Sec 3: Surd

University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

117

Pages:

2
Author:

MAlphen

SURD 1 9 Given that √𝑝 + 𝑞√8 = , where p and q are rational numbers, find the values 4−√8 [4] of p and q. 2 (i) The length of each side of an equilateral triangle is 6(√3 − 1) cm. Without the using a calculator, express the area of the triangle in the form (𝑎 + 𝑏√3) cm2 where a and b are integers. (ii) The triangle in part (i) is the base of a prism. Given that the volume of the prism is 9(√3 − 1)cm3, express the height of the prism in the [3] [4] form (𝑚 + 𝑛√3) cm where m and n are fractions in the simplest form. 3 Without using a calculator, find the integer value of 𝑎 and of 𝑏 for which the 𝑎+𝑏 solution of the equation 2𝑥√5 = 𝑥√2 + √18 is √ . 3 1 9 Solutions: √𝑝 + 𝑞√8 = 4−√8 9 𝑝 + 𝑞√8 = = = = = ( ) 4− √8 81 24−8√8 81 × 24+8√8 24−8√8 24+8√8 81(24+8√8) 64 243 81 + 8 √8 8 243 𝑝= Ans: 𝑝 = 2 243 8 or 8 𝑞= 81 8 𝑞= 81 8 [3] 2 Solutions: (i) Area =× 6(√3 − 1) × 6(√3 − 1) sin 60 ° √3 = 18(3 + 1 − 2√3) ( ) 2 = 9√3(4 − 2√3) = 36√3 − 54 (ii) height = 9(√3−1) = 9(√3−1) = = 36√3−54 36√3+54 × 36√3−54 36√3+54 9(108+54√3−36√3−54) 972 54+18√3 108 1 = 1 + √3 2 6 1 1 Ans: (i) 36√3 − 54 (ii) + √3 2 6 3 Solution: 𝑥(2√5 − √2) = √18 2√5+√2 √18 𝑥 = 2√5−√2 × 2√5+√2 = = 2√90+6 18 6√10+6 18 √10+1 3 = 𝑎 = 10, 𝑏 = 1 Ans: 𝑎 = 10, 𝑏 = 1