Olympiad Test: Algebraic Expressions - Free MCQ with solutions Class 7

The constant term in the expression 1 + x² + x is 1.

A constant has a fixed value, such as 4, 100, or -17. In algebraic expressions, we combine variables and constants using operations like addition, subtraction, multiplication, and division.

For example:

• The expression 4x + 5 is formed by multiplying the variable x by the constant 4 and then adding 5.
• The expression 10y - 20 is created by multiplying y by 10 and then subtracting 20.

We can also create expressions by combining variables with themselves or with other variables. For instance:

• x² is obtained by multiplying x by itself.
• 2y² is formed by multiplying y by itself and then multiplying by 2.
• 3x² - 5 involves first obtaining x², multiplying it by 3, and then subtracting 5.

In summary, the constant term in the expression is clearly defined as 1.

Coefficient refers to the numerical value associated with a variable in an expression. In the expression y - y³ + y², we can identify the coefficient of y³ as follows:

• The term involving y³ is -y³.
• The coefficient is the number in front of y³, which is -1.

Therefore, the coefficient of y³ in the expression is -1.

Subtracting z from y is written as y – z, which corresponds to option B.

- To find the value of  x + 4 for x = 2, substitute 2 into the expression:
x + 4 = 2 + 4
- Calculate the result:
2 + 4 = 6 
- The correct answer is option A: 6.

- When terms have the same algebraic factor, they are called like terms.
- Like terms have identical variables raised to the same powers, but they may have different coefficients.
- For example, in the expression 3x² + 5x - 2x², the terms 3x² and -2x² are like terms.

- An expression containing two unlike terms is called a binomial.
- Binomial comes from the prefix "bi-" meaning two, and "nomial" meaning terms.
- A binomial is part of polynomial expressions, which can have multiple terms.
- Examples of a binomial include expressions like 3x + 2 or a - b.
- Monomial refers to a single term, and trinomial refers to three terms.
- Thus, the correct answer is A: binomial.

- A variable is a symbol that represents a quantity which can change or take different values.

- In both mathematics and programming, variables act as placeholders for data that can vary.

- Examples of variables include x and y in algebra, which can represent any number.

- Unlike constants, which have a fixed value (e.g., 4, 100, -17), variables are flexible and adaptable.

- We create algebraic expressions by combining variables with constants using operations such as addition, subtraction, multiplication, and division.

• For example, the expression 4x + 5 is formed by multiplying x by the constant 4 and then adding 5.
• Similarly, 10y - 20 is obtained by multiplying y by 10 and then subtracting 20.

- Expressions can also be formed by combining variables with themselves or with other variables.

• The expression x² is created by multiplying x by itself.
• Other examples include 2y², 3x² - 5, and xy.

- In summary, variables are essential in forming algebraic expressions, allowing for a wide range of mathematical operations and representations.

Solution:

To simplify the expression 3x - 5a - x² + 9b with the values:

• x = 3
• a = -1
• b = -2

We substitute the values into the expression:

3(3) - 5(-1) - (3)² + 9(-2)

• Calculate each term:
• 3(3) = 9
• -5(-1) = 5
• (3)² = 9
• 9(-2) = -18

Now, combine the results:

9 + 5 - 9 - 18

Calculating step-by-step:

• 9 + 5 = 14
• 14 - 9 = 5
• 5 - 18 = -13

The final value is -13.

Solution:

To simplify the expression 2b - 8x + 4x² + 4a, we substitute the values:

• Let x = 3, a = -1, and b = -2.

Now, substituting these values into the expression:

• 2b 2(-2) = -4
• -8x: -8(3) = -24
• 4x²: 4(3²) = 4(9) = 36
• 4a: 4(-1) = -4

Now, combine these results:

• -4 - 24 + 36 - 4

Calculating this gives:

• -4 - 24 = -28
• -28 + 36 = 8
• 8 - 4 = 4

The final value of the expression is 4.

Simplifying the expression:

To simplify the expression 3a – 2b – ab – (a – b + ab) + 3ab + b – a, follow these steps:

• Start by rewriting the expression:
• 3a – 2b – ab – (a – b + ab) + 3ab + b – a
• Group similar terms together:
• 3a - a - a - 2b + b + b - ab - ab + 3ab
• Combine the terms:
• 3a - a - a = a
• -2b + b + b = 0
• -ab - ab + 3ab = ab
• Putting it all together gives:
• a + ab

The simplified expression is a + ab.

The expression 1.2ab – 2.4b + 3.6a consists of three terms:

• 1.2ab
• –2.4b
• 3.6a

Therefore, the number of terms in this expression is 3.

Charges of a taxi per km: $27

Fixed charge: $45

Distance hired: z km

• Charges for z km = $27 × z = $27z
• Algebraic expression for total fare = Charges for z km + Fixed charge
• Total fare = $27z + 45

The numerical coefficient refers to the constant part of a term that does not include any variables. In the expression provided, the coefficient of y² is -15.

Given:
7xy² − y² + 7x²y − 5x² − 3y² + 4y²x − 3y² + x²

Step-wise combining like terms:

1.Combine xy² terms
7xy² + 4xy² = 11xy²

2.Combine x²y terms
Only one term: 7x²y

3.Combine x² terms
−5x² + x² = −4x²

4.Combine y² terms
−y² − 3y² − 3y² = −7y²

Final simplified expression
7x²y + 11xy² − 4x² − 7y²

- Sita's current age is y years.
- Her brother's current age is twice Sita's age, which is 2y 
- To find her brother's age 2 years ago, subtract 2 from his current age: 2y - 2 

Thus, the correct answer is option A: 2y - 2

- The expression xyz is classified as a monomial.

- A monomial consists of a single term formed by multiplying numbers and variables, without any addition or subtraction.

- It contains only one term, regardless of the number of variables or constants present.

- Examples of monomials include: 7xy, -5m, 3z², and 4.

- In contrast, an expression with two unlike terms is called a binomial, while one with three terms is a trinomial.

• A binomial example: x + y.
• A trinomial example: x + y + 7.

- In general, any expression with one or more terms is referred to as a polynomial.

To simplify the expression −3(x + y) + 5(x − y), follow these steps:

• Expand the expression:

−3(x + y) + 5(x − y)

• Distribute the terms:

−3x − 3y + 5x − 5y

• Combine like terms:

(−3x + 5x) + (−3y − 5y) = 2x − 8y

Thus, the simplified expression is 2x − 8y.

Like terms are terms that share the same variable(s) raised to the same power(s).

In this case:

• 10pq and -7pq are like terms because they both contain pq with identical powers of p and q.

To identify like terms, follow these steps:

• Ignore the numerical coefficients and focus on the variables.
• Ensure the variables are the same.
• Check that the powers of each variable match.

Note that the numerical coefficients and the order of multiplication do not affect whether terms are like or unlike.

The phrase “the product of 3 and x” means 3x.

“Twice” indicates that we multiply the product by 2.

Therefore, “twice the product of 3 and x” can be expressed as:

2(3x)

To find the value of the expression:

We need to evaluate the expression 2a²b + 2ab² + ab with the values:

• a = 0
• b = -1

Substituting the values into the expression:

• 2(0)²(-1) + 2(0)(-1)² + (0)(-1)
• = 0 + 0 + 0
• = 0

The final result is 0, which is the required value.