question you must show all stages of your working.

A college offers only three courses: Construction, Design and Hospitality. Each student enrols on just one of these courses. In 2019, there was a total of 1110 students at this college.
There were 370 more students enrolled on Construction than Hospitality.
In 2020 the number of students enrolled on

• Construction increased by \(1.25 \%\)
• Design increased by \(2.5 \%\)
• Hospitality decreased by \(2 \%\)

In 2020, the total number of students at the college increased by \(0.27 \%\) to 2 significant figures.

1. 1. Define, for each course, a variable for the number of students enrolled on that course in 2019.
   2. Using your variables from part (a)(i), write down three equations that model this situation.
2. By forming and solving a matrix equation, determine how many students were enrolled on each of the three courses in 2019.

1 In this question you must show detailed reasoning. Solve the equation \(x ( 3 - \sqrt { 5 } ) = 24\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.

2 In this question you must show detailed reasoning. Solve the equation \(x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x\). Give your answer in the form \(a \sqrt { 5 } + b\), where \(a\) and \(b\) are integers to be determined.

8 In this question you must show detailed reasoning. Determine the coordinates of the point of intersection of the line with equation \(y = 2 x + 3\) and the curve with equation \(y ^ { 2 } - 4 x ^ { 2 } = 33\).

7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
Find the coordinates of the points of intersection of the line and the circle.

4 In this question you must show detailed reasoning. Solve the simultaneous equations \(\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 3\) \(\mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { 2 y } = 33\). Give your answer in an exact form.

5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$

2 The line \(A B\) has equation \(3 x + 5 y = 8\) and the point \(A\) has coordinates (6, -2).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the straight line which is perpendicular to \(A B\) and which passes through \(A\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 3\) at the point \(B\). Find the coordinates of \(B\).
3. The point \(C\) has coordinates \(( 2 , k )\) and the distance from \(A\) to \(C\) is 5 . Find the two possible values of the constant \(k\).

3

1. 1. Express \(x ^ { 2 } + 10 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Write down the coordinates of the vertex (minimum point) of the curve with equation \(y = x ^ { 2 } + 10 x + 19\).
   3. Write down the equation of the line of symmetry of the curve \(y = x ^ { 2 } + 10 x + 19\).
   4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 10 x + 19\).
2. Determine the coordinates of the points of intersection of the line \(y = x + 11\) and the curve \(y = x ^ { 2 } + 10 x + 19\).

1 The line \(A B\) has equation \(3 x + 5 y = 11\).

1. 1. Find the gradient of \(A B\).
   2. The point \(A\) has coordinates (2,1). Find an equation of the line which passes through the point \(A\) and which is perpendicular to \(A B\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 8\) at the point \(C\). Find the coordinates of \(C\).

1 Solve the simultaneous equations. $$\begin{array} { r } x ^ { 2 } + 8 x + y ^ { 2 } = 84 \\ x - y = 10 \end{array}$$