1.02g Inequalities: linear and quadratic in single variable

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

Solutions relying on calculator technology are not acceptable.
Using algebra, solve the inequality $$x ^ { 2 } - x > 20$$ writing your answer in set notation.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\). The curve \(C\) meets \(l\) at the points \(( 2,4 )\) and \(( 6,0 )\) as shown. The shaded region \(R\), shown shaded in Figure 1, is bounded by \(\mathrm { C } , l\) and the \(y\)-axis. Given that \(\mathrm { f } ( x )\) is a quadratic function in \(x\), use inequalities to define region \(R\).

7. (i) Given that \(a\) and \(b\) are integers such that $$a + b \text { is odd }$$ Use algebra to prove by contradiction that at least one of \(a\) and \(b\) is odd.
(ii) A student is trying to prove that $$( p + q ) ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 } \quad \text { where } p < 0$$ The student writes: $$\begin{gathered} \qquad \begin{array} { c } p ^ { 2 } + 2 p q + q ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 } \\ 2 p q < 12 p ^ { 2 } \\ \text { so as } p < 0 \quad 2 q < 12 p \\ q < 6 p \end{array} \end{gathered}$$ a. Identify the error made in the proof.
b. Write out the correct solution.

3. a. "If \(p\) and \(q\) are irrational numbers, where \(p \neq q , q \neq 0\), then \(\frac { p } { q }\) is also irrational." Disprove this statement by means of a counter example.
b. (i) Sketch the graph of \(y = | x | - 2\).
(ii) Explain why \(| x - 2 | \geq | x | - 2\) for all real values of \(x\).

1. Given that \(a\) is a positive constant,
   a. Sketch the graph with equation

$$y = | a - 2 x |$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and \(y\)-axis.
b. Solve the inequality \(| a - 2 x | > x + 2 a\)

1. (i) Given that \(p\) and \(q\) are integers such that

use algebra to prove by contradiction that at least one of \(p\) or \(q\) is even.
(ii) Given that \(x\) and \(y\) are integers such that

• \(x < 0\)
• \(( x + y ) ^ { 2 } < 9 x ^ { 2 } + y ^ { 2 }\) show that \(y > 4 x\)

6. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = \begin{cases} ( x - 2 ) ^ { 2 } + 1 & x \leqslant 2 \\ 4 x - 7 & x > 2 \end{cases}$$

1. Find the value of \(\operatorname { gg } ( 0 )\).
2. Find all values of \(x\) for which $$\mathrm { g } ( x ) > 28$$ The function h is defined by $$\mathrm { h } ( x ) = ( x - 2 ) ^ { 2 } + 1 \quad x \leqslant 2$$
3. Explain why h has an inverse but g does not.
4. Solve the equation $$\mathrm { h } ^ { - 1 } ( x ) = - \frac { 1 } { 2 }$$

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = k - | 2 x - 3 k |$$ stating
   $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 f \left( \frac { 1 } { 2 } x \right)$$

6. Complete the table below. The first one has been done for you. For each statement you must state if it is always true, sometimes true or never true, giving a reason in each case.

6 In this question you must show detailed reasoning.

1. Solve the inequality \(x ^ { 2 } + x - 6 > 0\), giving your answer in set notation.
2. Solve the equation \(x ^ { 3 } - 7 x ^ { \frac { 3 } { 2 } } - 8 = 0\).
3. Find the exact solution of the equation \(\left( 3 ^ { x } \right) ^ { 2 } = 3 \times 2 ^ { x }\).

1 Write the solution of the inequality \(( x - 2 ) ( x + 3 ) > 0\) using set notation.

1 The quadratic equation \(k x ^ { 2 } + 3 x + k = 0\) has no real roots. Determine the set of possible values of \(k\).

2

1. Factorise \(3 x ^ { 2 } - 19 x - 14\).
2. Solve the inequality \(3 x ^ { 2 } - 19 x - 14 < 0\).

15 A family is planning a holiday in Europe. They need to buy some euros before they go. The exchange rate, \(y\), is the number of euros they can buy per pound. They believe that the exchange rate may be modelled by the formula \(y = a t ^ { 2 } + b t + c\),
where \(t\) is the time in days from when they first check the exchange rate.
Initially, when \(t = 0\), the exchange rate is 1.14 .

1. Write down the value of \(c\). When \(t = 2 , y = 1.20\) and when \(t = 4 , y = 1.25\).
2. Calculate the values of \(a\) and \(b\). The family will only buy their euros when their model predicts an exchange rate of at least 1.29 .
3. Determine the range of values of \(t\) for which, according to their model, they will buy their euros.
4. Explain why the family's model is not viable in the long run.

1 Solve the inequality \(2 x + 5 < 6 x - 3\).

3

1. The diagram shows the line \(y = x + 5\) and the curve \(y = 8 - 2 x - x ^ { 2 }\). The shaded region is the finite region between the line and the curve. The curved part of the boundary is included in the region but the straight part is not included. Write down the inequalities that define the shaded region. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-04_846_716_1379_322} \section*{(b) In this question you must show detailed reasoning.} Solve the inequality \(8 - 2 x - x ^ { 2 } > x + 5\) giving your answer in exact form.

2 The straight line \(y = 5 - 2 x\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-04_705_773_881_239}

1. On the copy of the diagram in the Printed Answer Booklet, sketch the graph of \(y = | 5 - 2 x |\).
2. Solve the inequality \(| 5 - 2 x | < 3\).

1 Solve the inequality \(\frac { x } { 5 } > 6 - x\).

2 The graph of \(y = | 1 - x | - 2\) is shown in Fig. 2. \begin{figure}[h] \end{figure} Determine the set of values of \(x\) for which \(| 1 - x | > 2\).

13

1. Prove by induction that, for all integers \(n \geq 1\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ [4 marks]
   13
2. Hence, or otherwise, write down a factorised expression for the sum of the first \(2 n\) squares $$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
3. Use the formula in part (a) to write down a factorised expression for the sum of the first \(n\) even squares $$2 ^ { 2 } + 4 ^ { 2 } + 6 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
4. Hence, or otherwise, show that the sum of the first \(n\) odd squares is $$a n ( b n - 1 ) ( b n + 1 )$$ where \(a\) and \(b\) are rational numbers to be determined.

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\). \includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\). \includegraphics[max width=\textwidth, alt={}, center]{81f6fc30-982b-47b5-bab3-076cc0cc6563-5_479_816_406_596} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).

2

1. Factorise \(2 x ^ { 2 } - 5 x + 3\).
2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - 5 x + 3 < 0\).

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 4 x + 12 y + 15 = 0\).

1. Find:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. Explain why the circle lies entirely below the \(x\)-axis.
3. The point \(P\) with coordinates \(( 5 , k )\) lies outside the circle.
   1. Show that \(P C ^ { 2 } = k ^ { 2 } + 12 k + 45\).
   2. Hence show that \(k ^ { 2 } + 12 k + 20 > 0\).
   3. Find the possible values of \(k\).

7

1. 1. Express \(4 - 10 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = 4 - 10 x - x ^ { 2 }\).
2. The curve \(C\) has equation \(y = 4 - 10 x - x ^ { 2 }\) and the line \(L\) has equation \(y = k ( 4 x - 13 )\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
   2. Given that the curve \(C\) and the line \(L\) intersect in two distinct points, show that $$4 k ^ { 2 } + 33 k + 29 > 0$$
   3. Solve the inequality \(4 k ^ { 2 } + 33 k + 29 > 0\).