SPM 2023 Add Maths Paper 1, Question 13Indices, Surds & Logarithms

Solved on video in English and Bahasa Melayu · Form 4, Indices, Surds & Logarithms · 6:23 video

The question

(a)(i)

Express y in terms of x and/or e for (a)(i) 6 ln y = √5 ln x.

(a)(ii)

Express y in terms of x and/or e for (a)(ii) ln(x + y²) − 3 = 0.

(b)

Diagram 9 shows the length of the chord BC in the semicircle ABCD. It is given that BC = (x√2 − 1) cm, the diameter AD = (x + 2√2) cm, and the ratio of the radius of the semicircle to the length of the chord is 6 : 1. Find the value of x in the form a + b√2, such that a and b are rational numbers.

SPM 2023 Add Maths Paper 1, Question 13: the question as drawn in Eduly's video6:23

Part (a)(i)

Step 1

Apply the power law of logarithms to both sides: ln y⁶ = ln x^(√5).

Why this step

Each side must become one single log before we can drop the ln. The power law lifts the 6 and the √5 up as powers to do that.

2 more steps, each with the reason for it, in the full solution

Final answer

y = x^(√5/6)

Where students lose marks

Watch out: keep √5 as √5. Do not turn it into 1/5, which wrongly gives y = x¹/30).

Part (a)(ii)

4 more steps, each with the reason for it, in the full solution

Final answer

y = ±√(e³ − x)

Where students lose marks

Watch out: the question says "in terms of e", so keep e³ exact. Do not write 20.09.

Part (b)

10 more steps, each with the reason for it, in the full solution

Final answer

x = 60/287 + (146/287)√2

Where students lose marks

Watch out: show the rationalising step clearly. Multiply top and bottom by the conjugate, then simplify to the form a + b√2.

Watch every step of this question solved on video (6:23), with the reason behind each step and a box that checks your own answer.

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