OCR A Level Mathematics A H240/03 combined SAM

1 A Level Mathematics A. H240/03 Pure Mathematics and Mechanics Sample question paper Date Morning/Afternoon Time allowed: 2 hours You must have: Printed Answer Booklet You may use: a scientific or graphical calculator INSTRUCTIONS. en * 0 0 0 0 0 0 *. im Use black ink. HB pencil may be used for graphs and diagrams only. Complete the boxes provided on the Printed Answer Booklet with your name, centre number and candidate number. ec Answer all the questions. Write your answer to each question in the space provided in the Printed Answer Booklet.

2 Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s).. Sp Do not write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper . Final answers should be given to a degree of accuracy appropriate to the context. The acceleration due to gravity is denoted by g m s-2. Unless otherwise instructed, when a numerical value is needed, use g = INFORMATION. The total number of marks for this paper is 100. The marks for each question are shown in brackets [ ].

3 You are reminded of the need for clear presentation in your answers. The Printed Answer Booklet consists of 16 pages. The question paper consists of 8 pages. OCR 2018 H240/03 Turn over 603/1038/8 2. Formulae A Level Mathematics A (H240). Arithmetic series Sn 12 n(a l ) 12 n{2a (n 1)d}. Geometric series a(1 r n ). Sn . 1 r a S for r 1. 1 r Binomial series (a b)n a n n C1 a n 1b n C2 a n 2b2 n Cr a n r b r bn (n ) , n n! where n Cr n Cr . r r !(n r )! (1 x)n 1 nx . Differentiation n(n 1) 2. 2! x . n(n 1) (n r 1) r r! x . en x 1, n . im f ( x) f ( x).

4 Tan kx k sec2 kx sec x sec x tan x ec cotx cosec2 x cosec x cosec x cot x du dv v u Sp u dy Quotient rule y , dx 2 dx v dx v Differentiation from first principles f ( x h) f ( x ). f ( x) lim h 0 h Integration f ( x). f ( x). dx ln f ( x) c . 1. f ( x) f ( x) dx . n f ( x) n 1 c n 1. u dx dx uv v dx dx dv du Integration by parts Small angle approximations sin ,cos 1 12 2 , tan where is measured in radians OCR 2018 H240/03 . 3. Trigonometric identities sin( A B) sin A cos B cos Asin B. cos( A B) cos A cos B sin Asin B. tan A tan B. tan( A B) ( A B ( k 12 ) ).

5 1 tan A tan B. Numerical methods b b a Trapezium rule: a y dx 12 h{( y0 yn ) 2( y1 y2 yn 1 ) }, where h n f( xn ). The Newton-Raphson iteration for solving f( x) 0 : xn 1 xn . f ( xn ). Probability P( A B) P( A) P( B) P( A B). P( A B). P( A B) P( A)P( B | A) P( B)P( A | B ) or P( A | B) . P (B ). Standard deviation x x . n 2.. x2. n x 2 or f x x . f 2.. fx 2. f x2en im The binomial distribution n . If X ~ B(n, p) then P( X x) p x (1 p)n x , Mean of X is np, Variance of X is np(1 p). ec x . Hypothesis test for the mean of a normal distribution 2 X.

6 If X ~ N , 2 then X ~ N , and ~ N(0, 1). Sp n / n Percentage points of the normal distribution If Z has a normal distribution with mean 0 and variance 1 then, for each value of p, the table gives the value of z such that P(Z z) p. p z Kinematics Motion in a straight line Motion in two dimensions v u at v u at s ut 12 at 2 s ut 12 at 2. s 12 u v t s 12 u v t v2 u 2 2as s vt 12 at 2 s vt 12 at 2. OCR 2018 H240/03 Turn over 4. Section A: Pure Mathematics Answer all the questions 1 (i) If x 3 , find the possible values of 2 x 1 . [3]. (ii) Find the set of values of x for which 2 x 1 x 1.

7 Give your answer in set notation. [4]. 2 (i) Use the trapezium rule, with four strips each of width , to find an approximate value for 1. 1. dx . [3]. 0 1 x 2. (ii) Explain how the trapezium rule might be used to give a better approximation to the integral given in part (i). [1]. en 3 In this question you must show detailed reasoning. Given that 5sin 2 x 3cos x , where 0 x 90 , find the exact value of sin x . [4]. im 4 Show that, for a small angle , where is in radians, 1 cos 3cos2 1 52 2 . ec [4]. (i) Find the first three terms in the expansion of 1 px 3 in ascending powers of x.

8 1. 5 [3]. Sp (ii) Given that the expansion of 1 qx 1 px 3 is 1. 1 x 92 x 2 .. find the possible values of p and q. [5]. 6 A curve has equation y x 2 kx 4 x 1 where k is a constant. Given that the curve has a minimum point when x 2. find the value of k, show that the curve has a point of inflection which is not a stationary point. [7]. OCR 2018 H240/03 . 5. 7 (i) Find 5 x3 x 2 1 dx . [5]. (ii) Find tan 2 d . You may use the result tan d ln sec c . [5]. 8 In this question you must show detailed reasoning. C. D. 1. en 2. im 15 . 45 30 . A B.

9 1 E. ec The diagram shows triangle ABC. The angles CAB and ABC are each 45 , and angle ACB 90 . The points D and E lie on AC and AB respectively, such that AE DE 1, DB 2 and angle BED 90 . Sp Angle EBD 30 and angle DBC 15 . 2 6. (i) Show that BC . [3]. 2. 6 2. (ii) By considering triangle BCD, show that sin15 . [3]. 4. OCR 2018 H240/03 Turn over 6. Section B: Mechanics Answer all the questions 9 Two forces, of magnitudes 2 N and 5 N, act on a particle in the directions shown in the diagram below. 2N. 5N. (i) Calculate the magnitude of the resultant force on the particle.

10 [3]. (ii) Calculate the angle between this resultant force and the force of magnitude 5 N. [1]. 10. en A body of mass 20 kg is on a rough plane inclined at angle to the horizontal. The body is held at rest on the plane by the action of a force of magnitude P N acting up the plane in a direction parallel to a line of greatest slope of the plane. The coefficient of friction between the body and the plane is . im (i) When P 100 , the body is on the point of sliding down the plane. Show that g sin g cos 5 . [4]. ec (ii) When P is increased to 150, the body is on the point of sliding up the plane.