Questions C4 (1162 questions)

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OCR MEI C4 Q5
5
  1. Simplify \(\frac { x ^ { 3 } - x ^ { 2 } - 3 x - 9 } { x - 3 }\).
  2. Hence or otherwise solve the equation \(x ^ { 3 } - x ^ { 2 } - 3 x - 9 = 6 ( x - 3 )\).
OCR MEI C4 Q6
6 Prove that
  1. \(\frac { \sin 2 \theta } { 2 \tan \theta } + \sin ^ { 2 } \theta = 1\),
  2. \(\quad \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 45 ^ { \circ } \right)\).
OCR MEI C4 Q7
7 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x } { y }\) given that when \(x = 1 , y = 2\).
OCR MEI C4 Q8
8 Scientists predict the velocity ( \(v\) kilometres per minute) for the new "outer explorer" spacecraft over the first minute of its entry to the atmosphere of the planet Titan to be modelled by the equation: $$v = \frac { 5000 } { ( 1 + t ) ( 2 + t ) ^ { 2 } } , 0 \leq t \leq 1 \text { where } t \text { represents time in minutes. }$$
  1. Use a binomial expansion to expand \(( 1 + t ) ^ { - 1 }\) up to and including the term in \(t ^ { 2 }\).
  2. Use a binomial expansion to expand \(( 2 + t ) ^ { - 2 }\) up to and including the term in \(t ^ { 2 }\).
  3. Hence, or otherwise, show that \(v \approx 1250 \left( 1 - 2 t + \frac { 11 t ^ { 2 } } { 4 } \right)\).
  4. The displacement of the spacecraft can be found by calculating the area under the velocity time graph. Use the approximation found in part (iii) to estimate the displacement of the spacecraft over the first half minute.
  5. Write \(\frac { 1 } { ( 1 + t ) ( 2 + t ) ^ { 2 } }\) in partial fractions.
  6. The displacement of the spacecraft in the first \(T\) minutes is given by \(\int _ { 0 } ^ { T } v \mathrm {~d} t\) Calculate the exact value of the displacement of the spacecraft over the first half minute given by the model.
  7. On further investigation the scientists believe the original model may be valid for up to three minutes. Explain why the approximation in (iii) will be no longer be valid for this time interval.
OCR MEI C4 Q9
9 Two astronomers wish to model the path of motion of a particle in two dimensions.
Experimental results show that the position of the particle can be found using the parametric equations $$x = 2 \cos \theta - \sin \theta + 2 \quad y = \cos \theta + 2 \sin \theta - 1 \quad \left( 0 \leq \theta \leq 360 ^ { \circ } \right)$$ One astronomer uses trigonometry.
  1. Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants to be determined. Show also that, for the same values of \(R\) and \(\alpha\), $$\cos \theta + 2 \sin \theta = R \sin ( \theta + \alpha )$$
  2. Hence, or otherwise, show that the path of particle may be written in the form $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 5$$ Describe the path of the particle. The second astronomer sets up a first order differential equation with the condition that \(x = 4\) when \(y = 0\).
  3. Verify that the point with parameter \(\theta = 0\) has coordinates \(( 4,0 )\).
  4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Deduce that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x - 2 } { y + 1 }$$
  5. Solve this differential equation, using the condition that \(y = 0\) when \(x = 4\). Hence show that the two solutions give the same cartesian equation for the path of particle.
OCR C4 Q1
  1. Express
$$\frac { 2 x } { 2 x ^ { 2 } + 3 x - 5 } \div \frac { x ^ { 3 } } { x ^ { 2 } - x }$$ as a single fraction in its simplest form.
OCR C4 Q2
2. A curve has the equation $$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
OCR C4 Q3
3. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are $$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).
  1. Find the values of \(a\) and \(n\).
  2. Show that \(k = - 160\).
OCR C4 Q4
4. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1
5
- 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6
3
- 6 \end{array} \right)\) respectively. Find, in exact, simplified form,
  1. the cosine of \(\angle A O B\),
  2. the area of triangle \(O A B\),
  3. the shortest distance from \(A\) to the line \(O B\).
OCR C4 Q5
5. (i) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(ii) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.
OCR C4 Q6
6. (i) Find $$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$ (ii) Use the substitution \(u ^ { 2 } = x + 1\) to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$
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 \(y ^ { 2 } = \mathrm { f } ( x )\).