2006 - JAMB Mathematics Past Questions and Answers - page 4

Locus of point equidistant from a given straight is the perpendicular bisector of the straight line
∴Gradient of the line ax + by + c = 0
implies by = -ax - c
y = -^a/_bx - c
the gradient (m) = ^a/_b
∴Gradient of the perpendicular (m) = ^b/_a
If P(x,y) is the point intersection of the two lines equation of the perpendicular becomes
y - y₁ = m(x - x₁)
y - y₁ = ^b/_a(x - x₁)
ay - ay₁ = bx - bx₁
ay - bx + bx₁ - ay₁ = 0
ay - bx + b(x₁ - y₁) = 0
implies ay - bx + q

Each ∠of a regular polygon
\(=\frac{(n-2)180}{n}\\ =\frac{(6-2)180}{6}\\ =\frac{(4)180}{6}\)
= 120^o
ΔQSR is isosceles
∴ Q = S = 30^o
Also ΔTSR is isosceles
∴ T = R = 30^o
∠TSV + ∠VSR = 120
∠TSV + 30 = 120
∠TSV = 120 - 30
∠TSV = 90^o
∴∴VTS + ∴VST + ∴TVS = 180 (sum of ∠s of a Δ)
30 + 90 + ∠TVS = 180
120 + ∠TVS = 180
∠TVS = 180 - 120
∠TVS = 60^o

Locus of point equidistant from points P and q is a perpendicular bisector of the straight joining P and Q
∴PN = NQ

In ΔPST; cos 60 = ^ST/₈
ST = 8cos 60
ST = 8 x ¹/₂ = 4
TR = 10 cm (opp. sides of a rectangle PQRT)
SR = ST + TR
SR = 4 + 10
SR = 14 cm

Y = x implies y ≤ x
Y + x = 4 implies y = 4 – x
∴y = 4 – x
∴y ≤ x, y + x ≤ 4 and y ≥ 0

X ⊕ y = ^xy/₆ = ^xe/₆ = x
Xe = 6x
e = 6
X ⊕ y = ^xx1/₆ where x¹ = 20
Xx20/6 = 6
X = ³⁶/₂₀
X = ⁹/₅

P ∝ ¹/_q³
P = ^K/_q³
q³ = ^K/_P
q = ^K/_p^1/3
But q ∝ r²
q = Kr^2
K/_p^1/3 = Kr²
r² = ^K/_p^1/3 x ¹/_K
r² = ¹/_p^1/3
r = (¹/_p^1/3)²
r = ¹/_p^1/6
r ∝ ¹/_p^1/6
∴ r varies inversely as ⁶√P

\(x \times y = \frac{x^2 - y^2}{2xy}find -5 \times 3\\ =\frac{(x+y)(x-y)}{2xy}\\ =\frac{(-5+3)(-5-3)}{2(-5\times3)}\\ =\frac{-2 \times -8}{2(-5\times3)}\\ =\frac{-8}{15}\)

F(x) = Q x D + R
Kx³ + x² - 5x – 2 = Q(2x+1)+R
If 2x+1 = 0 implies x = -¹/₂
∴k(-¹/₂)³ + (-¹/₂)² -5(-¹/₂) -2
= Q(2(-¹/₂) + 1) +2
K(-¹/₈) + ¹/₄ + ⁵/₂ - 2 = Q(-1+1)+2
-^k/₈ + ¹/₄ + ⁵/₂ - 2 = 0+2
(-k+2+20-16) / 8 = 2
(-k+6) / 8 = 2
-k+6 = 2*8
-k+6 = 16
-k = 16-6
-k = 10
∴k = -10

\(\frac{x+4}{3}-\frac{x-3}{2} < 4\\ \frac{(2(x+4))-(3(x-3))}{6}<4\)
2(x+4) – 3(x-3) < 4 x 6
2x + 8 – 3x+9 < 24
-x + 17 < 24
-x < 24 – 17
-x < 7
X > -7