Mathematics Part 2

Polynomials Question 1.Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients: Solution:1. We have: x2 – 2x – 8 = x2 + 2 – 4x – 8= x(x + 2) – 4(x + 2)= (x + 2)(x – 4)The value of x2 – 2x – 8 is 0, when […]

Polynomials Question 1.The graphs of y = p(x) are given in figures below for some polynomials p(x). Find the number of zeroes of p(x), in each case. Solution:

Real Numbers Question 1.Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: Solution:We know that if the denominator of a rational number has no prime factors other than 2 or 5 (or both), then it is expressible as a terminating

Real Numbers Question 1.Prove that √5 is irrational. Solution:Let us assume, to the contrary, that √5 is rational.Squaring on both sides, we get5 = a2/b2 or 5b2 = a2 ………………. (1)This shows that a2 is divisible by 5.It follows that a is divisible by 5. …………………. (2)So, a = 5m for some integer m.Substituting a = 5m in (1), we get5b2 = (5m)2 = 25m2or b2 =

Real Numbers Question 1.Express each number as a product of its prime factors:(i) 140(ii) 156(iii) 3825(iv) 5005(v) 7429Solution:(i) We use the division method as shown below: ∴ 140 = 2 × 2 × 5 × 7 = 22 × 5 × 7 (ii) We use the division method as shown below: ∴ 156 = 2 ×

Real Numbers Question 1.Use Euclid’s division algorithm to find the HCF of: (i) 135 and 225(ii) 196 and 38220(iii) 867 and 255 Solution: (i) We have:Dividend = 225 and Divisor = 135The process can be exhibited as under: Hence, HCF (135, 225) = 45. (ii) We have:Dividend = 38220 and Divisor = 19638220 = 196