OCR H240/01 (Pure Mathematics) 2022 June

1
\includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-4_303_451_358_242} The diagram shows part of the curve \(y = \sqrt { x ^ { 2 } - 1 }\).

1. Use the trapezium rule with 4 intervals to find an estimate for \(\int _ { 1 } ^ { 3 } \sqrt { x ^ { 2 } - 1 } \mathrm {~d} x\). Give your answer correct to \(\mathbf { 3 }\) significant figures.
2. State whether the value from part (a) is an under-estimate or an over-estimate, giving a reason for your answer.
3. Explain how the trapezium rule could be used to obtain a more accurate estimate.

2

1. Given that \(a\) and \(b\) are real numbers, find a counterexample to disprove the statement that, if \(a > b\), then \(a ^ { 2 } > b ^ { 2 }\).
2. A student writes the statement that \(\sin x ^ { \circ } = 0.5 \Longleftrightarrow x ^ { \circ } = 30 ^ { \circ }\).
   1. Explain why this statement is incorrect.
   2. Write a corrected version of this statement.
3. Prove that the sum of four consecutive multiples of 4 is always a multiple of 8 .

3

1. In this question you must show detailed reasoning.
   Find the coordinates of the points of intersection of the curves with equations \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\).
2. The diagram shows the curves \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\). This diagram is repeated in the Printed Answer Booklet.
   \includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-5_377_542_603_322} On the diagram in the Printed Answer Booklet, draw the line \(y = 2 x - 2\).
3. Show on your diagram in the Printed Answer Booklet the region of the \(x - y\) plane within which all three of the following inequalities are satisfied.
   \(y \geqslant x ^ { 2 } - 2 x + 1 \quad y \leqslant - x ^ { 2 } + 6 x - 5 \quad y \leqslant 2 x - 2\)
   You should indicate the region for which all the inequalities hold by labelling the region \(R\).[1]

4

1. Write \(2 x ^ { 2 } + 6 x + 7\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are constants.
2. State the coordinates of the minimum point on the graph of \(y = 2 x ^ { 2 } + 6 x + 7\).
3. Hence deduce
   • the minimum value of \(2 \tan ^ { 2 } \theta + 6 \tan \theta + 7\),
   • the smallest positive value of \(\theta\), in degrees, for which the minimum value occurs.

5

1. The graph of \(y = 2 ^ { x }\) can be transformed to the graph of \(y = 2 ^ { x + 4 }\) either by a translation or by a stretch.
   1. Give full details of the translation.
   2. Give full details of the stretch.
2. In this question you must show detailed reasoning. Solve the equation \(\log _ { 2 } ( 8 x ) = 1 - \log _ { 2 } ( 1 - x )\).

6

1. Find the first four terms in the expansion of \(( 3 + 2 x ) ^ { 5 }\) in ascending powers of \(x\).
2. Hence determine the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 3 + 2 y + 4 y ^ { 2 } \right) ^ { 5 }\).

7 A curve has equation \(2 x ^ { 3 } + 6 x y - 3 y ^ { 2 } = 2\).
Show that there are no points on this curve where the tangent is parallel to \(y = x\).

8

1. Substance \(A\) is decaying exponentially such that its mass is \(m\) grams at time \(t\) minutes. Find the missing values of \(m\) and \(t\) in the following table.

   \(t\)	0	10		50
   \(m\)	1250	750	450

2. Substance \(B\) is also decaying exponentially, according to the model \(m = 160 \mathrm { e } ^ { - 0.055 t }\), where \(m\) grams is its mass after \(t\) minutes.
   1. Determine the value of \(t\) for which the mass of substance \(B\) is half of its original mass.
   2. Determine the rate of decay of substance \(B\) when \(t = 15\).
3. State whether substance \(A\) or substance \(B\) is decaying at a faster rate, giving a reason for your answer.

9 Use the substitution \(x = 2 \sin \theta\) to show that \(\int _ { 1 } ^ { \sqrt { 3 } } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 3 } \pi\).

10
\includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-7_545_659_255_244} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(O A\). The angle \(A O B\) is \(\theta\) radians. \(M\) is the mid-point of \(O A\). The ratio of areas \(O M B : M A B\) is 2:3.

1. Show that \(\theta = 1.25 \sin \theta\). The equation \(\theta = 1.25 \sin \theta\) has only one root for \(\theta > 0\).
2. This root can be found by using the iterative formula \(\theta _ { n + 1 } = 1.25 \sin \theta _ { n }\) with a starting value of \(\theta _ { 1 } = 0.5\).
   • Write down the values of \(\theta _ { 2 } , \theta _ { 3 }\) and \(\theta _ { 4 }\).
   • Hence find the value of this root correct to \(\mathbf { 3 }\) significant figures.
   • The diagram in the Printed Answer Booklet shows the graph of \(y = 1.25 \sin \theta\), for \(0 \leqslant \theta \leqslant \pi\).
   • Use this diagram to show how the iterative process used in (b) converges to this root.
   • State the type of convergence.
   • Draw a suitable diagram to show why using an iterative process with the formula \(\theta _ { n + 1 } = \sin ^ { - 1 } \left( 0.8 \theta _ { n } \right)\) does not converge to the root found in (b).

11 The gradient function of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x ^ { 2 } \ln x } { \mathrm { e } ^ { 3 y } }\).
The curve passes through the point (e, 1).

1. Find the equation of this curve, giving your answer in the form \(\mathrm { e } ^ { 3 y } = \mathrm { f } ( x )\).
2. Show that, when \(x = \mathrm { e } ^ { 2 }\), the \(y\)-coordinate of this curve can be written as \(y = a + \frac { 1 } { 3 } \ln \left( b \mathrm { e } ^ { 3 } + c \right)\), where \(a , b\) and \(c\) are constants to be determined.

12 A curve has parametric equations \(x = \frac { 1 } { t } , y = 2 t\). The point \(P\) is \(\left( \frac { 1 } { p } , 2 p \right)\).

1. Show that the equation of the tangent at \(P\) can be written as \(y = - 2 p ^ { 2 } x + 4 p\). The tangent to this curve at \(P\) crosses the \(x\)-axis at the point \(A\) and the normal to this curve at \(P\) crosses the \(x\)-axis at the point \(B\).
2. Show that the ratio \(P A : P B\) is \(1 : 2 p ^ { 2 }\). \section*{END OF QUESTION PAPER}