1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

8 The function f is such that \(\mathrm { f } ( x ) = \frac { 3 } { 2 x + 5 }\) for \(x \in \mathbb { R } , x \neq - 2.5\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and explain why f is a decreasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. A curve has the equation \(y = \mathrm { f } ( x )\). Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-5_609_897_255_625} The diagram shows part of the curve \(y = \frac { 1 } { ( 3 x + 1 ) ^ { \frac { 1 } { 4 } } }\). The curve cuts the \(y\)-axis at \(A\) and the line \(x = 5\) at \(B\).

1. Show that the equation of the line \(A B\) is \(y = - \frac { 1 } { 10 } x + 1\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

2 Points \(A , B\) and \(C\) have coordinates \(( 2,5 ) , ( 5 , - 1 )\) and \(( 8,6 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find the equation of the line through \(C\) perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\).

9 \includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-4_767_993_255_575} The diagram shows a quadrilateral \(A B C D\) in which the point \(A\) is ( \(- 1 , - 1\) ), the point \(B\) is ( 3,6 ) and the point \(C\) is (9,4). The diagonals \(A C\) and \(B D\) intersect at \(M\). Angle \(B M A = 90 ^ { \circ }\) and \(B M = M D\). Calculate

1. the coordinates of \(M\) and \(D\),
2. the ratio \(A M : M C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-3_602_751_255_696} The diagram shows a triangle \(A B C\) in which \(A\) has coordinates ( 1,3 ), \(B\) has coordinates ( 5,11 ) and angle \(A B C\) is \(90 ^ { \circ }\). The point \(X ( 4,4 )\) lies on \(A C\). Find

1. the equation of \(B C\),
2. the coordinates of \(C\).

7 The point \(A\) has coordinates ( \(- 1,6\) ) and the point \(B\) has coordinates (7,2).

1. Find the equation of the perpendicular bisector of \(A B\), giving your answer in the form \(y = m x + c\).
2. A point \(C\) on the perpendicular bisector has coordinates \(( p , q )\). The distance \(O C\) is 2 units, where \(O\) is the origin. Write down two equations involving \(p\) and \(q\) and hence find the coordinates of the possible positions of \(C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_636_811_255_667} The diagram shows a rectangle \(A B C D\) in which point \(A\) is ( 0,8 ) and point \(B\) is ( 4,0 ). The diagonal \(A C\) has equation \(8 y + x = 64\). Find, by calculation, the coordinates of \(C\) and \(D\).

6 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_465_663_1160_740} In the diagram, \(S\) is the point ( 0,12 ) and \(T\) is the point ( 16,0 ). The point \(Q\) lies on \(S T\), between \(S\) and \(T\), and has coordinates \(( x , y )\). The points \(P\) and \(R\) lie on the \(x\)-axis and \(y\)-axis respectively and \(O P Q R\) is a rectangle.

1. Show that the area, \(A\), of the rectangle \(O P Q R\) is given by \(A = 12 x - \frac { 3 } { 4 } x ^ { 2 }\).
2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.

3 The point \(A\) has coordinates \(( 3,1 )\) and the point \(B\) has coordinates \(( - 21,11 )\). The point \(C\) is the mid-point of \(A B\).

1. Find the equation of the line through \(A\) that is perpendicular to \(y = 2 x - 7\).
2. Find the distance \(A C\).

4 The line \(4 x + k y = 20\) passes through the points \(A ( 8 , - 4 )\) and \(B ( b , 2 b )\), where \(k\) and \(b\) are constants.

1. Find the values of \(k\) and \(b\).
2. Find the coordinates of the mid-point of \(A B\).

6 Points \(A , B\) and \(C\) have coordinates \(A ( - 3,7 ) , B ( 5,1 )\) and \(C ( - 1 , k )\), where \(k\) is a constant.

1. Given that \(A B = B C\), calculate the possible values of \(k\). The perpendicular bisector of \(A B\) intersects the \(x\)-axis at \(D\).
2. Calculate the coordinates of \(D\).

9 A curve passes through the point \(A ( 4,6 )\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 + 2 x ^ { - \frac { 1 } { 2 } }\). A point \(P\) is moving along the curve in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 3 units per minute.

1. Find the rate at which the \(y\)-coordinate of \(P\) is increasing when \(P\) is at \(A\).
2. Find the equation of the curve.
3. The tangent to the curve at \(A\) crosses the \(x\)-axis at \(B\) and the normal to the curve at \(A\) crosses the \(x\)-axis at \(C\). Find the area of triangle \(A B C\).

3 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-2_515_750_669_699} In the diagram \(O C A\) and \(O D B\) are radii of a circle with centre \(O\) and radius \(2 r \mathrm {~cm}\). Angle \(A O B = \alpha\) radians. \(C D\) and \(A B\) are arcs of circles with centre \(O\) and radii \(r \mathrm {~cm}\) and \(2 r \mathrm {~cm}\) respectively. The perimeter of the shaded region \(A B D C\) is \(4.4 r \mathrm {~cm}\).

1. Find the value of \(\alpha\).
2. It is given that the area of the shaded region is \(30 \mathrm {~cm} ^ { 2 }\). Find the value of \(r\). \(4 C\) is the mid-point of the line joining \(A ( 14 , - 7 )\) to \(B ( - 6,3 )\). The line through \(C\) perpendicular to \(A B\) crosses the \(y\)-axis at \(D\).

5 The line \(\frac { x } { a } + \frac { y } { b } = 1\), where \(a\) and \(b\) are positive constants, intersects the \(x\) - and \(y\)-axes at the points \(A\) and \(B\) respectively. The mid-point of \(A B\) lies on the line \(2 x + y = 10\) and the distance \(A B = 10\). Find the values of \(a\) and \(b\).

6 The points \(A ( 1,1 )\) and \(B ( 5,9 )\) lie on the curve \(6 y = 5 x ^ { 2 } - 18 x + 19\).

1. Show that the equation of the perpendicular bisector of \(A B\) is \(2 y = 13 - x\).
   The perpendicular bisector of \(A B\) meets the curve at \(C\) and \(D\).
2. Find, by calculation, the distance \(C D\), giving your answer in the form \(\sqrt { } \left( \frac { p } { q } \right)\), where \(p\) and \(q\) are integers.

11 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-18_428_857_260_644} The diagram shows the curve \(y = ( x - 1 ) ^ { \frac { 1 } { 2 } }\) and points \(A ( 1,0 )\) and \(B ( 5,2 )\) lying on the curve.

1. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
2. Find, showing all necessary working, the equation of the tangent to the curve which is parallel to \(A B\).
3. Find the perpendicular distance between the line \(A B\) and the tangent parallel to \(A B\). Give your answer correct to 2 decimal places.

3 Two points \(A\) and \(B\) have coordinates ( \(3 a , - a\) ) and ( \(- a , 2 a\) ) respectively, where \(a\) is a positive constant.

1. Find the equation of the line through the origin parallel to \(A B\).
2. The length of the line \(A B\) is \(3 \frac { 1 } { 3 }\) units. Find the value of \(a\).

4 Two points \(A\) and \(B\) have coordinates \(( - 1,1 )\) and \(( 3,4 )\) respectively. The line \(B C\) is perpendicular to \(A B\) and intersects the \(x\)-axis at \(C\).

1. Find the equation of \(B C\) and the \(x\)-coordinate of \(C\).
2. Find the distance \(A C\), giving your answer correct to 3 decimal places.

2 The point \(M\) is the mid-point of the line joining the points \(( 3,7 )\) and \(( - 1,1 )\). Find the equation of the line through \(M\) which is parallel to the line \(\frac { x } { 3 } + \frac { y } { 2 } = 1\).

2

1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 9 |\) and \(y = 5 x - 3\).
2. Solve the equation \(| 2 x - 9 | = 5 x - 3\).

2 \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-03_515_598_260_762} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { ( A - B ) x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.4,3.6 )\) and \(( 2.9,14.1 )\), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 3 significant figures.

2 \includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-03_515_598_260_762} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { ( A - B ) x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.4,3.6 )\) and \(( 2.9,14.1 )\), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 3 significant figures.

1 The variables \(x\) and \(y\) satisfy the equation \(a ^ { 2 y } = \mathrm { e } ^ { 3 x + k }\), where \(a\) and \(k\) are constants.
The graph of \(y\) against \(x\) is a straight line.

1. Use logarithms to show that the gradient of the straight line is \(\frac { 3 } { 2 \ln a }\).
2. Given that the straight line passes through the points \(( 0.4,0.95 )\) and \(( 3.3,3.80 )\), find the values of \(a\) and \(k\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-03_2723_33_99_21}

2 The variables \(x\) and \(y\) satisfy the relation \(3 ^ { y } = 4 ^ { x + 2 }\).

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. Find the exact value of the gradient of this line.
2. Calculate the \(x\)-coordinate of the point of intersection of this line with the line \(y = 2 x\), giving your answer correct to 2 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-08_558_785_258_680} The diagram shows a kite \(O A B C\) in which \(A C\) is the line of symmetry. The coordinates of \(A\) and \(C\) are \(( 0,4 )\) and \(( 8,0 )\) respectively and \(O\) is the origin.

1. Find the equations of \(A C\) and \(O B\).
2. Find, by calculation, the coordinates of \(B\).