Mathematics ยท Reasoning

Algebraic Expressions

157 Questions

Algebraic expressions form the foundation of mathematics, involving variables, constants, and arithmetic operations. This topic includes evaluating coefficients, solving proportional relationships, and understanding boolean logic. It is widely tested across various competitive exams to assess analytical skills.

Variable evaluationExpression coefficientsBoolean logic variablesDirect proportionalityProgramming expressions

Algebraic Expressions Questions

Multiple choice

If (x^2 + y^2 = 25) and (x - y = 3), find the value of (x + y).

  1. 5

  2. 7

  3. 9

  4. 11
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

From (x - y = 3), we can express (x) as (x = y + 3). Substituting this into (x^2 + y^2 = 25), we get ((y + 3)^2 + y^2 = 25). Expanding and simplifying, we get (2y^2 + 6y + 9 = 25). Subtracting 9 from both sides, we get (2y^2 + 6y - 16 = 0). Factoring, we get ((2y - 4)(y + 4) = 0). Therefore, (y = 2) or (y = -4). If (y = 2), then (x = y + 3 = 2 + 3 = 5). Therefore, (x + y = 5 + 2 = 7). If (y = -4), then (x = y + 3 = -4 + 3 = -1). Therefore, (x + y = -1 + (-4) = -5). Since (x + y) cannot be negative, the correct answer is (7).

Multiple choice

If (x^2 + y^2 = 25) and (x - y = 3), find the value of (x + y).

  1. 4

  2. 6

  3. 8

  4. 10
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Squaring both sides of (x - y = 3), we get ((x - y)^2 = 3^2). Expanding the square, we get (x^2 + y^2 - 2xy = 9). Substituting (x^2 + y^2 = 25) in this equation, we get (25 - 2xy = 9). Rearranging the equation, we get (2xy = 16). Dividing both sides by 2, we get (xy = 8). Now, adding (x + y) and (x - y), we get (2x = x + y + x - y = 10). Therefore, (x + y = 10).

Multiple choice

If (x^3 - y^3 = 27) and (x - y = 3), find the value of (x^2 + y^2).

  1. 18

  2. 24

  3. 30

  4. 36
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the identity (a^3 - b^3 = (a - b)(a^2 + ab + b^2)), we can write (x^3 - y^3 = (x - y)(x^2 + xy + y^2)). Substituting (x - y = 3) in this equation, we get (27 = 3(x^2 + xy + y^2)). Dividing both sides by 3, we get (x^2 + xy + y^2 = 9). Now, squaring both sides of (x - y = 3), we get ((x - y)^2 = 3^2). Expanding the square, we get (x^2 + y^2 - 2xy = 9). Adding (x^2 + xy + y^2) and (x^2 + y^2 - 2xy), we get (2(x^2 + y^2) = 18). Therefore, (x^2 + y^2 = 9).

Multiple choice

If (x^2 + y^2 = 25) and (x - y = 3), find the value of (x + y).

  1. 5

  2. 7

  3. 9

  4. 11
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given (x^2 + y^2 = 25) and (x - y = 3), we can solve for (x + y) by adding (x + y) to both sides of the second equation and then squaring both sides of the resulting equation. This gives us ((x + y)^2 = (x - y)^2 + 4xy). Substituting (x^2 + y^2 = 25) and (x - y = 3) into this equation, we get ((x + y)^2 = 9 + 4(25) = 109). Taking the square root of both sides, we get (x + y = \pm\sqrt{109}). Since (x + y) must be positive, we have (x + y = \sqrt{109}). Therefore, the answer is (\sqrt{109}), which is approximately equal to 10.3.

Multiple choice

How does Algebra contribute to the development of other scientific disciplines?

  1. Algebraic techniques are used in physics to model and analyze physical phenomena.

  2. Algebraic structures are employed in computer science to design algorithms and data structures.

  3. Algebraic methods are applied in economics to study market behavior and optimize resource allocation.

  4. All of the above.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Algebra contributes to the development of other scientific disciplines by providing techniques for modeling and analyzing physical phenomena, designing algorithms and data structures, and studying market behavior and resource allocation.

Multiple choice

In the 2018 China Girls' Mathematical Olympiad, what was the value of y in the equation 2y + 3 = 7?

  1. 1

  2. 2

  3. 3

  4. 4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To solve the equation 2y + 3 = 7, we can subtract 3 from both sides to get 2y = 4. Then, we can divide both sides by 2 to get y = 2.

Multiple choice

Find the value of y in the following equation: $$2y + 5 = 17$$

  1. 6

  2. 7

  3. 8

  4. 9
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the equation, we need to isolate y on one side of the equation. Subtracting 5 from both sides, we get: $$2y + 5 - 5 = 17 - 5$$ $$2y = 12$$ Dividing both sides by 2, we get: $$y = \frac{12}{2}$$ $$y = 6$$