1.02c Simultaneous equations: two variables by elimination and substitution

8 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-3_613_897_1311_623} The diagram shows part of the curve \(y = \frac { 2 } { 1 - x }\) and the line \(y = 3 x + 4\). The curve and the line meet at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\).
2. Find the length of the line \(A B\) and the coordinates of the mid-point of \(A B\).

7 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-3_534_895_255_625} The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48 m .

1. Find an expression for \(y\) in terms of \(x\).
2. Given that the area of the garden is \(A \mathrm {~m} ^ { 2 }\), show that \(A = 48 x - 8 x ^ { 2 }\).
3. Given that \(x\) can vary, find the maximum area of the garden, showing that this is a maximum value rather than a minimum value.

9 A line has equation \(y = k x + 6\) and a curve has equation \(y = x ^ { 2 } + 3 x + 2 k\), where \(k\) is a constant.

1. For the case where \(k = 2\), the line and the curve intersect at points \(A\) and \(B\). Find the distance \(A B\) and the coordinates of the mid-point of \(A B\).
2. Find the two values of \(k\) for which the line is a tangent to the curve.

4 The equation of a curve is \(y ^ { 2 } + 2 x = 13\) and the equation of a line is \(2 y + x = k\), where \(k\) is a constant.

1. In the case where \(k = 8\), find the coordinates of the points of intersection of the line and the curve.
2. Find the value of \(k\) for which the line is a tangent to the curve.

7 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-3_821_688_255_731}

1. The diagram shows part of the curve \(y = 11 - x ^ { 2 }\) and part of the straight line \(y = 5 - x\) meeting at the point \(A ( p , q )\), where \(p\) and \(q\) are positive constants. Find the values of \(p\) and \(q\).
2. The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \begin{cases} 11 - x ^ { 2 } & \text { for } 0 \leqslant x \leqslant p \\ 5 - x & \text { for } x > p \end{cases}$$ Express \(\mathrm { f } ^ { - 1 } ( x )\) in a similar way.

10 A straight line has equation \(y = - 2 x + k\), where \(k\) is a constant, and a curve has equation \(y = \frac { 2 } { x - 3 }\).

1. Show that the \(x\)-coordinates of any points of intersection of the line and curve are given by the equation \(2 x ^ { 2 } - ( 6 + k ) x + ( 2 + 3 k ) = 0\).
2. Find the two values of \(k\) for which the line is a tangent to the curve. The two tangents, given by the values of \(k\) found in part (ii), touch the curve at points \(A\) and \(B\).
3. Find the coordinates of \(A\) and \(B\) and the equation of the line \(A B\).

6 The points \(A ( 1,1 )\) and \(B ( 5,9 )\) lie on the curve \(6 y = 5 x ^ { 2 } - 18 x + 19\).

1. Show that the equation of the perpendicular bisector of \(A B\) is \(2 y = 13 - x\).
   The perpendicular bisector of \(A B\) meets the curve at \(C\) and \(D\).
2. Find, by calculation, the distance \(C D\), giving your answer in the form \(\sqrt { } \left( \frac { p } { q } \right)\), where \(p\) and \(q\) are integers.

9 A curve has equation \(y = 2 x ^ { 2 } - 3 x + 1\) and a line has equation \(y = k x + k ^ { 2 }\), where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve and the line meet.
2. State the value of \(k\) for which the line is a tangent to the curve and find the coordinates of the point where the line touches the curve.

5 The equation of a curve is \(3 x ^ { 2 } + 2 x y + y ^ { 2 } = 6\). It is given that there are two points on the curve where the tangent is parallel to the \(x\)-axis.

1. Show by differentiation that, at these points, \(y = - 3 x\).
2. Hence find the coordinates of the two points.

3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).

8 The points \(A , B\) and \(C\) have position vectors \(\overrightarrow { \mathrm { OA } } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } , \overrightarrow { \mathrm { OB } } = 5 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { \mathrm { OC } } = 8 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }\), where \(O\) is the origin. The line \(l _ { 1 }\) passes through \(B\) and \(C\).

1. Find a vector equation for \(l _ { 1 }\).
   The line \(l _ { 2 }\) has equation \(\mathbf { r } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\).
2. Find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(D\) on \(l _ { 2 }\) is such that \(\mathrm { AB } = \mathrm { BD }\). Find the position vector of \(D\). \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-13_58_1545_388_349}

3 The variables \(x\) and \(y\) satisfy the relation \(2 ^ { y } = 3 ^ { 1 - 2 x }\).

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
2. Find the exact \(x\)-coordinate of the point of intersection of this line with the line \(y = 3 x\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.

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

Solutions relying on calculator technology are not acceptable.

1. $$f ( x ) = ( x + \sqrt { 2 } ) ^ { 2 } + ( 3 x - 5 \sqrt { 8 } ) ^ { 2 }$$ Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 2 } + b x \sqrt { 2 } + c\) where \(a , b\) and \(c\) are integers to be found.
2. Solve the equation $$\sqrt { 3 } ( 4 y - 3 \sqrt { 3 } ) = 5 y + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are simplified fractions to be found.

4. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} Figure 1 shows a line \(l\) with equation \(x + y = 6\) and a curve \(C\) with equation \(y = 6 x - 2 x ^ { 2 } + 1\) The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 1, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).
   

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8. The line \(l _ { 1 }\) has equation $$2 x - 5 y + 7 = 0$$

1. Find the gradient of \(l _ { 1 }\) Given that
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(M\).
2. Using algebra and showing all your working, find the coordinates of \(M\).
   (Solutions relying on calculator technology are not acceptable.) Given that the diagonals of a square \(A B C D\) meet at \(M\),
3. find the coordinates of the point \(C\).

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

Solutions relying on calculator technology are not acceptable. Figure 1 shows the plan view of a flower bed.
The flowerbed is in the shape of a triangle \(A B C\) with

• \(A B = p\) metres
• \(A C = q\) metres
• \(B C = 2 \sqrt { 2 }\) metres
• angle \(B A C = 60 ^ { \circ }\)
  1. Show that

$$p ^ { 2 } + q ^ { 2 } - p q = 8$$ Given that side \(A C\) is 2 metres longer than side \(A B\), use algebra to find

1. the exact value of \(p\),
2. the exact value of \(q\). Using the answers to part (b),

calculate the exact area of the flower bed.

6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

1. Given that $$2 x y - 3 x ^ { 2 } = 50$$ and $$y - x ^ { 3 } + 6 x = 0$$ show that $$2 x ^ { 4 } - 15 x ^ { 2 } - 50 = 0$$
2. Hence solve the simultaneous equations $$\begin{aligned} 2 x y - 3 x ^ { 2 } & = 50 \\ y - x ^ { 3 } + 6 x & = 0 \end{aligned}$$ Give your answers in fully simplified surd form. \includegraphics[max width=\textwidth, alt={}, center]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-14_2257_52_312_1982}

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A rectangular sports pitch has length \(x\) metres and width \(y\) metres, where \(x > y\) Given that the perimeter of the pitch is 350 m ,

1. write down an equation linking \(x\) and \(y\) Given also that the area of the pitch is \(7350 \mathrm {~m} ^ { 2 }\)
2. write down a second equation linking \(x\) and \(y\)
3. hence find the value of \(x\) and the value of \(y\)

1. The curve \(C _ { 1 }\) has equation

$$y = x ^ { 2 } + k x - 9$$ and the curve \(C _ { 2 }\) has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where \(k\) is a constant.
Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at a single point \(P\)

1. show that $$k ^ { 2 } + 26 k + 169 = 0$$
2. Hence find the coordinates of \(P\)

6. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.} Figure 3 shows

• the line \(l\) with equation \(y - 5 x = 75\)
• the curve \(C\) with equation \(y = 2 x ^ { 2 } + x - 21\)

The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\), as shown in Figure 3 .

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = 2 x ^ { 3 } - k x ^ { 2 } + 14 x + 24$$ and \(k\) is a constant.

1. Find, in simplest form,
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { f } ^ { \prime \prime } ( x )\) The curve with equation \(y = \mathrm { f } ^ { \prime } ( x )\) intersects the curve with equation \(y = \mathrm { f } ^ { \prime \prime } ( x )\) at the points \(A\) and \(B\). Given that the \(x\) coordinate of \(A\) is 5
2. find the value of \(k\).
3. Hence find the coordinates of \(B\).

2. A tree was planted in the ground. Exactly 2 years after it was planted, the height of the tree was 1.85 m . Exactly 7 years after it was planted, the height of the tree was 3.45 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted in the ground, can be modelled by the equation $$H = a t + b$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\).
2. State, according to the model, the height of the tree when it was planted.
   

4. Use algebra to solve the simultaneous equations $$\begin{array} { r } y - 3 x = 4 \\ x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 4 \end{array}$$ You must show all stages of your working.

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

\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where \(p\) is a constant, has two distinct real roots.

1. Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
2. Hence, using algebra, find the range of possible values of \(p\)