NCERT Solutions for Class 9 Maths Chapter 2 — Exercise Set 2.2#
Evaluating a polynomial means substituting a specific value for the variable and simplifying. Word problems in this exercise translate everyday statements into a linear equation of the form $ax + b = c$, which is then solved for the unknown.
1. Find the value of the linear polynomial 5x − 3 if: #
Let $p(x) = 5x - 3$.
(i) x = 0
∴ $p(0) = -3$.
(ii) x = −1
We find $p(-1)$:
$$= 5(-1) - 3$$$$= -5 - 3$$
$$= -8$$
∴ $p(-1) = -8$.
(iii) x = 2
We find $p(2)$:
$$= 5(2) - 3$$$$= 10 - 3$$
$$= 7$$
∴ $p(2) = 7$.
2. Find the value of the quadratic polynomial 7s² − 4s + 6 if: #
Let $q(s) = 7s^{2} - 4s + 6$.
(i) s = 0
∴ $q(0) = 6$.
(ii) s = −3
We find $q(-3)$:
$$= 7(-3)^{2} - 4(-3) + 6$$$$= 7(9) + 12 + 6$$
$$= 63 + 12 + 6$$
$$= 81$$
∴ $q(-3) = 81$.
(iii) s = 4
We find $q(4)$:
$$= 7(4)^{2} - 4(4) + 6$$$$= 7(16) - 16 + 6$$
$$= 112 - 16 + 6$$
$$= 102$$
∴ $q(4) = 102$.
3. The present age of Salil’s mother is three times Salil’s present age. After 5 years, their ages will add up to 70 years. Find their present ages. #
Solution
Let Salil’s present age be $x$ years.
His mother is three times as old, so her present age is $3x$ years.
After 5 years, their ages will add up to 70 years, so:
$$(x + 5) + (3x + 5) = 70$$$$\Rightarrow 4x + 10 = 70$$
$$\Rightarrow 4x = 60$$
$$\Rightarrow x = 15$$
Mother’s present age $= 3 \times 15 = 45$
∴ Salil is 15 years old and his mother is 45 years old.
4. The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers. #
Solution
The ratio of the integers is 2:5, so let them be $2k$ and $5k$.
Their difference is 63, so:
$$5k - 2k = 63$$$$\Rightarrow 3k = 63$$
$$\Rightarrow k = 21$$
Smaller integer $= 2 \times 21 = 42$
Larger integer $= 5 \times 21 = 105$
∴ The two integers are 42 and 105.
5. Ruby has 3 times as many two-rupee coins as she has five rupee-coins. If she has a total ₹88, how many coins does she have of each type? #
Solution
Let the number of five-rupee coins be $x$.
She has 3 times as many two-rupee coins, so she has $3x$ two-rupee coins.
The coins are worth ₹88 in all, so:
$$5x + 2(3x) = 88$$$$\Rightarrow 5x + 6x = 88$$
$$\Rightarrow 11x = 88$$
$$\Rightarrow x = 8$$
Two-rupee coins $= 3 \times 8 = 24$
Check: $5(8) + 2(24) = 40 + 48 = 88$ ✓
∴ Ruby has 8 five-rupee coins and 24 two-rupee coins.
6. A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces? #
Solution
Let the shorter piece be $x$ feet long.
The longer piece is four times as long, so it is $4x$ feet long.
The two pieces make up the 300 feet fence, so:
$$x + 4x = 300$$$$\Rightarrow 5x = 300$$
$$\Rightarrow x = 60$$
Longer piece $= 4 \times 60 = 240$
∴ The pieces are 60 feet and 240 feet long.
7. If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle? #
Solution
Let the width be $w$ cm.
The length is three more than twice the width, so the length is $(2w + 3)$ cm.
The perimeter is 24 cm, and perimeter $= 2(\text{length} + \text{width})$, so:
$$2\bigl[(2w + 3) + w\bigr] = 24$$$$\Rightarrow 2(3w + 3) = 24$$
$$\Rightarrow 6w + 6 = 24$$
$$\Rightarrow 6w = 18$$
$$\Rightarrow w = 3$$
Length $= 2 \times 3 + 3 = 9$
∴ The rectangle is 9 cm long and 3 cm wide.