Cambridge A Level Mathematics 9709 — 2021 Oct/Nov Paper 1 · Variant 1

2 A curve has equation y kx2 2x and a line has equation y kx where k is a constant. = + −k = −2, Find the set of values of k for which the curve and line do not intersect. [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 2 ( ) [ ] 2 2 2 2 leading to 2 2 0 kx x k kx kx k x k + − = − + −+ − + = ( ) ( ) 2 2 4 2 −+ − −+ k k k DM1 Apply 2 4 − b ac ; allow 1 error but a, b and c must be correct for their quadratic. ( )( ) 2 5 12 4 or 2 2 4 − + −+ −+ − k k k k k A1 May be shown in quadratic formula. ( )( ) 2 5 2 −+ − + k k DM1 Solving a 3-term quadratic in k (all terms on one side) by factorising, use of formula or completing the square. Factors must expand to give their coefficient of k2. 2 2 5 < < k A1 WWW, accept two separate correct inequalities. If M0 for solving quadratic, SC B1 can be awarded for correct final answer. 5

More questions on Quadratics

3 Solve, by factorising, the equation 6 cos tan cos 4 tan 0, 1 1 −3 1 + 1 −2 = for [4] 0Å ≤1 ≤180Å. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 [ ] 3cos (2tan 1) 2(2tan 1) 0 θ θ θ − + − = [ ] (2tan 1) (3cos 2) 0 θ θ − + = [leading to 1 2 tan , cos ] 2 3 θ θ = = − M1 OE. At least 2 out of 4 products correct. 26.6º, 131.8º A1 A1 WWW. Must be 1 d.p. or better. Final A0 if extra solution within the interval. SC B1 No factorisation: Division by 2tanθ ‒1 leading to 131.8° or division by 3cos 2 θ + or similar leading to 26.6°. Alternative method for question 3 [ ] sin sin 6cos 3cos 4 2 0 cos cos θ θ θ θ θ θ     − + − =         [ ] 2 6cos sin 3cos 4sin 2cos 0 θ θ θ θ θ − + − = ( ) ( ) 2sin 3cos 2 cos 3cos 2 θ θ θ θ + − + [ ] 0 = M1 Using sin tan cos θ θ θ = and reaching a partial factorisation; condone sign errors. (2 ( ) sin cos ) 3cos 2 θ θ θ − + [ ] 0 = [leading to 1 2 tanθ , cos ] 2 3 θ = = − M1 At least 2 out of 4 products correct. 26.6º, 131.8º A1 A1 WWW. Must be 1 d.p. or better. Final A0 if extra solution within the interval. SC B1 No factorisation: Division by 2tanθ ‒1 leading to 131.8° or division by 3cos 2 θ + or similar leading to 26.6°. 4

More questions on Trigonometry

4 The first term of an arithmetic progression is a and the common difference is The first term of a geometric progression is 5a and the common ratio is The sum to infinity−4.of the geometric 4. progression is equal to the sum of the first eight terms of the−1arithmetic progression. (a) Find the value of a. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The kth term of the arithmetic progression is zero. (b) Find the value of k. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(a) ( ) 1 4 5 1 a −± B1 Use of correct formula for sum to infinity. ( ) 8 2 7 4 2 a  + −    *M1 Use of correct formula for sum of 8 terms and form equation; allow 1 error. 4 8 112 = − a a leading to [ ] 28 = a DM1 Solve equation to reach a value of a. 28 = a A1 Correct value. 4 4(b) ( )( ) 28 1 4 0 + − − = their k M1 Use of correct method with their a. [ ] 8 = k A1 2

More questions on Series

6 C A 6 cm B The diagram shows a metal plate ABC in which the sides are the straight line AB and the arcs AC and BC. The line AB has length 6 cm. The arc AC is part of a circle with centre B and radius 6 cm, and the arc BC is part of a circle with centre A and radius 6 cm. (a) Find the perimeter of the plate, giving your answer in terms of [3] π. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the area of the plate, giving your answer in terms of and 3. [4] π ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(a) Recognise that at least one of angles A, B, C is π 3 One arc 6 × their π 3 leading to two arcs 2 × 6 × their π 3 M1 SOI e.g. may see 2π or 4π. Use of correct formula for length of arc and multiply by 2. Perimeter = 6 + 4π A1 Must be exact value. Alternative method for question 6(a) Calculate circumference of whole circle = 12π B1 One arc 1 12π 6 × leading to two arcs 1 2 12π 6 × × M1 SOI e.g. may see 2π or 4π. Perimeter = 6 + 4π A1 Must be exact value. 3 Question Answer Marks Guidance 6(b) Sector = 2 1 π 6 2 3 their  × ×     M1 Use of correct formula for area of sector. SOI e.g. may see 6π or 12π. ( ) ( ) 2 2 1 π 1 π 6 6 sin 6π 2 3 2 3 their their       × × − × × +             6π 9 3 6π   = − +   M1 A1 M1 for attempt at strategy with values substituted: area of segment + area of sector A1 if correct (unsimplified). Area = 12π 9 3 − A1 Must be simplified exact value. Alternative method for question 6(b) Sector = 2 1 π 6 2 3 their  × ×     M1 Use of correct formula for area of sector. SOI e.g. may see 6π or 12π. ( ) 2 2 1 π 1 π 2 6 6 sin 2 3 2 3 their their         × × × − × ×                 M1 A1 M1 for attempt at strategy with values substituted: 2 × sector – triangle A1 if correct (unsimplified). Area = 12π 9 3 − A1 Must be simplified exact value. Alternative method for question 6(b) Sector = 2 1 π 6 2 3 their  × ×     M1 Use of correct formula for area of sector. SOI e.g. may see 6π or 12π. ( ) ( ) 2 2 1 π 1 π 2 6 6 sin 2 3 2 3 their their         × × × − × ×                 + ( ) 2 1 π 6 sin 2 3 their     × ×         12π 18 3 9 3   = − +   M1 A1 M1 for attempt at strategy with values substituted: 2 × segment + triangle A1 if correct (unsimplified). Area 6π 9 3 6π   = − +   = 12π 9 3 − A1 Must be simplified exact value. 4

More questions on Circular measure

7 A circle with centre 5, 2 passes through the point 7, 5 . (a) Find an equation of the circle. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The line y 5x intersects the circle at A and B. = −10 (b) Find the exact length of the chord AB. [7] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(a) ( ) ( ) 2 2 2 5 2 7 5 13 r   = − + − =   B1 2 13 = r or 13 = r Equation of circle is ( ) ( ) 2 2 5 2 13 − + − = x y B1 FT OE. FT on their 13 but LHS must be correct. 2 7(b) ( ) ( ) 2 2 5 5 10 2 13 x x − + − − = M1 Substitute 5 10 = − y x into their equation. [ ] 2 26 130 156 0 − + = x x A1 FT OE 3-term quadratic with all terms on one side. FT on their circle equation. [ ]( )( ) [ ] 26 2 3 0 − − = x x M1 Solve 3-term quadratic in x by factorising, using formula or completing the square. Factors must expand to give their coefficient of x2. (2, 0), (3, 5) A1 A1 Coordinates must be clearly paired; A1 for each correct point. A1 A0 available if two x or y values only. If M0 for solving quadratic, SC B2 can be awarded for correct coordinates, SC B1 if two x or y values only. ( ) ( ) ( ) 2 2 2 3 2 5 0 = − + − AB M1 SOI. Using their points to find length of AB. 26 = AB A1 ISW. Dependent on final M1 only.

More questions on Coordinate geometry