Questions — OCR C4 (310 questions)

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OCR C4 Q7
  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
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
  1. Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).
  2. Find the quotient and remainder when \(\left( x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 9 \right)\) is divided by \(\left( x ^ { 2 } + x - 6 \right)\).
  3. Differentiate each of the following with respect to \(x\) and simplify your answers.
    1. \(\cot x ^ { 2 }\)
    2. \(\frac { \sin x } { 3 + 2 \cos x }\)
    3. (i) Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
    4. Hence, or otherwise, show that for small \(x\),
    $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$
OCR C4 Q5
5.
\includegraphics[max width=\textwidth, alt={}]{027c173c-0afe-4773-8bb4-9b634858e1ff-1_556_816_1414_477}
The diagram shows the curve with parametric equations $$x = a \sqrt { t } , \quad y = a t ( 1 - t ) , \quad t \geq 0$$ where \(a\) is a positive constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.
  2. Show that the area of triangle \(O A B\) is \(a ^ { 2 }\).
OCR C4 Q6
6. Relative to a fixed origin, two lines have the equations $$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where \(s\) and \(t\) are scalar parameters.
  1. Show that the two lines intersect and find the position vector of the point where they meet.
  2. Find, in degrees to 1 decimal place, the acute angle between the lines.
OCR C4 Q7
7. At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k \mathrm { e } ^ { - 0.2 t }$$ where \(k\) is a positive constant,
  1. Find an expression for \(y\) in terms of \(k\) and \(t\). Given that two hours after being filled the depth of water in the tank is 1.6 metres,
  2. find the value of \(k\) to 4 significant figures. Given also that the hole in the tank is \(h \mathrm {~cm}\) above the base of the tank,
  3. show that \(h = 79\) to 2 significant figures.
OCR C4 Q8
8. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
  2. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  3. Find the coordinates of \(Q\).
OCR C4 Q9
9. (i) Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ into the integral $$\int \frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) } \mathrm { d } u$$ (ii) Express \(\frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) }\) in partial fractions.
(iii) Hence, evaluate $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers.
OCR C4 Q1
  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
2. Show that $$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$
OCR C4 Q3
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
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
  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
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
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
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
  1. Evaluate
$$\int _ { 0 } ^ { \pi } \sin x ( 1 + \cos x ) d x$$
OCR C4 Q2
  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
3. Find the exact value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x d x$$
OCR C4 Q4
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
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
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
  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
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
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.