PRC-2 · Chapter 1
Linear And Quadratic Equations MCQs with Answers
45 multiple-choice questions on Linear And Quadratic Equations for PRC-2 Quantitative Analysis for Business. Try each one before revealing the answer and explanation.
Practise this chapter interactivelyQuestion 1
Which of the following best describes an algebraic expression?
- A) A mathematical statement containing an equals sign.
- B) A combination of numbers, variables, and operation symbols without an equals sign.
- C) A value that makes an equation fundamentally true.
- D) A sequence of numbers following a specific constant rule.
Show answer & explanation
Answer: B) A combination of numbers, variables, and operation symbols without an equals sign.
An expression is a combination of numbers and variables and their related operation symbols. Unlike an equation, it does not include an equals sign and only forms mathematical phrases.
Question 2
What is the degree of the polynomial equation 4x^4 + 3x^3 - 2x = 0?
- A) 1
- B) 2
- C) 3
- D) 4
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Answer: D) 4
The degree of a polynomial equation is determined by the highest power (exponent) of the variable used in the expression. In this equation, the highest power is 4.
Question 3
Ahmed is four times as old as his daughter today. If 'x' represents Ahmed's age and 'y' represents his daughter's age, which linear equation correctly models this relationship?
- A) x = 4y
- B) y = 4x
- C) x = y + 4
- D) 4x = y
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Answer: A) x = 4y
If Ahmed (x) is four times as old as his daughter (y), you must multiply the daughter's age by 4 to equal Ahmed's current age, resulting in the equation x = 4y.
Question 4
What are the values of x and y in the following simultaneous linear equations: 2x + 3y = 20 and x + y = 8?
- A) x = 4, y = 4
- B) x = 6, y = 2
- C) x = 5, y = 3
- D) x = 2, y = 6
Show answer & explanation
Answer: A) x = 4, y = 4
Multiplying the second equation by 2 gives 2x + 2y = 16. Subtracting this from the first equation (2x + 3y = 20) yields y = 4. Substituting y = 4 back into x + y = 8 gives x = 4.
Question 5
Which of the following strictly defines a quadratic expression?
- A) An expression where the highest degree of the variable is one.
- B) An expression where the highest degree of the variable is two.
- C) An expression where the highest degree of the variable is three.
- D) An expression containing three different variables.
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Answer: B) An expression where the highest degree of the variable is two.
A quadratic expression (or second-degree expression) is defined as one in which the highest power (degree) of the variable is a square, or two.
Question 6
Find the roots of the quadratic equation x^2 - 5x + 6 = 0 using factorization.
- A) 2 and 3
- B) -2 and -3
- C) 1 and 6
- D) -1 and -6
Show answer & explanation
Answer: A) 2 and 3
The equation factors into (x - 2)(x - 3) = 0 because -2 and -3 multiply to positive 6 and add to -5. Therefore, the roots are x = 2 and x = 3.
Question 7
When using the quadratic formula to solve ax^2 + bx + c = 0, what does the expression under the square root sign represent?
- A) b^2 + 4ac
- B) a^2 - 4bc
- C) b^2 - 4ac
- D) 2a - bc
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Answer: C) b^2 - 4ac
The standard quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a. The core expression under the root, known as the discriminant, is b^2 - 4ac.
Question 8
What are the roots of the quadratic equation x^2 - 4x - 5 = 0?
- A) 5 and -1
- B) -5 and 1
- C) 4 and -5
- D) -4 and 5
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Answer: A) 5 and -1
By factoring the equation into (x - 5)(x + 1) = 0, we can determine that the roots satisfying the equation are x = 5 and x = -1.
Question 9
Which of the following is the core concept when solving a quadratic equation by completing the square?
- A) Multiplying the entire equation by the constant term.
- B) Making the coefficient of x^2 equal to zero.
- C) Creating a perfect square trinomial on one side of the equation.
- D) Eliminating the x term completely from the equation.
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Answer: C) Creating a perfect square trinomial on one side of the equation.
The method of completing the square involves manipulating the equation to solve for a perfect square, utilizing reference expansion formulas like (x + a)^2 = x^2 + 2ax + a^2.
Question 10
When formulating a linear equation from a word problem, which of the following is the recommended first step?
- A) Cancel out the largest value immediately.
- B) Write all information in terms of the largest value.
- C) Select the unknown value, typically assigning the smallest unknown value to X.
- D) Multiply all given numbers together to form a baseline.
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Answer: C) Select the unknown value, typically assigning the smallest unknown value to X.
The first systematic step in formulating a linear equation from word problems is to select the unknown value. If there are multiple values, always select the smallest value to be the unknown 'X'.
Question 11
A rectangular plot has a length that is 4 meters greater than its width. If the total area is 96 square meters, which of the following quadratic equations represents this scenario where 'w' is the width?
- A) w^2 + 4 = 96
- B) w^2 + 4w - 96 = 0
- C) 4w^2 = 96
- D) w(w - 4) = 96
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Answer: B) w^2 + 4w - 96 = 0
The length is (w + 4). The area is w(w + 4) = 96, which expands to w^2 + 4w = 96. Rearranging this yields the quadratic equation w^2 + 4w - 96 = 0.
Question 12
Assuming a linear cost relationship, if 5 notebooks cost Rs. 150 and 8 notebooks cost Rs. 240, what is the cost of 1 notebook?
- A) Rs. 20
- B) Rs. 25
- C) Rs. 30
- D) Rs. 35
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Answer: C) Rs. 30
The rate of change (slope) represents the cost per unit. (240 - 150) / (8 - 5) = 90 / 3 = 30. Therefore, each notebook costs Rs. 30.
Question 13
Solve the following linear equation for x: 3(x - 2) + 4 = 13.
- A) x = 3
- B) x = 4
- C) x = 5
- D) x = 6
Show answer & explanation
Answer: C) x = 5
First, subtract 4 from both sides: 3(x - 2) = 9. Then, divide by 3: (x - 2) = 3. Finally, add 2 to both sides to find x = 5.
Question 14
In the expression 5x^2 + 8x - 3, what are the distinct components separated by the mathematical operation symbols called?
- A) Degrees
- B) Variables
- C) Roots
- D) Terms
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Answer: D) Terms
An algebraic expression is separated by operations (addition or subtraction) into distinct components known as 'terms'.
Question 15
How many possible values (roots) does a standard quadratic equation of the form ax^2 + bx + c = 0 typically have?
- A) 1
- B) 2
- C) 3
- D) 4
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Answer: B) 2
Because the highest degree of a quadratic equation is two, solving it typically involves finding the two values (roots) of x that satisfy the equation.
Question 16
The product of two positive integers is 45 and when added together the result is 14. Find the integers.
- A) 3 and 15
- B) 5 and 9
- C) 6 and 8
- D) 4 and 10
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Answer: B) 5 and 9
Let the numbers be x and (14 - x). Their product is x(14 - x) = 45, which simplifies to the quadratic equation x^2 - 14x + 45 = 0. Factoring this yields (x - 5)(x - 9) = 0, giving the integers 5 and 9.
Question 17
Solve the following equation for the value of x: 10x - (200 / x) = 10.
- A) 5 and -4
- B) 4 and -5
- C) 10 and -2
- D) -10 and 2
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Answer: A) 5 and -4
Multiply the entire equation by x to remove the denominator: 10x^2 - 200 = 10x. Rearranging yields 10x^2 - 10x - 200 = 0. Dividing by 10 gives x^2 - x - 20 = 0. Factoring yields (x - 5)(x + 4) = 0, meaning x = 5 and x = -4.
Question 18
Two times the square of a positive number, when added to five times the same number, results in 33. What is the number?
- A) 2
- B) 3
- C) 4
- D) 5
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Answer: B) 3
The problem translates to the quadratic equation 2x^2 + 5x = 33. Rearranging gives 2x^2 + 5x - 33 = 0. Factoring yields (2x + 11)(x - 3) = 0. Since it must be a positive number, x = 3.
Question 19
A father distributes Rs. 2,000 amongst his three daughters in such a way that each one of them receives at least Rs. 200. What is the maximum possible mathematical difference between the amounts received by any two daughters?
- A) Rs. 1,400
- B) Rs. 1,600
- C) Rs. 1,800
- D) Rs. 1,200
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Answer: A) Rs. 1,400
To maximize the difference, one daughter must receive the absolute maximum while the others receive the absolute minimum. The two minimums are Rs. 200 each (total Rs. 400). The maximum one can receive is 2,000 - 400 = 1,600. The difference is 1,600 - 200 = Rs. 1,400.
Question 20
Solve the following linear equation for x: 4(x - 2) + 3 = 15.
- A) 4
- B) 5
- C) 6
- D) 7
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Answer: B) 5
First, subtract 3 from both sides: 4(x - 2) = 12. Then divide by 4: x - 2 = 3. Adding 2 to both sides results in x = 5.
Question 21
A company's revenue function is R = 60x and its cost function is C = 25x + 14,000. At what production level 'x' does the company break even?
- A) 300 units
- B) 350 units
- C) 400 units
- D) 450 units
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Answer: C) 400 units
Break-even occurs when Revenue equals Cost. 60x = 25x + 14,000. Subtracting 25x leaves 35x = 14,000. Dividing by 35 gives x = 400 units.
Question 22
What is the discriminant of the quadratic equation 3x^2 - 4x + 2 = 0?
- A) -8
- B) 8
- C) 16
- D) -24
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Answer: A) -8
The discriminant formula is b^2 - 4ac. Here, a = 3, b = -4, and c = 2. (-4)^2 - 4(3)(2) = 16 - 24 = -8. Because it is negative, this equation has no real roots.
Question 23
What is the sum of the roots for the quadratic equation 2x^2 - 10x + 12 = 0?
- A) -5
- B) 5
- C) -10
- D) 10
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Answer: B) 5
The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is calculated as -b / a. Here, -(-10) / 2 = 10 / 2 = 5.
Question 24
What is the product of the roots for the quadratic equation 2x^2 - 10x + 12 = 0?
- A) 6
- B) 12
- C) -6
- D) 5
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Answer: A) 6
The product of the roots of a quadratic equation ax^2 + bx + c = 0 is calculated as c / a. Here, 12 / 2 = 6.
Question 25
Find the values of x and y in the following simultaneous linear equations: 3x - 2y = 8 and x + y = 6.
- A) x = 3, y = 3
- B) x = 4, y = 2
- C) x = 5, y = 1
- D) x = 2, y = 4
Show answer & explanation
Answer: B) x = 4, y = 2
Multiply the second equation by 2 to get 2x + 2y = 12. Add this to the first equation (3x - 2y = 8) to eliminate y. This yields 5x = 20, so x = 4. Substituting x = 4 into x + y = 6 gives y = 2.
Question 26
What are the specific roots of the quadratic equation x^2 - 11x + 30 = 0?
- A) 5 and -6
- B) -5 and 6
- C) 5 and 6
- D) -5 and -6
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Answer: C) 5 and 6
To factor the equation, find two numbers that multiply to 30 and add to -11. These are -5 and -6. The factored form is (x - 5)(x - 6) = 0, meaning the roots are x = 5 and x = 6.
Question 27
A mother is currently twice as old as her daughter. Five years ago, she was three times as old as her daughter. How old is the daughter today?
- A) 10 years old
- B) 15 years old
- C) 20 years old
- D) 5 years old
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Answer: A) 10 years old
Let daughter's age = x, mother's age = 2x. Five years ago: 2x - 5 = 3(x - 5). This expands to 2x - 5 = 3x - 15. Solving for x yields x = 10. The daughter is 10 years old.
Question 28
A factory has a linear cost function. Producing 50 units costs Rs. 2,500, and producing 100 units costs Rs. 4,000. What is the variable cost per unit?
- A) Rs. 25
- B) Rs. 30
- C) Rs. 40
- D) Rs. 50
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Answer: B) Rs. 30
Variable cost is the slope of the linear cost equation. Slope = (Change in Cost) / (Change in Units) = (4000 - 2500) / (100 - 50) = 1500 / 50 = Rs. 30 per unit.
Question 29
The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, the fraction becomes 1/2. What is the original denominator?
- A) 3
- B) 4
- C) 6
- D) 8
Show answer & explanation
Answer: B) 4
Let denominator be x, so numerator is (x - 3). The new fraction is (x - 3 + 2) / (x + 2) = 1/2. This simplifies to (x - 1) / (x + 2) = 1/2. Cross multiplying gives 2x - 2 = x + 2. Solving yields x = 4.
Question 30
Find the roots of the quadratic equation 4x^2 - 8x - 5 = 0.
- A) 5/2 and -1/2
- B) -5/2 and 1/2
- C) 5 and -1
- D) -5 and 1
Show answer & explanation
Answer: A) 5/2 and -1/2
Splitting the middle term: 4x^2 - 10x + 2x - 5 = 0. Factoring by grouping gives 2x(2x - 5) + 1(2x - 5) = 0, which is (2x + 1)(2x - 5) = 0. Therefore, the roots are x = -1/2 and x = 5/2.
Question 31
Which of the following expressions is a quadratic equation?
- A) 2x + 5 = 11
- B) 3x^2 - 4x + 7 = 0
- C) 5x + 3y = 20
- D) x^3 - 8 = 0
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Answer: B) 3x^2 - 4x + 7 = 0
A quadratic equation is defined by having its highest power of the variable (x) as 2. Selection B follows the standard format ax^2 + bx + c = 0.
Question 32
Solve the linear equation for x: 3(x - 4) = 2(x + 1).
- A) x = 10
- B) x = 14
- C) x = 5
- D) x = 13
Show answer & explanation
Answer: B) x = 14
First, expand the brackets: 3x - 12 = 2x + 2. Then, move variables to one side and constants to the other: 3x - 2x = 12 + 2, resulting in x = 14.
Question 33
In the quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, what is the term (b^2 - 4ac) specifically called?
- A) The Coefficient
- B) The Denominator
- C) The Discriminant
- D) The Square Root Factor
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Answer: C) The Discriminant
The term b^2 - 4ac is the discriminant. It is used to determine the nature of the roots (whether they are real, equal, or imaginary).
Question 34
If the discriminant of a quadratic equation is exactly zero (b^2 - 4ac = 0), what can be concluded about its roots?
- A) The roots are real and distinct.
- B) The roots are real and equal (identical).
- C) The roots are imaginary.
- D) The equation has no solution.
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Answer: B) The roots are real and equal (identical).
When the discriminant is zero, the plus/minus part of the quadratic formula becomes zero, leaving only one unique value for x.
Question 35
What are the roots of the quadratic equation x^2 - 7x + 10 = 0?
- A) x = 2, 5
- B) x = -2, -5
- C) x = 1, 10
- D) x = 7, 10
Show answer & explanation
Answer: A) x = 2, 5
Factoring the equation (x - 2)(x - 5) = 0 gives roots x = 2 and x = 5. Note that -2 and -5 add to -7 and multiply to +10.
Question 36
An algebraic expression that contains exactly two terms is called a:
- A) Monomial
- B) Binomial
- C) Trinomial
- D) Polynomial
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Answer: B) Binomial
In algebra, 'Bi-' means two. Therefore, an expression with two terms (like x + 5) is a binomial.
Question 37
If two lines represented by linear equations have the exactly same slope but different y-intercepts, they are:
- A) Intersecting
- B) Perpendicular
- C) Parallel
- D) The same line
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Answer: C) Parallel
Parallel lines have the same gradient (slope), meaning they never cross each other regardless of how far they are extended.
Question 38
Solve the following system of linear equations for x and y: x + y = 10 and x - y = 2.
- A) x = 5, y = 5
- B) x = 6, y = 4
- C) x = 7, y = 3
- D) x = 8, y = 2
Show answer & explanation
Answer: B) x = 6, y = 4
Adding the two equations: (x + x) + (y - y) = 10 + 2 -> 2x = 12 -> x = 6. Substituting x=6 into the first equation: 6 + y = 10 -> y = 4.
Question 39
What is the degree of the polynomial expression: 4x^3 + 2x^2 - x + 1?
- A) 1
- B) 2
- C) 3
- D) 4
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Answer: C) 3
The degree of a polynomial is determined by the highest exponent of the variable. Here, the highest power is 3.
Question 40
In a coordinate geometry plane, the vertical axis is known as the:
- A) x-axis
- B) y-axis
- C) Origin
- D) Slope
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Answer: B) y-axis
The y-axis represents vertical movement (ordinate), while the x-axis represents horizontal movement (abscissa).
Question 41
If you multiply (x + 3) by (x - 3), the resulting expression is:
- A) x^2 + 6x + 9
- B) x^2 - 6x + 9
- C) x^2 - 9
- D) 2x
Show answer & explanation
Answer: C) x^2 - 9
This follows the 'difference of squares' rule: (a + b)(a - b) = a^2 - b^2. Thus, x^2 - 3^2 = x^2 - 9.
Question 42
A function that can be written in the form f(x) = mx + c is a:
- A) Quadratic Function
- B) Linear Function
- C) Cubic Function
- D) Exponential Function
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Answer: B) Linear Function
Linear functions represent straight lines on a graph and have a constant rate of change (m).
Question 43
For the quadratic equation ax^2 + bx + c = 0, the sum of the roots is equal to:
- A) c/a
- B) -b/a
- C) b/a
- D) √c/a
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Answer: B) -b/a
The Vieta's formulas state that for a quadratic equation, the sum of the roots is -b/a and the product of the roots is c/a.
Question 44
If x = 5 is a root of the equation x^2 + kx - 15 = 0, what is the value of k?
- A) 2
- B) -2
- C) 3
- D) -3
Show answer & explanation
Answer: B) -2
Substitute x = 5 into the equation: (5)^2 + 5k - 15 = 0 -> 25 + 5k - 15 = 0 -> 10 + 5k = 0 -> 5k = -10 -> k = -2.
Question 45
The point where the graph of an equation crosses the y-axis is called the:
- A) x-intercept
- B) y-intercept
- C) Gradient
- D) Vertex
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Answer: B) y-intercept
The y-intercept is found by setting x = 0 in the equation.