Exercise: 12-A
Q1: Identify monomials, binomials and trinomials from the following:
i. \(\frac{1}{2}a^2b^2c^2\)
Step 1: This expression has only one term.
Step 2: It is a product of variables and a constant.
Answer: Monomial
ii. \(7x \times y^2 \times z^3\)
Step 1: All parts are multiplied — one single term.
Step 2: It’s a product of constants and variables.
Answer: Monomial
iii. \(\frac{9x^3}{z}\)
Step 1: This is a single algebraic fraction (a single term).
Step 2: No addition or subtraction — only one term.
Answer: Monomial
iv. \(2x + 5\)
Step 1: Two terms separated by a plus sign.
Step 2: One term is algebraic, one is constant.
Answer: Binomial
v. \(\frac{a}{3} + \frac{b}{6}\)
Step 1: Two terms: one in \(a\), one in \(b\).
Step 2: Terms are added — two distinct algebraic parts.
Answer: Binomial
vi. \(xy + yz + zx\)
Step 1: Three terms: \(xy\), \(yz\) and \(zx\).
Step 2: Three terms separated by a plus sign.
Answer: Trinomial
vii. \(\frac{x^2 – 2y^2 + z^2}{3}\)
Step 1: Inside the numerator: three terms: \(x^2\), \(-2y^2\), \(z^2\).
Step 2: Dividing the whole trinomial by 3 doesn’t change the number of terms.
Answer: Trinomial
viii. \(8a \div 9b – 2a^2 \times b^2\)
Step 1: First term: \(8a \div 9b = \frac{8a}{9b}\)
Step 2: Second term: \(-2a^2b^2\)
Step 3: Two distinct terms with subtraction.
Answer: Binomial
ix. \(7x^3 + \frac{2y^2 + 1}{5}\)
Step 1: First term: \(7x^3\)
Step 2: Second term: \(\frac{2y^2 + 1}{5}\) — itself a binomial divided by 5.
Step 3: Entire expression contains three terms: \( 7x^3, \frac{2y^2}{5}, \frac{1}{5}\).
Answer: Trinomial
Q2: Write the numerical and literal coefficients of each of the following terms:
i. \(-7x^2y\)
Step 1: The number part of the term is \(-7\).
Step 2: The variable part (literal coefficient) is \(x^2y\).
Answer: Numerical coefficient: -7, Literal coefficient: \(x^2y\)
ii. \(\pi r^2\)
Step 1: The number \(\pi\) is treated as the numerical coefficient (constant).
Step 2: The variable part is \(r^2\).
Answer: Numerical coefficient: \(\pi\), Literal coefficient: \(r^2\)
iii. \(-\frac{3}{8}bcx\)
Step 1: The number part is \(-\frac{3}{8}\).
Step 2: The literal part is \(bcx\).
Answer: Numerical coefficient: \(-\frac{3}{8}\), Literal coefficient: \(bcx\)
iv. \(\frac{2a}{3}\)
Step 1: The term is written as \(\frac{2}{3} \cdot a\).
Step 2: So, the number part is \(\frac{2}{3}\).
Answer: Numerical coefficient: \(\frac{2}{3}\), Literal coefficient: \(a\)
v. \(5a^2 \times b \div 2c\)
Step 1: First rewrite the expression: \(\frac{5a^2b}{2c}\)
Step 2: The numerical coefficient is \(\frac{5}{2}\).
Step 3: The literal coefficient is \(\frac{a^2b}{c}\).
Answer: Numerical coefficient: \(\frac{5}{2}\), Literal coefficient: \(\frac{a^2b}{c}\)
vi. \(-\frac{7pq}{9xy}\)
Step 1: This is a fraction where the numerator has \(pq\), and the denominator has \(xy\).
Step 2: The numerical coefficient is \(-\frac{7}{9}\).
Step 3: The literal coefficient is \(\frac{pq}{xy}\).
Answer: Numerical coefficient: \(-\frac{7}{9}\), Literal coefficient: \(\frac{pq}{xy}\)
Q3: In \(-\frac{3}{5}x^3y^2z\), write down the coefficient of:
i. \(x^2\)
Step 1: Original term is: \(-\frac{3}{5}x^3y^2z\)
Step 2: We are finding the coefficient of \(x^2\).
Step 3: Divide the original term by \(x^2\):
\[
\frac{-\frac{3}{5}x^3y^2z}{x^2} = -\frac{3}{5}x y^2 z
\]
Answer: \(-\frac{3}{5} x y^2 z\)
ii. \(-yz\)
Step 1: Original term is: \(-\frac{3}{5}x^3y^2z\)
Step 2: We are finding the coefficient of \(-yz\).
Step 3: Divide the original term by \(-yz\):
\[
\frac{-\frac{3}{5}x^3y^2z}{-yz} = \frac{3}{5} x^3 y
\]
Answer: \(\frac{3}{5} x^3 y\)
iii. \(\frac{3}{5}xy\)
Step 1: Original term is: \(-\frac{3}{5}x^3y^2z\)
Step 2: We are finding the coefficient of \(\frac{3}{5}xy\).
Step 3: Divide the original term by \(\frac{3}{5}xy\):
\[
\frac{-\frac{3}{5}x^3y^2}{\frac{3}{5}xy} = -x^2 y
\]
Answer: \(-x^2 y\)
iv. \(-x^2y\)
Step 1: Original term is: \(-\frac{3}{5}x^3y^2z\)
Step 2: We are finding the coefficient of \(-x^2y\).
Step 3: Divide the original term by \(-x^2y\):
\[
\frac{-\frac{3}{5}x^3y^2z}{-x^2y} = \frac{3}{5} x y z
\]
Answer: \(\frac{3}{5} x y z\)
Q4: Identify the pairs of like terms:
i. \(\frac{x}{2},\ -\frac{x}{3}\)
Step 1: Both terms have the same literal part: \(x\).
Step 2: Only the numerical coefficients are different.
Answer: Like terms
ii. \(6a^2bc,\ 6ab^2c\)
Step 1: First term has variables: \(a^2bc\).
Step 2: Second term has variables: \(ab^2c\).
Step 3: Powers of variables are different, so these are not like terms.
Answer: Unlike terms
iii. \(6pq,\ -3qx\)
Step 1: First term has \(pq\), second has \(qx\).
Step 2: Literal parts are different.
Answer: Unlike terms
iv. \(8a^2,\ -\frac{2}{3}a^2\)
Step 1: Both terms have the same literal part: \(a^2\).
Step 2: Only numerical coefficients differ.
Answer: Like terms
v. \(2x,\ 2y\)
Step 1: First term has variable \(x\), second has \(y\).
Step 2: Literal parts are different.
Answer: Unlike terms
vi. \(3xy^2p,\ -8py^2x\)
Step 1: First term: \(xy^2p\), second term: \(py^2x\).
Step 2: Variables and powers are same, though the order is different.
Step 3: Like terms can have variables in any order.
Answer: Like terms
Q5: Which of the following expressions are polynomials?
i. \(1 – x\)
Answer:Yes, it is a polynomial of degree 1.
ii. \(3 + y + y^2\)
Answer:Yes, all powers of variable are non-negative integers.
iii. \(z + \sqrt{z}\)
Answer:No, \(\sqrt{z}\) is not a whole number power.
iv. \(x – \frac{1}{x}\)
Answer:No, \(\frac{1}{x}\) means negative power of x.
v. \(x^3 + x\sqrt{x} – x + 2\)
Answer:No, \(x\sqrt{x} = x^{1.5}\) is not an integer power.
vi. \(x^2 + y^2 + xy + x^2y^2\)
Answer:Yes, all terms have whole number powers.
vii. \(5\)
Answer:Yes, constant is also a polynomial (degree 0).
viii. \(\frac{1}{3}x^3 – x^4\)
Answer:Yes, coefficients can be rational numbers.
ix. \(x^2 + \sqrt{3}x + 5\)
Answer:Yes, \(\sqrt{3}\) is a constant coefficient, powers of variable are okay.
x. \(5x^2 + 6xy – 7\sqrt{y}\)
Answer:No, \(\sqrt{y}\) is not allowed in polynomials.
xi.\(6x^2\sqrt{y} – 3xy + 5\)
Answer:No, because of the term \(6x^2\sqrt{y}\).
Q6: Write the degree of each of the following polynomials:
i. \(2 – x\)
Step 1: Identify the term with the highest exponent of the variable.
Answer: 1
ii. \(3 – x^2 + x^3\)
Step 1: The highest power of x is 3.
Answer: 3
iii. \(5x^2 – 6x\)
Step 1: The highest power of x is 2.
Answer: 2
iv. \(2x^3 – 8x\)
Step 1: The term with the highest degree is \(2x^3\).
Answer: 3
v. \(1 – x + x^4 – 3x^2\)
Step 1: The term with the highest exponent is \(x^4\).
Answer: 4
vi. \(z^3 – z^4 + 2z^2 – 6\)
Step 1: The term with the highest exponent is \(-z^4\).
Answer: 4
vii. \(1 – y – y^2 + 3y^5\)
Step 1: The term with the highest exponent is \(3y^5\).
Answer: 5
viii. \(x^2 – \frac{x}{2}\)
Step 1: The term with the highest exponent is \(x^2\).
Answer: 2
ix. \(t^4 – t^3 + 2t – 3t^6\)
Step 1: The term with the highest exponent is \(-3t^6\).
Answer: 6
x. \(5\)
Step 1: A constant has degree 0.
Answer: 0
xi. \(9 – x^2\)
Step 1: The term with the highest power is \(-x^2\).
Answer: 2
xii. \(1 – x^3\)
Step 1: The highest power is 3 in \(-x^3\).
Answer: 3
Q7: Write the degree of each of the following polynomials:
i. \(xy+yz+zx+3xyz\)
Step 1: Identify the degree of each term:
– \(xy\) → degree = 1 + 1 = 2
– \(yz\) → degree = 1 + 1 = 2
– \(zx\) → degree = 1 + 1 = 2
– \(3xyz\) → degree = 1 + 1 + 1 = 3
Answer: 3
ii. \(a^2+b^2+c^2-3abc\)
Step 1: Identify the degree of each term:
– \(a^2\), \(b^2\), \(c^2\) → degree = 2
– \(3abc\) → degree = 1 + 1 + 1 = 3
Answer: 3
iii. \(2xy+3xy^2+5x^2y+7x^2y^2\)
Step 1: Identify the degree of each term:
– \(2xy\) → degree = 1 + 1 = 2
– \(3xy^2\) → degree = 1 + 2 = 3
– \(5x^2y\) → degree = 2 + 1 = 3
– \(7x^2y^2\) → degree = 2 + 2 = 4
Answer: 4
iv. \(a^5-b^5-2a^3b^3\)
Step 1: Identify the degree of each term:
– \(a^5\), \(b^5\) → degree = 5
– \(2a^3b^3\) → degree = 3 + 3 = 6
Answer: 6
v. \(x^2y+xy^2+5xy\)
Step 1: Identify the degree of each term:
– \(x^2y\) → degree = 2 + 1 = 3
– \(xy^2\) → degree = 1 + 2 = 3
– \(5xy\) → degree = 1 + 1 = 2
Answer: 3
vi. \(1+2x+5x^2y+6yz^2\)
Step 1: Identify the degree of each term:
– Constant term \(1\) → degree = 0
– \(2x\) → degree = 1
– \(5x^2y\) → degree = 2 + 1 = 3
– \(6yz^2\) → degree = 1 + 2 = 3
Answer: 3
Q8:
i. Find the value of \(4x^3 – 3x^2 + 5x – 6\), when \(x = 3\)
Step 1: Substitute \(x = 3\) in the expression
\(= 4(3)^3 – 3(3)^2 + 5(3) – 6\)
\(= 4 \times 27 – 3 \times 9 + 15 – 6\)
\(= 108 – 27 + 15 – 6\)
\(= 90\)
Answer: 90
ii. Find the value of \(x^3 – 8x^2 + 14x – 7\), when \(x = -1\)
Step 1: Substitute \(x = -1\) in the expression
\(= (-1)^3 – 8(-1)^2 + 14(-1) – 7\)
\(= -1 – 8(1) – 14 – 7\)
\(= -1 – 8 – 14 – 7\)
\(= -30\)
Answer: -30
Q9: If a = 4 and b = 5, find the value of \(a^3+b^3-3a^2b+3ab^2\).
Step 1: Identify the expression to evaluate:
\(a^3 + b^3 – 3a^2b + 3ab^2\)
Step 2: Substitute the values of \(a = 4\) and \(b = 5\):
\(= 4^3 + 5^3 – 3 \cdot 4^2 \cdot 5 + 3 \cdot 4 \cdot 5^2\)
Step 3: Simplify each term:
\(= 64 + 125 – 3 \cdot 16 \cdot 5 + 3 \cdot 4 \cdot 25\)
\(= 64 + 125 – 240 + 300\)
Step 4: Add and subtract:
\(= (64 + 125 + 300) – 240 = 489 – 240 = 249\)
Answer: 249
Q10: If x = 4, y = 3 and z = -2, find the value of:
i. \(x^2 + y^2 + z^2 + 2xyz\)
Step 1: Substitute the values:
\(= 4^2 + 3^2 + (-2)^2 + 2 \cdot 4 \cdot 3 \cdot (-2)\)
Step 2: Simplify each term:
\(= 16 + 9 + 4 + 2 \cdot 4 \cdot 3 \cdot (-2)\)
\(= 29 + 2 \cdot 4 \cdot 3 \cdot (-2)\)
Step 3: Evaluate the product:
\(= 29 + (-48) = -19\)Answer: -19
ii. \(x^3 + y^3 + z^3 – 3xyz\)
Step 1: Substitute the values:
\(= 4^3 + 3^3 + (-2)^3 – 3 \cdot 4 \cdot 3 \cdot (-2)\)
Step 2: Simplify each term:
\(= 64 + 27 – 8 – 3 \cdot 4 \cdot 3 \cdot (-2)\)
Step 3: Evaluate the expression:
\(= 91 – 8 + 72 = 155\)Answer: 155
Q11: State whether each of the following statements is true or false and rewrite the false statement after correcting them:
i. \(7x + y\) is the same as \(7xy\)
Step 1: \(7x + y\) is the sum of two terms, while \(7xy\) is the product of three variables.
Answer:False. Correct statement: \(7x + y\) is not equal to \(7xy\)
ii. \(3x \times 2p \div 5q\) is a trinomial.
Step 1: The expression has only one term after simplification: \(\frac{6xp}{5q}\).
Answer:False. Correct statement: It is a monomial, not a trinomial.
iii. \(2xy + 3yx + 8yz\) is a binomial.
Step 1: \(2xy + 3yx = 5xy\); so expression becomes \(5xy + 8yz\).
Step 2: Two terms remain → It is a binomial.
Answer:True.
iv. \(8p^2q^3r = 8pq^2r^3\)
Step 1: Check powers of each variable on both sides.
LHS: \(p^2 q^3 r\), RHS: \(p q^2 r^3\). They are not equal.
Answer:False. Correct statement: \(8p^2q^3r \ne 8pq^2r^3\)
v. \(3x^2y + 5yx^2\) is a monomial.
Step 1: Both terms are like terms. Combine: \(3x^2y + 5x^2y = 8x^2y\).
Answer:True.
vi. \(9p \times 6q + 5r\) is a trinomial.
Step 1: Simplify: \(9p \times 6q = 54pq\). Final expression: \(54pq + 5r\) → 2 terms.
Answer:False. Correct statement: It is a binomial.
vii. \(-3ab^2c + 4cab^2 + 9b^2ac\) is a trinomial.
Step 1: All terms are like term. Simplify: \(-3ab^2c+4cab^2+9b^2ac = 10ab^2c\).
Answer:False. It is monomial.
viii. \(6x^2yz + 5xy^2z – 2xyz^2\) is a trinomial.
Step 1: All three are distinct terms. Trinomial has 3 terms.
Answer:True.
ix. \(-7pq^2 + 7p^2q = 0\)
Step 1: Combine terms: They are not like terms. Their sum ≠ 0.
Answer:False. Correct statement: \(-7pq^2 + 7p^2q \ne 0\)
x. \(\frac{9x}{8y}\) is a binomial.
Step 1: Only one term → Monomial, not binomial.
Answer:False. Correct statement: \(\frac{9x}{8y}\) is a monomial.
