PhysicsAndMathsTutor.com
physicsandmathstutor.com June 2005 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (b) Solve, for 0 θ < 360, the equation 2 tan 2 θ + secθ = 1, giving your answers to 1 decimal place. (2) (6) 2 *n23494b0220*
physicsandmathstutor.com June 2005 5. (a) Using the identity cos(a + B) cosa cosb sin A sinb, prove that cos 2A 1 2 sin 2 A. (2) (b) Show that 2 sin 2θ 3 cos 2θ 3 sin θ + 3 sin θ (4 cos θ + 6 sin θ 3). (4) 1 (c) Express 4 cos θ + 6 sin θ in the form R sin(θ + α), where R > 0 and 0 < α < 2 π. (4) (d) Hence, for 0 θ < π, solve 2 sin 2θ = 3(cos 2θ + sin θ 1), giving your answers in radians to 3 significant figures, where appropriate. (5) 10 *n23494b01020*
physicsandmathstutor.com June 2005 Question 5 continued *n23494b01120* 11 Turn over
physicsandmathstutor.com January 2006 6. f(x) = 12 cos x 4 sin x. Given that f(x) = R cos(x + α), where R 0 and 0 α 90, (a) find the value of R and the value of α. (4) (b) Hence solve the equation 12 cos x 4 sin x = 7 for 0 x < 360, giving your answers to one decimal place. (c) (i) Write down the minimum value of 12 cos x 4 sin x. (5) (1) (ii) Find, to 2 decimal places, the smallest positive value of x for which this minimum value occurs. (2) 12 *N23495A01220*
physicsandmathstutor.com January 2006 Question 6 continued Q6 (Total 12 marks) *N23495A01320* 13 Turn over
physicsandmathstutor.com January 2006 7. (a) Show that cos 2x 1 (i) cos x sin x, x (n 4 )π, n Z, cos x+ sin x 1 (ii) (cos 2x sin 2x) cos 2 1 x cos x sin x. 2 (b) Hence, or otherwise, show that the equation cos 2θ 1 cosθ = cosθ + sinθ 2 can be written as sin 2θ = cos 2θ. (c) Solve, for 0 θ < 2π, sin 2θ = cos 2θ, giving your answers in terms of π. 2 (2) (3) (3) (4) 14 *N23495A01420*
physicsandmathstutor.com January 2006 Question 7 continued *N23495A01520* 15 Turn over
physicsandmathstutor.com June 2006 6. (a) Using sin 2 θ + cos 2 θ 1, show that cosec 2 θ cot 2 θ 1. (b) Hence, or otherwise, prove that cosec 4 θ cot 4 θ cosec 2 θ + cot 2 θ. (c) Solve, for 90 < θ < 180, cosec 4 θ cot 4 θ = 2 cot θ. (2) (2) (6) 14 *N23581A01424*
physicsandmathstutor.com June 2006 3 8. (a) Given that cos A =, where 270 < A < 360, find the exact value of sin 2A. 4 π π (b) (i) Show that cos 2x+ + cos 2x cos 2 x. 3 3 (5) (3) Given that (ii) show that π π y = x+ x+ + x 3 3 2 3sin cos 2 cos 2, dy sin 2 x. dx = (4) 22 *N23581A02224*
physicsandmathstutor.com June 2006 Question 8 continued Q8 (Total 12 marks) TOTAL FOR PAPER: 75 MARKS END 24 *N23581A02424*
physicsandmathstutor.com January 2007 1. (a) By writing sin 3θ as sin (2θ +θ, show that 3 sin3θ = 3sinθ 4sin θ. 3 (b) Given that sin θ =, find the exact value of sin3θ. 4 (5) (2) 2 *N23583A0224*
physicsandmathstutor.com January 2007 5. Figure 1 Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation y = cos x+ sin x. (a) Express the equation of the curve in the form y = Rsin(x + ), where R and are π constants, R 0and0 < <. 2 (4) (b) Find the values of x, 0 x < 2π, for which y = 1. (4) 12 *N23583A01224*
physicsandmathstutor.com January 2007 Question 5 continued Q5 (Total 8 marks) *N23583A01324* 13 Turn over
physicsandmathstutor.com January 2007 8. (i) Prove that 2 2 2 2 sec x cosec x tan x cot x. (3) (ii) Given that y = arccos x, 1x1 and 0 yπ, (a) express arcsin x in terms of y. (b) Hence evaluate arccos x + arcsin x. Give your answer in terms of π. (2) (1) 22 *N23583A02224*
physicsandmathstutor.com January 2007 Question 8 continued Q8 (Total 6 marks) TOTAL FOR PAPER: 75 MARKS END 24 *N23583A02424*
physicsandmathstutor.com June 2007 6. (a) Express 3 sin x + 2 cos x in the form R sin(x + α) where R > 0 and 0 < α < π 2. (4) (b) Hence find the greatest value of (3 sin x + 2 cos x) 4. (2) (c) Solve, for 0 < x < 2π, the equation 3 sin x + 2 cos x = 1, giving your answers to 3 decimal places. (5) 14 *N26109A01424*
physicsandmathstutor.com June 2007 Question 6 continued *N26109A01524* 15 Turn over
physicsandmathstutor.com June 2007 7. (a) Prove that sin θ cosθ + = 2 cosec 2 θ, θ 90 n. cosθ sin θ (4) (b) On the axes on page 20, sketch the graph of y = 2 cosec 2θ for 0 < θ < 360. (2) (c) Solve, for 0 < θ < 360, the equation sinθ cosθ + =3, cosθ sinθ giving your answers to 1 decimal place. (6) 18 *N26109A01824*
physicsandmathstutor.com June 2007 Question 7 continued y O 90 180 270 360 θ 20 *N26109A02024*
physicsandmathstutor.com January 2008 6. (a) Use the double angle formulae and the identity cos( A+ B) cos Acos B sin Asin B to obtain an expression for cos 3x in terms of powers of cos x only. (b) (i) Prove that (ii) Hence find, for 0 cos x 1 sin x + + π 2sec x, x ( 2n+ 1). 1 + sin x cos x 2 < x <2π, all the solutions of cos x 1 sin x + + = 4. 1+ sin x cos x (4) (4) (3) 14 *H26315RB01424*
physicsandmathstutor.com January 2008 Question 6 continued *H26315RB01524* 15 Turn over
2. physicsandmathstutor.com June 2008 f( x) = 5cosx+ 12sinx Given that f ( x) = Rcos( x α), where R > 0 and 0 α π, (a) find the value of R and the value of α to 3 decimal places. (4) (b) Hence solve the equation for 0 x < 2π. 5cos x+ 12sin x= 6 (5) (c) (i) Write down the maximum value of 5cos x+ 12sin x. (1) (ii) Find the smallest positive value of x for which this maximum value occurs. (2) 4 *N30745A0424*
physicsandmathstutor.com June 2008 Question 2 continued *N30745A0524* 5 Turn over
physicsandmathstutor.com June 2008 5. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + cot 2 θ cosec 2 θ. (b) Solve, for 0 θ < 180, the equation 2 cot 2 θ 9 cosec θ = 3, giving your answers to 1 decimal place. (2) (6) 16 *N30745A01624*
physicsandmathstutor.com January 2009 6. (a) (i) By writing 3θ = (2θ + θ), show that (ii) Hence, or otherwise, for Give your answers in terms of. sin 3θ = 3 sin θ 4 sin 3 θ. π 0 θ 3, solve 8 sin 3 θ 6 sin θ + 1 = 0. (4) (5) (b) Using sin( θ α) = sin θcosα cosθsin α, or otherwise, show that 1 sin15=. 4 (4) 16 *H31123A01628*
physicsandmathstutor.com January 2009 Question 6 continued *H31123A01728* 17 Turn over
physicsandmathstutor.com January 2009 8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ α), where R and α are constants, R > 0 and 0 < α < 90. (4) (b) Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs. (3) The temperature, f (t), of a warehouse is modelled using the equation f (t) = 10 + 3 cos(15t) + 4 sin(15t), where t is the time in hours from midday and 0 t < 24. (c) Calculate the minimum temperature of the warehouse as given by this model. (d) Find the value of t when this minimum temperature occurs. (2) (3) 24 *H31123A02428*
physicsandmathstutor.com January 2009 Question 8 continued Q8 END (Total 12 marks) TOTAL FOR PAPER: 75 MARKS *H31123A02728* 27
physicsandmathstutor.com June 2009 6. (a) Use the identity cos( A+ B) = cos Acos B sin Asin B, to show that 2 cos 2A = 1 2sin A (2) The curves C 1 and C 2 have equations C 1 : y = 3sin2x C 2 : y 2 = 4sin x 2 cos 2x (b) Show that the x-coordinates of the points where C 1 and C 2 intersect satisfy the equation 4cos2x + 3sin2x = 2 (3) (c) Express 4cos2x + 3sin2x in the form R cos (2x α), where R > 0 and 0 < α < 90, giving the value of α to 2 decimal places. (3) (d) Hence find, for 0 x < 180, all the solutions of 4cos2x + 3sin2x = 2 giving your answers to 1 decimal place. (4) 18 *H34264A01828*
physicsandmathstutor.com June 2009 Question 6 continued *H34264A01928* 19 Turn over
8. Solve physicsandmathstutor.com January 2010 cosec 2 2x cot 2x = 1 for 0 x 180. (7) 22 *N35381A02228*
physicsandmathstutor.com June 2010 1. (a) Show that sin 2θ = tanθ 1+ cos 2θ (2) (b) Hence find, for 180 θ< 180, all the solutions of 2 sin 2θ = 1 1+ cos 2θ Give your answers to 1 decimal place. (3) 2 *H35385A0228*
physicsandmathstutor.com June 2010 π 7. (a) Express 2 sin θ 1.5 cos θ in the form R sin (θ α), where R > 0 and 0 < α < 2. Give the value of α to 4 decimal places. (3) (b) (i) Find the maximum value of 2 sin θ 1.5 cos θ. (ii) Find the value of θ, for 0 θ<, at which this maximum occurs. (3) Tom models the height of sea water, H metres, on a particular day by the equation 4πt 4πt H = 6+ 2sin 1. 5cos 25 25, 0 t < 12, where t hours is the number of hours after midday. (c) Calculate the maximum value of H predicted by this model and the value of t, to 2 decimal places, when this maximum occurs. (3) (d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres. (6) 22 *H35385A02228*
physicsandmathstutor.com June 2010 Question 7 continued *H35385A02328* 23 Turn over
physicsandmathstutor.com January 2011 3. Find all the solutions of 2 cos 2 = 1 2 sin in the interval 0 360. (6) 8 *H35404RA0828*
physicsandmathstutor.com June 2011 6. (a) Prove that 1 cos 2θ = tan θ, θ 90n, n Z sin 2θ sin 2θ (4) (b) Hence, or otherwise, (i) show that tan 15 = 2 3, (3) (ii) solve, for 0 x 360, cosec 4x cot 4x = 1 (5) 12 *P38159A01224*
physicsandmathstutor.com June 2011 Question 6 continued *P38159A01324* 13 Turn over
physicsandmathstutor.com January 2012 5. Solve, for 0 180, 2cot 2 3 7 cosec 3 5 Give your answers in degrees to 1 decimal place. (10) 10 *P40084A01024*
physicsandmathstutor.com January 2012 ( )= + ( ) 6. f x x 2 3x 2 cos 1 x, 0 x 2 (a) Show that the equation f (x) = 0 has a solution in the interval 0.8 < x < 0.9 (2) The curve with equation y = f (x) has a minimum point P. (b) Show that the x-coordinate of P is the solution of the equation x x = 3 + ( 1 sin ) 2 2 (4) (c) Using the iteration formula x n xn + = + ( 1 3 sin ) 2 1, x0 = 2 2 find the values of x 1, x 2 and x 3, giving your answers to 3 decimal places. (3) (d) By choosing a suitable interval, show that the x-coordinate of P is 1.9078 correct to 4 decimal places. (3) 12 *P40084A01224*
physicsandmathstutor.com January 2012 Question 6 continued *P40084A01324* 13 Turn over
physicsandmathstutor.com January 2012 8. (a) Starting from the formulae for sin ( A + B ) and cos ( A + B ), prove that (b) Deduce that tan A+ tan B tan ( A+ B)= 1 tan Atan B π 1+ 3tanθ tan θ + = 6 3 tanθ (4) (3) (c) Hence, or otherwise, solve, for 0 θ π, tan ( tan ) tan ( ) 1+ 3 θ = 3 θ π θ Give your answers as multiples of (6) 20 *P40084A02024*
physicsandmathstutor.com January 2012 Question 8 continued Q8 (Total 13 marks) TOTAL FOR PAPER: 75 MARKS END 24 *P40084A02424*
physicsandmathstutor.com June 2012 2 5. (a) Express 4cosec 2 2θ cosec θ in terms of sin and cos. (2) (b) Hence show that 2 4cosec 2 2 2θ cosec θ = sec θ (c) Hence or otherwise solve, for 0 < <, giving your answers in terms of. 2 4cosec 2 2θ cosec θ = 4 (4) (3) 16 *P40686RA01632*
physicsandmathstutor.com June 2012 8. f( x) = 7 cos2x 24sin2x Given that f( x) = Rcos( 2x+ α ), where R 0 and 0< α < 90, (a) find the value of R and the value of. (3) (b) Hence solve the equation 7 cos 2x 24sin 2x= 12. 5 for 0 x180, giving your answers to 1 decimal place. 2 (c) Express 14cos x 48sin xcos x in the form acos 2x+ bsin 2x+ c, where a, b, and c are constants to be found. (5) (2) (d) Hence, using your answers to parts (a) and (c), deduce the maximum value of 2 14cos x 48sin xcos x (2) 28 *P40686RA02832*
physicsandmathstutor.com June 2012 Question 8 continued Q8 (Total 12 marks) TOTAL FOR PAPER: 75 MARKS END 32 *P40686RA03232*
physicsandmathstutor.com January 2013 4. (a) Express 6 cos+8sin in the form Rcos(), where R > 0 and 0 < α < π. 2 Give the value of to 3 decimal places. (b) p( ) = 4, 0θ 2π 12 + 6cos+ 8sin Calculate (i) the maximum value of p(), (4) (ii) the value of at which the maximum occurs. (4) 10 *P41486A01028*
physicsandmathstutor.com January 2013 6. (i) Without using a calculator, find the exact value of (sin 22.5 + cos 22.5 ) 2 You must show each stage of your working. (5) (ii) (a) Show that cos 2 + sin = 1 may be written in the form k sin 2 sin = 0, stating the value of k. (2) (b) Hence solve, for 0 < 360, the equation cos 2 + sin = 1 (4) 18 *P41486A01828*
physicsandmathstutor.com January 2013 Question 6 continued *P41486A01928* 19 Turn over
physicsandmathstutor.com June 2013 (R)
physicsandmathstutor.com June 2013 (R)
physicsandmathstutor.com June 2013 (R)
physicsandmathstutor.com June 2013 (R)
physicsandmathstutor.com June 2013 (R)
physicsandmathstutor.com June 2013 3. Given that (a) Show, without using a calculator, that 2cos(x + 50) = sin(x + 40) tan x = 1 3 tan (4) (b) Hence solve, for 0 < 360, 2cos(2 + 50) = sin(2 + 40) giving your answers to 1 decimal place. (4) 8 *P43016A0832*
physicsandmathstutor.com June 2013 8. B 3ms 1 V m s 1 7 m A 24 m Figure 2 Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s 1. Kate is 24 m ahead of John when she starts to cross the road from the fixed point A. John passes her as she reaches the other side of the road at a variable point B, as shown in Figure 2. Kate s speed is V ms 1 and she moves in a straight line, which makes an angle, 0 < < 150, with the edge of the road, as shown in Figure 2. You may assume that V is given by the formula V = 21, 0 < < 150 24sin θ + 7cosθ (a) Express 24sin + 7cos in the form Rcos( ), where R and are constants and where R > 0 and 0 < < 90, giving the value of to 2 decimal places. (3) Given that varies, (b) find the minimum value of V. (2) Given that Kate s speed has the value found in part (b), (c) find the distance AB. (3) Given instead that Kate s speed is 1.68 m s 1, (d) find the two possible values of the angle, given that 0 < < 150. (6) 28 *P43016A02832*
physicsandmathstutor.com June 2013 Question 8 continued Q8 (Total 14 marks) END TOTAL FOR PAPER: 75 MARKS 32 *P43016A03232*
Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2. Logarithms and exponentials e x ln a = a x Trigonometric identities sin ( A ± B ) = sin A cos B ± cos A sin B cos( A ± B ) = cos A cos B m sin A sin B tan A ± tan B tan ( A ± B ) = ( A ± B ( k + 2) 1m tan A tan B A + B A B sin A + sin B = 2 sin cos 2 2 A + B A B sin A sin B = 2cos sin 2 2 A + B A B cos A + cos B = 2 cos cos 2 2 A + B A B cos A cos B = 2 sin sin 2 2 1 π ) Differentiation f(x) tan kx sec x cot x cosec x f( x) g( x) f (x) k sec 2 kx sec x tan x cosec 2 x cosec x cot x f ( x )g( x ) f( x )g ( x ) (g( x)) 2 6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 Issue 1 September 2009
Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 Issue 1 September 2009 5 Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1. Cosine rule a 2 = b 2 + c 2 2bc cos A Binomial series 2 1 ) ( 2 2 1 n r r n n n n n b b a r n b a n b a n a b a + + + + + + = + K K (n N) where )!!(! C r n r n r n r n = = < + + + + + + = + n x x r r n n n x n n nx x r n 1, ( 2 1 1 ) ( 1 ) ( 2 1 1 ) ( 1 ) ( 1 2 K K K K R) Logarithms and exponentials a x x b b a log log log = Geometric series u n = ar n 1 S n = r r a n 1 ) ( 1 S = r a 1 for r < 1 Numerical integration The trapezium rule: b a x y d 21 h{(y 0 + y n ) + 2(y 1 + y 2 +... + y n 1 )}, where n a b h =
Core Mathematics C1 Mensuration Surface area of sphere = 4π r 2 Area of curved surface of cone = π r slant height Arithmetic series u n = a + (n 1)d S n = 2 1 n(a + l) = 2 1 n[2a + (n 1)d] 4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 September 2009