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

2 The polynomial \(x ^ { 3 } - 2 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, find the quadratic factor of \(\mathrm { p } ( x )\).

4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$

1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
2. Given that $$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$ state the value of \(3 ^ { y }\) and hence find \(y\) correct to 3 significant figures.

2

1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$

5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + 5 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quadratic factor of \(\mathrm { p } ( x )\).

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } - 3 a x + 4 a$$ where \(a\) is a constant.

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
2. When \(a\) has this value,
   1. factorise \(\mathrm { p } ( x )\) completely,
   2. find all the roots of the equation \(\mathrm { p } \left( x ^ { 2 } \right) = 0\).

3

1. Show that the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ can be written in the form $$2 \tan ^ { 2 } x + ( \sqrt { } 3 ) \tan x - 1 = 0$$
2. Hence solve the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

4 The equation of a curve is \(y = \frac { 1 + \mathrm { e } ^ { - x } } { 1 - \mathrm { e } ^ { - x } }\), for \(x > 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is always negative.
2. The gradient of the curve is equal to - 1 when \(x = a\). Show that \(a\) satisfies the equation \(\mathrm { e } ^ { 2 a } - 4 \mathrm { e } ^ { a } + 1 = 0\). Hence find the exact value of \(a\).

1

1. Show that the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\) can be written as a quadratic equation in \(x\).
2. Hence solve the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\), giving your answer correct to 3 significant figures.

3

1. Show that the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$ may be written as a quadratic equation in \(x\).
2. Hence find the value of \(x\) satisfying the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$

3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 4\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\). When \(a\) has this value,
2. factorise \(\mathrm { p } ( x )\),
3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

3 Express the equation \(\sec \theta = 3 \cos \theta + \tan \theta\) as a quadratic equation in \(\sin \theta\). Hence solve this equation for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

3 Express the equation \(\cot 2 \theta = 1 + \tan \theta\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 The polynomial \(4 x ^ { 4 } + a x ^ { 2 } + 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 2\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 28 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
3. State the number of roots of the equation \(\mathrm { p } \left( 2 ^ { y } \right) = 0\), justifying your answer. \includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799} The diagram shows part of the curve $$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$ The shaded region is bounded by the curve and the two axes.
4. Show that \(2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)\) can be expressed in the form $$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$ where the values of the constants \(k _ { 1 }\) and \(k _ { 2 }\) are to be determined.
5. Find the exact area of the shaded region.

4

1. Find the quotient when \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9\) is divided by ( \(2 x + 1\) ), and show that the remainder is 5 .
2. Show that the equation \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0\) has exactly one real root.

3 The polynomial \(4 x ^ { 3 } - 7 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x - 3\) ) is a factor of \(\mathrm { p } ( x )\).

1. Show that \(a = - 3\).
2. Hence, or otherwise, solve the equation \(\mathrm { p } ( x ) = 0\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } } { 3 } + \frac { 4 } { x _ { n } ^ { 2 } }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

5 The polynomial \(3 x ^ { 3 } + 8 x ^ { 2 } + a x - 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).

2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a x - 6\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

3 The polynomial \(4 x ^ { 3 } - 8 x ^ { 2 } + a x - 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

5 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 10 .

1. Find the value of \(a\) and hence verify that ( \(x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).

3

1. Show that the equation $$\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0$$ can be expressed as a quadratic equation in \(\mathrm { e } ^ { x }\).
2. Hence solve the equation \(\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0\), giving your answer correct to 3 decimal places.

5 By first expressing the equation $$\tan \theta \tan \left( \theta + 45 ^ { \circ } \right) = 2 \cot 2 \theta$$ as a quadratic equation in \(\tan \theta\), solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

3

1. Show that the equation \(\log _ { 3 } ( 2 x + 1 ) = 1 + 2 \log _ { 3 } ( x - 1 )\) can be written as a quadratic equation in \(x\).
2. Hence solve the equation \(\log _ { 3 } ( 4 y + 1 ) = 1 + 2 \log _ { 3 } ( 2 y - 1 )\), giving your answer correct to 2 decimal places.

3 Two particles \(P\) and \(Q\) are projected simultaneously with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a horizontal plane. Both particles subsequently pass at different times through the point \(A\) which has horizontal and vertically upward displacements from \(O\) of 40 m and 15 m respectively.

1. By considering the equation of the trajectory of a projectile, show that each angle of projection satisfies the equation \(\tan ^ { 2 } \theta - 8 \tan \theta + 4 = 0\).
2. Calculate the distance between the points at which \(P\) and \(Q\) strike the plane.