Algebraic Expressions & Identities MCQ Test — Class 8 CBSE

Solution:

The terms in the expression 5xy - 3y + 7xz are 5xy, -3y, and 7xz. Ravi correctly identified these terms (ignoring the sign for 3y in his list, but implicitly including it in the original expression).

The coefficient is the numerical factor of a term. For the term -3y, the numerical factor is -3, not 3. Ravi made an error in identifying the coefficient of y.

Solution:

A monomial has one term. 4x - 5y + 3 has three terms, so it's a trinomial, not a monomial.

A binomial has two terms. 7p²q has only one term, so it's a monomial, not a binomial.

x² + y² has two terms (x² and y²), making it a binomial. This statement is TRUE.

A polynomial cannot have terms with negative exponents or fractional exponents. These would make it not a polynomial.

Solution:

The original expression is (7x - 4y) - (3x - 2y).

When removing the parentheses from the second expression, the negative sign must be distributed to both terms inside: -(3x - 2y) becomes -3x + 2y.

The student wrote 7x - 4y - 3x - 2y, which shows that they incorrectly changed -2y to -2y instead of +2y. This is an incorrect distribution of the negative sign.

The correct simplification would be 7x - 4y - 3x + 2y = (7x - 3x) + (-4y + 2y) = 4x - 2y.

Solution:

The Distributive Property states that a × (b + c) = (a × b) + (a × c).

When multiplying a monomial (like 'a') by a polynomial (like 'b + c'), the monomial is distributed to each term of the polynomial.

For example, 2x(3x + 5) = (2x × 3x) + (2x × 5) = 6x² + 10x. This demonstrates the distributive property.

Solution:

When multiplying two binomials, (a + b) and (c + d), you take each term from the first binomial and multiply it by the entire second binomial.

So, the term 'a' from the first binomial multiplies (c + d), giving a(c + d).

And the term 'b' from the first binomial also multiplies (c + d), giving b(c + d).

These two results are then added together: a(c + d) + b(c + d).

Solution:

An equation is a statement of equality between two expressions that holds true for some specific values of the variables.

An identity, on the other hand, is an equality that is true for all possible values of the variables involved, as long as both sides of the equality are defined.

For example, (x + y)² = x² + 2xy + y² is an identity because it holds true no matter what real numbers you substitute for x and y.

Solution:

The algebraic identity for (a + b)² is a² + 2ab + b².

In the given expression (2x + 3y)², 'a' is 2x and 'b' is 3y.

Applying the identity, it should be (2x)² + 2(2x)(3y) + (3y)².

The student only wrote (2x)² + (3y)², completely missing the middle term, which is 2ab or 2(2x)(3y) = 12xy.

Solution:

Given the expression (5p - 2q)² and the identity (a - b)² = a² - 2ab + b².

Here, a = 5p and b = 2q.

Substitute these values into the identity: (5p)² - 2(5p)(2q) + (2q)².

Calculate each part: (5p)² = 25p², 2(5p)(2q) = 20pq, and (2q)² = 4q². So the expansion is 25p² - 20pq + 4q².

Solution:

The given expression is x² + 10x + 25. This looks like a perfect square trinomial, which follows the identity a² + 2ab + b².

Compare x² + 10x + 25 with a² + 2ab + b²:

We can see that a² = x², so a = x.

We can see that b² = 25, so b = 5.

Now check the middle term: 2ab = 2(x)(5) = 10x. This matches the middle term of the given expression.

Therefore, x² + 10x + 25 is the expansion of (x + 5)².

Solution:

The formula for the perimeter of a rectangle is P = 2 × (length + width).

Given length (L) = (3x + 2) units and width (W) = (2x - 1) units.

First, find the sum of length and width: L + W = (3x + 2) + (2x - 1).

Combine like terms: (3x + 2x) + (2 - 1) = 5x + 1.

Now, multiply the sum by 2 for the perimeter: P = 2 × (5x + 1) = 10x + 2.