1.02c Simultaneous equations: two variables by elimination and substitution

6. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$\begin{gathered} a b = 25 \\ \log _ { 4 } a - \log _ { 4 } b = 3 \end{gathered}$$ Show each step of your working, giving exact values for \(a\) and \(b\).

14. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale Figure 2 shows part of the line \(l\) with equation \(y = 2 x - 3\) and part of the curve \(C\) with equation \(y = x ^ { 2 } - 2 x - 15\) The line \(l\) and the curve \(C\) intersect at the points \(A\) and \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). In Figure 2, the shaded region \(R\) is bounded by the line \(l\), the curve \(C\) and the positive \(x\)-axis.
2. Use integration to calculate an exact value for the area of \(R\).

13. The curve \(C\) has equation $$y = 3 x ^ { 2 } - 4 x + 2$$ The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)

1. Show that \(l _ { 1 }\) has equation $$x + 2 y - 3 = 0$$ The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
2. By solving simultaneous equations, determine the coordinates of the point \(Q\). Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
3. Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when $$k ^ { 2 } - 16 k + 40 = 0$$
4. Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the line \(l\) with equation \(y = 8 - x\) and part of the curve \(C\) with equation \(y = 14 + 3 x - 2 x ^ { 2 }\) The line \(l\) and the curve \(C\) intersect at the point \(A\) and the point \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the coordinate axes, the line \(l\), and the curve \(C\).
2. Use algebraic integration to calculate the exact area of \(R\).

14. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 3, is bounded by the straight line \(l\) with equation \(y = 4 x + 3\) and the curve \(C\) with equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 2 x + 3 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(A\) on the \(y\)-axis and \(l\) meets \(C\) again at the point \(B\), as shown in Figure 3.

1. Use algebra to find the coordinates of \(A\) and \(B\).
2. Use integration to find the area of the shaded region \(R\).

14. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph of \(y = g ( x ) , - 3 \leqslant x \leqslant 4\) and part of the line \(l\) with equation \(y = \frac { 1 } { 2 } x\) The graph of \(y = \mathrm { g } ( x )\) consists of three line segments, from \(P ( - 3,4 )\) to \(Q ( 0,4 )\), from \(Q ( 0,4 )\) to \(R ( 2,0 )\) and from \(R ( 2,0 )\) to \(S ( 4,10 )\). The line \(l\) intersects \(y = \mathrm { g } ( x )\) at the points \(A\) and \(B\) as shown in Figure 4.

1. Use algebra to find the \(x\) coordinate of the point \(A\) and the \(x\) coordinate of the point \(B\). Show each step of your working and give your answers as exact fractions.
2. Sketch the graph with equation $$y = \frac { 3 } { 2 } g ( x ) , \quad - 3 \leqslant x \leqslant 4$$ On your sketch show the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.

1. In a large theatre there are 20 rows of seats.

The number of seats in the first row is \(a\), where \(a\) is a constant. In the second row the number of seats is \(( a + d )\), where \(d\) is a constant. In the third row the number of seats is \(( a + 2 d )\), and on each subsequent row there are \(d\) more seats than on the previous row. The number of seats in each row forms an arithmetic sequence. The total number of seats in the first 10 rows is 395

1. Use this information to show that \(10 a + 45 d = 395\) The total number of seats in the first 18 rows is 927
2. Use this information to write down a second simplified equation relating \(a\) and \(d\).
3. Solve these equations to find the value of \(a\) and the value of \(d\).
4. Find the number of seats in the 20th row of the theatre.

2. Use algebra to solve the simultaneous equations $$\begin{aligned} x + y & = 5 \\ x ^ { 2 } + x + y ^ { 2 } & = 51 \end{aligned}$$ You must show all stages of your working.
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11. \begin{figure}[h] \end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 2.

1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 2.
2. Use calculus to find the exact area of \(R\).

4. Solve the simultaneous equations $$\begin{gathered} x + y = 2 \\ x ^ { 2 } + 2 y = 12 \end{gathered}$$

4. Solve the simultaneous equations $$\begin{aligned} & y = x - 2 , \\ & y ^ { 2 } + x ^ { 2 } = 10 . \end{aligned}$$

5. Solve the simultaneous equations $$\begin{array} { r } y - 3 x + 2 = 0 \\ y ^ { 2 } - x - 6 x ^ { 2 } = 0 \end{array}$$

3. Solve the simultaneous equations $$\begin{gathered} x - 2 y - 1 = 0 \\ x ^ { 2 } + 4 y ^ { 2 } - 10 x + 9 = 0 \end{gathered}$$

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

6. (a) By eliminating \(y\) from the equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ show that $$x ^ { 2 } + 4 x - 8 = 0$$ (b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ giving your answers in the form \(a \pm b \sqrt { } 3\), where \(a\) and \(b\) are integers.

4. Solve the simultaneous equations $$\begin{aligned} x + y & = 2 \\ 4 y ^ { 2 } - x ^ { 2 } & = 11 \end{aligned}$$

11. \begin{figure}[h] \end{figure} The line \(y = x + 2\) meets the curve \(x ^ { 2 } + 4 y ^ { 2 } - 2 x = 35\) at the points \(A\) and \(B\) as shown in Figure 2.

1. Find the coordinates of \(A\) and the coordinates of \(B\).
2. Find the distance \(A B\) in the form \(r \sqrt { 2 }\) where \(r\) is a rational number.

1. Given the simultaneous equations

$$\begin{aligned} & 2 x + y = 1 \\ & x ^ { 2 } - 4 k y + 5 k = 0 \end{aligned}$$ where \(k\) is a non zero constant,

1. show that $$x ^ { 2 } + 8 k x + k = 0$$ Given that \(x ^ { 2 } + 8 k x + k = 0\) has equal roots,
2. find the value of \(k\).
3. For this value of \(k\), find the solution of the simultaneous equations.

11. \begin{figure}[h] \end{figure} A sketch of part of the curve \(C\) with equation $$y = 20 - 4 x - \frac { 18 } { x } , \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has an \(x\) coordinate equal to 2

1. Show that the equation of the normal to \(C\) at \(A\) is \(y = - 2 x + 7\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3 .
2. Use algebra to find the coordinates of \(B\).

5. Solve the simultaneous equations $$\begin{gathered} y + 4 x + 1 = 0 \\ y ^ { 2 } + 5 x ^ { 2 } + 2 x = 0 \end{gathered}$$

5. (a) Show that eliminating \(y\) from the equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ produces the equation $$x ^ { 2 } + 8 x - 1 = 0$$ (b) Hence solve the simultaneous equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ giving your answers in the form \(a + b \sqrt { } 17\), where \(a\) and \(b\) are integers.

8. The straight line \(l _ { 1 }\) has equation \(y = 3 x - 6\).
The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point (6, 2).

1. Find an equation for \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   (3)
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\).
2. Use algebra to find the coordinates of \(C\).
   (3)
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) cross the \(x\)-axis at the points \(A\) and \(B\) respectively.
3. Calculate the exact area of triangle \(A B C\).
   (4) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \end{tabular} & Leave blank
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   8. continued	Leave blank

   \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

8. Figure 2 Figure 2 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = 10 x - x ^ { 2 } - 8 & x > 0 \\ C _ { 2 } : y = x ^ { 3 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\).

1. Verify that the point \(A\) has coordinates (1, 1)
2. Use algebra to find the coordinates of the point \(B\) The finite region \(R\) is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
3. Use calculus to find the exact area of \(R\) \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-23_936_759_118_582} \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-26_2674_1948_107_118}

8. \begin{figure}[h] \end{figure} The line with equation \(y = 3 x + 20\) cuts the curve with equation \(y = x ^ { 2 } + 6 x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
2. Use calculus to find the exact area of \(S\).

8. \begin{figure}[h] \end{figure} The line with equation \(y = x + 5\) cuts the curve with equation \(y = x ^ { 2 } - 3 x + 8\) at the points \(A\) and \(B\), as shown in Fig. 2.

1. Find the coordinates of the points \(A\) and \(B\).
2. Find the area of the shaded region between the curve and the line, as shown in Fig. 2.