Questions — OCR (4907 questions)

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OCR C4 Q4
8 marks Moderate -0.8
4.
  1. Express $$\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$$ as a single fraction in its simplest form.
  2. 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. \(f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. 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.
  3. 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. Simplify $$\frac { x ^ { 2 } + 7 x + 12 } { 2 x ^ { 2 } + 9 x + 4 }$$
  2. 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.
  1. Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
  2. 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.
  1. Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
  2. State the set of values of \(x\) for which your expansion is valid.
  3. 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.
  1. 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 \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
  2. Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
  3. Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).
OCR C4 Q7
12 marks Standard +0.3
7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
  2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
  4. Find the position vector of \(C\).
OCR C4 Q8
13 marks Standard +0.8
8. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after \(t\) hours is denoted by \(x\). In a model, it is assumed that the rate at which the crop is destroyed is proportional to \(x ( 1 - x )\). A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,
  1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )\),
  2. find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.
OCR C4 Q1
4 marks Moderate -0.3
  1. \(\mathrm { f } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 }\).
Find the values of the constants \(A , B , C\) and \(D\) such that $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$
OCR C4 Q2
6 marks Standard +0.3
  1. Use the substitution \(u = 1 - x ^ { \frac { 1 } { 2 } }\) to find
$$\int \frac { 1 } { 1 - x ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$$
OCR C4 Q3
7 marks Standard +0.3
  1. A curve has the equation
$$4 \cos x + 2 \sin y = 3$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y\).
  2. Find an equation for the tangent to the curve at the point ( \(\frac { \pi } { 3 } , \frac { \pi } { 6 }\) ), giving your answer in the form \(a x + b y = c\), where \(a\) and \(b\) are integers.