1998 - JAMB Mathematics Past Questions and Answers - page 1

\(\frac{11 + 2}{6x^2 - x - 1}\) = \(\frac{11 + 2}{3x + 1}\)

= \(\frac{A}{3x + 1}\) + \(\frac{B}{2x - 1}\)

11x = 2 = A(2x - 1) + B(3x + 1)

put x = \(\frac{1}{2}\)

= -\(\frac{-5}{3}\)

= -\(\frac{-5}{3}\)A \(\to\) A = 1

∴ \(\frac{11x +2}{6x^2 - x - 1}\) = \(\frac{1}{3x + 1}\) + \(\frac{3}{2x - 1}\)

\(\frac{1}{3x}\) + \(\frac{1}{2}\) > \(\frac{1}{4x}\)

= \(\frac{2 + 3x}{6x}\) > \(\frac{1}{4x}\)

= 4(2 + 3x) > 6x = 12x² - 2x = 0

= 2x(6x - 1) > 0 = x(6x - 1) > 0

Case 1 (-, -) = x < 0, 6x -1 < 0

= x < 0, x < \(\frac{1}{6}\) = x < \(\frac{1}{6}\) (solution)

Case 2 (+, +) = x > 0, 6x -1 > 0 = x > 0, x > \(\frac{1}{6}\)

Combining solutions in cases(1) and (2)

= x > 0, x < \(\frac{1}{6}\) = 0 < x < \(\frac{1}{6}\)

2p - 10 = \(\frac{p + 1 + 1 - 4P^2}{2}\) (Arithmetic mean)

= 2(2p - 100 = p + 2 - 4P²)

= 4p - 20 = p + 2 - 4p²

= 4p² + 3p - 22 = 0

= (p - 2)(4p + 11) = 0

∴ p = 2 or -\(\frac{4}{11}\)

Let the G.p be a, ar, ar², S³ = \(\frac{1}{2}\)S

a + ar + ar² = \(\frac{1}{2}\)(\(\frac{a}{1 - r}\))

2(1 + r + r)(r - 1) = 1

= 2r³ = 3

= r³ = \(\frac{3}{2}\)

r(\(\frac{3}{2}\))^{\(\frac{1}{3}\)} = \(\sqrt{\frac{3}{2}}\)

x \(\ast\) y = xy - y - x, x \(\ast\) 3 = 3x - 3 - x = 2x - 3

2 \(\ast\) x = 2x - x - 2 = x - 2

∴ 2x - 3 = x - 2

x = -2 + 3

= 1

\(\begin{vmatrix} x & 1 & 0 \ 1-x & 2 & 3 \ 1 & 1+x & 4\end{vmatrix}\) = x\(\begin{vmatrix}2 & 3 \ 1+x & 4\end{vmatrix}\) - \(\begin{vmatrix}1-x & 3 \ 1 & 4\end{vmatrix}\) = 0

= x[8 - 3(1 + x)] - [4(1 - x)-3] - 0 = x[5 - 3x] - [1 - 4x]

= 5x - 3x² -1 + 4x

= -3x² + 9X - 1

PQ = \(\begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}\)\(\begin{pmatrix} u & 4+u \ -2v & v \end{pmatrix}\)

= \(\begin{pmatrix} (2u-6v & 2(4+u) +3v)\ 4u-10v & 4(4+u)+5v \end{pmatrix}\)

= \(\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\)

2u - 6v = 1.....(i)

4u - 10v = 0.......(ii)

2(4 + u) + 3v = 0......(iii)

4(4 + u) + 5v = 1......(iv)

2u - 6v = 1 .....(i) x 2

4u - 10v = 0......(ii) x 1

\(\frac{\text{4u - 12v = 0}}{\text{-4u - 10v = 0}}\)

-2v = 2 = v = -1

2u - 6(-1) = 1 = 2u = 5

u = -\(\frac{5}{2}\)

∴ (U, V) = (-\(\frac{5}{2}\) - 1)

V = \(\pi r^2 h\)

= 123.2 x 1000

= \(\frac{22}{7}\) x 28 x 28 x h

= 123200 = 88 x 28h

= 2464h

∴ h = \(\frac{123200}{2464}\)

= 50cm

N.B: 1 litre = 1 dm³

= 1000cm³