Cambridge A Level Mathematics 9709 — 2012 Oct/Nov Paper 1 · Variant 2

x + 3 2 A function f is such that f(x) = + 1, for x ≥−3. Find 2 (i) f −1(x) in the form ax2 + bx + c, where a, b and c are constants, [3] (ii) the domain of f −1. [1]

Mark scheme: q → 2( x − )1 2 − 3 and "÷2". → 2 x 2 −x4 − 1 A1 co [3] (ii) domain of f –1 is ≥ 1. B1 co. condone >1 [1] 3 (i) A = 2400 − 20(60 − 2x) −x(40 − x) − 30x → A = x² −30x + 1200. M1 Needs attempts at all areas (could be trapezium − triangle) A1 co answer given [2] dA (ii) = 2x − 30 or (x − 15)² + 975 dx B1 co - either method okay = 0 when x = 15 or Min at x = 15 → A = 975 . M1 Sets differential to 0 + solution. co A1 co. [3] 4 x 2 y = + k 4 y = x k x 2 x (i) = + k → kx 2 − 4 x − 4 k 2 = 0 M1 Eliminates x or y completely.

More questions on Functions

7 The position vectors of the points A and B, relative to an origin O, are given by 1 k −−→ −−→ OA = 0 ! and OB = −k !, 2 2k where k is a constant. (i) In the case where k = 2, calculate angle AOB. [4] −−→ (ii) Find the values of k for which AB is a unit vector. [4]

Mark scheme: 1 7  2 a   x −x   Term in x5 is 7C3 × (x²)4 × (−a/x)³ B1 Allow on own or in an expansion. This term isolated M1 Correct term in x5 selected. Equated to −280 → a = 2. A1 Equated to −280 [3] 2 x + 3 (i) f(x) = + 1 , for x ≥ −3 2 M1 Attempt at x as subject and removes +1 Make x the subject or interchanges x,y M1 Squares both sides and deals with "+3" 2

More questions on Vectors

9 y B (0, 3) 9 y = 2 x + 3 C A (3, 1) x O 9 The diagram shows part of the curve y = crossing the y-axis at the point B (0, 3). The point 2x + 3, A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C. (i) Find the equation of the tangent to the curve at A. [4] (ii) Determine, showing all necessary working, whether C is nearer to B or to O. [1] (iii) Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through 360◦about the x-axis. [4]

Mark scheme: → y 1 = 9 ( x )3 [4] (normal →max 2/4, no calculus 0/4) (ii) Meets the y-axis when x = 0, y = 1⅔ B1 Sets x to 0 in his tangent. This is nearer to B than to O. [1] The 1⅔ and part (i) must be correct.

More questions on Differentiation