1.02g Inequalities: linear and quadratic in single variable

6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.

1. The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
2. The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
3. Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
4. Hence determine the possible values of the width of the garden. \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
   1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
   2. Write down:
      1. the coordinates of \(C\);
      2. the radius of the circle.
      3. Sketch the circle.
      4. A line has equation \(y = k x + 6\), where \(k\) is a constant.
         1. Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
         2. The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
         3. Hence find the values of \(k\) for which the line is a tangent to the circle.

7 Solve each of the following inequalities:

1. \(3 ( x - 1 ) > 3 - 5 ( x + 6 )\);
2. \(\quad x ^ { 2 } - x - 6 < 0\).

7

1. 1. Express \(2 x ^ { 2 } - 20 x + 53\) in the form \(2 ( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Use your result from part (a)(i) to explain why the equation \(2 x ^ { 2 } - 20 x + 53 = 0\) has no real roots.
2. The quadratic equation \(( 2 k - 1 ) x ^ { 2 } + ( k + 1 ) x + k = 0\) has real roots.
   1. Show that \(7 k ^ { 2 } - 6 k - 1 \leqslant 0\).
   2. Hence find the possible values of \(k\).

7 Solve each of the following inequalities:

1. \(\quad 2 ( 4 - 3 x ) > 5 - 4 ( x + 2 )\);
2. \(\quad 2 x ^ { 2 } + 5 x \geqslant 12\).

7 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve \(C\) at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 20 x - 6 x ^ { 2 } - 16$$

1. 1. Show that \(y\) is increasing when \(3 x ^ { 2 } - 10 x + 8 < 0\).
   2. Solve the inequality \(3 x ^ { 2 } - 10 x + 8 < 0\).
2. The curve \(C\) passes through the point \(P ( 2,3 )\).
   1. Verify that the tangent to the curve at \(P\) is parallel to the \(x\)-axis.
   2. The point \(Q ( 3 , - 1 )\) also lies on the curve. The normal to the curve at \(Q\) and the tangent to the curve at \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
      (7 marks)

7 The quadratic equation $$( 2 k - 7 ) x ^ { 2 } - ( k - 2 ) x + ( k - 3 ) = 0$$ has real roots.

1. Show that \(7 k ^ { 2 } - 48 k + 80 \leqslant 0\).
2. Find the possible values of \(k\).

8 Solve the following inequalities:

1. \(\quad 3 ( 1 - 2 x ) - 5 ( 3 x + 2 ) > 0\)
2. \(\quad 6 x ^ { 2 } \leqslant x + 12\) [0pt] [4 marks]

8 A curve has equation \(y = x ^ { 2 } + ( 3 k - 4 ) x + 13\) and a line has equation \(y = 2 x + k\), where \(k\) is a constant.

1. Show that the \(x\)-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
2. Given that the line and the curve do not intersect:
   1. show that \(9 k ^ { 2 } - 32 k - 16 < 0\);
   2. find the possible values of \(k\).
      \includegraphics[max width=\textwidth, alt={}]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-18_1657_1714_1050_153}

8 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).

2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.

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

1. \(6 x - 7 < 2 x + 3\),
2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).

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

1. \(6 x - 7 < 2 x + 3\),
2. \(\quad 2 x ^ { 2 } - 11 x + 5 < 0\),
3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).

5. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.

4. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,

1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
2. form a quadratic inequality in \(x\).
3. by solving your inequalities, find the set of possible values of \(x\).

2. Find the set of values of \(x\) for which $$( x - 1 ) ( x - 2 ) < 20$$

1. Solve the inequality

$$x ( 2 x + 1 ) \leq 6 .$$

7. Given that the equation $$4 x ^ { 2 } - k x + k - 3 = 0$$ where \(k\) is a constant, has real roots,

1. show that $$k ^ { 2 } - 16 k + 48 \geq 0 ,$$
2. find the set of possible values of \(k\),
3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value.

4. (a) Solve the inequality $$4 ( x - 2 ) < 2 x + 5$$ (b) Find the value of \(y\) such that $$4 ^ { y + 1 } = 8 ^ { 2 y - 1 } .$$

3 Solve $$x ^ { 2 } \geqslant | 5 x - 6 |$$ [5 marks]

5 A ball is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\) above the horizontal so as to hit a point \(P\) on a wall. The ball travels in a vertical plane through the point of projection. During the motion, the horizontal and upward vertical displacements of the ball from the point of projection are \(x\) metres and \(y\) metres respectively.

1. Show that, during the flight, the equation of the trajectory of the ball is given by $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)$$
2. The ball is projected from a point 1 metre vertically below and \(R\) metres horizontally from the point \(P\).
   1. By taking \(g = 10 \mathrm {~ms} ^ { - 2 }\), show that \(R\) satisfies the equation $$5 R ^ { 2 } \tan ^ { 2 } \alpha - u ^ { 2 } R \tan \alpha + 5 R ^ { 2 } + u ^ { 2 } = 0$$
   2. Hence, given that \(u\) and \(R\) are constants, show that, for \(\tan \alpha\) to have real values, \(R\) must satisfy the inequality $$R ^ { 2 } \leqslant \frac { u ^ { 2 } \left( u ^ { 2 } - 20 \right) } { 100 }$$
   3. Given that \(R = 5\), determine the minimum possible speed of projection.

4 A curve has equation $$y = \frac { 6 x } { x - 1 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve and the two asymptotes.
3. Solve the inequality $$\frac { 6 x } { x - 1 } < 3$$

5 A curve has equation $$y = \frac { x } { x ^ { 2 } - 1 }$$

1. Write down the equations of the three asymptotes to the curve.
2. Sketch the curve.
   (You are given that the curve has no stationary points.)
3. Solve the inequality $$\frac { x } { x ^ { 2 } - 1 } > 0$$

6 A curve has equation $$y = \frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the equations of the three asymptotes of this curve.
   2. State the coordinates of the points at which the curve intersects the \(x\)-axis.
   3. Sketch the curve.
      (You are given that the curve has no stationary points.)
2. Hence, or otherwise, solve the inequality $$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$

7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$

7

1. 1. Write down the equations of the two asymptotes of the curve \(y = \frac { 1 } { x - 3 }\).
   2. Sketch the curve \(y = \frac { 1 } { x - 3 }\), showing the coordinates of any points of intersection with the coordinate axes.
   3. On the same axes, again showing the coordinates of any points of intersection with the coordinate axes, sketch the line \(y = 2 x - 5\).
2. 1. Solve the equation $$\frac { 1 } { x - 3 } = 2 x - 5$$
   2. Find the solution of the inequality $$\frac { 1 } { x - 3 } < 2 x - 5$$ □ \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-08_367_197_2496_155}