CBSE Class 8 Mathematics Algebraic Expressions And Identities Worksheet Set 02

Question. Find the degree of the polynomial 5x³ - 6x³y + 4y² - 8.
(A) 3
(B) 4
(C) 2
(D) 6
Answer : B

Question. What is the measure of the third side of a triangle given that its two sides are a² - 2a + 1 and 3a² - Sa + 3 and has a perimeter 6a² - 4a + 9?
(A) 2a² - 3a - 5
(B) 2a² + 3a - 5
(C) 2a² + 3a + 5
(D) 2a² - 3a + 5
Answer : C

Question. Two sides of a rectangle are (3p + 5q) units and (Sp- 7q) units. What is its area in sq. units?
(A) 15p² + 4pq - 35q²
(B) 15p² - 4pq + 35q²
(C) 15p² + 14pq + 35q²
(D) 15p² - 4pq - 35q²
Answer : A

Question. The difference of x⁴ + 2x² - 3x + 7 and another polynomial is x³ + x² + x - 1. What is the other polynomial ?
(A) x⁴ - x³ + x² - 4x + 8
(B) x³ + x² - 4x + 8
(C) x⁴ - x³ + x² + 4x - 8
(D) x⁴ - x³ - x² + 4x - 8
Answer : A

Question. Find the side of a square field of area

Answer : D

Question. Given that x - 1/x = 5 , find the value of x² + 1/x²
(A) - 27
(B) 23
(C) 27
(D) - 23
Answer : C

Question. Find the missing term in the following expression.
(3x/4 - 4y/3)² = 9x²/16 +....... + 16y²/9
(A) 2xy
(B) -2xy
(C) 12xy
(D) -12xy
Answer : B

Question. The sum of two expressions is x³ -x² + 3x- 2. If one of them is x² + 5x- 6, what is the other ?
(A) x³ - 2x² - 2x - 4
(B) x³ + 3x² - 2x + 4
(C) x³ - 2x² - 2x + 4
(D) -x³ - 2x² + 2x - 4
Answer : C

Question. If (3x - 4)(5x + 7) = 15x² - ax- 28, what is the value of 'a'?
(A) 1
(B) -1
(C) - 2
(D) 2
Answer : B

Question. Find the product of (x² + 3x + 5) and (x 2 -1).
(A) x⁴ + 3x³ - 4x² - 3x - 5
(B) x⁴ + 3x³ + 4x² - 3x - 5
(C) x⁴ + 3x³ + 4x² + 3x - 5
(D) x⁴ - 3x³ - 4x² + 3x - 5
Answer : B

Question. What is the difference obtained when x³ - xy² + 5x²y - y³ is subtracted from - y³ - 6x²y - xy² + x³ ?
(A) 2y³ - 8x²y + 3xy²- 2x³
(B) 2x³ - 2xy² - x 2y - 2y³
(C) - 11x 2y
(D) 11x²y
Answer : C

Question. If 3x- 7y = 10 and xy = - 1, find the value of 9x² + 49y²
(A) 58
(B) 142
(C) 1 04
(D) -104
Answer : A

Question. What is the va1ue of 7.83 x 7.83 - 1.17 x 1.17/6.66 ?
(A) 9
(B) 6.66
(C) 1.176
(D) -9
Answer : A

Question. What must be added to x³ + 3x - 8 to get 3x³ + x 2 + 6 ?
(A) 2x³ + x² - 3x + 14
(B) 2x² + x² + 14
(C) 2x³ + x² - 6x -14
(D) -2x³ - x² + 6x - 14
Answer : A

Question. If x² + 1/x² = 79, what is the value of X + 1/x ?
(A) 7
(B) 9
(C) J79
(D) -9
Answer : B

Question. Evaluate the expression 81p² + 16q² -72pq when P = 2/3 and q = 3/4.
(A) -16
(B) 9
(C) -9
(D) 16
Answer : B

Question. The area A, of a solid is given by the formula A = πr² + 2n rh + πrl. Find the value of A given that n = 3.142, r = 10, h = 3 and l = 4.
(A) 629
(B) 628.4
(C) 630
(D) 631
Answer : B

Question. If the perimeter of the figure given is 57 em, find the perimeter of the triangle in the figure.

(A) 30 cm
(B) 45 cm
(C) 39 cm
(D) 3 cm

Answer : B
 

ALGEBRAIC EXPRESSIONS

Question 1. The degree of the expression \(2x - 4xy^2 + 5x^2y^3\)
Answer: The degree of a polynomial is the highest sum of the exponents of the variables in any single term.
For term \(2x\): exponent of \(x\) is 1, degree = 1
For term \(-4xy^2\): sum of exponents = \(1 + 2 = 3\)
For term \(5x^2y^3\): sum of exponents = \(2 + 3 = 5\)
The highest degree among all terms is 5.
In simple words: The degree is the highest total power of variables in any single term.

Exam Tip: Always add the exponents of all variables present in each term before finding the maximum.

Question 2. The number of terms in a binomial is
Answer: A binomial is an algebraic expression consisting of exactly 2 unlike terms.
In simple words: Bi means two, so a binomial always has 2 terms.

Exam Tip: Remember prefixes: Monomial (1), Binomial (2), Trinomial (3), Polynomial (many).

Question 3. 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.
\((2x^2 - 3x + 4) - P = x^2 + 2x - 1\)
\(P = (2x^2 - 3x + 4) - (x^2 + 2x - 1)\)
\(P = 2x^2 - 3x + 4 - x^2 - 2x + 1\)
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: Be very careful with signs when distributing the negative sign across brackets.

Question 4. The product of \(3x^2\) and \(-5xy^3\) is
Answer: Multiply the numerical coefficients and use the laws of exponents for variables with the same base:
\(= (3 \times -5) \cdot (x^2 \cdot x) \cdot y^3\)
\(= -15x^{2+1}y^3\)
\(= -15x^3y^3\)
In simple words: Multiply numbers together and add the powers of matching letters.

Exam Tip: When multiplying powers with the same base, add their exponents: \(x^a \cdot x^b = x^{a+b}\).

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

Question 6. The product of \((x + y)(x - y)(x^2 + y^2)\) is
Answer: First, apply the algebraic identity \((a + b)(a - b) = a^2 - b^2\) on the first two terms:
\((x + y)(x - y) = x^2 - y^2\)
Now, multiply this result by the remaining term \((x^2 + y^2)\):
\(= (x^2 - y^2)(x^2 + y^2)\)
Apply 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 in succession.

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

Question 7. What will be the remainder when \(3x^2 - 2x + 1\) is divided by \(x + 5\)?
Answer: Using the Remainder Theorem, let \(x + 5 = 0\), which gives \(x = -5\).
Substitute \(x = -5\) into the polynomial \(P(x) = 3x^2 - 2x + 1\):
\(P(-5) = 3(-5)^2 - 2(-5) + 1\)
\(P(-5) = 3(25) + 10 + 1\)
\(P(-5) = 75 + 10 + 1\)
\(P(-5) = 86\)
In simple words: Set the divisor to zero to find the value of x, then plug that value into the polynomial to get the remainder directly.

Exam Tip: The Remainder Theorem is much faster than long division for linear divisors.

Question 8. What will be the remainder when \(x^6 - 3x^2 - 10\) is divided by \(x^3 - 1\)?
Answer: Let \(P(x) = x^6 - 3x^2 - 10\). Rewrite the polynomial in terms of \(x^3\):
\(P(x) = (x^3)^2 - 3x^2 - 10\)
Set the divisor \(x^3 - 1 = 0\), which gives \(x^3 = 1\).
Substitute \(x^3 = 1\) into the expression:
Remainder \(= (1)^2 - 3x^2 - 10\) - wait, let us substitute properly using synthetic substitution or polynomial division since \(x^2\) remains.
Using long division or division algorithm:
\(x^6 - 3x^2 - 10 = (x^3 - 1)(x^3 + 1) - 3x^2 - 9\)
Thus, the remainder is \(-3x^2 - 9\).
In simple words: Express powers of x in terms of the divisor and find the leftover expression.

Exam Tip: When the divisor is not strictly linear, substitute the highest power block or perform algebraic substitution.

Question 9. What must be added to \(y^3 - y^2 - 3y - 2\) to get \(y^2 - 5y + 6\)?
Answer: Let the required expression be Q.
\((y^3 - y^2 - 3y - 2) + Q = y^2 - 5y + 6\)
\(Q = (y^2 - 5y + 6) - (y^3 - y^2 - 3y - 2)\)
\(Q = y^2 - 5y + 6 - y^3 + y^2 + 3y + 2\)
Grouping like terms and arranging in descending order of powers:
\(Q = -y^3 + (y^2 + y^2) + (-5y + 3y) + (6 + 2)\)
\(Q = -y^3 + 2y^2 - 2y + 8\)
In simple words: Subtract the starting polynomial from the desired result polynomial.

Exam Tip: Always write polynomial answers in standard form (highest power to lowest power).

Question 9. Find the value of \(a\) if \((2x - 5)(x + 3) = 2x^2 + x + a\)?
Answer: Expand the left side of the equation:
\((2x - 5)(x + 3) = 2x(x + 3) - 5(x + 3)\)
\(= 2x^2 + 6x - 5x - 15\)
\(= 2x^2 + x - 15\)
Compare this with the right side: \(2x^2 + x + a\)
\(2x^2 + x - 15 = 2x^2 + x + a\)
Equating the constant terms on both sides:
\(a = -15\)
In simple words: Expand both sides and compare coefficients of corresponding terms.

Exam Tip: If two polynomials are identical, their corresponding coefficients must be equal.

Question 10. When \((3y + 1)(2y - 3) = 6y^2 + ay - 3\), what will be the value of \(a\)?
Answer: Expand the left side:
\((3y + 1)(2y - 3) = 3y(2y - 3) + 1(2y - 3)\)
\(= 6y^2 - 9y + 2y - 3\)
\(= 6y^2 - 7y - 3\)
Compare with the right side: \(6y^2 + ay - 3\)
\(6y^2 - 7y - 3 = 6y^2 + ay - 3\)
Equating the coefficients of \(y\):
\(a = -7\)
In simple words: Multiply out the brackets on the left and match the middle term coefficient with a.

Question 11. If \(x = 3\), \(y = 2\), then \((x - y)(x^2 + xy + y^2)\) is
Answer: Method 1: Use the algebraic identity \((x - y)(x^2 + xy + y^2) = x^3 - y^3\).
Substitute \(x = 3\) and \(y = 2\):
\(= 3^3 - 2^3\)
\(= 27 - 8\)
\(= 19\)
Method 2: Substitute directly into the expression:
\((3 - 2)(3^2 + (3)(2) + 2^2) = (1)(9 + 6 + 4) = 1 \times 19 = 19\)
In simple words: Simplify the expression using the cube difference formula first, then plug in the numbers.

Exam Tip: Using algebraic identities before substituting values makes calculations much simpler.

Question 12. Add (i) \(3m^2 + 8m - 4\) and \(6m^2 - m + 7\) (ii) \(a^2 + 2ab\) and \(-3ab - d^2\)?
Answer:
(i) \((3m^2 + 8m - 4) + (6m^2 - m + 7)\)
\(= (3m^2 + 6m^2) + (8m - m) + (-4 + 7)\)
\(= 9m^2 + 7m + 3\)

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

Exam Tip: Grouping like terms vertically or horizontally ensures no term is missed.

Question 13. Simplify (i) \(4x - (-8y - 3x)\) (ii) \(-(49 - a^2) - 9a^2\)?
Answer:
(i) \(4x - (-8y - 3x)\)
\(= 4x + 8y + 3x\)
\(= (4x + 3x) + 8y\)
\(= 7x + 8y\)

(ii) \(-(49 - a^2) - 9a^2\)
\(= -49 + a^2 - 9a^2\)
\(= -49 - 8a^2\)
In simple words: Distribute the negative sign carefully into every term inside the brackets.

Exam Tip: A negative sign outside a bracket changes the sign of every term inside when opened.

Question 14. Subtract \(72pq - 30q^2 - 15p^2\) from \(30p^2 + 40q^2 - 12pq\)?
Answer: Write the expression to subtract from first:
\(= (30p^2 + 40q^2 - 12pq) - (72pq - 30q^2 - 15p^2)\)
\(= 30p^2 + 40q^2 - 12pq - 72pq + 30q^2 + 15p^2\)
Group like terms together:
\(= (30p^2 + 15p^2) + (40q^2 + 30q^2) + (-12pq - 72pq)\)
\(= 45p^2 + 70q^2 - 84pq\)
In simple words: Remember "subtract A from B" means writing B - A.

Exam Tip: Pay close attention to the order in subtraction statements ("subtract X from Y" means Y comes first).

Question 15. 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 all its three sides.
Perimeter \(= (3x + 1) + (4x + 2) + 5x\)
Combine like terms:
Perimeter \(= (3x + 4x + 5x) + (1 + 2)\)
Perimeter \(= 12x + 3\)
In simple words: Add all three side lengths together to get the total perimeter.

Exam Tip: Perimeter always has the same linear unit dimensions as the individual sides.