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

1

1. Show that the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ can be expressed in the form $$2 \cos ^ { 2 } x + 5 \cos x - 3 = 0$$
2. Hence solve the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

9 A geometric progression has first term \(a\) and common ratio \(r\), and the terms are all different. The first, second and fourth terms of the geometric progression form the first three terms of an arithmetic progression.

1. Show that \(r ^ { 3 } - 2 r + 1 = 0\).
2. Given that the geometric progression converges, find the exact value of \(r\).
3. Given also that the sum to infinity of this geometric progression is \(3 + \sqrt { 5 }\), find the value of the integer \(a\).

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x + 10\).

1. Use the factor theorem to find a factor of \(\mathrm { f } ( x )\).
2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form.

8 Two cubic polynomials are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + ( a - 3 ) x + 2 b , \quad \mathrm {~g} ( x ) = 3 x ^ { 3 } + x ^ { 2 } + 5 a x + 4 b$$ where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have a common factor of ( \(x - 2\) ), show that \(a = - 4\) and find the value of \(b\).
2. Using these values of \(a\) and \(b\), factorise \(\mathrm { f } ( x )\) fully. Hence show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have two common factors.

4

1. Show that the equation $$\sin x - \cos x = \frac { 6 \cos x } { \tan x }$$ can be expressed in the form $$\tan ^ { 2 } x - \tan x - 6 = 0 .$$
2. Hence solve the equation \(\sin x - \cos x = \frac { 6 \cos x } { \tan x }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 12 - 22 x + 9 x ^ { 2 } - x ^ { 3 }\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
2. Show that ( \(3 - x\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
4. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in simplified surd form where appropriate.

10 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10.

12 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).

1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).

4 The acute angles \(\alpha\) and \(\beta\) are such that $$2 \cot \alpha = 1 \text { and } 24 + \sec ^ { 2 } \beta = 10 \tan \beta \text {. }$$

1. State the value of \(\tan \alpha\) and determine the value of \(\tan \beta\).
2. Hence find the exact value of \(\tan ( \alpha + \beta )\).

3 Given that \(z = 6\) is a root of the cubic equation \(z ^ { 3 } - 10 z ^ { 2 } + 37 z + p = 0\), find the value of \(p\) and the other roots.

2 Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.

2 You are given that \(z = \frac { 3 } { 2 }\) is a root of the cubic equation \(2 z ^ { 3 } + 9 z ^ { 2 } + 2 z - 30 = 0\). Find the other two roots.

3 A particle \(P\), of mass \(m\), is placed at the highest point of a fixed solid smooth sphere with centre \(O\) and radius \(a\). The particle \(P\) is given a horizontal speed \(u\) and it moves in part of a vertical circle, with centre \(O\), on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the surface of the sphere, the speed of \(P\) is \(v\) and the reaction of the sphere on \(P\) has magnitude \(R\). Show that \(R = m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the sphere at the instant when \(v = 2 u\). Find \(u\) in terms of \(a\) and \(g\).

4 Justify the statement that the equation in line 83, $$\frac { \phi } { 1 } = \frac { 1 } { \phi - 1 }$$ has the solution \(\phi = \frac { 1 \pm \sqrt { 5 } } { 2 }\).

3 In this question you must show detailed reasoning.
Find the two real roots of the equation \(x ^ { 4 } - 5 = 4 x ^ { 2 }\). Give the roots in an exact form.

3 The function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 17\) for \(x \geqslant k\), where \(k\) is a constant.

1. Given that \(\mathrm { f } ^ { - 1 } ( x )\) exists, state the least possible value of \(k\).
2. Evaluate \(\mathrm { ff } ( 5 )\).
3. Solve the equation \(\mathrm { f } ( x ) = x\).
4. Explain why your solution to part (c) is also the solution to the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\).

1

1. Express \(2 x ^ { 2 } - 12 x + 23\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Use your result to show that the equation \(2 x ^ { 2 } - 12 x + 23 = 0\) has no real roots.
3. Given that the equation \(2 x ^ { 2 } - 12 x + k = 0\) has repeated roots, find the value of the constant \(k\).

7 In this question you must show detailed reasoning.
Use the substitution \(u = 6 x ^ { 2 } + x\) to solve the equation \(36 x ^ { 4 } + 12 x ^ { 3 } + 7 x ^ { 2 } + x - 2 = 0\).

3 A publisher has to choose the price at which to sell a certain new book. The total profit, \(\pounds t\), that the publisher will make depends on the price, \(\pounds p\). He decides to use a model that includes the following assumptions.

• If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
• If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.

The graphs below show two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how model A is inconsistent with one of the assumptions given above.
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k \left( 12 p - p ^ { 2 } \right)\), and find the value of the constant \(k\).
3. The publisher needs to make a total profit of at least \(\pounds 6400\). Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie.
4. Comment briefly on how realistic model B may be in the following cases.

4 In this question you must show detailed reasoning.
The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + 2 .$$

1. Write down expressions for
   1. \(\mathrm { fg } ( x )\),
   2. \(\operatorname { gf } ( x )\).
   3. Hence find the values of \(x\) for which \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24\).

Find, using algebra, all real solutions to the equation

1. \(16 a ^ { 2 } = 2 \sqrt { a }\)
2. \(b ^ { 4 } + 7 b ^ { 2 } - 18 = 0\)

Given that \(k\) is a positive constant and \(\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4\)

1. show that \(3 k + 5 \sqrt { k } - 12 = 0\)
2. Hence, using algebra, find any values of \(k\) such that $$\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$$

4. Figure 1 Figure 1 shows a sketch of triangle \(A B C\) with \(A B = ( x + 2 ) \mathrm { cm } , B C = ( 3 x + 10 ) \mathrm { cm }\), \(A C = 7 x \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = \theta ^ { \circ }\)

1. 1. Show that \(17 x ^ { 2 } - 35 x - 48 = 0\)
   2. Hence find the value of \(x\).
2. Hence find the value of \(\theta\) giving your answer to one decimal place.

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

Using the substitution \(u = \sqrt { x }\) or otherwise, solve $$6 x + 7 \sqrt { x } - 20 = 0$$

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

Solutions relying on calculator technology are not acceptable.

1. Using algebra, find all solutions of the equation $$3 x ^ { 3 } - 17 x ^ { 2 } - 6 x = 0$$
2. Hence find all real solutions of $$3 ( y - 2 ) ^ { 6 } - 17 ( y - 2 ) ^ { 4 } - 6 ( y - 2 ) ^ { 2 } = 0$$