Simplify the following: (i) (ii) (iii) Note: Assume that the denominators are not equal to 0.

We need to find how many multiples of 4 are between 10 and 250. Step 1 — Identify the arithmetic progression First, we find the smallest multiple of 4 greater than 10. We divide 10 by 4. So, the next multiple of 4 is . This is our first term, . Next, we find the largest multiple of 4 less than 250. We divide 250 by 4. So, the largest multiple of 4 is . This is our last term, . The common difference, , for multiples of 4 is 4. So, our arithmetic progression (AP) is: . Here are the values for our AP: > Step 2 — Calculate the number of terms We use the formula for the -th term of an AP. The formula is . Let's substitute our values into the formula. Now, we solve for . We divide both sides by 4. Now, we add 1 to both sides. > Answer > (i) There are 60 multiples of 4 between 10 and 250.

Exploring Algebraic Identities | EOT

Question 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.

Solution

We need to find how many multiples of 4 are between 10 and 250.

Step 1 — Identify the arithmetic progression

First, we find the smallest multiple of 4 greater than 10. We divide 10 by 4. $10 \div 4 = 2 \text{ remainder } 2$ So, the next multiple of 4 is $4 \times 3 = \textbf{12}$ . This is our first term, $a$ .

Next, we find the largest multiple of 4 less than 250. We divide 250 by 4. $250 \div 4 = 62 \text{ remainder } 2$ So, the largest multiple of 4 is $4 \times 62 = \textbf{248}$ . This is our last term, $t_n$ .

The common difference, $d$ , for multiples of 4 is 4. So, our arithmetic progression (AP) is: $12, 16, 20, \dots, 248$ .

Here are the values for our AP: $a = \textbf{12}$ $d = \textbf{4}$ $t_n = \textbf{248}$

$\boxed{\text{First term } a = 12, \text{ Common difference } d = 4, \text{ Last term } t_n = 248}$

Diagram 1

Step 2 — Calculate the number of terms

We use the formula for the $n$ -th term of an AP. The formula is $t_n = a + (n-1)d$ . Let's substitute our values into the formula.

$248 = 12 + (n - 1)4$

Now, we solve for $n$ . $248 - 12 = (n - 1)4$

$236 = (n - 1)4$

We divide both sides by 4. $\frac{236}{4} = n - 1$

$59 = n - 1$

Now, we add 1 to both sides. $n = 59 + 1$

$n = 60$

$\boxed{\text{Number of multiples } n = 60}$

Answer

(i) There are 60 multiples of 4 between 10 and 250.

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.

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

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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 $x = 2y + 6$