Surds Questions and Answers - Form 3 Topical Mathematics

Surds

1. Simplify; ³/_{(√7 − 2)} + ¹/_√7  leaving the answer in the form a + b c , where a, b and c are rational numbers 
2. Given that:-
   2+√5  −  3 + √5 = a+b√5
   2-√5        2+√5
   Find the values of a and b where a and b are rational numbers
3.  If:
     14       −    14    = a√7 + b√2
   7 - √12     7 + √2
   Find the values of a and b, where a and b are rational numbers *
4. Rationalize the denominator
    2 − √2  
   (√2 − 1)³
   and express your answer in the form of a + c√2
5. The figure below is a right pyramid with a rectangular base ABCD and vertex V. 
   
   O is the centre of the base and M is a point on OV such that OM = ¹/₃ OV, AB = 8 cm, BC = 6 cm and VA = VB=VD = VC = 15 cm. Find ;
   1. The height OV of the pyramid.
   2. The angle between the plane BMC and base ABCD.
6. Find the value of y which satisfies the equation
   Log ₁₀5 – 2 + log₁₀ (2y + 10) = log₁₀ (y – 4)
7. Simplify the expression
   √ 3 - √2 giving your answer in the for of a + b√c.
   √3 + √2

Answers

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5. (2 – 3 ) (3 + 2)
   3(2)2 – 2)2
   3x2 – 3 + 2 – 2
   9x2 – 4x3
   6 – 3 + 2 – 6
   18 – 12 = 6
6. 1. Or = 16² - 5²
      = √(256 – 25)
      = 15.198 cm
   2. tan θ= ^5.066/₄ = 1.2665
      ∴ θ51.710
7. Log₁₀5 – log₁₀10² + log₁₀(2y + 10) = log₁₀(y – 4)
   Log₁₀{^{5 (2y + 10)}/_10²} = log ₁₀ (y-4)
   10y + 50 = 100 y – 400
   90 y = 450
   y = 5
8.