1.01a Proof: structure of mathematical proof and logical steps

5

1. Prove that the following statement is not true. \(m\) is an odd number greater than \(1 \Rightarrow m ^ { 2 } + 4\) is prime.
2. By considering separately the case when \(n\) is odd and the case when \(n\) is even, prove that the following statement is true. \(n\) is a positive integer \(\Rightarrow n ^ { 2 } + 1\) is not a multiple of 4 .

8

1. Prove that the following statement is not true. $$p \text { is a positive integer } \Rightarrow 2 ^ { p } \geqslant p ^ { 2 }$$
2. Prove that the following statement is true. \(m\) and \(n\) are consecutive positive odd numbers \(\Rightarrow m n + 1\) is the square of an even number

3 Prove that, when \(n\) is an even number, \(n ^ { 3 } + 4\) is a multiple of 4 but not a multiple of 8 .

3 Without using a calculator, prove that \(3 \sqrt { 2 } > 2 \sqrt { 3 }\).

15 The circle \(x ^ { 2 } + y ^ { 2 } + 2 x - 14 y + 25 = 0\) has its centre at the point \(C\). The line \(7 y = x + 25\) intersects the circle at points A and B . Prove that triangle ABC is a right-angled triangle.

12 Lines 5 and 6 outline the stages in a proof that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\). Starting from \(( a - b ) ^ { 2 } \geqslant 0\), give a detailed proof of the inequality of arithmetic and geometric means.

3 An infinite sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by \(a _ { \mathrm { n } } = \frac { \mathrm { n } } { \mathrm { n } + 1 }\), for all positive integers \(n\).

1. Find the limit of the sequence.
2. Prove that this is an increasing sequence.

12 With the aid of a suitable diagram, show that the three triangles referred to in line 26 have the areas given in line 27 .

15 The expression given in line 34 is used to calculate \(\sum _ { r = 1 } ^ { 6 } \frac { 1 } { r }\).
Show that the error in the result is less than \(1.5 \%\) of the true value.

13 Substitute appropriate values of \(t _ { 1 }\) and \(t _ { 2 }\) to verify that \(t _ { 1 } t _ { 2 }\) gives the correct value for the \(y\)-coordinate of the point of intersection of the tangents at the points A and B in Fig. \(\mathbf { C 1 . }\)

15

1. Show that, for the curve \(y = a x ^ { 2 } + b x + c\), the equation of the tangent at the point with \(x\)-coordinate \(t\) is \(\mathrm { y } = ( 2 \mathrm { at } + \mathrm { b } ) \mathrm { x } - \mathrm { at } ^ { 2 } + \mathrm { c }\).
2. Hence show that for the curve with equation \(y = a x ^ { 2 } + b x + c\), the tangents at two points, \(P\) and Q , on the curve cross at a point which has \(x\)-coordinate equal to the mean of the \(x\)-coordinates of points P and Q , as given in lines 11 to 14 .

6. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011.

9. (a) Prove that the sum of the first \(n\) terms of an arithmetic series with first term \(a\) and common difference \(d\) is given by $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A novelist begins writing a new book. She plans to write 16 pages during the first week, 18 during the second and so on, with the number of pages increasing by 2 each week. Find, according to her plan,
(b) how many pages she will write in the fifth week,
(c) the total number of pages she will write in the first five weeks.
(d) Using algebra, find how long it will take her to write the book if it has 250 pages.

8 Prove that, for all values of \(x\), the value of the expression $$( 3 \sin x + \cos x ) ^ { 2 } + ( \sin x - 3 \cos x ) ^ { 2 }$$ is an integer and state its value.

7

1. Explain why \(n ( n + 1 )\) is a multiple of 2 when \(n\) is an integer.
2. 1. Given that $$\mathrm { f } ( n ) = n \left( n ^ { 2 } + 5 \right)$$ show that \(\mathrm { f } ( k + 1 ) - \mathrm { f } ( k )\), where \(k\) is a positive integer, is a multiple of 6 .
   2. Prove by induction that \(\mathrm { f } ( n )\) is a multiple of 6 for all integers \(n \geqslant 1\).

6

1. Show that \(\frac { 1 } { ( k + 2 ) ! } - \frac { k + 1 } { ( k + 3 ) ! } = \frac { 2 } { ( k + 3 ) ! }\).
2. Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { r \times 2 ^ { r } } { ( r + 2 ) ! } = 1 - \frac { 2 ^ { n + 1 } } { ( n + 2 ) ! }$$ (6 marks)
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1. The normal to the parabola \(y ^ { 2 } = 4 a x\) at the point \(P \left( a p ^ { 2 } , 2 a p \right)\) passes through the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\).

The line \(O P\) is perpendicular to the line \(O Q\), where \(O\) is the origin.
Prove that \(p ^ { 2 } = 2\)

1. \(P\) and \(Q\) are two distinct points on the ellipse described by the equation \(x ^ { 2 } + 4 y ^ { 2 } = 4\)

The line \(l\) passes through the point \(P\) and the point \(Q\).
The tangent to the ellipse at \(P\) and the tangent to the ellipse at \(Q\) intersect at the point \(( r , s )\).
Show that an equation of the line \(l\) is $$4 s y + r x = 4$$

11 In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).

4 Prove that the sum of the squares of any two consecutive integers is of the form \(4 k + 1\), where \(k\) is an integer.

12

1. By first writing \(\tan 3 \theta\) as \(\tan ( 2 \theta + \theta )\), show that \(\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }\).
2. Hence show that there are always exactly two different values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) which satisfy the equation \(3 \tan 3 \theta = \tan \theta + k\),
   where \(k\) is a non-zero constant. \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

4

1. Find $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 6 r \right)$$ expressing your answer in the form $$k n ( n + 1 ) ( n + p ) ( n + q )$$ where \(k\) is a fraction and \(p\) and \(q\) are integers.
2. It is given that $$S = \sum _ { r = 1 } ^ { 1000 } \left( r ^ { 3 } - 6 r \right)$$ Without calculating the value of \(S\), show that \(S\) is a multiple of 2008 .

7 It is given that any integer can be expressed in the form \(3 m + r\), where \(m\) is an integer and \(r\) is 0,1 or 2 . Use this fact to answer the following.

1. By considering the different values of \(r\), prove that the square of any integer cannot be expressed in the form \(3 n + 2\), where \(n\) is an integer.
2. Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different. By considering the different values of \(r\), determine the probability that the sum of these three integers is divisible by 3 .

4 Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$

9

1. Express \(n ^ { 3 } - n\) as a product of three factors. 9
2. Given that \(n\) is a positive integer, prove that \(n ^ { 3 } - n\) is a multiple of 6