1.02g Inequalities: linear and quadratic in single variable

11. The straight line \(l\) has equation \(y = m x - 2\), where \(m\) is a constant. The curve \(C\) has equation \(y = 2 x ^ { 2 } + x + 6\) The line \(l\) does not cross or touch the curve \(C\).

1. Show that \(m\) satisfies the inequality $$m ^ { 2 } - 2 m - 63 < 0$$
2. Hence find the range of possible values of \(m\).

11. The equation \(5 x ^ { 2 } + 6 = k \left( 13 x ^ { 2 } - 12 x \right)\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } + 13 k - 5 > 0$$
2. Find the set of possible values for \(k\).

1. The circle \(C\) has equation

$$( x - 3 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 30$$ Write down

1. 1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). Given that the point \(P\) with coordinates \(( 6 , k )\), where \(k\) is a constant, lies inside circle \(C\), (b) show that $$k ^ { 2 } + 8 k - 5 < 0$$
2. Hence find the exact set of values of \(k\) for which \(P\) lies inside \(C\). \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-34_2256_52_315_1978}

11. The equation \(7 x ^ { 2 } + 2 k x + k ^ { 2 } = k + 7\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } - 7 k - 49 < 0$$
2. Find the range of possible values for \(k\).

8. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
2. Hence find the set of possible values of \(k\).

5. The equation \(2 x ^ { 2 } - 3 x - ( k + 1 ) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\).

8. The equation $$x ^ { 2 } + k x + 8 = k$$ has no real solutions for \(x\).

1. Show that \(k\) satisfies \(k ^ { 2 } + 4 k - 32 < 0\).
2. Hence find the set of possible values of \(k\).

7. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 5 k + 4 > 0 .$$
2. Hence find the set of possible values of \(k\).

8. The equation \(x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies $$k ^ { 2 } + 2 k - 3 > 0$$
2. Find the set of possible values of \(k\).

3. Find the set of values of \(x\) for which

1. \(4 x - 5 > 15 - x\)
2. \(x ( x - 4 ) > 12\)

9. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
2. Hence find the set of possible values of \(k\).

6. Find the set of values of \(x\) for which

1. \(3 ( 2 x + 1 ) > 5 - 2 x\),
2. \(2 x ^ { 2 } - 7 x + 3 > 0\),
3. both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\).

7. The equation \(x ^ { 2 } + k x + ( k + 3 ) = 0\), where \(k\) is a constant, has different real roots.

1. Show that \(k ^ { 2 } - 4 k - 12 > 0\).
2. Find the set of possible values of \(k\).

Given that the equation \(2 q x ^ { 2 } + q x - 1 = 0\), where \(q\) is a constant, has no real roots,

1. show that \(q ^ { 2 } + 8 q < 0\).
2. Hence find the set of possible values of \(q\).

4. Find the set of values of \(x\) for which

1. \(4 x - 3 > 7 - x\)
2. \(2 x ^ { 2 } - 5 x - 12 < 0\)
3. both \(4 x - 3 > 7 - x\) and \(2 x ^ { 2 } - 5 x - 12 < 0\)

3. Find the set of values of \(x\) for which

1. \(3 ( x - 2 ) < 8 - 2 x\)
2. \(( 2 x - 7 ) ( 1 + x ) < 0\)
3. both \(3 ( x - 2 ) < 8 - 2 x\) and \(( 2 x - 7 ) ( 1 + x ) < 0\)

5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3 \\ & a _ { n + 1 } = 2 a _ { n } - c \quad ( n \geqslant 1 ) \end{aligned}$$ where \(c\) is a constant.

1. Write down an expression, in terms of \(c\), for \(a _ { 2 }\)
2. Show that \(a _ { 3 } = 12 - 3 c\) Given that \(\sum _ { i = 1 } ^ { 4 } a _ { i } \geqslant 23\)
3. find the range of values of \(c\).

1. A rectangular room has a width of \(x \mathrm {~m}\).

The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m ,

1. show that \(x > 2.8\) Given also that the area of the room is less than \(21 \mathrm {~m} ^ { 2 }\),
2. 1. write down an inequality, in terms of \(x\), for the area of the room.
   2. Solve this inequality.
3. Hence find the range of possible values for \(x\).

5. Find the set of values of \(x\) for which

1. \(2 ( 3 x + 4 ) > 1 - x\)
2. \(3 x ^ { 2 } + 8 x - 3 < 0\)

6. \begin{figure}[h] \end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres.
Given that the perimeter of the garden is greater than 40 m ,

1. show that \(x > 1.7\) Given that the area of the garden is less than \(120 \mathrm {~m} ^ { 2 }\),
2. form and solve a quadratic inequality in \(x\).
3. Hence state the range of the possible values of \(x\).

1. The equation

$$( p - 1 ) x ^ { 2 } + 4 x + ( p - 5 ) = 0 , \text { where } p \text { is a constant }$$ has no real roots.

1. Show that \(p\) satisfies \(p ^ { 2 } - 6 p + 1 > 0\)
2. Hence find the set of possible values of \(p\).

8. The straight line with equation \(y = 3 x - 7\) does not cross or touch the curve with equation \(y = 2 p x ^ { 2 } - 6 p x + 4 p\), where \(p\) is a constant.

1. Show that \(4 p ^ { 2 } - 20 p + 9 < 0\)
2. Hence find the set of possible values of \(p\).
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1. Solve the inequality \(10 + x ^ { 2 } > x ( x - 2 )\).
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