Questions — OCR (4619 questions)

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OCR FP3 2007 June Q6
6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\frac { x - 3 } { 2 } = \frac { y - 4 } { - 1 } = \frac { z + 1 } { 1 } \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 2 }$$ respectively.
  1. Find the equation of the plane \(\Pi _ { 1 }\) which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\), giving your answer in the form r.n \(= p\).
  2. Find the equation of the plane \(\Pi _ { 2 }\) which contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\), giving your answer in the form r.n \(= p\).
  3. Find the distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
  4. State the relationship between the answer to part (iii) and the lines \(l _ { 1 }\) and \(l _ { 2 }\).
OCR FP3 2007 June Q8
8
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tan x = \cos ^ { 3 } x$$ expressing \(y\) in terms of \(x\) in your answer.
  2. Find the particular solution for which \(y = 2\) when \(x = \pi\).
OCR FP3 2007 June Q9
9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)
  1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
  2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
    (a) The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
    (b) The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
    (c) The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\). 4
OCR FP3 Specimen Q1
1 Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = x ,$$ giving \(y\) in terms of \(x\) in your answer.
OCR FP3 Specimen Q2
2 The set \(S = \{ a , b , c , d \}\) under the binary operation * forms a group \(G\) of order 4 with the following operation table.
\(*\)\(a\)\(b\)\(c\)\(d\)
\(a\)\(d\)\(a\)\(b\)\(c\)
\(b\)\(a\)\(b\)\(c\)\(d\)
\(c\)\(b\)\(c\)\(d\)\(a\)
\(d\)\(c\)\(d\)\(a\)\(b\)
  1. Find the order of each element of \(G\).
  2. Write down a proper subgroup of \(G\).
  3. Is the group \(G\) cyclic? Give a reason for your answer.
  4. State suitable values for each of \(a , b , c\) and \(d\) in the case where the operation \(*\) is multiplication of complex numbers.
OCR FP3 Specimen Q3
3 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(\mathbf { r } \cdot ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) = 1\) and \(\mathbf { r } \cdot ( 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) = 3\) respectively. Find
  1. the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), correct to the nearest degree,
  2. the equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
OCR FP3 Specimen Q4
4 In this question, give your answers exactly in polar form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  1. Express \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
  2. Find the cube roots of \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
  3. Sketch an Argand diagram showing the positions of the cube roots found in part (ii). Hence, or otherwise, prove that the sum of these cube roots is zero.
OCR FP3 Specimen Q5
5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\frac { x - 5 } { 1 } = \frac { y - 1 } { - 1 } = \frac { z - 5 } { - 2 } \quad \text { and } \quad \frac { x - 1 } { - 4 } = \frac { y - 11 } { - 14 } = \frac { z - 2 } { 2 } .$$
  1. Find the exact value of the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Find an equation for the plane containing \(l _ { 1 }\) and parallel to \(l _ { 2 }\) in the form \(a x + b y + c z = d\).
OCR FP3 Specimen Q6
6 The set \(S\) consists of all non-singular \(2 \times 2\) real matrices \(\mathbf { A }\) such that \(\mathbf { A Q } = \mathbf { Q A }\), where $$\mathbf { Q } = \left( \begin{array} { l l } 1 & 1
0 & 1 \end{array} \right)$$
  1. Prove that each matrix \(\mathbf { A }\) must be of the form \(\left( \begin{array} { l l } a & b
    0 & a \end{array} \right)\).
  2. State clearly the restriction on the value of \(a\) such that \(\left( \begin{array} { l l } a & b
    0 & a \end{array} \right)\) is in \(S\).
  3. Prove that \(S\) is a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.)
OCR FP3 Specimen Q7
7
  1. Prove that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), then \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\).
  2. Express \(\cos ^ { 6 } \theta\) in terms of cosines of multiples of \(\theta\), and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 6 } \theta \mathrm {~d} \theta$$
OCR FP3 Specimen Q8
8
  1. Find the value of the constant \(k\) such that \(y = k x ^ { 2 } \mathrm { e } ^ { - 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 \mathrm { e } ^ { - 2 x }$$
  2. Find the solution of this differential equation for which \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
  3. Use the differential equation to determine the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 0\). Hence prove that \(0 < y \leqslant 1\) for \(x \geqslant 0\).
OCR C1 Q5
5
  1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7 .$$
  2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
  3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{c532661c-8a94-483a-a921-b35d5c0a0188-04_754_810_1053_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.
  1. Calculate the radius of the circle and the coordinates of the centre.
  2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
  3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).
OCR C1 2005 January Q1
1
  1. Express \(11 ^ { - 2 }\) as a fraction.
  2. Evaluate \(100 ^ { \frac { 3 } { 2 } }\).
  3. Express \(\sqrt { 50 } + \frac { 6 } { \sqrt { 3 } }\) in the form \(a \sqrt { } 2 + b \sqrt { } 3\), where \(a\) and \(b\) are integers.
OCR C1 2005 January Q2
2 Given that \(2 x ^ { 2 } - 12 x + p = q ( x - r ) ^ { 2 } + 10\) for all values of \(x\), find the constants \(p , q\) and \(r\).
OCR C1 2005 January Q3
3
  1. The curve \(y = 5 \sqrt { } x\) is transformed by a stretch, scale factor \(\frac { 1 } { 2 }\), parallel to the \(x\)-axis. Find the equation of the curve after it has been transformed.
  2. Describe the single transformation which transforms the curve \(y = 5 \sqrt { } x\) to the curve \(y = ( 5 \sqrt { } x ) - 3\).
OCR C1 2005 January Q4
4 Solve the simultaneous equations $$x ^ { 2 } - 3 y + 11 = 0 , \quad 2 x - y + 1 = 0$$
OCR C1 2005 January Q5
5 On separate diagrams,
  1. sketch the curve \(y = \frac { 1 } { x }\),
  2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
  3. sketch the curve \(y = - \sqrt { } x\).
OCR C1 2005 January Q6
6
  1. Calculate the discriminant of \(- 2 x ^ { 2 } + 7 x + 3\) and hence state the number of real roots of the equation \(- 2 x ^ { 2 } + 7 x + 3 = 0\).
  2. The quadratic equation \(2 x ^ { 2 } + ( p + 1 ) x + 8 = 0\) has equal roots. Find the possible values of \(p\).
OCR C1 2005 January Q7
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = \frac { 1 } { 2 } x ^ { 4 } - 3 x\),
  2. \(y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )\),
  3. \(y = \sqrt [ 5 ] { x }\).
OCR C1 2005 January Q8
8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is \(x\) metres.
  1. The perimeter of the playground is greater than 64 m . Write down a linear inequality in \(x\).
  2. The area of the playground is less than \(299 \mathrm {~m} ^ { 2 }\). Show that \(( x - 13 ) ( x + 23 ) < 0\).
  3. By solving the inequalities in parts (i) and (ii), determine the set of possible values of \(x\).
OCR C1 2005 January Q9
9
  1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
  2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
  3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
  4. Deduce the value of \(k\).
OCR C1 2005 January Q10
10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.
  1. Calculate the gradient of \(D E\).
  2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
  3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
  4. Calculate the length of \(D F\).
  5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).
OCR C1 2006 January Q1
1 Solve the equations
  1. \(x ^ { \frac { 1 } { 3 } } = 2\),
  2. \(10 ^ { \prime } = 1\),
  3. \(\left( y ^ { - 2 } \right) ^ { 2 } = \frac { 1 } { 81 }\).
OCR C1 2006 January Q2
2
  1. Simplify \(( 3 x + 1 ) ^ { 2 } - 2 ( 2 x - 3 ) ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 3 \right) \left( x ^ { 2 } - 2 x + 1 \right)$$