Questions — OCR C4 (310 questions)

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OCR C4 Q4
4. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( - 1 , - 3 )\).
OCR C4 Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{5840974b-b08a-4818-9a59-97b2d3ce9890-1_469_809_1777_484} The diagram shows the curve with equation \(y = x \sqrt { 1 - x } , 0 \leq x \leq 1\).
Use the substitution \(u ^ { 2 } = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac { 4 } { 15 }\).
OCR C4 Q6
6. The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac { \mathrm { d } n } { \mathrm {~d} t } = \mathrm { e } ^ { 0.5 t } - 5 , \quad t \geq 0$$
  1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue.
  2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation.
  3. Explain why this model would not be appropriate for large values of \(t\).
OCR C4 Q7
7. (i) Show that ( \(2 x + 3\) ) is a factor of ( \(\left. 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\) and hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (ii) Show that $$\int _ { 2 } ^ { 5 } \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } \mathrm {~d} x = \ln k$$ where \(k\) is an integer.
OCR C4 Q8
8. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.
  1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
  2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
  3. Find the coordinates of \(D\).
  4. Find the area of triangle \(O A B\) to 3 significant figures.
OCR C4 Q9
9. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$
  1. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
  2. find a cartesian equation for the curve.
  3. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
  4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 Q1
  1. Express
$$\frac { 5 x } { ( x - 4 ) ( x + 1 ) } + \frac { 3 } { ( x - 2 ) ( x + 1 ) }$$ as a single fraction in its simplest form.
OCR C4 Q2
2. A curve has the equation $$x ^ { 2 } + 2 x y ^ { 2 } + y = 4$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 Q3
3. Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin 2 x \cos x d x$$
OCR C4 Q4
  1. A curve has parametric equations
$$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 }$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).
  1. Find the value of the parameter \(t\) at \(P\).
  2. Show that the tangent to the curve at \(P\) has the equation $$y = 2 x + 1$$
OCR C4 Q5
  1. (i) Express \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }\) as a sum of partial fractions.
    (ii) Hence find the series expansion of \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  2. Use the substitution \(x = 2 \tan u\) to show that
$$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } d x = \frac { 1 } { 2 } ( 4 - \pi )$$
OCR C4 Q7
  1. A straight road passes through villages at the points \(A\) and \(B\) with position vectors \(( 9 \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k } )\) and ( \(4 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.
The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + t ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(t\) is a scalar parameter.
  1. Find the position vector of \(C\). Given that 1 unit on each coordinate axis represents 200 metres,
  2. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\).
OCR C4 Q8
8. (i) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
(ii) Show that $$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where \(c\) is an arbitrary constant.
\includegraphics[max width=\textwidth, alt={}, center]{1e93a786-6105-4c69-a79a-a5f6e6c4aa0a-2_554_784_1484_507} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \tan x\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 3 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(iii) Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).
OCR C4 Q9
9. An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = k P$$ where \(k\) is a constant.
  1. Find an expression for \(P\) in terms of \(k\) and \(t\). Given that after one week there are 360 insects in the colony,
  2. find the value of \(k\) to 3 significant figures. Given also that after two and three weeks there are 440 and 600 insects respectively,
  3. comment on suitability of the modelling assumption. An alternative model assumes that $$\frac { \mathrm { d } P } { \mathrm {~d} t } = P ( 0.4 - 0.25 \cos 0.5 t )$$
  4. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation.
  5. Compare the suitability of the two models.
OCR C4 Q1
1 Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\).
OCR C4 Q2
2 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < { } _ { 2 } ^ { 1 } \pi\).-
Hence solve the equation \(4 \cos \theta - \sin \theta = 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).
OCR C4 Q3
3 Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).
  1. Fig. 8.1 shows a regular 12 -sided polygon inscribed in a circle of radius 1 unit, centre \(\mathrm { O } . \mathrm { AB }\) is one of the sides of the polygon. C is the midpoint of AB . Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-2_457_422_457_936} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
    \end{figure} (A) Show that \(\mathrm { AB } = 2 \sin 15 ^ { \circ }\).
    (B) Use a double angle formula to express \(\cos 30 ^ { \circ }\) in terms of \(\sin 15 ^ { \circ }\). Using the exact value of \(\cos 30 ^ { \circ }\), show that \(\sin 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }\).
    (C) Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6 \sqrt { 2 - \sqrt { 3 } }\).
  2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-2_450_416_1562_940} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} (A) Show that \(\mathrm { DE } = 2 \tan 15 ^ { \circ }\).
    (B) Let \(t = \tan 15 ^ { \circ }\). Use a double angle formula to express \(\tan 30 ^ { \circ }\) in terms of \(t\). Hence show that \(t ^ { 2 } + 2 \sqrt { 3 } t - 1 = 0\).
    (C) Solve this equation, and hence show that \(\pi < 12 ( 2 - \sqrt { 3 } )\).
  3. Use the results in parts (i)( \(C\) ) and (ii)( \(C\) ) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form.
OCR C4 Q4
4 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).
OCR C4 Q5
5 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).
OCR C4 Q6
6 Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR C4 Q7
7 Fig. 1 shows part of the graph of \(y = \sin x \quad \sqrt { 3 } \cos x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-3_452_613_1187_745} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Express \(\quad \sqrt { } \quad\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi\).
Hence write down the exact coordinates of the turning point P .
OCR C4 Q1
1 Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
Hence find the range of the function \(\mathbf { f } ( \theta )\), where $$f ( \theta ) = 7 + 3 \cos \theta + 4 \sin \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi .$$ Write down the greatest possible value of \(\frac { 1 } { 7 + 3 \cos \theta + 4 \sin \theta }\).
OCR C4 Q2
2 Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = 3 \sin x + 2 \cos x , 0 \leqslant x \leqslant \pi$$
OCR C4 Q3
3 Show that \(\frac { \sin 2 \theta } { 1 + \cos 2 \theta } = \tan \theta\).
OCR C4 Q5
5 Solve the equation \(2 \sin 2 \theta + \cos 2 \theta = 1\), for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).