1.02k Simplify rational expressions: factorising, cancelling, algebraic division

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }\), where \(a > \frac { 5 } { 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the point of intersection with the \(y\)-axis and labelling the asymptotes.
4. 1. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|\).
   2. On your sketch in part (i), draw the line \(\mathrm { y } = \mathrm { a }\).
   3. It is given that \(\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }\) for \(- 5 - \sqrt { 14 } < x < - 3\) and \(- 5 + \sqrt { 14 } < x < 3\). Find the value of \(a\).

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that there is no point on \(C\) for which \(1 < y < 5\).
3. Find the coordinates of the intersections of \(C\) with the axes, and sketch \(C\).
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 } \right|\).

7 The curve \(C\) has equation \(\mathrm { y } = \frac { 4 \mathrm { x } + 5 } { 4 - 4 \mathrm { x } ^ { 2 } }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The curve \(C\) has equation \(y = \frac { 5 x ^ { 2 } } { 5 x - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\).
4. Sketch the curve with equation \(y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } - \mathrm { x } } { \mathrm { x } + 1 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the exact coordinates of the stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { x ^ { 2 } - x } { x + 1 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x } { x + 1 } \right| < 6\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - x - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\), giving your answers correct to 1 decimal place.
3. Sketch \(C\), stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
5. Find the set of values for which \(\frac { 1 } { \mathrm { f } ( x ) } < \mathrm { f } ( x )\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } } { x + 1 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\).
3. Sketch \(C\).
4. Find the coordinates of any stationary points on the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
5. Sketch the curve with equation \(y = \frac { 1 } { f ( x ) }\) and find, in exact form, the set of values for which $$\frac { 1 } { \mathrm { f } ( x ) } > \mathrm { f } ( x ) .$$ If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } + 1 }\).

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote.
2. Show that \(1 < y \leqslant 3\) for all real values of \(x\).
3. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-12_2718_42_107_2007} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-13_2720_40_106_18}
4. Sketch \(C\), stating the coordinates of any intersections with the axes and labelling the asymptote.
5. Sketch the curve with equation \(y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 }\) and find the set of values of \(x\) for which \(\frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 } < \frac { 1 } { 2 }\).

7 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\).

1. State the equations of the asymptotes of \(C\).
2. Show that \(y \leqslant \frac { 25 } { 12 }\) at all points on \(C\).
3. Find the coordinates of any stationary points of \(C\).
4. Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.
5. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right| < 2\).

5

1. Find the quotient and remainder when \(2 x ^ { 3 } + x ^ { 2 } - 8 x\) is divided by ( \(2 x + 1\) ).
2. Hence find the exact value of \(\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x\), giving the answer in the form \(\ln \left( k \mathrm { e } ^ { a } \right)\) where \(k\) and \(a\) are constants.

4 The cubic polynomial \(2 x ^ { 3 } - 5 x ^ { 2 } + a x + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 6 . Find the values of \(a\) and \(b\).

2 The polynomial \(x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3\) is denoted by \(\mathrm { p } ( x )\).

1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
2. Find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 1\).

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 \(\mathrm { p } ( x ) = 0\).

7 The polynomial \(2 x ^ { 3 } - 4 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 4 , and that when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by ( \(x ^ { 2 } - 2\) ).

3 The polynomial \(x ^ { 4 } - 4 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 4\) is denoted by \(\mathrm { p } ( x )\).

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - 3 x + 2\).
2. Hence solve the equation \(\mathrm { p } ( x ) = 0\).

4

1. The polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\) the remainder is 14 , and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 24 . Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + 2 x - 8\) and hence solve the equation \(\mathrm { p } ( x ) = 0\).

6

1. Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( \(x ^ { 2 } - x + 4\) ).
2. It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by ( \(x ^ { 2 } - x + 4\) ), the remainder is zero. Find the values of the constants \(p\) and \(q\).
3. When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.

1 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - 6 x + 11$$ Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(\left( x ^ { 2 } + 2 \right)\).

2 Find the quotient and remainder when \(2 x ^ { 4 } - 27\) is divided by \(x ^ { 2 } + x + 3\).

8

1. Find the quotient and remainder when \(8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7\) is divided by \(4 x ^ { 2 } + 1\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7 } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\).

3 The polynomial \(2 x ^ { 4 } + a x ^ { 3 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - x + 1\) the remainder is \(3 x + 2\). Find the values of \(a\) and \(b\).

1 Find the quotient and remainder when \(x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 12 x + 27\) is divided by \(x ^ { 2 } + 5\).

1 Find the quotient and remainder when \(2 x ^ { 4 } + 1\) is divided by \(x ^ { 2 } - x + 2\).

1 Find the exact coordinates of the points on the curve \(y = \frac { x ^ { 2 } } { 1 - 3 x }\) at which the gradient of the tangent is equal to 8 .

9

1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).