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

9 A line has equation \(y = k x + 6\) and a curve has equation \(y = x ^ { 2 } + 3 x + 2 k\), where \(k\) is a constant.

1. For the case where \(k = 2\), the line and the curve intersect at points \(A\) and \(B\). Find the distance \(A B\) and the coordinates of the mid-point of \(A B\).
2. Find the two values of \(k\) for which the line is a tangent to the curve.

4

1. Show that the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) can be expressed as $$4 \sin ^ { 2 } \theta - 15 \sin \theta - 4 = 0$$
2. Hence solve the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

8 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

1. In the case where \(a = 6\) and \(b = - 8\), find the range of f .
2. In the case where \(a = 5\), the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots, then \(a ^ { 2 } < 4 ( b - a )\).

7

1. Show that the equation \(\frac { 1 } { \cos \theta } + 3 \sin \theta \tan \theta + 4 = 0\) can be expressed as $$3 \cos ^ { 2 } \theta - 4 \cos \theta - 4 = 0$$ and hence solve the equation \(\frac { 1 } { \cos \theta } + 3 \sin \theta \tan \theta + 4 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-3_581_773_1400_721} The diagram shows part of the graph of \(y = a \cos x - b\), where \(a\) and \(b\) are constants. The graph crosses the \(x\)-axis at the point \(C \left( \cos ^ { - 1 } c , 0 \right)\) and the \(y\)-axis at the point \(D ( 0 , d )\). Find \(c\) and \(d\) in terms of \(a\) and \(b\).

1 Showing all necessary working, solve the equation \(4 x - 11 x ^ { \frac { 1 } { 2 } } + 6 = 0\).

9 A curve has equation \(y = 2 x ^ { 2 } - 3 x + 1\) and a line has equation \(y = k x + k ^ { 2 }\), where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve and the line meet.
2. State the value of \(k\) for which the line is a tangent to the curve and find the coordinates of the point where the line touches the curve.

6 A straight line has gradient \(m\) and passes through the point ( \(0 , - 2\) ). Find the two values of \(m\) for which the line is a tangent to the curve \(y = x ^ { 2 } - 2 x + 7\) and, for each value of \(m\), find the coordinates of the point where the line touches the curve.

9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 2 } + 8 x + 1 \quad \text { for } x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 2 x - k \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(k\) is a constant.

1. Find the value of \(k\) for which the line \(y = \mathrm { g } ( x )\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
2. In the case where \(k = - 9\), find the set of values of \(x\) for which \(\mathrm { f } ( x ) < \mathrm { g } ( x )\).
3. In the case where \(k = - 1\), find \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) and solve the equation \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = 0\).
4. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants, and hence state the least value of \(\mathrm { f } ( x )\).

2

1. Find the quotient when \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x\) is divided by \(x ^ { 2 } + 5 x + 6\), and show that the remainder is 18 .
2. Hence solve the equation \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x - 18 = 0\).

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where \(a\) and \(b\) are constants.It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) ,and that the remainder is 24 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) .

1. Find the values of \(a\) and \(b\) . \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
3. Solve the equation \(\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and \(y = \frac { 27 } { 2 x + 5 }\) for \(x \geqslant 0\).
   The curves meet at the point \(( 2,3 )\).
   Region \(A\) is bounded by the curve \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and the straight lines \(x = 0 , x = 2\) and \(y = 0\).
   Region \(B\) is bounded by the two curves and the straight line \(x = 0\).

2 The polynomial \(x ^ { 4 } - 9 x ^ { 2 } - 6 x - 1\) is denoted by \(\mathrm { f } ( x )\).

1. Find the value of the constant \(a\) for which $$f ( x ) \equiv \left( x ^ { 2 } + a x + 1 \right) \left( x ^ { 2 } - a x - 1 \right)$$
2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in an exact form.

4

1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$3 \tan ^ { 2 } x - 10 \tan x + 3 = 0$$
2. Hence solve the equation $$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 5 } \left( 4 x _ { n } + \frac { 306 } { x _ { n } ^ { 4 } } \right)$$ with initial value \(x _ { 1 } = 3\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
2. State an equation satisfied by \(\alpha\), and hence show that the exact value of \(\alpha\) is \(\sqrt [ 5 ] { 306 }\).

3 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 13 x - 6\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).

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

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

1. Use this iteration to calculate \(\alpha\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
2. State an equation which is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

4 The polynomial \(x ^ { 3 } - x ^ { 2 } + a x + b\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 1\) ) 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, factorise \(\mathrm { p } ( x )\).

4 The cubic polynomial \(a x ^ { 3 } + b x ^ { 2 } - 3 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\).

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

3

1. Show that the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\) can be written in the form $$6 \tan ^ { 2 } x - 5 \tan x + 1 = 0$$
2. Hence solve the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

5

1. Given that \(y = 2 ^ { x }\), show that the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ can be written in the form $$y ^ { 2 } - 4 y + 3 = 0$$
2. Hence solve the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ giving the values of \(x\) correct to 3 significant figures where appropriate.

3 Given that \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\), find the possible values of \(\mathrm { e } ^ { x }\) and hence solve the equation \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\) correct to 3 significant figures.

4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + x _ { n } + 9 } { \left( x _ { n } + 1 \right) ^ { 2 } }$$ with \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Find the value of \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. Determine the exact value of \(\alpha\).

6

1. Use the factor theorem to show that ( \(x + 2\) ) is a factor of the expression $$6 x ^ { 3 } + 13 x ^ { 2 } - 33 x - 70$$ and hence factorise the expression completely.
2. Deduce the roots of the equation $$6 + 13 y - 33 y ^ { 2 } - 70 y ^ { 3 } = 0$$

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + 14 x + a + 1$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).

1. Use the factor theorem to find the value of \(a\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Hence, without using a calculator, solve the equation \(\mathrm { f } ( 2 x ) = 3 \mathrm { f } ( x )\).

5 The polynomial \(x ^ { 4 } + 5 x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that \(x ^ { 2 } - x + 3\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and factorise \(\mathrm { p } ( x )\) completely.
2. Hence state the number of real roots of the equation \(\mathrm { p } ( x ) = 0\), justifying your answer.