CBSE Class 8 Mathematics Revision Assignment Set 04

Q1 The degree of the expression 2x-4xy^2+5x^2y^3
Answer: The degree of an algebraic expression is the highest sum of the powers of variables in any individual term.
Term 1: \(2x \rightarrow\) degree is 1
Term 2: \(-4xy^2 \rightarrow\) degree is \(1 + 2 = 3\)
Term 3: \(5x^2y^3 \rightarrow\) degree is \(2 + 3 = 5\)
The highest degree among all terms is 5.
In simple words: The degree is the highest total power of variables in any single term, which is 5 here.

Exam Tip: Always add the exponents of all variables present in each term to find its degree, rather than just taking the highest single exponent.

Q2 The number of terms in a binomial is
Answer: A binomial is an algebraic expression consisting of exactly 2 unlike terms connected by plus or minus signs.
In simple words: A binomial always has 2 terms.

Exam Tip: Remember the prefixes: monomial = 1 term, binomial = 2 terms, trinomial = 3 terms, multinomial = 2 or more terms.

Q3 What must be subtracted from 2x^2-3x+4 to get x^2+2x-1?
Answer: Let the required expression to be subtracted be P.
Then, \((2x^2 - 3x + 4) - P = x^2 + 2x - 1\)
\(P = (2x^2 - 3x + 4) - (x^2 + 2x - 1)\)
\(P = 2x^2 - 3x + 4 - x^2 - 2x + 1\)
Grouping like terms:
\(P = (2x^2 - x^2) + (-3x - 2x) + (4 + 1)\)
\(P = x^2 - 5x + 5\)
In simple words: Subtract the target expression from the given expression to find what is missing.

Exam Tip: When subtracting polynomials, be extremely careful to distribute the negative sign to every term inside the parentheses.

Q4 The product of 3x^2 and -5xy^3 is
Answer: Multiply the numerical coefficients and add the exponents of like bases using laws of exponents (\(x^m \cdot x^n = x^{m+n}\)):
\(= (3 \times -5) \cdot (x^2 \cdot x^1) \cdot (y^3)\)
\(= -15x^{2+1}y^3\)
\(= -15x^3y^3\)
In simple words: Multiply the numbers together and add the powers of matching variables.

Exam Tip: Do not forget to multiply the signs correctly (positive multiplied by negative gives negative).

Q5 The product of (x+3) and (2x-1) is
Answer: Using distributive multiplication (FOIL method):
\(= x(2x - 1) + 3(2x - 1)\)
\(= 2x^2 - x + 6x - 3\)
Combining like terms:\(\)
\(= 2x^2 + 5x - 3\)
In simple words: Multiply each term of the first bracket by each term of the second bracket and combine like terms.

Exam Tip: Combine the middle like terms carefully by checking their signs.

Q6 The product of (x+y)(x-y)(x^2+y^2) is
Answer: Using the algebraic identity \((a+b)(a-b) = a^2 - b^2\) on the first two factors:
\(= [(x + y)(x - y)](x^2 + y^2)\)
\(= (x^2 - y^2)(x^2 + y^2)\)
Applying the identity again where \(a = x^2\) and \(b = y^2\):
\(= (x^2)^2 - (y^2)^2\)
\(= x^4 - y^4\)
In simple words: Use the difference of squares formula twice to simplify the product quickly.

Exam Tip: Recognizing standard algebraic identities saves time and avoids lengthy expansion errors.

Q7 What will be the remainder when 3x^2-2x+1 is divided by x+5?
Answer: Using the Remainder Theorem, let \(f(x) = 3x^2 - 2x + 1\).
Set the divisor \(x + 5 = 0\), which gives \(x = -5\).
Substitute \(x = -5\) into \(f(x)\):
Remainder \(= f(-5) = 3(-5)^2 - 2(-5) + 1\)
\(= 3(25) + 10 + 1\)
\(= 75 + 10 + 1\)
\(= 86\)
In simple words: Substitute \(x = -5\) into the polynomial to find the remainder directly without long division.

Exam Tip: The Remainder Theorem states that the remainder of the division of polynomial \(f(x)\) by \((x - a)\) is equal to \(f(a)\).

Q8 What will be the remainder when x^6-3x^2-10 is divided by x^3-1?
Answer: Let \(f(x) = x^6 - 3x^2 - 10\). We can rewrite \(f(x)\) in terms of \(x^3\):
\(f(x) = (x^3)^2 - 3(x^2) - 10\)
We are dividing by \(x^3 - 1\), so put \(x^3 = 1\):
Remainder \(= (1)^2 - 3(\text{not applicable directly unless substituted correctly}) \dots\)
Let us use polynomial substitution or long division structure:
Let \(y = x^3\). Then the expression becomes \(y^2 - 3x^2 - 10\). Better yet, use algebraic substitution: substitute \(x^3 = 1\).
Notice that \(x^6 = (x^3)^2\). When divided by \(x^3 - 1\):
Substitute \(x^3 = 1\):
\((1)^2 - 3x^2 - 10\)
Since the divisor is \(x^3 - 1\), let us perform division by noting \(x^6 - 3x^2 - 10 = (x^3 - 1)(x^3 + 1) - 3x^2 - 9\)...
Wait, let's substitute \(x^3 = 1\) directly into \(x^6 - 3x^2 - 10\):
\((1)^2 - 3x^2 - 10 = 1 - 3x^2 - 10 = -3x^2 - 9\).
Let us verify by long division or relation: \(x^6 - 3x^2 - 10 = (x^3 - 1)(x^3 + 1) - 3x^2 - 9\).
Thus, the remainder is \(-3x^2 - 9\).
In simple words: Replace \(x^3\) with 1 wherever possible to find the remainder.

Exam Tip: For divisors involving higher powers like \(x^3 - 1\), substitute \(x^3 = 1\) into the polynomial to get the remainder directly.

Q9 What must be added to y^3-y^2-3y-2 to get y^2-5y+6?
Answer: Let the required expression to be added be P.
Then, \((y^3 - y^2 - 3y - 2) + P = y^2 - 5y + 6\)
\(P = (y^2 - 5y + 6) - (y^3 - y^2 - 3y - 2)\)
\(P = y^2 - 5y + 6 - y^3 + y^2 + 3y + 2\)
Rearranging in descending order of powers and combining like terms:
\(P = -y^3 + (y^2 + y^2) + (-5y + 3y) + (6 + 2)\)
\(P = -y^3 + 2y^2 - 2y + 8\)
In simple words: Subtract the original polynomial from the target polynomial to find the missing expression.

Exam Tip: Arrange all polynomials in standard descending order of powers before adding or subtracting.

Q9 Find the value of a if (2x-5)(x+3)=2x^2+x+a?
Answer: Expand the left-hand side:
\((2x - 5)(x + 3) = 2x(x + 3) - 5(x + 3)\)
\(= 2x^2 + 6x - 5x - 15\)
\(= 2x^2 + x - 15\)
Compare this with the right-hand side \(2x^2 + x + a\):
\(2x^2 + x - 15 = 2x^2 + x + a\)
Comparing constant terms on both sides:
\(a = -15\)
In simple words: Expand the left side and compare the constant terms to find \(a\).

Exam Tip: When two polynomial expressions are identically equal, their corresponding coefficients must be equal.

Q10 When (3y+1)(2y-3)=6y^2+ay-3, what will be the value of a?
Answer: Expand the left-hand side:
\((3y + 1)(2y - 3) = 3y(2y - 3) + 1(2y - 3)\)
\(= 6y^2 - 9y + 2y - 3\)
\(= 6y^2 - 7y - 3\)
Compare with the right-hand side \(6y^2 + ay - 3\):
\(6y^2 - 7y - 3 = 6y^2 + ay - 3\)
Comparing the coefficients of \(y\):
\(a = -7\)
In simple words: Multiply the brackets on the left and match the coefficient of \(y\) with \(a\).

Exam Tip: Pay close attention to the signs when combining like terms during expansion.

Q11 If x=3, y=2, then (x-y)(x^2+xy+y^2) is
Answer: Method 1 (Using algebraic identity):
We know that \((x - y)(x^2 + xy + y^2) = x^3 - y^3\)
Substitute \(x = 3\) and \(y = 2\):
\(= 3^3 - 2^3\)
\(= 27 - 8\)
\(= 19\)
Method 2 (Direct substitution):
\((3 - 2)(3^2 + (3)(2) + 2^2) = (1)(9 + 6 + 4) = 1 \times 19 = 19\)
In simple words: Simplify using the formula \(x^3 - y^3\) or substitute the values directly.

Exam Tip: Using algebraic identities often makes substitution much faster and less prone to arithmetic errors.

Q12 Add (i) 3m^2+8m-4 and 6m^2-m+7 (ii) a^2+2ab and -3ab-d^2
Answer:
(i) \((3m^2 + 8m - 4) + (6m^2 - m + 7)\)
Group like terms:
\(= (3m^2 + 6m^2) + (8m - m) + (-4 + 7)\)
\(= 9m^2 + 7m + 3\)

(ii) \((a^2 + 2ab) + (-3ab - d^2)\)
Group like terms:
\(= a^2 + (2ab - 3ab) - d^2\)
\(= a^2 - ab - d^2\)
In simple words: Combine terms that have the exact same variables and powers.

Exam Tip: Grouping terms in columns or brackets helps prevent missing any terms.

Q13 Simplify (i) 4x-(-8y-3x) (ii)-(49-a^2)-9a^2
Answer:
(i) \(4x - (-8y - 3x)\)
Distribute the negative sign:
\(= 4x + 8y + 3x\)
Combine like terms:
\(= (4x + 3x) + 8y\)
\(= 7x + 8y\)

(ii) \(-(49 - a^2) - 9a^2\)
Distribute the negative sign:
\(= -49 + a^2 - 9a^2\)
Combine like terms:
\(= -49 + (1 - 9)a^2\)
\(= -49 - 8a^2\)
In simple words: Open the brackets by changing the signs inside, then combine similar terms.

Exam Tip: A negative sign outside a bracket reverses every sign inside that bracket.

Q14 Subtract 72pq-30q^2-15p^2 from 30p^2+40q^2-12pq
Answer: Set up the subtraction expression (from-term minus what is being subtracted):
\(= (30p^2 + 40q^2 - 12pq) - (72pq - 30q^2 - 15p^2)\)
Distribute the negative sign to all terms in the second bracket:
\(= 30p^2 + 40q^2 - 12pq - 72pq + 30q^2 + 15p^2\)
Group like terms together:
\(= (30p^2 + 15p^2) + (40q^2 + 30q^2) + (-12pq - 72pq)\)
\(= 45p^2 + 70q^2 - 84pq\)
In simple words: Write the second expression first, then subtract each term of the first expression by changing its sign.

Exam Tip: The phrase "Subtract A from B" always means write B first: \(B - A\).

Q15 The sides of a triangle are 3x+1, 4x+2 and 5x. Find the perimeter of the triangle.
Answer: The perimeter of a triangle is the sum of the lengths of all its three sides.
Perimeter \(= (3x + 1) + (4x + 2) + 5x\)
Group like terms:
\(= (3x + 4x + 5x) + (1 + 2)\)
\(= 12x + 3\)
In simple words: Add all three side lengths together and combine like terms.

Exam Tip: Perimeter is a linear measure, so the degree of the resulting expression should be 1.