Questions — OCR C4 (317 questions)

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OCR C4 Q7
12 marks Standard +0.8
  1. During a chemical reaction, a compound is being made from two other substances.
At time \(t\) hours after the start of the reaction, \(x \mathrm {~g}\) of the compound has been produced. Assuming that \(x = 0\) initially, and that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 2 ( x - 6 ) ( x - 3 )$$
  1. show that it takes approximately 7 minutes to produce 2 g of the compound.
  2. Explain why it is not possible to produce 3 g of the compound.
OCR C4 Q8
12 marks Standard +0.3
8. \includegraphics[max width=\textwidth, alt={}, center]{85427816-dcf1-49af-8d68-f4e88fc7d8f1-3_497_784_246_461} The diagram shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point \(P\) on the curve has coordinates \(( 1 , \sqrt { 6 } )\).
  1. Find the value of \(\theta\) at \(P\).
  2. Show that the normal to the curve at \(P\) passes through the origin.
  3. Find a cartesian equation for the curve.
OCR C4 Q1
5 marks Standard +0.3
  1. A curve has the equation
$$x ^ { 2 } ( 2 + y ) - y ^ { 2 } = 0$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 Q2
5 marks Moderate -0.3
2. Show that $$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$
OCR C4 Q3
7 marks Standard +0.8
3.
\includegraphics[max width=\textwidth, alt={}]{47c69f14-a336-4255-87fc-64ff1d2ee5e1-1_556_858_904_557}
The diagram shows the curve with equation \(y = 2 \sin x + \operatorname { cosec } x , 0 < x < \pi\).
The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 2 }\) is rotated through four right angles about the \(x\)-axis. Show that the volume of the solid formed is \(\frac { 1 } { 2 } \pi ( 4 \pi + 3 \sqrt { 3 } )\).
OCR C4 Q4
8 marks Moderate -0.8
4. (i) Express $$\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$$ as a single fraction in its simplest form.
(ii) Simplify $$\frac { x ^ { 3 } - 8 } { 3 x ^ { 2 } - 8 x + 4 }$$
OCR C4 Q5
11 marks Moderate -0.3
  1. A bath is filled with hot water which is allowed to cool. The temperature of the water is \(\theta ^ { \circ } \mathrm { C }\) after cooling for \(t\) minutes and the temperature of the room is assumed to remain constant at \(20 ^ { \circ } \mathrm { C }\).
Given that the rate at which the temperature of the water decreases is proportional to the difference in temperature between the water and the room,
  1. write down a differential equation connecting \(\theta\) and \(t\). Given also that the temperature of the water is initially \(37 ^ { \circ } \mathrm { C }\) and that it is \(36 ^ { \circ } \mathrm { C }\) after cooling for four minutes,
  2. find, to 3 significant figures, the temperature of the water after ten minutes. Advice suggests that the temperature of the water should be allowed to cool to \(33 ^ { \circ } \mathrm { C }\) before a child gets in.
  3. Find, to the nearest second, how long a child should wait before getting into the bath.
OCR C4 Q6
12 marks Standard +0.3
6. A curve has parametric equations $$x = 3 \cos ^ { 2 } t , \quad y = \sin 2 t , \quad 0 \leq t < \pi$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 } { 3 } \cot 2 t\).
  2. Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
  3. Show that the tangent to the curve at the point where \(t = \frac { \pi } { 6 }\) has the equation $$2 x + 3 \sqrt { 3 } y = 9$$
  4. Find a cartesian equation for the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR C4 Q7
12 marks Standard +0.8
7. Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } - 4 \\ 1 \\ 3 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ 6 \\ 1 \end{array} \right)\) respectively.
  1. Find a vector equation for the line \(l _ { 1 }\) which passes through \(A\) and \(B\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3 \\ - 7 \\ 9 \end{array} \right) + t \left( \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right)$$
  2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
  3. Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
OCR C4 Q9
Standard +0.3
9 \end{array} \right) + t \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (ii) Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
(iii) Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
8. \(f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\).
(i) Express \(\mathrm { f } ( x )\) in partial fractions.
(ii) Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(iii) State the set of values of \(x\) for which your expansion is valid.
OCR C4 Q1
4 marks Moderate -0.8
  1. Evaluate
$$\int _ { 0 } ^ { \pi } \sin x ( 1 + \cos x ) d x$$
OCR C4 Q2
5 marks Moderate -0.8
  1. (i) Simplify
$$\frac { x ^ { 2 } + 7 x + 12 } { 2 x ^ { 2 } + 9 x + 4 }$$ (ii) Express $$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 }$$ as a single fraction in its simplest form.
OCR C4 Q3
5 marks Standard +0.3
3. Find the exact value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x d x$$
OCR C4 Q4
6 marks Standard +0.3
4.
\includegraphics[max width=\textwidth, alt={}]{23bd8979-9ba6-4e77-a3d1-88feb5e5a5b3-1_444_728_1425_536}
The diagram shows the curve with parametric equations $$x = t + \sin t , \quad y = \sin t , \quad 0 \leq t \leq \pi$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis.
OCR C4 Q5
7 marks Moderate -0.3
5. Given that \(y = - 2\) when \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR C4 Q6
9 marks Standard +0.3
6. (i) Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
(ii) Using the substitution \(u = x ^ { 2 } + 4\), evaluate $$\int _ { 0 } ^ { 2 } \frac { 5 x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x$$
OCR C4 Q7
10 marks Standard +0.3
  1. A curve has the equation
$$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point \(P\) on the curve has coordinates \(( - 1,3 )\).
  1. Show that the normal to the curve at \(P\) has the equation \(y = 2 - x\).
  2. Find the coordinates of the point where the normal to the curve at \(P\) meets the curve again.
OCR C4 Q8
13 marks Standard +0.8
8. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors \(( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )\) and ( \(7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(B C\).
  2. Show that one possible position vector for \(C\) is \(( \mathbf { i } + 3 \mathbf { j } )\) and find the other. Assuming that \(C\) has position vector \(( \mathbf { i } + 3 \mathbf { j } )\),
  3. find the area of triangle \(A B C\), giving your answer in the form \(k \sqrt { 5 }\).
OCR C4 Q9
13 marks Standard +0.3
9. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$ where \(k\) is an integer to be found.
  3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
OCR C4 Q1
4 marks Standard +0.3
1. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 }$$ Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$
OCR C4 Q2
7 marks Standard +0.0
  1. A curve has the equation
$$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find an equation for the tangent to the curve at the point \(( 2 , - 2 )\).
OCR C4 Q3
8 marks Standard +0.3
3. Find
  1. \(\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x\),
  2. \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).
OCR C4 Q4
8 marks Standard +0.3
4. \includegraphics[max width=\textwidth, alt={}, center]{c7b867af-0730-459e-9c76-15eb07b9e476-1_465_976_1539_388} The diagram shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$
  1. Find a cartesian equation for the curve. The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\).
  2. Using integration, with the substitution \(x = \tan u\), find the area of the shaded region.
OCR C4 Q5
9 marks Moderate -0.3
5. (i) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.
(iii) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.
OCR C4 Q6
11 marks Standard +0.3
6. (i) Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(ii) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(iii) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).