2010 Y6 ACS Mathematics HL Mock Paper 1

Anglo-Chinese School (Independent)/ Mathematics Department / IBDP Mathematics HL / Mock Exam / Paper 1 Page 1 of 4 Anglo-Chinese School (Independent) IBDP Mathematics Higher Level Mathematics HL Mock Exam Paper 1 SECTION A Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 6] NOT IN SYLLABUS Let 𝐴 = (2 6 𝑘 −1)and 𝐵 = ( ℎ 3 −3 7), where ℎ and 𝑘 are integers. Given that det 𝐴 = det 𝐵 and that det 𝐴𝐵 = 256ℎ, (a) show that ℎ satisfies the equation 49ℎ2 − 130ℎ + 81 = 0; [2 marks] (b) hence find the value of 𝑘. [4 marks] 2. [Maximum mark:6] Find the area between the curves 𝑦 = 2 − 3𝑥 + 𝑥2 and 𝑦 = 2 + 𝑥 − 𝑥2. 3. [Maximum mark: 6] The random variable 𝑇 has the probability density function 𝑓(𝑡) = 𝜋 4 cos (𝜋𝑡 2 ), − 𝑎 ≤ 𝑡 ≤ 𝑎. Find (a) the value of 𝑎; [2 marks] (b) an expression for the cumulative distribution function 𝐹(𝑥). [2 marks] (c) the interquartile range. [2 marks] 4. [Maximum mark: 6] The marks in an IB Mathematics HL exam are distributed normally with mean μ and standard deviation σ. If the cut off score for a 7 is a mark of 80%, and 10% of students get a 7, and the cut off score for a 6 is a mark of 65% and 30% of students get a 6 or 7, find the mean and standard deviation of the marks in this exam. Give your answers correct to two significant figures. 5. [Maximum mark: 6] Find the exact value of arcsin(sin 4) 4 + arccos(cos 3) 3 + arctan(tan 2) 2 + arccot(cot 1) 1 . 6. [Maximum mark:6] Given that 𝑧 = (𝑏 + 𝑖)2, where 𝑏 is real and positive, find the exact value of 𝑏 when arg 𝑧 = 𝜋 3.

Anglo-Chinese School (Independent)/ Mathematics Department / IBDP Mathematics HL / Mock Exam / Paper 1 Page 2 of 4 7. [Maximum mark: 6] Let 𝑋 be a random variable. By expanding the expression 𝐸 [(𝑋 − 𝐸(𝑋)) 2 ] show that 𝐸(𝑋2) ≥ (𝐸(𝑋)) 2 . 8. [Maximum mark: 6] Find an equation of the plane containing the two lines 𝑥 − 1 = 1 − 𝑦 2 = 𝑧 − 2 and 𝑥 + 1 3 = 2 − 𝑦 3 = 𝑧 + 2 5 . 9. [Maximum mark: 6] (a) State the transformations from graph 𝑦 = 𝑎 sin(𝑥 + 𝑏) + 𝑐 to graph 𝑦 = sin 𝑥. [3 marks] (b) The graph below represents 𝑦 = 𝑎 sin(𝑥 + 𝑏) + 𝑐, where 𝑎,𝑏, and 𝑐 are constant. Find values for 𝑎, 𝑏 and 𝑐. [3 marks] 10. [Maximum mark: 6] Given that (1 − 2𝑥)5(1 + 3𝑥)4 = 𝑎 + 𝑏𝑥 + 𝑐𝑥2 + ⋯, find the values for 𝑎, 𝑏 and 𝑐.

Anglo-Chinese School (Independent)/ Mathematics Department / IBDP Mathematics HL / Mock Exam / Paper 1 Page 3 of 4 Section B Answer all the questions on the answer sheets provided. Please start each question on a new page. 11. [Maximum mark: 13] (a) The function 𝑓 is defined by 𝑓(𝑥) = (𝑥 + 2)2 − 3. The function 𝑔 is defined by 𝑔(𝑥) = 𝑎𝑥 + 𝑏, where 𝑎 and 𝑏 are constants. Find the value of 𝑎, 𝑎 > 0 and the corresponding value of 𝑏, such that 𝑓(𝑔(𝑥)) = 4𝑥2 + 6𝑥 − 3 4. [8 marks] (b) The functions ℎ and 𝑘 are defined by ℎ(𝑥) = 5𝑥 + 2 and 𝑘(𝑥) = 𝑐𝑥2 − 𝑥 + 2 respectively. Find the value of 𝑐 such that ℎ(𝑘(𝑥)) = 0 has equal roots. [5 marks] 12. [Maximum mark: 14] (a) Sketch the graphs of 𝑦 = 𝑥 + 1 𝑥 − 1 and 𝑦 = |3𝑥 − 5|. State intercepts, asymptotes, maximum and minimum points of each graph clearly if there is any. [8 marks] (b) Hence, find the range(s) of 𝑥 for which 𝑥 + 1 𝑥 − 1 < |3𝑥 − 5| where 𝑥 ≠ 1. [6 marks] 13. [Maximum mark: 16] (a) Using Mathematical Induction, prove that 𝑑𝑛 𝑑𝑥𝑛 (cos 𝑥) = cos (𝑥 + 𝑛𝜋 2 ), for all positive integer values 𝑛. [7 marks] (b) Solve the equation sin 4𝑥 = cos 𝑥 for – 2𝜋 3 < 𝑥 < 𝜋 4. [9 marks] 14. [Maximum mark: 17] Let 𝐴 be the point (2, −1,0), 𝐵 the point (3,0,1) and 𝐶 the point (1, 𝑚, 2), where 𝑚 ∈ ℤ, 𝑚 < 0. (a) Given that 𝐴𝐵̂𝐶 = arccos √2 3 , show that 𝑚 = −1. [6 marks] (b) Determine the Cartesian equation of the plane 𝐴𝐵𝐶. [4 marks] (c) The line 𝐿 is perpendicular to plane 𝐴𝐵𝐶 and passes through 𝐴. Find a vector equation of 𝐿. [3 marks] (d) The point 𝐷 (6, −7,2) lies on 𝐿. Find the volume of the pyramid 𝐴𝐵𝐶𝐷. [4 marks]

Anglo-Chinese School (Independent)/ Mathematics Department / IBDP Mathematics HL / Mock Exam / Paper 1 Page 4 of 4 1 ,24 ,3 −=+−== ckba  2 1 ,3 == ba Answers 1(b) k = -3 2. 8/3 3(a) a = 1 or 1 + 4k (b) 𝐹(𝑥) = 1 2 (sin ( 𝑥𝜋 2 ) + 1) (c) 2/3 4. μ=55 σ=-20 5. 2 − 𝜋 4 6. √3. 8. −7𝑥 − 2𝑦 + 3𝑧 = −3 or 𝒓 = ( 1 1 2 ) + 𝜇 ( 1 −2 1 ) + 𝜆 ( 3 −3 5 ) 9(a) • Translate the graph along negative 𝑦-axis by 𝑐; • Translate the graph along positive 𝑥-axis by 𝑏; • Stretch vertically by factor 1 𝑎. (b) 10. 𝑎 = 1, 𝑏 = 2, 𝑐 = −26 11(a) (b) 𝑐 = 5 48 12. 𝑥 < 1 𝑜𝑟 𝑥 > 9+√33 6 13(b) 𝑥 = − 𝜋 2 , − 3𝜋 10 , 𝜋 10 , 𝜋 6 14(a)(i) 𝐵𝐴⃗⃗⃗⃗⃗ ⋅ 𝐵𝐶⃗⃗⃗⃗⃗ = 1 − 𝑚 and show m = -1 (b) −2𝑥 + 3𝑦 − 𝑧 = −7 (c) √14 2 (d)(i) 𝑟 = ( 2 −1 0 ) + 𝜆 ( −2 3 −1 ) (ii) 14/3