Questions — OCR (4619 questions)

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OCR C4 2006 January Q8
11 marks Moderate -0.3
8
  1. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - x } { y - 3 }$$ giving the particular solution that satisfies the condition \(y = 4\) when \(x = 5\).
  2. Show that this particular solution can be expressed in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where the values of the constants \(a , b\) and \(k\) are to be stated.
  3. Hence sketch the graph of the particular solution, indicating clearly its main features.
OCR C4 2006 January Q9
13 marks Standard +0.3
9 Two lines have vector equations $$\mathbf { r } = \left( \begin{array} { r } 4 \\ 2 \\ - 6 \end{array} \right) + t \left( \begin{array} { r } - 8 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2 \\ a \\ - 2 \end{array} \right) + s \left( \begin{array} { r } - 9 \\ 2 \\ - 5 \end{array} \right) ,$$ where \(a\) is a constant.
  1. Calculate the acute angle between the lines.
  2. Given that these two lines intersect, find \(a\) and the point of intersection.
OCR C4 2007 January Q1
3 marks Easy -1.2
1 It is given that $$f ( x ) = \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } - 4 x } \quad \text { for } x \neq 0 , x \neq 4$$ Express \(\mathrm { f } ( x )\) in its simplest form.
OCR C4 2007 January Q3
6 marks Moderate -0.3
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
5 marks Moderate -0.3
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
7 marks Standard +0.3
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
7 marks Moderate -0.3
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
8 marks Standard +0.8
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
10 marks Standard +0.8
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
10 marks Moderate -0.3
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
11 marks Moderate -0.3
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
4 marks Moderate -0.5
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
5 marks Moderate -0.8
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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
8 marks Standard +0.3
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
8 marks Standard +0.3
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 marks Standard +0.3
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 marks Standard +0.3
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
11 marks Standard +0.3
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
4 marks Moderate -0.3
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
5 marks Moderate -0.5
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
7 marks Standard +0.3
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
7 marks Standard +0.2
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
8 marks Standard +0.3
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.