Questions C3 (1200 questions)

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OCR MEI C3 Q6
6
  1. Multiply out \(\left( 3 ^ { n } + 1 \right) \left( 3 ^ { n } - 1 \right)\).
  2. Hence prove that if \(n\) is a positive integer then \(3 ^ { 2 n } - 1\) is divisible by 8 .
OCR MEI C3 Q7
7 State whether the following statements are true or false; if false, provide a counterexample.
  1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
  2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
  3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational.
OCR MEI C3 Q8
8
  1. Disprove the following statement.
    'If \(p > q\), then \(\frac { 1 } { p } < \frac { 1 } { q }\).
  2. State a condition on \(p\) and \(q\) so that the statement is true.
  3. Show that
OCR MEI C3 Q11
11 Use the method of exhaustion to prove the following result.
No 1 - or 2 -digit perfect square ends in \(2,3,7\) or 8
State a generalisation of this result.
OCR MEI C3 Q12
12 Prove that the following statement is false.
For all integers \(n\) greater than or equal to \(1 , n ^ { 2 } + 3 n + 1\) is a prime number.
OCR MEI C3 Q13
13 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
  1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
  2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).
OCR C3 2009 January Q1
1 Find
  1. \(\int 8 \mathrm { e } ^ { - 2 x } \mathrm {~d} x\),
  2. \(\int ( 4 x + 5 ) ^ { 6 } \mathrm {~d} x\).
OCR C3 2009 January Q2
2
  1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 4 } ^ { 12 } \ln x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Deduce an approximation to \(\int _ { 4 } ^ { 12 } \ln \left( x ^ { 10 } \right) \mathrm { d } x\).
OCR C3 2009 January Q3
3
  1. Express \(2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta }\) in terms of \(\sec \theta\).
  2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta } = 4$$
OCR C3 2009 January Q4
4 For each of the following curves, find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the exact \(x\)-coordinate of the stationary point:
  1. \(y = \left( 4 x ^ { 2 } + 1 \right) ^ { 5 }\),
  2. \(y = \frac { x ^ { 2 } } { \ln x }\).
OCR C3 2009 January Q5
5 The mass, \(M\) grams, of a certain substance is increasing exponentially so that, at time \(t\) hours, the mass is given by $$M = 40 \mathrm { e } ^ { k t }$$ where \(k\) is a constant. The following table shows certain values of \(t\) and \(M\).
\(t\)02163
\(M\)80
  1. In either order,
    (a) find the values missing from the table,
    (b) determine the value of \(k\).
  2. Find the rate at which the mass is increasing when \(t = 21\).
OCR C3 2009 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_627_689_264_726} The function f is defined for all real values of \(x\) by $$f ( x ) = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$ The graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) meet at the point \(P\), and the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) meets the \(x\)-axis at \(Q\) (see diagram).
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and determine the \(x\)-coordinate of the point \(Q\).
  2. State how the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related geometrically, and hence show that the \(x\)-coordinate of the point \(P\) is the root of the equation $$x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$
  3. Use an iterative process, based on the equation \(x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }\), to find the \(x\)-coordinate of \(P\), giving your answer correct to 2 decimal places.
OCR C3 2009 January Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_419_700_1809_721} The diagram shows the curve \(y = \mathrm { e } ^ { k x } - a\), where \(k\) and \(a\) are constants.
  1. Give details of the pair of transformations which transforms the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { k x } - a\).
  2. Sketch the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\).
  3. Given that the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\) passes through the points \(( 0,13 )\) and \(( \ln 3,13 )\), find the values of \(k\) and \(a\).
OCR C3 2009 January Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-4_538_702_264_719} The diagram shows the curve with equation $$y = \frac { 6 } { \sqrt { x } } - 3$$ The point \(P\) has coordinates \(( 0 , p )\). The shaded region is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = p\). The shaded region is rotated completely about the \(y\)-axis to form a solid of volume \(V\).
  1. Show that \(V = 16 \pi \left( 1 - \frac { 27 } { ( p + 3 ) ^ { 3 } } \right)\).
  2. It is given that \(P\) is moving along the \(y\)-axis in such a way that, at time \(t\), the variables \(p\) and \(t\) are related by $$\frac { \mathrm { d } p } { \mathrm {~d} t } = \frac { 1 } { 3 } p + 1 .$$ Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} t }\) at the instant when \(p = 9\).
OCR C3 2009 January Q9
9
  1. By first expanding \(\cos ( 2 \theta + \theta )\), prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$
  2. Hence prove that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
  3. Show that the only solutions of the equation $$1 + \cos 6 \theta = 18 \cos ^ { 2 } \theta$$ are odd multiples of \(90 ^ { \circ }\).
OCR C3 2010 January Q1
1 Find \(\int \frac { 10 } { ( 2 x - 7 ) ^ { 2 } } \mathrm {~d} x\).
OCR C3 2010 January Q2
2 The angle \(\theta\) is such that \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
  1. Given that \(\theta\) satisfies the equation \(6 \sin 2 \theta = 5 \cos \theta\), find the exact value of \(\sin \theta\).
  2. Given instead that \(\theta\) satisfies the equation \(8 \cos \theta \operatorname { cosec } ^ { 2 } \theta = 3\), find the exact value of \(\cos \theta\).
OCR C3 2010 January Q3
3
  1. Find, in simplified form, the exact value of \(\int _ { 10 } ^ { 20 } \frac { 60 } { x } \mathrm {~d} x\).
  2. Use Simpson's rule with two strips to find an approximation to \(\int _ { 10 } ^ { 20 } \frac { 60 } { x } \mathrm {~d} x\).
  3. Use your answers to parts (i) and (ii) to show that \(\ln 2 \approx \frac { 25 } { 36 }\).
OCR C3 2010 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{18ad8019-1dc8-44bb-8a25-eaf5e05465ab-2_444_1249_1233_447} The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 - \sqrt [ 3 ] { x + 1 }$$ The diagram shows the graph of \(y = \mathrm { f } ( x )\).
  1. Evaluate \(\mathrm { ff } ( - 126 )\).
  2. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) = | \mathrm { f } ( x ) |\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  4. State how the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related geometrically. The equation of a curve is \(y = \left( x ^ { 2 } + 1 \right) ^ { 8 }\).
OCR C3 2010 January Q6
6 Given that $$\int _ { 0 } ^ { \ln 4 } \left( k \mathrm { e } ^ { 3 x } + ( k - 2 ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x = 185$$ find the value of the constant \(k\).
  1. Leaking oil is forming a circular patch on the surface of the sea. The area of the patch is increasing at a rate of 250 square metres per hour. Find the rate at which the radius of the patch is increasing at the instant when the area of the patch is 1900 square metres. Give your answer correct to 2 significant figures.
  2. The mass of a substance is decreasing exponentially. Its mass now is 150 grams and its mass, \(m\) grams, at a time \(t\) years from now is given by $$m = 150 \mathrm { e } ^ { - k t } ,$$ where \(k\) is a positive constant. Find, in terms of \(k\), the number of years from now at which the mass will be decreasing at a rate of 3 grams per year.
    1. The curve \(y = \sqrt { x }\) can be transformed to the curve \(y = \sqrt { 2 x + 3 }\) by means of a stretch parallel to the \(y\)-axis followed by a translation. State the scale factor of the stretch and give details of the translation.
    2. It is given that \(N\) is a positive integer. By sketching on a single diagram the graphs of \(y = \sqrt { 2 x + 3 }\) and \(y = \frac { N } { x ^ { 3 } }\), show that the equation $$\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }$$ has exactly one real root.
    3. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) has the property that $$x _ { n + 1 } = N ^ { \frac { 1 } { 3 } } \left( 2 x _ { n } + 3 \right) ^ { - \frac { 1 } { 6 } }$$ For certain values of \(x _ { 1 }\) and \(N\), it is given that the sequence converges to the root of the equation \(\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }\).
OCR C3 2010 January Q9
9 The value of \(\tan 10 ^ { \circ }\) is denoted by \(p\). Find, in terms of \(p\), the value of
  1. \(\tan 55 ^ { \circ }\),
  2. \(\tan 5 ^ { \circ }\),
  3. \(\tan \theta\), where \(\theta\) satisfies the equation \(3 \sin \left( \theta + 10 ^ { \circ } \right) = 7 \cos \left( \theta - 10 ^ { \circ } \right)\).
OCR C3 2011 January Q1
1 Solve the equation \(| 3 x + 4 a | = 5 a\), where \(a\) is a positive constant.
OCR C3 2011 January Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_538_1061_388_541} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ( - 7 ) = 0\) and that there are stationary points at \(( - 2 , - 6 )\) and \(( 0,0 )\). Sketch the curve with equation \(y = - 4 \mathrm { f } ( x + 3 )\), indicating the coordinates of the stationary points.
OCR C3 2011 January Q3
3 A giant spherical balloon is being inflated in a theme park. The radius of the balloon is increasing at a rate of 12 cm per hour. Find the rate at which the surface area of the balloon is increasing at the instant when the radius is 150 cm . Give your answer in \(\mathrm { cm } ^ { 2 }\) per hour correct to 2 significant figures.
[0pt] [Surface area of sphere \(= 4 \pi r ^ { 2 }\).]
OCR C3 2011 January Q4
4
  1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).