# Further Mathematics Questions and Answers

Test your knowledge of advanced level mathematics with this aptitude test. This test comprises Further Maths questions and answers from past JAMB and WAEC examinations.

1.

If $$(x - 5)$$ is a factor of $$x^3 - 4x^2 - 11x + 30$$, find the remaining factors.

A.

$$(x + 3) and (x - 2)$$

B.

$$(x - 3) and (x + 2)$$

C.

$$(x - 3) and (x - 2)$$

D.

$$(x + 3) and (x + 2)$$

(x - 5) is a factor of $$x^3 - 4x^2 - 11x + 30$$. To find the remaining factors, let's draw out $$(x - 5)$$ from the parent expression.

$$x^3 - 4x^2 - 11x + 30 = x^3 - 5x^2 + x^2 - 5x - 6x + 30$$

$$= x^2(x - 5) + x(x - 5) - 6(x - 5) = (x - 5)(x^2 + x - 6)$$

∴ To find the remaining factors, we factorize $$(x2 + x - 6)$$

$$x^2 + x - 6 = x^2 + 3x - 2x - 6$$

$$= x(x + 3) - 2(x + 3) = (x + 3)(x - 2)$$

∴ The other two factors are $$(x + 3) and (x - 2)$$

ALTERNATIVELY
$$∴ x^2 + x - 6 = (x + 3) and (x - 2)$$

2.

In how many ways can four Mathematicians be selected from six ?

A.

90

B.

60

C.

15

D.

360

$$=^6C_4$$

$$=\frac{6!}{(4!\times2!)}$$

$$=\frac{6\times5}{2\times1}$$

= 15

3.

Find the coefficient of the $$6^{th}term$$ in the binomial expansion of $$(1 - \frac{2x}{3})10$$ in ascending powers of $$x$$.

A.

$$-\frac{896x^6}{9}$$

B.

$$-\frac{896x^5}{9}$$

C.

$$-\frac{896x^5}{27}$$

D.

$$-\frac{896x^6}{27}$$

rth term of a binomial expansion =$$^nC_r-1 a^{n-(r-1)}b^{r-1}$$

$$n = 10,r = 6 \therefore r-1=5$$

6th term =$$^{10}C_5 1^{10} - 5 (-\frac{2}{3}x)^5$$

$$=252*1*-\frac{32x^5}{243}=-\frac{896x^5}{27}$$

4.

If m and ( m + 4) are the roots of $$4x^2 - 4x - 15 = 0$$, find the equation whose roots are 2 m and (2 m + 8)

A.

$$x^2+8x-15=0$$

B.

$$x^2-2x-15=0$$

C.

$$x^2-8x-15=0$$

D.

$$x^2+2x+15=0$$

$$x^2-$$(sum of roots)$$x+$$(product of roots) = $$0$$

$$4x^2-4x-15=0$$

Divide through by 4

$$=x^2-x-\frac{15}{4}=0$$

$$=x^2-x+(-\frac{15}{4})=0$$

$$=x^2-(1)x+(-\frac{15}{4})=0$$

sum of roots =1

= m + (m + 4) = 1
=2m+4=1

=2m=-3

=m=-$$\frac{3}{2}$$

The equation whose roots are 2m and 2m+8

2m=2×-$$\frac{3}{2}=-3$$and $$2m+8=2×-\frac{3}{2}+8=5$$

$$=x^2-(-3+5)x+(-3)(5)=0$$

$$=x^2-2x+(-15)=0$$

$$∴x^2-2x-15=0$$

5.

Given that $$p = \begin{bmatrix} x&4\\3&7\end{bmatrix} Q =\begin{bmatrix} x&3\\1&2x\end{bmatrix}$$ and the determinant of $$Q$$ is three more than that of $$P$$ , find the values of $$x$$

A.

$$-2,\frac{3}{2}$$

B.

$$2,\frac{3}{2}$$

C.

$$-2,-\frac{3}{2}$$

D.

$$2-,\frac{3}{2}$$

$$p = \begin{bmatrix} x&4\\3&7\end{bmatrix}, Q =\begin{bmatrix} x&3\\1&2x\end{bmatrix}$$
$$p = \begin{bmatrix} x&4\\3&7\end{bmatrix}=7x-12, Q =\begin{bmatrix} x&3\\1&2x\end{bmatrix}=2x^2-3$$