Questions C2 (1410 questions)

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OCR C2 2005 June Q5
5 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and that the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) is 16 .
  1. Find the values of \(a\) and \(b\).
  2. Hence verify that \(\mathrm { f } ( 2 ) = 0\), and factorise \(\mathrm { f } ( x )\) completely.
OCR C2 2005 June Q6
6
  1. Find the binomial expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 } \mathrm {~d} x\).
OCR C2 2005 June Q7
7
  1. Evaluate \(\log _ { 5 } 15 + \log _ { 5 } 20 - \log _ { 5 } 12\).
  2. Given that \(y = 3 \times 10 ^ { 2 x }\), show that \(x = a \log _ { 10 } ( b y )\), where the values of the constants \(a\) and \(b\) are to be found.
OCR C2 2005 June Q8
8 The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9 . The amount pumped in 2001 was 100000 barrels.
  1. Calculate the amount pumped in 2004. It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.
  2. Calculate in which year the amount pumped will fall below 5000 barrels.
  3. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production.
OCR C2 2005 June Q9
9
    1. Write down the exact values of \(\cos \frac { 1 } { 6 } \pi\) and \(\tan \frac { 1 } { 3 } \pi\) (where the angles are in radians). Hence verify that \(x = \frac { 1 } { 6 } \pi\) is a solution of the equation $$2 \cos x = \tan 2 x$$
    2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2 x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2 x$$
    1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.)
    2. State with a reason whether this approximation is an underestimate or an overestimate.
OCR C2 2006 June Q1
1 Find the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).
OCR C2 2006 June Q2
2 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = 1 - u _ { n } \text { for } n \geqslant 1 .$$
  1. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Find \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).
OCR C2 2006 June Q3
3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }\), and the curve passes through the point (4,5). Find the equation of the curve.
OCR C2 2006 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-2_634_670_1123_740} The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
  2. Use integration to find the area of the shaded region bounded by the line and the curve.
OCR C2 2006 June Q5
5 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
  2. \(\sin 2 x = - \cos 2 x\).
OCR C2 2006 June Q6
6
  1. John aims to pay a certain amount of money each month into a pension fund. He plans to pay \(\pounds 100\) in the first month, and then to increase the amount paid by \(\pounds 5\) each month, i.e. paying \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, etc. If John continues making payments according to this plan for 240 months, calculate
    (a) how much he will pay in the final month,
    (b) how much he will pay altogether over the whole period.
  2. Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with \(\pounds 100\) in the first month. Her monthly payments will form a geometric progression, and she will pay \(\pounds 1500\) in the final month. Calculate how much Rachel will pay altogether over the whole period.
OCR C2 2006 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-3_476_1018_1000_566} The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the arc \(D C\).
  1. Show that the length of \(A C\) is 7.90 cm , correct to 3 significant figures.
  2. Find the area of the shaded segment.
  3. Find the perimeter of the shaded segment.
OCR C2 2006 June Q8
8 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 10\) is denoted by \(\mathrm { f } ( x )\). It is given that, when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 12 . It is also given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. Divide \(\mathrm { f } ( x )\) by ( \(x + 2\) ) to find the quotient and the remainder.
OCR C2 2006 June Q9
9
  1. Sketch the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), and state the coordinates of any point where the curve crosses an axis.
  2. Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), the axes, and the line \(x = 2\).
  3. The point \(P\) on the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) has \(y\)-coordinate equal to \(\frac { 1 } { 6 }\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$
OCR C2 2007 June Q1
1 A geometric progression \(\mathrm { u } _ { 1 } , \mathrm { u } _ { 2 } , \mathrm { u } _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 .$$
  1. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
OCR C2 2007 June Q2
2 Expand \(\left( x + \frac { 2 } { x } \right) ^ { 4 }\) completely, simplifying the terms.
OCR C2 2007 June Q3
3 U se logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures.
OCR C2 2007 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{e429080f-8634-46bc-b451-7b13b871e518-2_543_857_1155_644} The diagram shows the curve \(\mathrm { y } = \sqrt { 4 \mathrm { X } + 1 }\).
  1. Use the trapezium rule, with strips of width 0.5 , to find an approximate value for the area of the region bounded by the curve \(y = \sqrt { 4 x + 1 }\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 3\). Give your answer correct to 3 significant figures.
  2. State with a reason whether this approximation is an under-estimate or an over-estimate.
OCR C2 2007 June Q5
5
  1. Show that the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1$$ can be expressed in the form $$3 \sin ^ { 2 } \theta + \sin \theta - 2 = 0$$
  2. Hence solve the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1 ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2007 June Q6
6
    1. Find \(\int x \left( x ^ { 2 } - 4 \right) d x\)
    2. Hence evaluate \(\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x\).
  2. Find \(\int \frac { 6 } { x ^ { 3 } } d x\)
OCR C2 2007 June Q7
7
  1. In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
  2. In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio.
OCR C2 2007 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{e429080f-8634-46bc-b451-7b13b871e518-3_300_744_1046_703} The diagram shows a triangle \(A B C\), where angle \(B A C\) is 0.9 radians. \(B A D\) is a sector of the circle with centre A and radius AB .
  1. The area of the sector \(B A D\) is \(16.2 \mathrm {~cm} ^ { 2 }\). Show that the length of \(A B\) is 6 cm .
  2. The area of triangle \(A B C\) is twice the area of sector \(B A D\). Find the length of \(A C\).
  3. Find the perimeter of the region \(B C D\).
OCR C2 2007 June Q9
9 The polynomial \(f ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4 .$$
  1. (a) Show that ( \(\mathrm { x } + 1\) ) is a factor of \(\mathrm { f } ( \mathrm { x } )\).
    (b) Hence find the exact roots of the equation \(f ( x ) = 0\).
  2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(f ( x ) = 0\).
    (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.
OCR C2 Specimen Q1
1 Expand \(( 1 - 2 x ) ^ { 4 }\) in ascending powers of \(x\), simplifying the coefficients.
  1. Find \(\int \frac { 1 } { x ^ { 2 } } \mathrm {~d} x\).
  2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } }\). Find the equation of the curve, given that it passes through the point \(( 1,3 )\).
OCR C2 Specimen Q3
3
  1. Express each of the following in terms of \(\log _ { 2 } x\) :
    1. \(\log _ { 2 } \left( x ^ { 2 } \right)\),
    2. \(\log _ { 2 } \left( 8 x ^ { 2 } \right)\).
  2. Given that \(y ^ { 2 } = 27\), find the value of \(\log _ { 3 } y\).