Trig Functions, Equations & Identities May a. [2 marks] Let. For what values of x does Markscheme (M1)

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1 Trig Functions, Equations & Identities May a. Let. For what values of x does () 1b. [5 marks] not exist? Simplify the expression. EITHER OR [5 marks] 2a. 1 In the triangle ABC,, AB = BC + 1. Show that cos. R1 2b. [8 marks] By squaring both sides of the equation in part (a), solve the equation to find the angles in the triangle. angles in the triangle are Note: Accept answers in radians. [8 marks] 2c. Apply Pythagoras theorem in the triangle ABC to find BC, hence show that.

2 2d. [4 marks] Hence, or otherwise, calculate the length of the perpendicular from B to [AC]. EITHER OR [4 marks] 3a. [1 mark] The function is defined for. Write down the coordinates of the minimum point on the graph of f. [1 mark] 3b. The points, lie on the graph of. Find p q. 3c. [4 marks] Find the coordinates of the point, on, where the gradient of the graph is 3. ()() () Coordinates are [4 marks] 3d. [7 marks] Find the coordinates of the point of intersection of the normals to the graph at the points P Q. () gradient at P is so gradient of normal at P is () gradient at Q is 4 so gradient of normal at Q is equation of normal at P is () () equation of normal at Q is () Note: Award the previous two even if the gradients are incorrect in where are coordinates of P Q (or in with c determined using coordinates of P Q. intersect at Note: Award N2 for 3.79 without other working. [7 marks] 5. Show that. 2

3 METHOD 1 consider right h side Note: Award for recognizing the need for single angles for recognizing. Note: Award for correct numerator for correct denominator. 6. In the diagram below, AD is perpendicular to BC. CD = 4, BD = 2 AD = 3.. Find the exact value of. METHOD 1 (may be seen on diagram) () () () Note: If only the two cosines are correctly given award ()()(A0). Use of () (substituting) Use of amd N1 (may be seen on diagram) () () 3

4 Use of () () N1 9. [5 marks] The diagram below shows a curve with equation, defined for. The point lies on the curve is the maximum point. (a) Show that k = 6. (b) Hence, find the values of a b. (a) N0 (b) METHOD 1 maximum N2 N2 Note: Award for. [5 marks] 10. [5 marks] (a) Show that. (b) Hence, or otherwise, find the value of. (a) METHOD 1 let so, 4

5 for, if so, METHOD 3 an appropriate sketch e.g. correct reasoning leading to (b) METHOD 1 R1 () Note: Only one of the previous two marks may be implied. let N1 () as () (R1) Note: Only one of the previous two marks may be implied. so, METHOD 3 N1 for, () so, () Note: Only one of the previous two marks may be implied. N1 METHOD 4 an appropriate sketch 5

6 e.g. correct reasoning leading to R1 [5 marks] 12. [20 marks] (a) Show that. (b) Hence prove, by induction, that for all. (c) Solve the equation. (a) (b) if n = 1 so LHS = RHS the statement is true for n = 1 R1 assume true for n = k Note: Only award if the word true appears. Do not award for let n = k only. Subsequent marks are independent of this. so if n = k + 1 then so if true for n = k, then also true for n = k + 1 as true for n = 1 then true for all R1 Note: Final R1 is independent of previous work. [12 marks] (c) but this is impossible 6

7 for not including any answers outside the domain R1 Note: Award the first for correctly obtaining or equivalent subsequent marks as appropriate including the answers. Total [20 marks] 13. [4 marks] The graph below shows. Find the value of a, the value of b the value of c. period () [4 marks] 14. If x satisfies the equation, show that, where a, b. () dividing by rearranging rationalizing the denominator 15a. Given that, where, find p. attempt at use of 7

8 Note: the value of p needs to be stated for the final mark. 15b. Hence find the value of. 16a. Solve the equation, where, expressing your answer(s) to the nearest degree. attempting to solve for or for u where or for x graphically. () EITHER () OR () THEN Note: Award ()()A0 for. Note: Award ()()A0 for radians. 16b. Find the exact values of satisfying the equation. attempting to solve for or for where. () 17a. Show that. () Note: Award for use of double angle formulae. 17b. Hence find the value of in the form, where. () 19a. Use the identity to prove that. 8

9 positive as R1 19b. Find a similar expression for. () 19c. [4 marks] Hence find the value of. [4 marks] 20. Given that, find. using () ()() using () Note: Award this for decomposition of cos 4x using double angle formula anywhere in the solution. 22a. Sketch the graph of for. Note: Award for correct shape for correct domain range. 9

10 22b. Solve for. attempting to find any other solutions Note: Award () if at least one of the other solutions is correct (in radians or degrees) or clear use of symmetry is seen. Note: Award for all other three solutions correct no extra solutions. Note: If working in degrees, then max A0A [7 marks] The first three terms of a geometric sequence are. (a) Find the common ratio r. (b) Find the set of values of x for which the geometric series converges. Consider. (c) Show that the sum to infinity of this series is. (a) Note: Accept. [1 mark] (b) EITHER OR THEN (c) Note: Award for correct numerator for correct denominator. Total [7 marks] 24a. Consider the following functions:,, Sketch the graph of. 10

11 Note: for correct shape, for asymptotic behaviour at. 24b. Find an expression for the composite function state its domain. domain of is equal to the domain of 24c. [7 marks] Given that, (i) find in simplified form; (ii) show that for. (i) () (ii) METHOD 1 f is a constant R1 when from diagram hence METHOD 3 R1 11

12 denominator = 0, so R1 [7 marks] 24d. Nigel states that is an odd function Tom argues that is an even function. (i) State who is correct justify your answer. (ii) Hence find the value of for. (i) Nigel is correct. METHOD 1 is an odd function is an odd function composition of two odd functions is an odd function sum of two odd functions is an odd function R1 therefore f is an odd function. R1 (ii) 12