Mathematical Solutions - Polynomials

Solution.1
(a) 9 a4 b3−3 a3 b4
6 a3 b3−2 a2 b4
= 3 a3 b3 ( 3 a−b )
2 a2 b3 ( 3 a−b )
= 3 a3 b3 ( a−2 b−2 )
2
= 3 a
2
(b) 3 √2− √5
2 √5+ √2
= 3 √ 2− √ 5
2 √ 5+ √ 2 × 2 √ 5− √ 2
2 √ 5− √ 2
= 3 √ 2 .2 √ 5−3 √ 2 . √ 2−2 √ 5 . √ 5+ √ 5 . √ 2
20−2
= 6 √10−6−10+ √10
18
= 7 √ 10−16
18
(c) (9 x )3 / 4 x1 /3
√ 27 x3
= (9 x )1 /2 ¿ ¿
= 3 x1/ 2 ¿ ¿
= ¿ ¿
(d) 3
2 x−3 - 4
x+ 4 = 0
= 3 ( x + 4 ) −4 (2 x −3)
( 2 x−3 ) (x−4 )

= 3 ( x + 4 ) −4 (2 x −3)
( 2 x−3 ) (x−4 )
= 3 x +12−8 x +12
2 x2 +8 x−3 x−12
= −5 x +24
2 x2 +5 x−12
= −5 x +24
2 x ( x+ 4 )−3( x +4 )
= −5 x+ 24
(2 x−3)(x+ 4)
(e) M = w
2 (L –l2)
2M = w (L –l2)
2M = wL – wl2
wl2 = wL – 2M
l = √ L− 2 M
w
Solution.2
(a) The solution of the student is showing in the form of an equation without any
elaboration, hence it is difficult to follow. So the better way to solve the question is to
elaborate the answer step by step.
(b) Given that,
Diameter of the circle = 2 m;
Length of the rectangle = 3m;
Circumference of the circle = diameter × π
= 2 × π = 2π
Total length of the rectangle = 3+3 = 6m

So overall circumference = 2π +6
Car is driven around the track = 25 times
Hence overall covered distance = 25 (2π +6 ¿
= 25(2( 22
7 )+6)
= 307.14m
Solution.3
(a) y = -3x-2
Points = (-2, 4)
x = -2, y = 4
By substituting the values of x and y in the given equation;
4 = - 3(-2) – 2
4 = 6 -2
4=4
LHS = RHS
Hence the point (-2, 4) satisfies the equation y = -3x – 2.
(b) Gradient = 2
Points = (1, 8)
y = mx +c
8= 2(1) +c
C = 6
By substituting the value of c in equation;
y = 2x + 6
(c) Graph for two lines: