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1-1 standard form and indices

IB Mathematical test paper on indices<br>

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1-1 standard form and indices

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  1. IB DP Maths 2 hours 19 questions 1.1 Number Toolkit 1.1.1 Standard Form / 1.1.2 Laws of Indices Medium (7 questions) /36 Scan here to return to the course or visit savemyexams.com Hard (6 questions) /36 Very Hard (6 questions) /32 Total Marks /104 © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 1

  2. Medium Questions 1 (a) Let Q =  30sin 2a , where a=45° and b=2. 8b Calculate the exact value of Q. (1 mark) (b) Give your answer from part (a) correct to (i) two decimal places (ii) two significant figures. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 2

  3. 4x 2 (a) Let  R= 6 cos5y, where x=1.25 and y=36°. Write the angle of y in radians. (1 mark) (b) Find the value of R. Give your answer as a fraction. (2 marks) (c) Give your answer from part (b) to (i) one decimal place (ii) three significant figures. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 3

  4. Consider the numbers a=4.14×106 and b=2.54×10−7. 3 (a) a b 3 Calculate C =10⎛⎜⎜ ⎞⎟⎟ ⎠ . Give your answer correct to the ⎝ (i) nearest integer (ii) three significant figures. (3 marks) (b) Give your answer to part (a) (i) in the form a×10k, where 1≤a≤10 and k ∈ℤ. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 4

  5. 4 (a) A cylinder has radius of 12.7 cm and height of 14.4 cm. Calculate the volume of the cylinder correct to (i) one decimal place (ii) three significant figures (iii) the nearest integer. (3 marks) (b) Write your answer to part (a) (ii) in the form a×10k, where 1≤a≤10 and k ∈ℤ. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 5

  6. A rectangular field has length, L, of 25.2 m and width, W, of 21.4 m, each correct to 1 decimal place. 5 (a) Calculate the lower and upper bound for L (i) W (ii) (2 marks) (b) Calculate the lower and upper bound for the perimeter, P area, A, of the field. (i) (ii) (4 marks) Calculate the following, giving your answer in the form a×10k, where 1≤a≤10 and  k ∈ℤ 6 6.2 ×10−5 (i) 4×( ) 4 ×105−( 5×104 (ii) ( ) ) 4321−1( 1.2×10−1 (iii) ( ) ) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 6

  7. (6 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 7

  8. 7 (a) Consider the following four numbers. 5 -1 2 b = 0.0272 ×10 d = 2.72 × 10 a = 0.272 c = e(10e) Write down the number that is in the form a×10k, where 1≤a≤10 and k ∈ℤ  (i) (ii) the largest of these numbers. (2 marks) Find the value of a+b−c+d. (b) (i) Give your answer to part (b)(i) in the form a×10k, where 1≤a≤10 and k ∈ℤ (ii)                      . (4 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 8

  9. Hard Questions ( 4 sin 2q−2 ( ) ) 6 tanq+2 ) r+s2 , where q=π 1 (a) Let P = 6, r=6 and s=2. 10( Calculate the exact value of P. (4 marks) (b) Give your answer from part (a) correct to (i) two decimal places. (ii) two significant figures. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 9

  10. tanx 2cos2x+y⎛⎜⎜ ⎞⎟⎟ ⎠ ( ) 2−z ) , where x=π ⎝ 2 (a) Let W = 2, y= −1 and z=2. 5sinx+z2 10( Find the value of W. Give your answer as a fraction. (4 marks) (b) Give your answer from part (a) to (i) three decimal places. (ii) three significant figures. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 10

  11. 3 (a) A prism has a cross sectional area of 5.50×103cm2 and volume of 4.40×104cm3. Calculate the length of the prism. (3 marks) (b) The cross-sectional area of the prism is in the shape of a trapezium and its parallel sides measure 2 m and 2.4 m. Calculate the height of the trapezium. Give your answer in cm. (3 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 11

  12. 4 (a) Mary has found the exact answer for R is  45 16. Write down the exact answer of as a decimal. (3 marks) (b) Give your answer from part (a) correct to (i) three decimal places (ii) one significant figure. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 12

  13. 3 1 2, where  0≤a≤90° and 0≤b ≤90°. 5 (a) It is given that sin a =  and sin b = 2 Find the size of the angles a and b. (2 marks) sin a sin bcm. (b) A circle has radius r equal to Find the exact value of the area of the circle, giving your answer in terms of π. (4 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 13

  14. 6 (a) A medium rare steak should have an internal temperature of 55°C to 56°C. Max decides to go to 10 different steak houses, he measures the internal temperature of a medium rare steak at each establishment and records the following: 51.0,   52.1,  62.9,  49.0,  59.8,  50.2,  54.3,  47.7,  48.6,   65.4 Find the mean internal temperature of Max’s recordings. (1 mark) (b) Max goes to 5 more steak houses and calculates the mean of all 15 restaurants to be  55.2°C. Calculate the mean internal temperature from the 5 additional steak houses Max went to. (3 marks) (c) Max records one last steak that has an internal temperature of T°C. Calculate the interval of T such that the mean internal temperature for all 16 steaks is within the temperature range for a medium rare steak. (3 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 14

  15. Very Hard Questions Consider the numbers  a=11 2, b=(5+6π), c= 2, d=6(π−1). 1 (a) Giving your answer to 1 decimal place, calculate the value of a (i) b (ii) c (iii) d (iv) (2 marks) (b) Points P and Q have coordinates ( a, b  and ( ) ) c, d  respectively. The formula for the distance, d, between two points with coordinates ( ( ) x2,y2 is given in your formula booklet. x1, y1 and ) (x1−x2)2+(y1−y2)2 d= Using your answers from part (a), calculate the distance, d, between points P and Q. Give your answer correct to 1 decimal place. (2 marks) pq−2r3 and T = pqr−1, where p=sinπ 2 Let Y= ( ) 3, q= 3, r=2. Find the exact value of YT. © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 15

  16. (5 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 16

  17. 3 (a) Point A has coordinates (−1, 7) and point B has coordinates (11, 12). x1, y1 and  The formula for the distance, d, between two points with coordinates ( ( ) x2, y2 is given in your formula booklet. ) 2+( 2 d = x1−x2 y1−y2 ( ) ) Calculate the distance between points A and B. (3 marks) (b) The formula for the coordinates of the midpoint of a line segment with endpoints  ( ) x1, y1 and ( ) x2, y2 is given in your formula booklet. x1+x2 2 y1+y2 2 ⎛⎜⎜⎜ ⎝ ⎞⎟⎟⎟ ⎠ , Calculate the midpoint of the line segment with endpoints A and B. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 17

  18. c2tan212d−1( ) a sin24b ( 4 (a) Let S = ( and d = 5° ) ) a +c−cos 48b , where a = 16, b = 7.5°, c = 3 2 2 Note: sin  θ = (sin θ) Find the value of S, giving your answer as a fraction. (2 marks) a +c2−2sin 54d (b) Let X =  (a3) −a−c Find the value of X, giving your answer as a fraction. (2 marks) (c) Calculate the value of SX, giving your answer as a fraction. (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 18

  19. Consider the numbersp=2.41×104 and q=4.12×105. 5 (a) Giving your answers in the form a×10k, where 1≤a<10, k ∈ℤ, calculate p+q (i) p−q (ii) q−p (iii) p q (iv) (4 marks) (b) The formula for the distance, d, between two points with coordinates (x1, y1) and ( ) x2, y2 is given in your formula booklet. (x1−x2)2+(y1−y2)2 d= Using your answers to part (a), estimate the distance between points A(p+q, p−q) and B⎛⎜⎜ ⎝ ⎠ q q−p,p ⎞⎟⎟ . (2 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 19

  20. 6 The mean height of the four tallest students in a classroom is 176 cm and the mean height of the six tallest students is 165 cm. The fifth tallest student is 4 cm taller than the sixth tallest student. Find the heights of the fifth and sixth tallest students. (6 marks) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 20

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