Analysis and Approaches SL — Questionbank

Exponents & Logs - Number & Algebra

Logarithms, Exponent & Logarithms Laws, Solving Exponential & Logarithmic Equations …

Question 1

NO CALCULATOR

Mild

[Maximum mark: 6]

Nickel in the asteroid 433 Eros is estimated to be worth 9457 quadrillion euros (EUR).
(Note: 1 quadrillion =10¹⁵)

(a) Write this value in standard scientific notation, in the form. a × 10^k, where 1 ≤ a < 10, k ∈ Z.

Ashley assumes that the asteroid is approximately spherical with a radius of 146 km.

[2]

(b) Using this assumption, calculate Ashley’s estimate of the asteroid’s volume, in km³

The actual volume of the asteroid is known to be. 9.72 x 10⁶

[2]

(c) Calculate the percentage error in James’s estimate compared to the actual volume.

[2]

NO CALCULATOR

Mild

[Maximum mark: 4]

A sphere with diameter 12 104 000 metres can model the shape of the Venus.


(a) Use this model to calculate the circumference of Venus in kilometres. Round your answer to three significant figures	[3]
(b) Write your answer from (a) in the form a × 10^k, where 1 ≤ a < 10 , k ∈ Z.	[1]

NO CALCULATOR

Spicy

[Maximum mark: 16]

Consider the arithmetic sequence

$b, m, n,\ldots$

where $b, m, n \neq 0$


(a) Show that 2m − n = b Now consider the geometric sequence b, r, u, … where b, r, u ≠ 0	[2]
(b) Show that r² = bu The first term of both sequences is b. It is given that n = u = 9	[1]
(c) Show that m > 3 Consider the case where b = 25, r > 0 and n = u = 9	[2]
(d.i) Write down the first four terms of the arithmetic sequence	[2]
(d.ii) Write down the first four terms of the geometric sequence A new arithmetic sequence (v_k)is formed by adding together corresponding terms of the two sequences: v_k = (term from arithmetic sequence) + ln (term from geometric sequence). It is given that v₁ = 25 + ln 25, v₂ = 17 + ln 15,	[2]
(e.i) Find the common difference of the sequence (v_k) in terms of ln 3 and ln 5	[2]
(e.ii) Show that $\sum_{k = 1}^{10}v_{k} = - 110 - 25\ln 5 + 45\ln 3$	[4]