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

1. A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 14 y = 40\).

The line \(l\) has equation \(y = x + k\), where \(k\) is a constant.
a. Show that the \(x\)-coordinate of the points where \(C\) and \(l\) intersect are given by the solutions to the equation $$2 x ^ { 2 } + ( 2 k - 20 ) x + k ^ { 2 } - 14 k - 40 = 0$$ b. Hence find the two values of \(k\) for which \(l\) is a tangent to \(C\).

15. The first three terms of a geometric series where \(\theta\) is a constant are $$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$ a. Show that \(4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0\) Given that \(\theta\) lies in the interval \(90 ^ { \circ } < \theta < 180 ^ { \circ }\),
b. Find the value of \(\theta\).
c. Hence prove that this series is convergent.
d. Find \(S _ { \infty }\), in the form \(a ( 1 - \sqrt { 3 } )\)

1. In an arithmetic series, the first term is \(a\) and the common difference is \(d\). Show that $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$
2. James saves money over a number of weeks to buy a printer that costs \(\pounds 64\) He saves \(\pounds 10\) in week \(1 , \pounds 9.20\) in week \(2 , \pounds 8.40\) in week 3 and so on, so that the weekly amounts he saves form an arithmetic sequence. Given that James takes \(n\) weeks to save exactly \(\pounds 64\)

1. show that $$n ^ { 2 } - 26 n + 160 = 0$$
2. Solve the equation $$n ^ { 2 } - 26 n + 160 = 0$$
3. Hence state the number of weeks James takes to save enough money to buy the printer, giving a brief reason for your answer.

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + 5 x ^ { 2 } - 10 x + 4 a \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.
Given ( \(x - a\) ) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a \left( 4 a ^ { 2 } + 5 a - 6 \right) = 0$$
2. Hence
   1. find the value of \(a\)
   2. use algebra to find the exact solutions of the equation $$f ( x ) = 3$$

1. The functions f and g are defined by

$$\begin{array} { l l } f ( x ) = 4 - 3 x ^ { 2 } & x \in \mathbb { R } \\ g ( x ) = \frac { 5 } { 2 x - 9 } & x \in \mathbb { R } , x \neq \frac { 9 } { 2 } \end{array}$$

1. Find fg(2)
2. Find \(\mathrm { g } ^ { - 1 }\)
3. 1. Find \(\mathrm { gf } ( x )\), giving your answer as a simplified fraction.
   2. Deduce the range of \(\operatorname { gf } ( x )\). The function h is defined by $$h ( x ) = 2 x ^ { 2 } - 6 x + k \quad x \in \mathbb { R }$$ where \(k\) is a constant.
4. Find the range of values of \(k\) for which the equation $$\mathrm { f } ( x ) = \mathrm { h } ( x )$$ has no real solutions.

6. $$f ( x ) = - 3 x ^ { 3 } + 8 x ^ { 2 } - 9 x + 10 , \quad x \in \mathbb { R }$$

1. 1. Calculate f(2)
   2. Write \(\mathrm { f } ( x )\) as a product of two algebraic factors. Using the answer to (a)(ii),
2. prove that there are exactly two real solutions to the equation $$- 3 y ^ { 6 } + 8 y ^ { 4 } - 9 y ^ { 2 } + 10 = 0$$
3. deduce the number of real solutions, for \(7 \pi \leqslant \theta < 10 \pi\), to the equation $$3 \tan ^ { 3 } \theta - 8 \tan ^ { 2 } \theta + 9 \tan \theta - 10 = 0$$

1. Given that

$$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$

1. show that $$3 x ^ { 2 } - 13 x - 30 = 0$$
2. 1. Write down the roots of the equation $$3 x ^ { 2 } - 13 x - 30 = 0$$
   2. Hence state which of the roots in part (b)(i) is not a solution of $$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$ giving a reason for your answer.

1. Jamie takes out an interest-free loan of \(\pounds 8100\)

Jamie makes a payment every month to pay back the loan.
Jamie repays \(\pounds 400\) in month \(1 , \pounds 390\) in month \(2 , \pounds 380\) in month 3 , and so on, so that the amounts repaid each month form an arithmetic sequence.

1. Show that Jamie repays \(\pounds 290\) in month 12 After Jamie's \(N\) th payment, the loan is completely paid back.
2. Show that \(N ^ { 2 } - 81 N + 1620 = 0\)
3. Hence find the value of \(N\).

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$\begin{aligned} u _ { n + 1 } & = k u _ { n } - 5 \\ u _ { 1 } & = 6 \end{aligned}$$ where \(k\) is a positive constant.
Given that \(u _ { 3 } = - 1\)

1. show that $$6 k ^ { 2 } - 5 k - 4 = 0$$
2. Hence
   1. find the value of \(k\),
   2. find the value of \(\sum _ { r = 1 } ^ { 3 } u _ { r }\)

1. A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2 x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
2. find the two possible values of \(r\), giving your answers as fully simplified surds.

8

1. Show that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\), where \(k\) is a constant, can be expressed in the form \(x ^ { 2 } - 8 k x + 8 = 0\).
2. Given that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\) has only one real root, find the value of this root.

5

1. Show that the equation \(2 \cos x \tan ^ { 2 } x = 3 ( 1 + \cos x )\) can be expressed in the form $$5 \cos ^ { 2 } x + 3 \cos x - 2 = 0$$ \section*{(b) In this question you must show detailed reasoning.} Hence solve the equation $$2 \cos 3 \theta \tan ^ { 2 } 3 \theta = 3 ( 1 + \cos 3 \theta ) ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(120 ^ { \circ }\), correct to \(\mathbf { 1 }\) decimal place where appropriate.

6 In this question you must show detailed reasoning.
You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 3 x + 1\).

1. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\).

3 The velocity-time graph for the motion of a particle is shown below. The velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(\mathrm { v } = - \mathrm { t } ^ { 2 } + 6 \mathrm { t } - 6\) where \(0 \leqslant t \leqslant 5\). \includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-3_860_979_1082_239}

1. Find the times at which the velocity is \(2 \mathrm {~ms} ^ { - 1 }\).
2. Write down the greatest speed of the particle.

2

1. Find the discriminant of the equation \(3 x ^ { 2 } - 2 x + 5 = 0\).
2. Use your answer to part (a) to find the number of real roots of the equation \(3 x ^ { 2 } - 2 x + 5 = 0\).

7 Determine the exact distance between the two points at which the line through ( 4,5 ) and ( \(6 , - 1\) ) meets the curve \(y = 2 x ^ { 2 } - 7 x + 1\).

9 The gradient of a curve is given by \(\frac { d y } { d x } = e ^ { x } - 4 e ^ { - x }\).

1. Show that the \(x\)-coordinate of any point on the curve at which the gradient is 3 satisfies the equation \(\left( e ^ { x } \right) ^ { 2 } - 3 e ^ { x } - 4 = 0\).
2. Hence show that there is only one point on the curve at which the gradient is 3 , stating the exact value of its \(x\)-coordinate.
3. The curve passes through the point \(( 0,0 )\). Show that when \(x = 1\) the curve is below the \(x\)-axis.

11 The first three terms of a geometric sequence are \(5 k - 2,3 k - 6 , k + 2\), where \(k\) is a constant.

1. Show that \(k\) satisfies the equation \(k ^ { 2 } - 11 k + 10 = 0\).
2. When \(k\) takes the smaller of the two possible values, find the sum of the first 20 terms of the sequence.
3. When \(k\) takes the larger of the two possible values, find the sum to infinity of the sequence.

4 In this question you must show detailed reasoning.
Find the coordinates of the points where the curve \(y = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6\) crosses the \(x\)-axis.

2 As part of a training exercise an army recruit of mass 75 kg falls a vertical distance of 5 m before landing on a mat of thickness 1.2 m . The army recruit sinks a distance of \(x \mathrm {~m}\) into the mat before instantaneously coming to rest. The mat can be modelled as a spring of natural length 1.2 m and modulus of elasticity 10800 N and the army recruit can be modelled as a particle falling vertically with an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(x\) satisfies the equation \(300 x ^ { 2 } - 49 x - 255 = 0\).
2. Calculate the value of \(x\).
3. Ignoring the possible effect of air resistance, make

6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm . \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561} A square of side \(x \mathrm {~cm}\) is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height \(x \mathrm {~cm}\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid the box can hold is given by $$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that any stationary values of \(V\) must occur when \(x ^ { 2 } - 11 x + 18 = 0\).
   3. Solve the equation \(x ^ { 2 } - 11 x + 18 = 0\).
   4. Explain why there is only one value of \(x\) for which \(V\) is stationary.
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence determine whether the stationary value is a maximum or minimum.

8 A circle has centre \(C ( 3 , - 8 )\) and radius 10 .

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis.
3. The tangent to the circle at the point \(A\) has gradient \(\frac { 5 } { 2 }\). Find an equation of the line \(C A\), giving your answer in the form \(r x + s y + t = 0\), where \(r , s\) and \(t\) are integers.
4. The line with equation \(y = 2 x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.

6

1. A curve has equation \(y = 8 - 4 x - 2 x ^ { 2 }\).
   1. Find the values of \(x\) where the curve crosses the \(x\)-axis, giving your answer in the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
   2. Sketch the curve, giving the value of the \(y\)-intercept.
2. A line has equation \(y = k ( x + 4 )\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the line with the curve \(y = 8 - 4 x - 2 x ^ { 2 }\) satisfy the equation $$2 x ^ { 2 } + ( k + 4 ) x + 4 ( k - 2 ) = 0$$
   2. Find the values of \(k\) for which the line is a tangent to the curve \(y = 8 - 4 x - 2 x ^ { 2 }\).
      [0pt] [3 marks]

2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.