Exploring Algebraic Identities | EOT

Question 5

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

(i) $25a^2 - 30ab + 9b^2$

(ii) $36s^2 - 49t^2$

Solution

We will use the properties of a geometric progression.

Step 1 — Set up equations

Let the first term of the GP be $a$ . Let the common ratio be $r$ . The GP terms are $a, ar, ar^2, ar^3, ar^4, \dots$ . The sum of the first two terms is $\mathbf{-4}$ .

$a + ar = -4$

$a(1 + r) = -4 \quad \text{(Equation 1)}$

The fifth term is $\mathbf{4}$ times the third term. The fifth term is $ar^4$ . The third term is $ar^2$ .

$ar^4 = 4ar^2 \quad \text{(Equation 2)}$

$\boxed{\text{Two equations are formed}}$

Step 2 — Find the common ratio

Let's solve Equation 2 for $r$ . We divide both sides by $ar^2$ . Note that $a$ cannot be zero. If $a$ were zero, $a+ar$ would be zero. But $a+ar$ is $\mathbf{-4}$ .

$ar^4 = 4ar^2$

$r^2 = 4$

$r = \pm\sqrt{4}$

$\boxed{r = 2 \text{ or } r = -2}$

Step 3 — Find the first term

We will use Equation 1 for each $r$ value.

Case 1: $r = 2$ Substitute $r = \mathbf{2}$ into Equation 1.

$a(1 + 2) = -4$

$3a = -4$

$\boxed{a = -4/3}$

Case 2: $r = -2$ Substitute $r = \mathbf{-2}$ into Equation 1.

$a(1 - 2) = -4$

$-a = -4$

$\boxed{a = 4}$

Step 4 — Write the GPs

We have two possible pairs of $a$ and $r$ .

For $a = -4/3$ and $r = 2$ : The GP terms are $a, ar, ar^2, \dots$ .

$-4/3, (-4/3)(2), (-4/3)(2^2), \dots$

$\boxed{-4/3, -8/3, -16/3, -32/3, \dots}$

For $a = 4$ and $r = -2$ : The GP terms are $a, ar, ar^2, \dots$ .

$4, (4)(-2), (4)(-2)^2, \dots$

$\boxed{4, -8, 16, -32, \dots}$

Answer

(i) The first possible GP is $\mathbf{-4/3, -8/3, -16/3, -32/3, \dots}$ . (ii) The second possible GP is $\mathbf{4, -8, 16, -32, \dots}$ .

More questions in EOT

Use suitable identities to find the following products:

(i) $(-3x + 4)^2$

(ii) $(2s + 7)(2s - 7)$

(iii) $\left(p^2 + \frac{1}{2}\right)\left(p^2 - \frac{1}{2}\right)$

(iv) $(2n + 7)(2n - 7)$

(v) $(s - 2t)(s^2 + 2st + 4t^2)$

(vi) $\left(\frac{1}{2r} - 4r\right)^2$

(vii) $(-3m + 4k - l)^2$

(viii) $\left(x - \frac{1}{3}y\right)^3$

(ix) $\left(\frac{7}{2}k - \frac{2}{3}m\right)^3$

Find the values using suitable identities:

(i) $17 \times 21$

(ii) $104 \times 96$

(iii) $24 \times 16$

(iv) $147^3$

(v) $199^3$

(vi) $127^3$

(vii) $(-107)^3$

(viii) $(-299)^3$

Factor the following algebraic expressions:

(i) $4 y^2 + 1 + \frac{1}{16 y^2}$

(ii) $9m^2 - \frac{1}{25n^2}$

(iii) $27b^3 - \frac{1}{64b^3}$

(iv) $x^2 + \frac{5x}{6} + \frac{1}{6}$

(v) $27u^3 - \frac{1}{125} - \frac{27u^2}{5} + \frac{9u}{25}$

(vi) $64y^3 + \frac{1}{125}z^3$

(vii) $p^3 + 27q^3 + r^3 - 9pqr$

(viii) $9m^2 - 12m + 4$

(ix) $9x^3 - \frac{8}{3}y^3 + \frac{z^3}{3} + 6xyz$

(x) $4x^2 + 9y^2 + 36z^2 + 12xz + 36yz + 24xy$

(xi) $27u^3 - \frac{1}{216} - \frac{9u^2}{2} + \frac{u}{4}$

Simplify the following:

(i) $\frac{4x^2 + 4x + 1}{4x^2 - 1}$

(ii) $\frac{9(3a^3 - 24b^3)}{9a^2 - 36b^2}$

(iii) $\frac{s^3 + 125t^3}{s^2 - 2st - 35t^2}$

Note: Assume that the denominators are not equal to 0.

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

(i) $25a^2 - 30ab + 9b^2$

(ii) $36s^2 - 49t^2$

Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.

(i) $6a^2 - 24b^2$

(ii) $3ps^2 - 15ps + 12p$

The village playground is shaped as a square of side 40 metres. A path of width $s$ metres is created around the playground for people to walk. Find an expression for the area of the path in terms of $s$ .

If a number plus its reciprocal equals $\frac{10}{3}$ , find the number.

A rectangular pool has area $2x^2 + 7x + 3$ square hastas. If its width is $2x + 1$ hastas, find its length. Hasta was a unit used to measure length.

Q10

If both $x - 2$ and $x - \frac{1}{2}$ are factors of $px^2 + 5x + r$ , show that $p = r$ .

Q11

If $a + b + c = 5$ and $ab + bc + ca = 10$ , then prove that $a^3 + b^3 + c^3 - 3abc = -25$ .

Q12

By factoring the expression, check that $n^3 - n$ is always divisible by 6 for all natural numbers $n$ . Give reasons.

Q13

Find the value of

(i) $x^3 + y^3 - 12xy + 64$ , when $x + y = -4$

(ii) $x^3 - 8y^3 - 36xy - 216$ , when $x = 2y + 6$