Questions — OCR C2 (296 questions)

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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 \(\log _ { 3 } y\).
OCR C2 Specimen Q4
4 Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher’s forecasts for
  1. the number of copies that will be sold in the 20th week after publication,
  2. the total number of copies sold during the first 20 weeks after publication,
  3. the total number of copies that will ever be sold.
OCR C2 Specimen Q5
5
  1. Show that the equation \(15 \cos ^ { 2 } \theta ^ { \circ } = 13 + \sin \theta ^ { \circ }\) may be written as a quadratic equation in \(\sin \theta ^ { \circ }\).
  2. Hence solve the equation, giving all values of \(\theta\) such that \(0 \leqslant \theta \leqslant 360\).
OCR C2 Specimen Q6
6 The diagram shows triangle \(A B C\), in which \(A B = 3 \mathrm {~cm} , A C = 5 \mathrm {~cm}\) and angle \(A B C = 2.1\) radians. Calculate
  1. angle \(A C B\), giving your answer in radians,
  2. the area of the triangle. An arc of a circle with centre \(A\) and radius 3 cm is drawn, cutting \(A C\) at the point \(D\).
  3. Calculate the perimeter and the area of the sector \(A B D\).
OCR C2 Specimen Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{73d67b39-3611-4afb-9470-2f813115abb5-3_460_709_1114_708} The diagram shows the curves \(y = - 3 x ^ { 2 } - 9 x + 30\) and \(y = x ^ { 2 } + 3 x - 10\).
  1. Verify that the curves intersect at the points \(A ( - 5,0 )\) and \(B ( 2,0 )\).
  2. Show that the area of the shaded region between the curves is given by \(\int _ { - 5 } ^ { 2 } \left( - 4 x ^ { 2 } - 12 x + 40 \right) \mathrm { d } x\).
  3. Hence or otherwise show that the area of the shaded region between the curves is \(228 \frac { 2 } { 3 }\).
OCR C2 Specimen Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{73d67b39-3611-4afb-9470-2f813115abb5-4_415_714_287_678} The diagram shows the curve \(y = 1.25 ^ { x }\).
  1. A point on the curve has \(y\)-coordinate 2. Calculate its \(x\)-coordinate.
  2. Use the trapezium rule with 4 intervals to estimate the area of the shaded region, bounded by the curve, the axes, and the line \(x = 4\).
  3. State, with a reason, whether the estimate found in part (ii) is an overestimate or an underestimate.
  4. Explain briefly how the trapezium rule could be used to find a more accurate estimate of the area of the shaded region.
OCR C2 Specimen Q9
9 The cubic polynomial \(x ^ { 3 } + a x ^ { 2 } + b x - 6\) is denoted by \(\mathrm { f } ( x )\).
  1. The remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\). Show that \(b = - 4\).
  2. Given also that ( \(x - 1\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  3. With these values of \(a\) and \(b\), express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
  4. Hence determine the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), explaining your reasoning.
OCR C2 Q2
2. \(f ( x ) = x ^ { 3 } + k x - 20\). Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),
  1. find the value of the constant \(k\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\).
OCR C2 Q3
3. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { x } - x ^ { 2 }$$ and that \(y = \frac { 2 } { 3 }\) when \(x = 1\), find the value of \(y\) when \(x = 4\).
OCR C2 Q4
4. A geometric progression has third term 36 and fourth term 27. Find
  1. the common ratio,
  2. the fifth term,
  3. the sum to infinity.
OCR C2 Q5
5. (i) Solve the equation $$\log _ { 2 } ( 6 - x ) = 3 - \log _ { 2 } x$$ (ii) Find the smallest integer \(n\) such that $$3 ^ { n - 2 } > 8 ^ { 250 }$$
OCR C2 Q6
  1. \(f ( x ) = \cos 2 x , 0 \leq x \leq \pi\).
    1. Sketch the curve \(y = \mathrm { f } ( x )\).
    2. Write down the coordinates of any points where the curve \(y = \mathrm { f } ( x )\) meets the coordinate axes.
    3. Solve the equation \(\mathrm { f } ( x ) = 0.5\), giving your answers in terms of \(\pi\).
    4. (i) Find
    $$\int \left( x + 5 + \frac { 3 } { \sqrt { x } } \right) \mathrm { d } x$$
  2. Evaluate $$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } d x$$