Questions C4 (1162 questions)

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OCR C4 2007 January Q3
3 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k }\).
  1. Find the length of \(A B\).
  2. Use a scalar product to find angle \(O A B\).
OCR C4 2007 January Q4
4 Use the substitution \(u = 2 x - 5\) to show that \(\int _ { \frac { 5 } { 2 } } ^ { 3 } ( 4 x - 8 ) ( 2 x - 5 ) ^ { 7 } \mathrm {~d} x = \frac { 17 } { 72 }\).
OCR C4 2007 January Q5
5
  1. Expand \(( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 1 - 3 \left( x + x ^ { 3 } \right) \right) ^ { - \frac { 1 } { 3 } }\).
OCR C4 2007 January Q6
6
  1. Express \(\frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } }\) in the form \(\frac { A } { x - 3 } + \frac { B } { ( x - 3 ) ^ { 2 } }\), where \(A\) and \(B\) are constants.
  2. Hence find the exact value of \(\int _ { 4 } ^ { 10 } \frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.
OCR C4 2007 January Q7
7 The equation of a curve is \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\). Show that there are two stationary points on the curve and find their coordinates.
OCR C4 2007 January Q8
8 The parametric equations of a curve are \(x = 2 t ^ { 2 } , y = 4 t\). Two points on the curve are \(P \left( 2 p ^ { 2 } , 4 p \right)\) and \(Q \left( 2 q ^ { 2 } , 4 q \right)\).
  1. Show that the gradient of the normal to the curve at \(P\) is \(- p\).
  2. Show that the gradient of the chord joining the points \(P\) and \(Q\) is \(\frac { 2 } { p + q }\).
  3. The chord \(P Q\) is the normal to the curve at \(P\). Show that \(p ^ { 2 } + p q + 2 = 0\).
  4. The normal at the point \(R ( 8,8 )\) meets the curve again at \(S\). The normal at \(S\) meets the curve again at \(T\). Find the coordinates of \(T\).
OCR C4 2007 January Q9
9
  1. Find the general solution of the differential equation $$\frac { \sec ^ { 2 } y } { \cos ^ { 2 } ( 2 x ) } \frac { d y } { d x } = 2$$
  2. For the particular solution in which \(y = \frac { 1 } { 4 } \pi\) when \(x = 0\), find the value of \(y\) when \(x = \frac { 1 } { 6 } \pi\).
OCR C4 2007 January Q10
10 The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5 \mathbf { i } + 2 \mathbf { j } - 9 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k }\) respectively.
  1. Find a vector equation for the line \(P Q\). The position vector of the point \(T\) is \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\).
  2. Write down a vector equation for the line \(O T\) and show that \(O T\) is perpendicular to \(P Q\). It is given that \(O T\) intersects \(P Q\).
  3. Find the position vector of the point of intersection of \(O T\) and \(P Q\).
  4. Hence find the perpendicular distance from \(O\) to \(P Q\), giving your answer in an exact form.
OCR C4 2008 January Q1
1 Find the angle between the vectors \(\mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) and \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
OCR C4 2008 January Q2
2
  1. Express \(\frac { x } { ( x + 1 ) ( x + 2 ) }\) in partial fractions.
  2. Hence find \(\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x\).
OCR C4 2008 January Q3
3 When \(x ^ { 4 } - 2 x ^ { 3 } - 7 x ^ { 2 } + 7 x + a\) is divided by \(x ^ { 2 } + 2 x - 1\), the quotient is \(x ^ { 2 } + b x + 2\) and the remainder is \(c x + 7\). Find the values of the constants \(a , b\) and \(c\).
OCR C4 2008 January Q4
4 Find the equation of the normal to the curve $$x ^ { 3 } + 4 x ^ { 2 } y + y ^ { 3 } = 6$$ at the point \(( 1,1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C4 2008 January Q5
5 The vector equations of two lines are $$\mathbf { r } = ( 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) + s ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = ( 2 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) + t ( 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k } ) .$$ Prove that the two lines are
  1. perpendicular,
  2. skew.
OCR C4 2008 January Q6
6
  1. Expand \(( 1 + a x ) ^ { - 4 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  2. The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 1 + b x ) ( 1 + a x ) ^ { - 4 }\) are 1 and - 2 respectively. Given that \(a > 0\), find the values of \(a\) and \(b\).
OCR C4 2008 January Q8
8 Water flows out of a tank through a hole in the bottom and, at time \(t\) minutes, the depth of water in the tank is \(x\) metres. At any instant, the rate at which the depth of water in the tank is decreasing is proportional to the square root of the depth of water in the tank.
  1. Write down a differential equation which models this situation.
  2. When \(t = 0 , x = 2\); when \(t = 5 , x = 1\). Find \(t\) when \(x = 0.5\), giving your answer correct to 1 decimal place.
OCR C4 2008 January Q9
9 The parametric equations of a curve are \(x = t ^ { 3 } , y = t ^ { 2 }\).
  1. Show that the equation of the tangent at the point \(P\) where \(t = p\) is $$3 p y - 2 x = p ^ { 3 } .$$
  2. Given that this tangent passes through the point ( \(- 10,7\) ), find the coordinates of each of the three possible positions of \(P\).
OCR C4 2008 January Q10
10
  1. Use the substitution \(x = \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
  2. Find the exact value of $$\int _ { 1 } ^ { 3 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$ 4
OCR C4 2005 June Q1
1 Find the quotient and the remainder when \(x ^ { 4 } + 3 x ^ { 3 } + 5 x ^ { 2 } + 4 x - 1\) is divided by \(x ^ { 2 } + x + 1\).
OCR C4 2005 June Q2
2 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \cos x \mathrm {~d} x\), giving your answer in an exact form.
OCR C4 2005 June Q3
3 The line \(L _ { 1 }\) passes through the points \(( 2 , - 3,1 )\) and \(( - 1 , - 2 , - 4 )\). The line \(L _ { 2 }\) passes through the point ( \(3,2 , - 9\) ) and is parallel to the vector \(4 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\).
  1. Find an equation for \(L _ { 1 }\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
  2. Prove that \(L _ { 1 }\) and \(L _ { 2 }\) are skew.
OCR C4 2005 June Q4
4
  1. Show that the substitution \(x = \tan \theta\) transforms \(\int \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\) to \(\int \cos ^ { 2 } \theta \mathrm {~d} \theta\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\).
    \(5 A B C D\) is a parallelogram. The position vectors of \(A , B\) and \(C\) are given respectively by $$\mathbf { a } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { c } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } .$$
  3. Find the position vector of \(D\).
  4. Determine, to the nearest degree, the angle \(A B C\).
OCR C4 2005 June Q6
6 The equation of a curve is \(x y ^ { 2 } = 2 x + 3 y\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - y ^ { 2 } } { 2 x y - 3 }\).
  2. Show that the curve has no tangents which are parallel to the \(y\)-axis.
OCR C4 2005 June Q7
7 A curve is given parametrically by the equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
  2. Show that the equation of the tangent at the point \(P \left( 4 , - \frac { 1 } { 2 } \right)\) is $$x - 16 y = 12$$
  3. Find the value of the parameter at the point where the tangent at \(P\) meets the curve again.
OCR C4 2005 June Q8
8
  1. Given that \(\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { 1 + x } + \frac { B } { 2 + x } + \frac { C } { ( 2 + x ) ^ { 2 } }\), find \(A , B\) and \(C\).
  2. Hence or otherwise expand \(\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  3. State the set of values of \(x\) for which the expansion in part (ii) is valid.
OCR C4 2005 June Q9
9 Newton's law of cooling states that the rate at which the temperature of an object is falling at any instant is proportional to the difference between the temperature of the object and the temperature of its surroundings at that instant. A container of hot liquid is placed in a room which has a constant temperature of \(20 ^ { \circ } \mathrm { C }\). At time \(t\) minutes later, the temperature of the liquid is \(\theta ^ { \circ } \mathrm { C }\).
  1. Explain how the information above leads to the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 ) ,$$ where \(k\) is a positive constant.
  2. The liquid is initially at a temperature of \(100 ^ { \circ } \mathrm { C }\). It takes 5 minutes for the liquid to cool from \(100 ^ { \circ } \mathrm { C }\) to \(68 ^ { \circ } \mathrm { C }\). Show that $$\theta = 20 + 80 \mathrm { e } ^ { - \left( \frac { 1 } { 5 } \ln \frac { 5 } { 3 } \right) t }$$
  3. Calculate how much longer it takes for the liquid to cool by a further \(32 ^ { \circ } \mathrm { C }\).