Cambridge A Level Mathematics 9709 — 2011 Oct/Nov Paper 2 · Variant 1

1 Solve the inequality |4 - 5x| < 3. [3]

Mark scheme: 1 EITHER State or imply non-modular inequality (4 – 5x)2 < 32, or corresponding equation or pair of linear equations M1 1 7 Obtain critical values and A1 5 5 1 7 State correct answer < x < A1 5 5 1 OR State one critical value, e.g. x = , by solving a linear equation (or inequality) 5 or from a graphical method or by inspection B1 State the other critical value correctly B1 1 7 State correct answer < x < B1 [3] 5 5

More questions on Quadratics

6 2 5 2 Show that dx ln 3. [5] 4x 1 = ä 2 +

Mark scheme: 2 Integrate and obtain term of the form kln(4x + 1) M1 1 State correct term ln( 4 x + )1 A1 2 Substitute limits correctly M1 Use law for the logarithm of a quotient or a power M1 Obtain given answer correctly A1 [5] 1 2

More questions on Integration

3 y 1 1 x O 4 p 2 p The diagram shows the part of the curve y 1 tan 2x for 0 Find the x-coordinates of the 2 2π. = ≤x ≤1 points on this part of the curve at which the gradient is 4. [5]

Mark scheme: 1 3 Obtain derivative of the form k sec2 2x, where k = 1 or k = M1 2 Obtain correct derivative sec2 2x A1 Use correct method for solving sec2 2x = 4 M1 1 Obtain answer x = π (or 0.524 radians) A1 6 1 Obtain answer x = π (or 1.05 radians) and no others in range A1 [5] 3

More questions on Differentiation

4 Solve the equation 32x 10 0, giving your answers correct to 3 significant figures. [5] −7(3x) + =

Mark scheme: 4 Carry out recognizable solution method for quadratic in 3x M1 Obtain 3x = 5 and 3x = 2 A1 Use logarithmic method to solve an equation of the form 3x = k, where k > 0 M1 State answer 1.46 A1 State answer 0.631 A1 [5] 1

More questions on Logarithmic and exponential functions

5 The polynomial 4x3 ax2 9x 9, where a is a constant, is denoted by It is given that when is divided by + +the remainder+ is 10. p(x). p(x) (2x −1) (i) Find the value of a and hence verify that is a factor of [3] (x −3) p(x). (ii) When a has this value, solve the equation 0. [4] p(x) =

Mark scheme: 1 5 (i) Substitute x = and equate to 10 M1 2 Obtain answer a = –16 A1 Either show that f(3) = 0 or divide by (x – 3) obtaining a remainder of zero B1 [3] (ii) At any stage state that x = 3 is a solution B1 Attempt division by (x – 3) reaching a partial quotient of 4x2 + kx M1 Obtain quadratic factor 4x2 – 4x – 3 A1 3 1 Obtain solutions x = and x = – A1 2 2 S.C. M1A1√ if value of ‘a’ incorrect [4] GCE AS/A LEVEL – October/November 2011 9709 21 3 2

More questions on Quadratics

6 (i) Verify by calculation that the cubic equation x3 5x 0 −2x2 + −3 = has a root that lies between x 0.7 and x 0.8. [2] = = (ii) Show that this root also satisfies an equation of the form ax2 3 x , + = x2 b + where the values of a and b are to be found. [2] (iii) With these values of a and b, use the iterative formula ax2n 3 + xn+1 = x2n b + to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

Mark scheme: 6 (i) Consider sign of x3 – 2x2 + 5x – 3 at x = 0.7 and x = 0.8 M1 Complete the argument correctly with appropriate calculations A1 [2] (ii) Rearrange equation to given equation or vice versa B1 State a = 2 and b = 5 B1 [2] (iii) Use the iterative formula correctly at least once M1 Obtain final answer 0.74 A1 Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.735, 0.745) B1 [3]

More questions on Numerical solution of equations

7 The parametric equations of a curve are x e3t, y t2et 3. = = + dy (i) Show that . [4] t(t + 2) dx = 3e2t (ii) Show that the tangent to the curve at the point is parallel to the x-axis. [2] (1, 3) (iii) Find the exact coordinates of the other point on the curve at which the tangent is parallel to the x-axis. [2]

Mark scheme: 7 (i) Use product rule to differentiate y M1 Obtain correct derivative in any form in t for y A1 dy dy dx Use = ÷ M1 dx dt dt Obtain given answer correctly A1 [4] (ii) State t = 0 M1 dy State that = 0 and make correct conclusion A1 [2] dx (iii) Substitute t = –2 into equation for x or y M1 Obtain (e–6, 4e–2 + 3) A1 [2]

More questions on Differentiation

8 (i) By first expanding show that cos(2x + x), cos 3x cos3x cos x. ≡4 −3 [5] (ii) Hence show that 16π cos3x dx 12.5 ã 0 (2 −cos x) = [5]

Mark scheme: 8 (i) Make relevant use of the cos(A + B) formula M1* Make relevant use of the cos 2A and sin 2A formulae M1* Obtain a correct expression in terms of cos x and sin x A1 Use sin2 x = 1 – cos2 x to obtain an expression in terms of cos x M1(dep*) Obtain given answer correctly A1 [5] 1 1 (ii) Replace integrand by cos 3 x + cos x , or equivalent B1 2 2 1 1 Integrate, obtaining sin 3 x + sin x , or equivalent B1 + B1√ 6 2 Use limits correctly M1 Obtain given answer A1 [5]

More questions on Trigonometry