1.02f Solve quadratic equations: including in a function of unknown

5. Ben is saving for the deposit for a house over a period of 60 months. Ben saves \(\pounds 100\) in the first month and in each subsequent month, he saves \(\pounds 5\) more than the previous month, so that he saves \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, and so on, forming an arithmetic sequence.

1. Find the amount Ben saves in the 40th month.
2. Find the total amount Ben saves over the 60 -month period. Lina is also saving for a deposit for a house.
   Lina saves \(\pounds 600\) in the first month and in each subsequent month, she saves \(\pounds 10\) less than the previous month, so that she saves \(\pounds 590\) in the second month, \(\pounds 580\) in the third month, and so on, forming an arithmetic sequence. Given that, after \(n\) months, Lina will have saved exactly \(\pounds 18200\) for her deposit,
3. form an equation in \(n\) and show that it can be written as $$n ^ { 2 } - 121 n + 3640 = 0$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible value for \(n\).

5. A company that owned a silver mine

• extracted 480 tonnes of silver from the mine in year 1
• extracted 465 tonnes of silver from the mine in year 2
• extracted 450 tonnes of silver from the mine in year 3
  and so on, forming an arithmetic sequence.
  1. Find the mass of silver extracted in year 14

After a total of 7770 tonnes of silver was extracted, the company stopped mining. Given that this occurred at the end of year \(N\),

show that $$N ^ { 2 } - 65 N + 1036 = 0$$

Hence, state the value of \(N\).

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$ can be written as $$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
2. Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) $$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$

4.

1. Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$
2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ giving your answers to 1 decimal place where appropriate.
   

7. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = x ^ { 2 } - 5 x + 4\). It cuts the \(x\)-axis at the points \(L\) and \(M\) as shown in Figure 2.

1. Find the coordinates of the point \(L\) and the point \(M\).
2. Show that the point \(N ( 5,4 )\) lies on \(C\).
3. Find \(\int \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\). The finite region \(R\) is bounded by \(L N , L M\) and the curve \(C\) as shown in Figure 2.
4. Use your answer to part (c) to find the exact value of the area of \(R\).
   \section*{LU}

6. Given that $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

1. Show that $$x ^ { 2 } - 34 x + 225 = 0$$
2. Hence, or otherwise, solve the equation $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

7.

1. Given that $$2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1$$ show that \(x ^ { 2 } - 16 x + 64 = 0\).
2. Hence, or otherwise, solve \(2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1\).
   

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) \mathrm { e } ^ { x ^ { 2 } }$$ The curve cuts the \(x\)-axis at points \(A\) and \(B\) as shown in Figure 2 .

1. Calculate the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\), giving your answers to 3 decimal places.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve has a minimum turning point at the point \(P\) as shown in Figure 2.
3. Show that the \(x\) coordinate of \(P\) is the solution of $$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$
4. Use the iteration formula $$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places. The \(x\) coordinate of \(P\) is \(\alpha\).
5. By choosing a suitable interval, prove that \(\alpha = - 2.43\) to 2 decimal places.

4. $$\mathrm { f } ( x ) = 25 x ^ { 2 } \mathrm { e } ^ { 2 x } - 16 , \quad x \in \mathbb { R }$$

1. Using calculus, find the exact coordinates of the turning points on the curve with equation \(y = \mathrm { f } ( x )\).
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \pm \frac { 4 } { 5 } \mathrm { e } ^ { - x }\) The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 0.5\) to 1 decimal place.
3. Starting with \(x _ { 0 } = 0.5\), use the iteration formula $$x _ { n + 1 } = \frac { 4 } { 5 } \mathrm { e } ^ { - x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
4. Give an accurate estimate for \(\alpha\) to 2 decimal places, and justify your answer.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = 3 + \sqrt { x + 2 } , \quad x \geqslant - 2$$

1. State the range of g .
2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
3. Find the exact value of \(x\) for which $$\mathrm { g } ( x ) = x$$
4. Hence state the value of \(a\) for which $$\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )$$

7.

1. Find $$\int ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } d x$$ giving your answer in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-13_727_1177_596_370} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = ( 2 x - 1 ) ^ { \frac { 3 } { 4 } } , \quad x \geqslant \frac { 1 } { 2 }$$ The curve \(C\) cuts the line \(y = 8\) at the point \(P\) with coordinates \(( k , 8 )\), where \(k\) is a constant.
2. Find the value of \(k\). The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(y = 8\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the exact value of the volume of the solid generated.

1. $$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.
(5)

1. $$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.

4. The hyperbola \(H\) has equation $$x y = 3$$ The point \(Q ( 1,3 )\) is on \(H\).

1. Find the equation of the normal to \(H\) at \(Q\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
   (5) The normal at \(Q\) intersects \(H\) again at the point \(R\).
2. Find the coordinates of \(R\).
   (5)

$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b ,$$ where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\).
2. Hence, or otherwise, show that the roots of $$x ^ { 2 } - 8 x - 29 = 0$$ are \(c \pm d \sqrt { } 5\), where \(c\) and \(d\) are integers to be found.

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants.

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.

1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).
   

7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)

1. Find a cartesian equation for the ellipse \(E\). The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
2. Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line \(l\) is a tangent to \(E\),
3. show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found. The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
4. Show that the area of triangle \(O A B\), where \(O\) is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
5. Find the minimum area of triangle \(O A B\).

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   Q7

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.

7

1. Solve the equation \(x ^ { 2 } - 8 x + 11 = 0\), giving your answers in simplified surd form.
2. Hence sketch the curve \(y = x ^ { 2 } - 8 x + 11\), labelling the points where the curve crosses the axes.
3. Solve the equation \(y - 8 y ^ { \frac { 1 } { 2 } } + 11 = 0\), giving your answers in the form \(p \pm q \sqrt { 5 }\).

4 Solve the equation \(x ^ { \frac { 2 } { 3 } } + 3 x ^ { \frac { 1 } { 3 } } - 10 = 0\).

6

1. Solve the equation \(x ^ { 2 } + 8 x + 10 = 0\), giving your answers in simplified surd form.
2. Sketch the curve \(y = x ^ { 2 } + 8 x + 10\), giving the coordinates of the point where the curve crosses the \(y\)-axis.
3. Solve the inequality \(x ^ { 2 } + 8 x + 10 \geqslant 0\).

1 Solve the inequality \(x ^ { 2 } - 6 x - 40 \geqslant 0\).

4 Solve the equation \(x ^ { 6 } + 26 x ^ { 3 } - 27 = 0\).

6 By using the substitution \(y = ( x + 2 ) ^ { 2 }\), find the real roots of the equation $$( x + 2 ) ^ { 4 } + 5 ( x + 2 ) ^ { 2 } - 6 = 0$$