Distributivity and Algebra | FIO

Question 14

Verify which of the following statements are true.

(i) $(k + 1)(k + 2) - (k + 3)$ is always 2.

(ii) $(2q + 1)(2q - 3)$ is a multiple of 4.

(iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8.

(iv) $(6n + 2)^2 - (4n + 3)^2$ is 5 less than a square number.

Solution

Understand the Question

To determine whether each algebraic statement is True or False, we expand and simplify the algebraic expressions using standard algebraic identities like $(a + b)(a - b) = a^2 - b^2$ and $(a + b)^2 = a^2 + 2ab + b^2$ , then test general integer properties and counterexamples.

(i) $(k + 1)(k + 2) - (k + 3)$ is always 2.

Step 1 · Expand and Simplify the Expression

Expand the given expression:

\begin{aligned} (k + 1)(k + 2) - (k + 3) &= (k^2 + 2k + k + 2) - (k + 3) \\ &= (k^2 + 3k + 2) - k - 3 \\ &= k^2 + 3k + 2 - k - 3 \\ &= k^2 + 2k - 1 \end{aligned}

Testing with $k = 0$ : $0^2 + 2(0) - 1 = -1 \neq 2$

Since the result depends on $k$ and is not always $2$ , the statement is False.

Answer

(i) False

(ii) $(2q + 1)(2q - 3)$ is a multiple of 4.

Step 1 · Expand and Check Divisibility by 4

Expand the given expression:

\begin{aligned} (2q + 1)(2q - 3) &= (2q)(2q) + (2q)(-3) + (1)(2q) + (1)(-3) \\ &= 4q^2 - 6q + 2q - 3 \\ &= 4q^2 - 4q - 3 \\ &= 4(q^2 - q) - 3 \end{aligned}

For any integer $q$ , letting $m = q^2 - q$ , the expression is of the form $4m - 3$ , which is $3$ less than a multiple of $4$ .

Testing with $q = 1$ : $(2(1) + 1)(2(1) - 3) = (3)(-1) = -3$

Since $-3$ is not a multiple of $4$ , the statement is False.

Answer

(ii) False

(iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8.

Step 1 · Verify Squares of Even Numbers

Let an even number be $2n$ , where $n$ is an integer: $(2n)^2 = 4n^2$

Since $n^2$ is an integer, $4n^2$ is a multiple of $4$ .

Step 2 · Verify Squares of Odd Numbers

Let an odd number be $2n + 1$ , where $n$ is an integer:

\begin{aligned} (2n + 1)^2 &= (2n)^2 + 2(2n)(1) + 1^2 \\ &= 4n^2 + 4n + 1 \\ &= 4n(n + 1) + 1 \end{aligned}

Since $n$ and $n + 1$ are consecutive integers, their product $n(n + 1)$ is always even. Let $n(n + 1) = 2m$ for some integer $m$ :

\begin{aligned} 4n(n + 1) + 1 &= 4(2m) + 1 \\ &= 8m + 1 \end{aligned}

This is $1$ more than a multiple of $8$ .

Since both parts are true, the statement is True.

Answer

(iii) True

(iv) $(6n + 2)^2 - (4n + 3)^2$ is 5 less than a square number.

Step 1 · Simplify and Check for Square Form

Using the difference of squares identity $a^2 - b^2 = (a - b)(a + b)$ :

\begin{aligned} (6n + 2)^2 - (4n + 3)^2 &= ((6n + 2) - (4n + 3))((6n + 2) + (4n + 3)) \\ &= (6n + 2 - 4n - 3)(6n + 2 + 4n + 3) \\ &= (2n - 1)(10n + 5) \\ &= 5(2n - 1)(2n + 1) \\ &= 5((2n)^2 - 1^2) \\ &= 5(4n^2 - 1) \\ &= 20n^2 - 5 \end{aligned}

For $20n^2 - 5$ to be $5$ less than a square number, $20n^2$ must be a perfect square. But $20n^2 = 4 \times 5 \times n^2$ , which is not a perfect square for any non-zero integer $n$ because $5$ is not a square.

Testing with $n = 1$ : $20(1)^2 - 5 = 15$

Adding $5$ gives $15 + 5 = 20$ , which is not a perfect square. Therefore, the statement is False.

Answer

(iv) False

Common Mistakes

Sign Errors with Parentheses: Forgetting to distribute the negative sign in $-(k + 3)$ , incorrectly writing $-k + 3$ instead of $-k - 3$ .
Incomplete Divisibility Check: In $(2q + 1)(2q - 3) = 4(q^2 - q) - 3$ , assuming it is divisible by $4$ because of the $4(q^2 - q)$ term, while ignoring the non-zero remainder $-3$ .
Consecutive Integer Property: Overlooking that $n(n + 1)$ is always an even number because one of two consecutive integers must be even.

More questions in FIO

Observe the multiplication grid below. Each number inside the grid is formed by multiplying two numbers. If the middle number of a $3 \times 3$ frame is given by the expression $pq$ , as shown in the figure, write the expressions for the other numbers in the grid.

Expand the following products.

(i) $(3 + u) (v - 3)$

(ii) $\dfrac{2}{3} (15 + 6a)$

(iii) $(10a + b) (10c + d)$

(iv) $(3 - x) (x - 6)$

(v) $(-5a + b) (c + d)$

(vi) $(5 + z) (y + 9)$

Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

Expand:

(i) $(a + ab - 3b^2)(4 + b)$

(ii) $(4y + 7)(y + 11z - 3)$

Expand:

(i) $(a - b)(a + b)$

(ii) $(a - b)(a^2 + ab + b^2)$

(iii) $(a - b)(a^3 + a^2b + ab^2 + b^3)$

Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

Which is greater: $(a - b)^2$ or $(b - a)^2$ ? Justify your answer.

Express 100 as the difference of two squares.

Find $406^2$ , $72^2$ , $145^2$ , $1097^2$ , and $124^2$ using the identities you have learnt so far.

Do Patterns 1 and 2 hold only for counting numbers? Do they hold for negative integers as well? What about fractions? Justify your answer.

Q10

Compute these products using the suggested identity.

(i) $46^2$ using Identity 1A for $(a + b)^2$

(ii) $397 \times 403$ using Identity 1C for $(a + b)(a - b)$

(iii) $91^2$ using Identity 1B for $(a - b)^2$

(iv) $43 \times 45$ using Identity 1C for $(a + b)(a - b)$

Q11

Use either a suitable identity or the distributive property to find each of the following products.

(i) $(p - 1)(p + 11)$

(ii) $(3a - 9b)(3a + 9b)$

(iii) $-(2y + 5)(3y + 4)$

(iv) $(6x + 5y)^2$

(v) $\left(2x - \dfrac{1}{2}\right)^2$

(vi) $(7p) \times (3r) \times (p + 2)$

Q12

For each statement identify the appropriate algebraic expression(s).

(i) Two more than a square number.

$2 + s \qquad (s + 2)^2 \qquad s^2 + 2 \qquad s^2 + 4 \qquad 2s^2 \qquad 2^2s$

(ii) The sum of the squares of two consecutive numbers

$m^2 + n^2 \qquad (m + n)^2 \qquad m^2 + 1 \qquad m^2 + (m + 1)^2$ $m^2 + (m - 1)^2 \qquad (m + (m + 1))^2 \qquad (2m)^2 + (2m + 1)^2$

Q13

Consider any 2 by 2 square of numbers in a calendar, as shown in the figure.

Find products of numbers lying along each diagonal — $4 \times 12 = 48$ , $5 \times 11 = 55$ . Do this for the other 2 by 2 squares. What do you observe about the diagonal products? Explain why this happens.

Hint: Label the numbers in each 2 by 2 square as shown in the diagram.

Q14

Verify which of the following statements are true.

(i) $(k + 1)(k + 2) - (k + 3)$ is always 2.

(ii) $(2q + 1)(2q - 3)$ is a multiple of 4.

(iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8.

(iv) $(6n + 2)^2 - (4n + 3)^2$ is 5 less than a square number.

Q15

A number leaves a remainder of 3 when divided by 7, and another number leaves a remainder of 5 when divided by 7. What is the remainder when their sum, difference, and product are divided by 7?

Q16

Choose three consecutive numbers, square the middle one, and subtract the product of the other two. Repeat the same with other sets of numbers. What pattern do you notice? How do we write this as an algebraic equation? Expand both sides of the equation to check that it is a true identity.

Q17

What is the algebraic expression describing the following steps—add any two numbers. Multiply this by half of the sum of the two numbers? Prove that this result will be half of the square of the sum of the two numbers.

Q18

Which is larger? Find out without fully computing the product.

(i) $14 \times 26$ or $16 \times 24$

(ii) $25 \times 75$ or $26 \times 74$

Q19

A tiny park is coming up in Dhauli. The plan is shown in the figure. The two square plots, each of area $g^2 \text{ sq. ft.}$ , will have a green cover. All the remaining area is a walking path $w \text{ ft.}$ wide that needs to be tiled. Write an expression for the area that needs to be tiled.

Q20

For each pattern shown below,

(i) Draw the next figure in the sequence. (ii) How many basic units are there in Step 10? (iii) Write an expression to describe the number of basic units in Step $y$ .