Legacy C1 — Core Mathematics 1: Edexcel A‑Level Maths questions

Edexcel A-Level Maths past-paper questions on L.C1 Legacy C1 — Core Mathematics 1, 2010–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 158 questions on Legacy C1 — Core Mathematics 1, as a list

June 2018, Legacy C1, Q1 (5 marks): (i) Simplify 48 6 3 − Write your answer in the form a 3, where a is an integer to be found. (2) (ii) Solve the equation 36 x – 3 = 81 Write your answer as a rat
June 2018, Legacy C1, Q2 (7 marks): Given y x x x = − + 3 6 4, > 0 (a) find y x d ∫ , simplifying each term. (3) (b) (i) Find d d y x (ii) Hence find the value of x such that d d y x = 0 (4)
June 2018, Legacy C1, Q3 (6 marks): f ( x ) = x 2 – 10x + 23 (a) Express f ( x ) in the form ( x + a )2 + b, where a and b are constants to be found. (2) (b) Hence, or otherwise, find the exact so
June 2018, Legacy C1, Q4 (7 marks): Each year, Andy pays into a savings scheme. In year one he pays in £600. His payments increase by £120 each year so that he pays £720 in year two, £840 in year
June 2018, Legacy C1, Q5 (5 marks): y (2, 7) (0, 4) O (5, 0) x y = 1 Figure 1 Figure 1 shows the sketch of a curve with equation y = f ( x ), x ∈ . The curve crosses the y‑axis at (0, 4) and cros
June 2018, Legacy C1, Q6 (7 marks): A sequence a1 , a2 , a3 ,… is defined by a1 = 4 an +1 = a a n n + 1 , n  1, n ∈  (a) Find the values of a2 , a3 and a4 Write your answers as simplified fracti
June 2018, Legacy C1, Q7 (8 marks): The equation 20x 2 = 4k x – 13k x 2 + 2, where k is a constant, has no real roots. (a) Show that k satisfies the inequality 2k 2 + 13k + 20 < 0 (4) (b) Find the
June 2018, Legacy C1, Q8 (8 marks): Not to scale y x O B A C D E (12, 5) l1 l2 Figure 2 Figure 2 shows the straight line l1 with equation 4 y = 5 x + 12 (a) State the gradient of l1 (1) The line l
June 2018, Legacy C1, Q9 (12 marks): The curve C has equation y = f ( x), where f ′ ( x) = ( x – 3)(3x + 5) Given that the point P (1, 20) lies on C, (a) find f ( x), simplifying each term. (5) (b)
June 2018, Legacy C1, Q10 (10 marks): y x O A B C Not to scale Figure 3 Figure 3 shows a sketch of part of the curve C with equation y x x x = + − 1 2 27 12 0 , > The point A lies on C and has coord
June 2017, Legacy C1, Q1 (4 marks): Find 2 1 4 5 5 3 x x x − − ⎛ ⎝⎜ ⎞ ⎠⎟ ∫ d giving each term in its simplest form. (4)
June 2017, Legacy C1, Q2 (5 marks): Given + 4 4, x y x = + x 0 find the value of d d y x when x = 8, writing your answer in the form a 2 , where a is a rational number. (5)
June 2017, Legacy C1, Q3 (6 marks): A sequence a1, a2, a3,… is defined by a1 = 1 a k a a n n n n + = + 1 1 1 ( ) , . where k is a positive constant. (a) Write down expressions for a2 and a3 in ter
June 2017, Legacy C1, Q4 (7 marks): A company, which is making 140 bicycles each week, plans to increase its production. The number of bicycles produced is to be increased by d each week, starting
June 2017, Legacy C1, Q5 (8 marks): f(x) = x2 – 8x + 19 (a) Express f(x) in the form (x + a)2 + b, where a and b are constants. (2) The curve C with equation y = f(x) crosses the y-axis at the poi
June 2017, Legacy C1, Q6 (6 marks): (a) Given y = 2x, show that 2௘x௘± ௘x) + 8 = 0 can be written in the form 2y௘2 – 17y + 8 = 0 (2) (b) Hence solve 2௘x+1± ௘x) + 8 = 0 (4)
June 2017, Legacy C1, Q7 (9 marks): The curve C has equation y = f(x), x 0, where Iމ ( ) = 30 + x x 6 5 2 − x Given that the point P(4, –8) lies on C, (a) find the equation of the tangent to C at
June 2017, Legacy C1, Q8 (8 marks): Not to scale y x l2 l1 S O T P Figure 1 The straight line l1, shown in Figure 1, has equation 5y = 4x + 10 The point P with x coordinate 5 lies on l1 The straig
June 2017, Legacy C1, Q9 (11 marks): (a) On separate axes sketch the graphs of (i) y = –3x + c, where c is a positive constant, (ii) y x = + 1 5 On each sketch show the coordinates of any point at
June 2017, Legacy C1, Q10 (11 marks): y x O Figure 2 Figure 2 shows a sketch of part of the curve y = f(x), x \, where f(x) = (2x – 5)2(x + 3) (a) Given that (i) the curve with equation y = f(x) –
June 2016, Legacy C1, Q1 (4 marks): Find 2 4 3 4x x x − + ⎛ ⎝⎜ ⎞ ⎠⎟ ∫ √ d giving each term in its simplest form. (4)
June 2016, Legacy C1, Q2 (2 marks): Express 93x + 1 in the form 3y, giving y in the form ax + b, where a and b are constants. (2)
June 2016, Legacy C1, Q3 (5 marks): (a) Simplify 50 18 − giving your answer in the form a 2, where a is an integer. (2) (b) Hence, or otherwise, simplify 12 3 50 18 − giving your answer in the for
June 2016, Legacy C1, Q4 (5 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point A at (–2, 4) and a minimum point B at (3, –8) and pa
June 2016, Legacy C1, Q5 (6 marks): Solve the simultaneous equations y + 4x + 1 = 0 y2 + 5x2 + 2x = 0 (6)
June 2016, Legacy C1, Q6 (6 marks): A sequence a1, a2, a3, … is defined by a1 = 4, an + 1 = 5 – kan, n . 1 where k is a constant. (a) Write down expressions for a2 and a3 in terms of k. (2) Find (
June 2016, Legacy C1, Q7 (6 marks): Given that y x x x x x = + + − 3 6 2 7 3 0 2 1 3 3 √ , find d d y x. Give each term in your answer in its simplified form. (6)
June 2016, Legacy C1, Q8 (8 marks): The straight line with equation y = 3x – 7 does not cross or touch the curve with equation y = 2px2 – 6px + 4p, where p is a constant. (a) Show that 4p2 – 20p +
June 2016, Legacy C1, Q9 (11 marks): On John’s 10th birthday he received the first of an annual birthday gift of money from his uncle. This first gift was £60 and on each subsequent birthday the gi
June 2016, Legacy C1, Q10 (12 marks): Figure 2 The points P (0, 2) and Q (3, 7) lie on the line l1, as shown in Figure 2. The line l2 is perpendicular to l1, passes through Q and crosses the x-axis
June 2016, Legacy C1, Q11 (10 marks): The curve C has equation y = 2x3 + kx2 + 5x + 6, where k is a constant. (a) Find d d y x (2) The point P, where x = –2, lies on C. The tangent to C at the point
June 2015, Legacy C1, Q1 (5 marks): Simplify (a) (2 2 √5) (1) (b) √2 √5 −3√2 2 giving your answer in the form a b + √, where a and b are integers. (4)
June 2015, Legacy C1, Q2 (7 marks): Solve the simultaneous equations y – 2x – 4 = 0 4x2 + y2 + 20x = 0 (7)
June 2015, Legacy C1, Q3 (6 marks): Given that y = 4x3 – 5 2x , x 0, find in their simplest form (a) d d y x (3) (b) y x d ∫ (3)
June 2015, Legacy C1, Q4 (8 marks): (i) A sequence U1, U2, U3, ... is defined by Un+2 = 2Un+1 – Un, n . 1 U1 = 4 and U2 = 4 Find the value of (a) U3 (1) (b) Un n=∑ 1 20 (2) (ii) Another sequence V
June 2015, Legacy C1, Q5 (7 marks): The equation (p – 1)x2 + 4x + (p – 5) = 0, where p is a constant has no real roots. (a) Show that p satisfies p2 – 6p + 1 0 (3) (b) Hence find the set of possib
June 2015, Legacy C1, Q6 (10 marks): The curve C has equation y x x x x = + − ≠ ( )( ) , 2 4 3 2 0 (a) Find d d y x in its simplest form. (5) (b) Find an equation of the tangent to C at the point w
June 2015, Legacy C1, Q7 (5 marks): Given that y = 2x, (a) express 4x in terms of y. (1) (b) Hence, or otherwise, solve 8(4x) – 9(2x) + 1 = 0 (4)
June 2015, Legacy C1, Q8 (10 marks): (a) Factorise completely 9x – 4x3 (3) (b) Sketch the curve C with equation y = 9x – 4x3 Show on your sketch the coordinates at which the curve meets the x-axis.
June 2015, Legacy C1, Q9 (7 marks): Jess started work 20 years ago. In year 1 her annual salary was £17000. Her annual salary increased by £1500 each year, so that her annual salary in year 2 was
June 2015, Legacy C1, Q10 (10 marks): A curve with equation y = f(x) passes through the point (4, 9). Given that f ′ √ √ ( ) , x x x x = − + 3 2 9 4 2 0 (a) find f(x), giving each term in its simple
June 2014, Legacy C1, Q1 (3 marks): Find ∫(8x3 + 4) dx giving each term in its simplest form. (3)
June 2014, Legacy C1, Q2 (4 marks): (a) Write down the value of 32 1 5 (1) (b) Simplify fully (32x5) −2 5 (3)
June 2014, Legacy C1, Q3 (7 marks): Find the set of values of x for which (a) 3x – 7 > 3 – x (2) (b) x2 – 9x - 36 (4) (c) both 3x – 7 > 3 – x and x2 – 9x - 36 (1)
June 2014, Legacy C1, Q4 (5 marks): Figure 1 Figure 1 shows a sketch of the curve C with equation y x = + 1 1, x 0 The curve C crosses the x-axis at the point A. (a) State the x coordinate of th
June 2014, Legacy C1, Q5 (5 marks): A sequence of numbers a1, a2, a3… is defined by an + 1 = 5an – 3, n . 1 Given that a2 = 7, (a) find the value of a1 (2) (b) Find the value of ar r =∑ 1 4 (3)
June 2014, Legacy C1, Q6 (5 marks): 6 (a) Write 80 in the form c, where c is a positive constant. (1) A rectangle R has a length of (1 + 5) cm and an area of 80 cm2. (b) Calculate the width o
June 2014, Legacy C1, Q7 (7 marks): Differentiate with respect to x, giving each answer in its simplest form. (a) (1 – 2x)2 (3) (b) x x x 5 2 6 2 + √ (4)
June 2014, Legacy C1, Q8 (9 marks): In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 com
June 2014, Legacy C1, Q9 (10 marks): Figure 2 The line l1, shown in Figure 2 has equation 2x + 3y = 26 The line l2 passes through the origin O and is perpendicular to l1 (a) Find an equation for th
June 2014, Legacy C1, Q10 (10 marks): A curve with equation y = f(x) passes through the point (4, 25). Given that f ƍ(x) = 3 8 x2 – 10x −1 2 + 1, x > 0 (a) find f(x), simplifying each term. (5) (b)
June 2014, Legacy C1, Q11 (10 marks): Given that f(x) = 2x2 + 8x + 3 (a) find the value of the discriminant of f(x). (2) (b) Express f(x) in the form p(x + q)2 + r where p, q and r are integers to b
June 2014, Legacy C1 (01R), Q1 (3 marks): Factorise fully 25x – 9x3 (3)
June 2014, Legacy C1 (01R), Q2 (4 marks): (a) Evaluate 81 3 2 (2) (b) Simplify fully x x 2 1 2 2 4 − ⎛ ⎝⎜ ⎞ ⎠⎟ (2)
June 2014, Legacy C1 (01R), Q3 (5 marks): A sequence a1, a2, a3,… is defined by an + 1 = 4an – 3, n . 1 a1 = k, where k is a positive integer. (a) Write down an expression for a2 in terms of k. (1) Give
June 2014, Legacy C1 (01R), Q4 (6 marks): Given that y x x = + 2 6 5 √ , x 0, find in their simplest form (a) d d y x (3) (b) y x d ∫ (3)
June 2014, Legacy C1 (01R), Q5 (4 marks): Solve the equation 10 8 6 2 + = x x √ √ Give your answer in the form a√b where a and b are integers. (4)
June 2014, Legacy C1 (01R), Q6 (9 marks): Figure 1 Figure 1 shows the plan of a garden. The marked angles are right angles. The six edges are straight lines. The lengths shown in the diagram are given i
June 2014, Legacy C1 (01R), Q7 (9 marks): Figure 2 Figure 2 shows a right angled triangle LMN. The points L and M have coordinates (–1, 2) and (7, –4) respectively. (a) Find an equation for the straight
June 2014, Legacy C1 (01R), Q8 (7 marks): d d y x x x x = + − 6 1 2 √, x 0 Given that y = 37 at x = 4, find y in terms of x, giving each term in its simplest form. (7)
June 2014, Legacy C1 (01R), Q9 (9 marks): The curve C has equation y x = + 1 3 8 2 The line L has equation y = 3x + k, where k is a positive constant. (a) Sketch C and L on separate diagrams, showing th
June 2014, Legacy C1 (01R), Q10 (8 marks): Xin has been given a 14 day training schedule by her coach. Xin will run for A minutes on day 1, where A is a constant. She will then increase her running time
June 2014, Legacy C1 (01R), Q11 (11 marks): Figure 3 A sketch of part of the curve C with equation y x x = − − 20 4 18 , x 0 is shown in Figure 3. Point A lies on C and has an x coordinate equal to 2 (a)
June 2013, Legacy C1, Q1 (4 marks): Simplify 7 5 5 1 + − giving your answer in the form a b + 5 , where a and b are integers. (4)
June 2013, Legacy C1, Q2 (4 marks): Find ∫ √ 10 4 3 4x x x − − ⎛ ⎝⎜ ⎞ ⎠⎟ x d giving each term in its simplest form. (4)
June 2013, Legacy C1, Q3 (5 marks): (a) Find the value of 8 5 3 (2) (b) Simplify fully 2 4 1 2 3 2 x x ⎛ ⎝ ⎞ ⎠ (3)
June 2013, Legacy C1, Q4 (7 marks): A sequence a1, a2, a3, ... is defined by a1 = 4 an +1 = k (an + 2), for n 1 where k is a constant. (a) Find an expression for a2 in terms of k. (1) Given that
June 2013, Legacy C1, Q5 (6 marks): Find the set of values of x for which (a) 2(3x + 4) > 1 – x (2) (b) 3x2 + 8x – 3 < 0 (4)
June 2013, Legacy C1, Q6 (7 marks): The straight line L1 passes through the points (–1, 3) and (11, 12). (a) Find an equation for L1 in the form ax + by + c = 0, where a, b and c are integers. (4)
June 2013, Legacy C1, Q7 (7 marks): A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each wee
June 2013, Legacy C1, Q8 (6 marks): Figure 1 Figure 1 shows a sketch of the curve with equation y = f(x) where f(x) = (x + 3)2 (x – 1), x . The curve crosses the x-axis at (1, 0), touches it at
June 2013, Legacy C1, Q9 (10 marks): f'( ) ( ) , x x x x = − ≠ 3 0 2 2 2 (a) Show that f'(x) = 9x–2 + A + Bx2, where A and B are constants to be found. (3) (b) Find f"(x). (2) Given that the point
June 2013, Legacy C1, Q10 (8 marks): Given the simultaneous equations 2x + y = 1 x2 – 4ky + 5k = 0 where k is a non zero constant, (a) show that x2 + 8kx + k = 0 (2) Given that x2 + 8kx + k = 0 has
June 2013, Legacy C1, Q11 (11 marks): Figure 2 Figure 2 shows a sketch of the curve H with equation y x = + 3 4 , x (a) Give the coordinates of the point where H crosses the x-axis. (1) (b) Giv
June 2013, Legacy C1 (01R), Q1 (4 marks): Given y=x°+4x+ 1, find the value of 2 when x =3 * (4)
June 2013, Legacy C1 (01R), Q2 (4 marks): Express ao 27 in the form kV3, where & is an integer. (4)
June 2013, Legacy C1 (01R), Q3 (4 marks): Find » 4 3x° —— | dx x giving each term in its simplest form. (4)
June 2013, Legacy C1 (01R), Q4 (5 marks): The line L, has equation 4x + 2y—3 =0 (a) Find the gradient of L,. (2) The line L, is perpendicular to L, and passes through the point (2, 5). (b) Find the equa
June 2013, Legacy C1 (01R), Q5 (5 marks): Solve (a) 2»=8 qd) (b) 2*x 4**!1=8 (4)
June 2013, Legacy C1 (01R), Q6 (9 marks): Asequence x,, x,,x,... is defined by x,=1 X.,=%,P-kx, nol where k is a constant, k #0 (a) Find an expression for x, in terms of k. qd) (b) Show that x, = 1 — 3k
June 2013, Legacy C1 (01R), Q7 (9 marks): Each year, Abbie pays into a savings scheme. In the first year she pays in £500. Her payments then increase by £200 each year so that she pays £700 in the secon
June 2013, Legacy C1 (01R), Q8 (8 marks): A rectangular room has a width of x m. The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m, (a) sho
June 2013, Legacy C1 (01R), Q9 (8 marks): y (0,4) y= fe) (-1, 0) O (2, 0) x Figure 1 Figure 1 shows a sketch of the curve C with equation y = f(x). The curve C passes through the point (—1, 0) and touch
June 2013, Legacy C1 (01R), Q10 (10 marks): A curve has equation y = f(x). The point P with coordinates (9, 0) lies on the curve. Given that f(x)=x ; x>0 (a) find f(x). (6) (b) Find the x-coordinates of t
June 2013, Legacy C1 (01R), Q11 (9 marks): y yHxt2 x2 + 4y? — 2x = 35 > A QO x <i Figure 2 The line y =x + 2 meets the curve x* + 4y? — 2x = 35 at the points A and B as shown in Figure 2. (a) Find the co
January 2013, Legacy C1, Q1 (3 marks): Factorise completely x x −4 3 (3)
January 2013, Legacy C1, Q2 (2 marks): Express 82 3 x+ in the form 2y , stating y in terms of x. (2)
January 2013, Legacy C1, Q3 (6 marks): (i) Express 5 8 1 2 −√ ( ) + √ ( ) in the form a b + √2, where a and b are integers. (3) (ii) Express √ + √ 80 30 5 in the form c √5, where c is an integer. (3)
January 2013, Legacy C1, Q4 (5 marks): A sequence u u u 1 2 3 , , , ... satisfies u u n n n + = − 1 2 1 1 , Given that u 2 9 = , (a) find the value of u 3 and the value of u 4 , (2) (b) evaluate u
January 2013, Legacy C1, Q5 (7 marks): The line l1 has equation y x = − + 2 3 The line l 2 is perpendicular to l1 and passes through the point (5, 6). (a) Find an equation for l 2 in the form ax by c
January 2013, Legacy C1, Q6 (12 marks): Figure 1 shows a sketch of the curve with equation y x x = ≠ 2 0 , The curve C has equation y x x = − ≠ 2 5 0 , , and the line l has equation y x = + 4 2 (a) Sk
January 2013, Legacy C1, Q7 (8 marks): Lewis played a game of space invaders. He scored points for each spaceship that he captured. Lewis scored 140 points for capturing his first spaceship. He score
January 2013, Legacy C1, Q8 (6 marks): d d y x x x x x = − + − ≠ 3 3 4 5 2 0 , Given that y = 7 at x = 1, find y in terms of x, giving each term in its simplest form. (6)
January 2013, Legacy C1, Q9 (7 marks): The equation k x x k + ( ) + + = 3 6 5 2 , where k is a constant, has two distinct real solutions for x. (a) Show that k satisfies k k 2 2 24 0 − − < (4) (b) He
January 2013, Legacy C1, Q10 (7 marks): 4 8 3 2 2 x x a x b c + + ≡ + ( ) + (a) Find the values of the constants a, b and c. (3) (b) On the axes on page 27, sketch the curve with equation y x x = + +
January 2013, Legacy C1, Q11 (12 marks): The curve C has equation y x x x = − √ + 2 8 5 0 , (a) Find d d y x , giving each term in its simplest form. (3) The point P on C has x-coordinate equal to 1
June 2012, Legacy C1, Q1 (4 marks): Find 6 2 5 2 2 x x x + + ⎛ ⎝⎜ ⎞ ⎠⎟ ∫ d giving each term in its simplest form. (4)
June 2012, Legacy C1, Q2 (4 marks): (a) Evaluate , giving your answer as an integer. (2) (b) Simplify fully 25 4 4 1 2 x ⎛ ⎝⎜ ⎞ ⎠⎟ − (2) 32 3 5 ( )
June 2012, Legacy C1, Q3 (5 marks): Show that can be written in the form a + b, where a and b are integers. (5) 2 12 8 √ √ ( ) −( )
June 2012, Legacy C1, Q4 (6 marks): (a) Find d d y x giving each term in its simplest form. (4) (b) Find d d 2 2 y x (2) y x x x = − + − 5 6 2 3 3 4 3
June 2012, Legacy C1, Q5 (7 marks): A sequence of numbers a1, a2, a3 ... is defined by a1 = 3 an +1 = 2an – c (n 1) where c is a constant. (a) Write down an expression, in terms of c, for a2 (1)
June 2012, Legacy C1, Q6 (10 marks): A boy saves some money over a period of 60 weeks. He saves 10p in week 1, 15p in week 2, 20p in week 3 and so on until week 60. His weekly savings form an arith
June 2012, Legacy C1, Q7 (8 marks): The point P (4, –1) lies on the curve C with equation y = f(x), x > 0, and f'( ) x x x = − + 1 2 6 3 √ (a) Find the equation of the tangent to C at the point P,
June 2012, Legacy C1, Q8 (8 marks): 4x – 5 – x 2 = q – (x + p) 2 where p and q are integers. (a) Find the value of p and the value of q. (3) (b) Calculate the discriminant of 4x – 5 – x 2 (2) (c)
June 2012, Legacy C1, Q9 (15 marks): The line L1 has equation 4y + 3 = 2x The point A (p, 4) lies on L1 (a) Find the value of the constant p. (1) The line L2 passes through the point C (2, 4) and i
June 2012, Legacy C1, Q10 (8 marks): x y A C (3, 27) O Figure 1 Figure 1 shows a sketch of the curve C with equation y = f(x) where f (x) = x2(9 – 2x) There is a minimum at the origin, a maximum at
January 2012, Legacy C1, Q1 (6 marks): Given that y x x = + 4 1 2 6 , find in their simplest form (a) d d y x (3) (b) y x d ∫ (3)
January 2012, Legacy C1, Q2 (6 marks): (a) Simplify √ + √ 32 18 giving your answer in the form a √2 , where a is an integer. (2) (b) Simplify √ + √ + √ 32 18 3 2 giving your answer in the form b c √+
January 2012, Legacy C1, Q3 (6 marks): Find the set of values of x for which (a) 4 5 15 x x − − (2) (b) x x ( ) −4 12 (4)
January 2012, Legacy C1, Q4 (6 marks): A sequence x x x 1 2 3 , , ,... is defined by x1 1 = x ax n n + = + 1 5, n .1 where a is a constant. (a) Write down an expression for x2 in terms of a. (1) (b)
January 2012, Legacy C1, Q5 (8 marks): The curve C has equation y x x = − ( ) 5 and the line L has equation 2 5 4 y x = + (a) Use algebra to show that C and L do not intersect. (4) (b) In the space o
January 2012, Legacy C1, Q6 (8 marks): Figure 1 The line l1 has equation 2 3 12 0 x y − + = (a) Find the gradient of l1 . (1) The line l1 crosses the x-axis at the point A and the y-axis at the point
January 2012, Legacy C1, Q7 (5 marks): A curve with equation y x = f ( ) passes through the point (2, 10). Given that ′ = − + f ( ) x x x 3 3 5 2 find the value of f ( )1 . (5)
January 2012, Legacy C1, Q8 (10 marks): The curve C1 has equation y x x = + 2 2 ( ) (a) Find d d y x (2) (b) Sketch C1, showing the coordinates of the points where C1 meets the x-axis. (3) (c) Find th
January 2012, Legacy C1, Q9 (9 marks): A company offers two salary schemes for a 10-year period, Year 1 to Year 10 inclusive. Scheme 1: Salary in Year 1 is £P. Salary increases by £(2T) each year, fo
January 2012, Legacy C1, Q10 (11 marks): Figure 2 Figure 2 shows a sketch of the curve C with equation y x x = − ≠ 2 1 0 , The curve crosses the x-axis at the point A. (a) Find the coordinates of A. (1
June 2011, Legacy C1, Q1 (3 marks): Find the value of (a) 25 1 2 (1) (b) 25 3 2 − (2)
June 2011, Legacy C1, Q2 (7 marks): Given that 5 3 1 2 7 , 0, y x x x = + + ≠ find, in their simplest form, (a) d d y x , (3) (b) d y x ∫ . (4)
June 2011, Legacy C1, Q3 (5 marks): The points P and Q have coordinates (–1, 6) and (9, 0) respectively. The line l is perpendicular to PQ and passes through the mid-point of PQ. Find an equation
June 2011, Legacy C1, Q4 (7 marks): Solve the simultaneous equations 2 2 2 4 11 x y y x + = − = (7)
June 2011, Legacy C1, Q5 (7 marks): A sequence 1 2 3 , , ,... a a a is defined by 1 1 , 5 3, 1 n n a k a a n + = = + . , where k is a positive integer. (a) Write down an expression for 2a in terms
June 2011, Legacy C1, Q6 (7 marks): Given that 5 2 6 3 x x x + ˲ can be written in the form 6 3 , p q x x + (a) write down the value of p and the value of q. (2) Given that 5 2 d 6 3 , d y x x x x
June 2011, Legacy C1, Q7 (6 marks): where k is a real constant. (a) Find the discriminant of f (x) in terms of k. (2) (b) Show that the discriminant of f (x) can be expressed in the form 2 ( ) , k
June 2011, Legacy C1, Q8 (10 marks): (3, –1) y C x O 6 Figure 1 Figure 1 shows a sketch of the curve C with equation f ( ). y x = The curve C passes through the origin and through (6, 0). The curve
June 2011, Legacy C1, Q9 (9 marks): (a) Calculate the sum of all the even numbers from 2 to 100 inclusive, 2 + 4 + 6 + ...... + 100 (3) (b) In the arithmetic series k + 2k + 3k + ...... + 100 k is
June 2011, Legacy C1, Q10 (14 marks): The curve C has equation 2 ( 1)( 3) y x x = + + (a) Sketch C, showing the coordinates of the points at which C meets the axes. (4) (b) Show that 2 d 3 14 15. d
January 2011, Legacy C1, Q1 (4 marks): (a) Find the value of 1 4 16 − (2) (b) Simplify x x 2 1 4 4 – ( ) (2)
January 2011, Legacy C1, Q2 (5 marks): Find 1 3 5 2 (12 3 4 ) d x x x x − + ∫ giving each term in its simplest form. (5)
January 2011, Legacy C1, Q3 (4 marks): Simplify 5 2 3 3 1 − − ˲ ˲ giving your answer in the form 3, p q + ˲ where p and q are rational numbers. (4)
January 2011, Legacy C1, Q4 (5 marks): A sequence 1 2 3 , , ,... a a a is defined by 1 1 2 3 n n a a a c + = = − where c is a constant. (a) Find an expression for 2a in terms of c. (1) Given that 3 1
January 2011, Legacy C1, Q5 (7 marks): Figure 1 Figure 1 shows a sketch of the curve with equation f ( ) y x = where f ( ) 2 x x x = − , 2 x ≠ The curve passes through the origin and has two asymptot
January 2011, Legacy C1, Q6 (7 marks): An arithmetic sequence has first term a and common difference d. The sum of the first 10 terms of the sequence is 162. (a) Show that 10 45 162 a d + = (2) Given
January 2011, Legacy C1, Q7 (5 marks): The curve with equation f ( ) y x = passes through the point ( 1,0). − Given that 2 f ( ) 12 8 1 x x x ′ = − + find f ( ). x (5)
January 2011, Legacy C1, Q8 (7 marks): The equation 2 ( 3) (3 2 ) 0, x k x k + − + − = where k is a constant, has two distinct real roots. (a) Show that k satisfies 2 2 3 0 k k + − (3) (b) Find the s
January 2011, Legacy C1, Q9 (11 marks): The line 1 L has equation 2 3 0, y x k − − = where k is a constant. Given that the point A (1,4) lies on 1 L , find (a) the value of k, (1) (b) the gradient of
January 2011, Legacy C1, Q10 (8 marks): (a) On the axes below, sketch the graphs of (i) ( 2)(3 ) y x x x = + − (ii) 2 y x = − showing clearly the coordinates of all the points where the curves cross t
January 2011, Legacy C1, Q11 (12 marks): The curve C has equation 3 2 3 1 8 9 30, 2 y x x x = − + + 0 x (a) Find d d y x . (4) (b) Show that the point P(4, 8) − lies on C. (2) (c) Find an equation of t
June 2010, Legacy C1, Q1 (2 marks): Write √(75)–√(27) in the form k √x, where k and x are integers. (2)
June 2010, Legacy C1, Q2 (4 marks): Find ∫ (8 6 5)dx 3 1 2 x x – + giving each term in its simplest form. (4)
June 2010, Legacy C1, Q3 (6 marks): Find the set of values of x for which (a) 3(x – 2) < 8 – 2x (2) (b) (2x – 7)(1 + x) < 0 (3) (c) both 3(x – 2) < 8 – 2x and (2x – 7)(1 + x) < 0 (1)
June 2010, Legacy C1, Q4 (6 marks): (a) Show that x2 +6x + 11 can be written as (x + p)2 +q where p and q are integers to be found. (2) (b) In the space at the top of page 7, sketch the curve with
June 2010, Legacy C1, Q5 (4 marks): A sequence of positive numbers is defined by a a a n n + + = 1 2 1 3 2 d √O P, n . 1, (a) Find a2 and a3, leaving your answers in surd form. (2) (b) Show that a
June 2010, Legacy C1, Q6 (7 marks): Figure 1 Figure 1 shows a sketch of the curve with equation y = f (x). The curve has a maximum point A at (–2, 3) and a minimum point B at (3, – 5). On separate
June 2010, Legacy C1, Q7 (6 marks): Given that 2 3 3 2 8 4 x y x x x + = − + √ , 0 x > find d d y x . (6)
June 2010, Legacy C1, Q8 (8 marks): (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax+by+c=0, where a, b and c are integers. (3) (b) Find the length
June 2010, Legacy C1, Q9 (8 marks): A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays £a for their first day, £(a + d ) for their second day, £(a + 2d )
June 2010, Legacy C1, Q10 (15 marks): (a) On the axes below sketch the graphs of (i) y = x (4 – x) (ii) y = x2 (7 – x) showing clearly the coordinates of the points where the curves cross the coordi
June 2010, Legacy C1, Q11 (9 marks): The curve C has equation y = f(x), x > 0, where d 5 3 2 d y x x x = − − √ Given that the point P (4, 5) lies on C, find (a) f(x), (5) (b) an equation of the tan
January 2010, Legacy C1, Q1 (3 marks): Given that y x x = + + 4 1 3 3, find x y d d . (3)
January 2010, Legacy C1, Q2 (6 marks): (a) Expand and simplify (7 + √5)(3 – √5). (3) (b) Express 7 5 3 5 + √ + √ in the form a b + √5, where a and b are integers. (3)
January 2010, Legacy C1, Q3 (5 marks): The line 1l has equation 0 2 5 3 = − + y x (a) Find the gradient of 1l . (2) The line 2l is perpendicular to 1l and passes through the point (3, 1). (b) Find th
January 2010, Legacy C1, Q4 (7 marks): d d y x x x x = + − 5 1 2 G , √ x > 0 Given that y = 35 at x = 4, find y in terms of x, giving each term in its simplest form. (7)
January 2010, Legacy C1, Q5 (7 marks): Solve the simultaneous equations y x − + = 3 2 0 y x x 2 2 6 0 − − = (7)
January 2010, Legacy C1, Q6 (8 marks): The curve C has equation y x x x x = + − > ( )( ) , 3 8 0 (a) Find x y d d in its simplest form. (4) (b) Find an equation of the tangent to C at the point where
January 2010, Legacy C1, Q7 (9 marks): Jill gave money to a charity over a 20-year period, from Year 1 to Year 20 inclusive. She gave £150 in Year 1, £160 in Year 2, £170 in Year 3, and so on, so tha
January 2010, Legacy C1, Q8 (7 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point (−2, 5) and an asymptote y = 1, as shown in Figure 1
January 2010, Legacy C1, Q9 (13 marks): (a) Factorise completely x x 3 4 − (3) (b) Sketch the curve C with equation y x x = − 3 4 , showing the coordinates of the points at which the curve meets the x
January 2010, Legacy C1, Q10 (10 marks): f(x) = + + + x kx k 2 4 3 11 ( ), where k is a constant. (a) Express f(x) in the form ( ) x p q + + 2 , where p and q are constants to be found in terms of k. (

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