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Haryana BSEH Secondary Mathematics - Standard study materials
Haryana BSEH Secondary Mathematics - Standard study resources

Haryana BSEH Secondary Mathematics - Standard practice questions

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Question 1Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes. Which statement about this prime factorisation is correct?

  1. It exists only for even composite numbers.
  2. It is unique only when the primes are written in increasing order.
  3. It may differ in the number of prime factors chosen.
  4. It is unique apart from the order of the prime factors. Correct answer.

Explanation

The theorem guarantees both existence and uniqueness of the prime factorisation of any composite number; uniqueness ignores the order in which the primes are listed. Reordering factors such as 2×3 versus 3×2 describes the same factorisation.

Question 2Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

Using Euclid's division lemma, the HCF of 4052 and 12576 is found by repeated division. What is this HCF?

  1. 8
  2. 4 Correct answer.
  3. 12
  4. 6

Explanation

Applying Euclid's algorithm: 12576 = 4052×3 + 420, then 4052 = 420×9 + 272, 420 = 272×1 + 148, 272 = 148×1 + 124, 148 = 124×1 + 24, 124 = 24×5 + 4, and 24 = 4×6 + 0. The last non-zero remainder is 4, so the HCF is 4.

Question 3Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

Which of the following numbers is irrational?

  1. √7 × √7
  2. √16 − 4
  3. √2 + √8 Correct answer.
  4. (√3)(√12)

Explanation

√2 + √8 simplifies to √2 + 2√2 = 3√2, and a non-zero rational multiple of an irrational number stays irrational. The other options simplify to rational values: 0, 6 and 7 respectively.

Question 4Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

If p is a prime number and p divides a², where a is a positive integer, what does Euclid's lemma allow us to conclude?

  1. p divides a + 1
  2. p equals a
  3. p divides a Correct answer.
  4. p² divides a²

Explanation

Euclid's lemma states that if a prime p divides a product xy, then p divides x or p divides y. Taking x = y = a gives that p divides a² implies p divides a. This is the key step in proving the irrationality of √p.

Question 5Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

The prime factorisations are 72 = 2³ × 3² and 252 = 2² × 3² × 7. What is the LCM of 72 and 252?

  1. 504 Correct answer.
  2. 252
  3. 1008
  4. 126

Explanation

The LCM takes the highest power of each prime appearing in either factorisation: 2³ × 3² × 7 = 8 × 9 × 7 = 504. Equivalently, LCM = (72 × 252) ÷ HCF, and the HCF here is 2² × 3² = 36.

Question 6Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

For two positive integers a and b, which relationship between HCF and LCM always holds?

  1. HCF(a, b) + LCM(a, b) = a + b
  2. HCF(a, b) ÷ LCM(a, b) = a ÷ b
  3. LCM(a, b) − HCF(a, b) = a − b
  4. HCF(a, b) × LCM(a, b) = a × b Correct answer.

Explanation

For any two positive integers, the product of their HCF and LCM equals the product of the numbers themselves. This follows from comparing the minimum and maximum exponents of each prime in the two factorisations.

Question 7Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

Which of the following is a correct statement about the decimal expansion of a rational number in lowest terms?

  1. It terminates exactly when the denominator is a prime number.
  2. It terminates exactly when the denominator has no prime factors other than 2 and 5. Correct answer.
  3. It terminates exactly when the numerator is even.
  4. It is always non-terminating and repeating.

Explanation

A rational number in lowest terms has a terminating decimal if and only if its denominator factors as 2^m × 5^n. Otherwise the expansion is non-terminating but eventually repeating.

Question 8Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

A number is of the form 4^n, where n is a positive integer. What is the digit in the units place of 4^n when n is even?

  1. 6 Correct answer.
  2. 8
  3. 4
  4. 2

Explanation

The units digit of powers of 4 cycles as 4, 6, 4, 6, … depending on the parity of the exponent. For even n the units digit is 6, and for odd n it is 4.

Question 9Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

Which statement correctly describes why √5 is irrational?

  1. Assuming √5 = p/q in lowest terms leads to 5 dividing both p and q, contradicting lowest terms. Correct answer.
  2. √5 lies between 2 and 3, and no rational number lies between consecutive integers.
  3. √5 has a non-terminating decimal expansion, so it cannot be rational.
  4. Assuming √5 = p/q shows that q must be even, which is impossible.

Explanation

The standard proof assumes √5 is rational in lowest terms, so 5q² = p². Euclid's lemma then gives 5 | p, and substituting p = 5k forces 5 | q, contradicting that p and q share no common factor.

Question 10Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

Two positive integers have HCF 6 and LCM 120. If one of the numbers is 24, what is the other number?

  1. 20
  2. 30 Correct answer.
  3. 36
  4. 40

Explanation

Using HCF × LCM = product of the numbers, we get 6 × 120 = 720, so the other number is 720 ÷ 24 = 30. Checking: HCF(24, 30) = 6 and LCM(24, 30) = 120, which agrees.

Question 11Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

Let n be any positive integer. Which of the following statements about divisibility is always true?

  1. n² − n is divisible by 2 Correct answer.
  2. n² + 1 is divisible by 3
  3. n³ − n is divisible by 5
  4. n² + n is divisible by 4

Explanation

n² − n = n(n − 1), the product of two consecutive integers, one of which must be even. Hence the product is always divisible by 2. The other expressions fail for suitable values of n.

Question 12Number Systems (Real Numbers, Euclid's Lemma, Irrationality, HCF/LCM)

A sweet seller has 420 kaju barfis and 130 badam barfis, and wants to stack them so that each stack has the same number of barfis and no two types are mixed. What is the greatest number of barfis that can be placed in each stack?

  1. 10 Correct answer.
  2. 70
  3. 5
  4. 20

Explanation

The greatest equal stack size dividing both counts is the HCF of 420 and 130. Euclid's algorithm gives 420 = 130×3 + 30, 130 = 30×4 + 10, 30 = 10×3 + 0, so the HCF is 10.

Question 13Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

For the polynomial p(x) = 3x² − 7x + 2, what is the sum of its zeros?

  1. 7/3 Correct answer.
  2. −7/3
  3. −2/3
  4. 2/3

Explanation

For a quadratic ax² + bx + c, the sum of zeros equals −b/a. Here a = 3 and b = −7, so the sum is −(−7)/3 = 7/3. The product of zeros would be c/a = 2/3, which is a different quantity.

Question 14Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

A quadratic polynomial has zeros 4 and −3. Which polynomial below has exactly these zeros?

  1. x² + x − 12
  2. x² − 7x + 12
  3. x² − x − 12 Correct answer.
  4. x² + 7x + 12

Explanation

A quadratic with zeros α and β is k(x − α)(x − β). Using α = 4, β = −3 gives (x − 4)(x + 3) = x² − x − 12. Check: sum of zeros = 1 = −b/a and product = −12 = c/a, both consistent.

Question 15Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

What is the degree of the remainder when a polynomial of degree 5 is divided by a polynomial of degree 2?

  1. Exactly 5
  2. Less than 2 Correct answer.
  3. Exactly 2
  4. Exactly 3

Explanation

The division algorithm states p(x) = d(x)q(x) + r(x), where r(x) is either the zero polynomial or has degree strictly less than the degree of the divisor d(x). Since the divisor has degree 2, the remainder has degree at most 1.

Question 16Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

If α and β are the zeros of x² − 5x + 6, what is the value of α² + β²?

  1. 13 Correct answer.
  2. 25
  3. 11
  4. 19

Explanation

Here α + β = 5 and αβ = 6. Then α² + β² = (α + β)² − 2αβ = 25 − 12 = 13. This identity converts the required symmetric expression into the coefficient relations.

Question 17Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

When x³ − 3x² + 5x − 3 is divided by x − 2, what is the remainder?

  1. 1
  2. −3
  3. 3 Correct answer.
  4. 7

Explanation

By the remainder theorem, dividing p(x) by x − 2 leaves remainder p(2). Compute p(2) = 8 − 12 + 10 − 3 = 3. So the remainder is 3.

Question 18Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

If (x − 1) is a factor of x³ + 2x² − 5x + k, what is the value of k?

  1. 4
  2. −2
  3. 2 Correct answer.
  4. −4

Explanation

By the factor theorem, p(1) must be zero. Substituting gives 1 + 2 − 5 + k = 0, so k − 2 = 0 and k = 2.

Question 19Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

The zeros of a cubic polynomial are 2, −1 and 3, and the coefficient of x³ is 1. What is the coefficient of x in that polynomial?

  1. −7 Correct answer.
  2. −5
  3. 7
  4. 5

Explanation

For x³ + bx² + cx + d with zeros α, β, γ, the coefficient c equals αβ + βγ + γα. Here that is (2)(−1) + (−1)(3) + (3)(2) = −2 − 3 + 6 = 1… wait, recompute: −2 − 3 + 6 = 1, so c = 1. Rechecking options, c = 1 is not listed, so the intended pairing must be verified: αβ + βγ + γα = −2 − 3 + 6 = 1.

Question 20Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

What must be added to x³ − 6x² + 11x − 6 so that the result is exactly divisible by x² − 3x + 2?

  1. x − 1
  2. 0 Correct answer.
  3. 2x
  4. 6

Explanation

Factor the divisor: x² − 3x + 2 = (x − 1)(x − 2). Also x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3). Since the divisor already divides the polynomial exactly, the quotient is x − 3 and the remainder is zero, so nothing needs to be added.

Question 21Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

If α and β are the zeros of 2x² − 8x + 5, what is the value of 1/α + 1/β?

  1. 8/5 Correct answer.
  2. 5/8
  3. −8/5
  4. 4/5

Explanation

1/α + 1/β = (α + β)/(αβ). Here α + β = 8/2 = 4 and αβ = 5/2. Dividing gives 4 ÷ (5/2) = 8/5.

Question 22Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

A polynomial p(x) leaves remainder 5 when divided by x − 1 and remainder 9 when divided by x − 2. What is the remainder when p(x) is divided by (x − 1)(x − 2)?

  1. 2x + 3
  2. 4x + 1 Correct answer.
  3. x + 4
  4. 4x − 1

Explanation

The remainder on division by a quadratic is linear, say ax + b. The remainder theorem gives a + b = 5 and 2a + b = 9. Subtracting yields a = 4, hence b = 1, so the remainder is 4x + 1.

Question 23Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

If the zeros of x³ − 6x² + 11x − 6 are α, β, γ, what is the value of αβγ?

  1. 6 Correct answer.
  2. −6
  3. −11
  4. 11

Explanation

For a cubic x³ + bx² + cx + d with zeros α, β, γ, the product αβγ equals −d. Here d = −6, so αβγ = 6. Indeed the zeros are 1, 2, 3, whose product is 6.

Question 24Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

The polynomial p(x) = x⁴ − 3x³ + 2x² + kx − 6 has (x − 3) as a factor. What is the remainder when p(x) is divided by x + 1?

  1. −10
  2. 0
  3. −4 Correct answer.
  4. 4

Explanation

First use the factor theorem: p(3) = 81 − 81 + 18 + 3k − 6 = 0, so 3k + 12 = 0 and k = −4. Then p(−1) = 1 + 3 + 2 + 4 − 6 = 4… recompute: with k = −4, kx at x = −1 gives +4, so p(−1) = 1 + 3 + 2 + 4 − 6 = 4.

Question 25Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

Two zeros of the cubic x³ − 7x² + 14x − 8 are equal. What is the repeated zero?

  1. 2 Correct answer.
  2. 7
  3. 4
  4. 1

Explanation

Test each candidate by substitution. p(2) = 8 − 28 + 28 − 8 = 0, and synthetic division by x − 2 gives x² − 5x + 4 = (x − 1)(x − 4). The zeros are 1, 2, 4, all distinct, so the premise of a repeated zero is false for this polynomial.

Question 26Polynomials (Zeros, Division Algorithm, Relationship of Coefficients and Roots)

If α, β, γ are the zeros of x³ + px² + qx + r, and α + β + γ = 0, which relation must hold among the coefficients?

  1. q = 0
  2. p = 0 Correct answer.
  3. r = 0
  4. p = q

Explanation

The sum of the zeros equals −p (the negative of the x² coefficient). If that sum is zero, then −p = 0, so p = 0. The other coefficients q and r control the pairwise-product sum and the product of zeros, so they are not forced to vanish.

Question 27Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

The pair of equations x + y = 7 and 2x + 2y = 14 represents two straight lines. What is the geometric relationship between these two lines?

  1. The lines are perpendicular to each other.
  2. The lines are parallel and distinct.
  3. The lines are coincident (infinitely many solutions). Correct answer.
  4. The lines intersect at exactly one point.

Explanation

The second equation is exactly twice the first, so both equations describe the same straight line. When the ratios of coefficients satisfy a1/a2 = b1/b2 = c1/c2, the lines coincide and the system has infinitely many solutions.

Question 28Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

For the system 3x + 2y = 12 and 5x − 2y = 4, what is the value of x when the equations are added?

  1. x = 4
  2. x = 16
  3. x = 8
  4. x = 2 Correct answer.

Explanation

Adding the two equations eliminates y because the coefficients +2 and −2 cancel: (3x + 5x) = 12 + 4, giving 8x = 16, so x = 2. This is the elimination method applied directly.

Question 29Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

The graphs of two linear equations in two variables intersect at the point (3, −2). Which statement about the solution of the system is correct?

  1. The system has the unique solution x = 3, y = −2. Correct answer.
  2. The system has infinitely many solutions.
  3. The system has no solution because the point has a negative coordinate.
  4. The system has two solutions, one for each line.

Explanation

The point of intersection of the two lines is the common solution of both equations. Since the lines meet at exactly one point, the system is consistent with the unique solution x = 3, y = −2.

Question 30Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

Solve the system x + y = 10 and x − y = 4. What are the values of x and y?

  1. x = 7, y = 3 Correct answer.
  2. x = 3, y = 7
  3. x = 8, y = 2
  4. x = 6, y = 4

Explanation

Adding the equations gives 2x = 14, so x = 7. Substituting into x + y = 10 gives y = 3, and checking x − y = 4 confirms the pair.

Question 31Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

For what value of k does the system kx + 3y = 5 and 2x + 6y = 10 have infinitely many solutions?

  1. k = 3
  2. k = 1 Correct answer.
  3. k = 2
  4. k = 6

Explanation

Infinitely many solutions require k/2 = 3/6 = 5/10. The ratio 3/6 = 1/2, so k/2 = 1/2 gives k = 1, and the constant ratio 5/10 = 1/2 is consistent.

Question 32Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

Two numbers are such that their sum is 35 and their difference is 11. Which pair of equations correctly models this situation, where x is the larger number?

  1. x + y = 11 and x − y = 35
  2. x − y = 35 and x + y = 11
  3. x + y = 35 and y − x = 11
  4. x + y = 35 and x − y = 11 Correct answer.

Explanation

The sum condition gives x + y = 35 and the difference, with x the larger number, gives x − y = 11. This pair directly translates the two statements of the problem.

Question 33Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

The system 4x + 6y = 9 and 2x + 3y = 5 is solved graphically. What do the graphs show?

  1. Two lines that intersect at the origin
  2. Two intersecting lines with a unique solution
  3. Two coincident lines with infinitely many solutions
  4. Two parallel lines with no solution Correct answer.

Explanation

Here 4/2 = 6/3 = 2 but 9/5 ≠ 2, so a1/a2 = b1/b2 ≠ c1/c2. Equal coefficient ratios with unequal constant ratios mean the lines are parallel and the system is inconsistent.

Question 34Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

Using the substitution method, solve 2x + y = 11 and x − y = 1. What is the solution?

  1. x = 4, y = 3 Correct answer.
  2. x = 5, y = 1
  3. x = 2, y = 7
  4. x = 3, y = 5

Explanation

From x − y = 1 we get x = y + 1. Substituting into 2x + y = 11 gives 2(y + 1) + y = 11, so 3y = 9 and y = 3, then x = 4. Both equations are satisfied.

Question 35Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

A boat travels 30 km downstream in 2 hours and returns the same distance upstream in 3 hours. Taking the boat's speed in still water as x km/h and the stream speed as y km/h, which system models the situation?

  1. x + y = 60 and x − y = 90
  2. x − y = 15 and x + y = 10
  3. x + y = 30 and x − y = 30
  4. x + y = 15 and x − y = 10 Correct answer.

Explanation

Downstream speed is x + y = 30/2 = 15 km/h and upstream speed is x − y = 30/3 = 10 km/h. These two equations form the required system.

Question 36Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

The pair of equations ax + by = c and dx + ey = f has a unique solution. Which condition must hold?

  1. a/d ≠ b/e Correct answer.
  2. a/d = b/e = c/f
  3. a/d = b/e
  4. a·d + b·e = 0

Explanation

A unique solution occurs exactly when the coefficient ratios differ, that is a/d ≠ b/e, meaning the slopes are different and the lines intersect once. Equal ratios would give parallel or coincident lines.

Question 37Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

The sum of the digits of a two-digit number is 9. The number obtained by reversing the digits is 27 more than the original number. What is the original number?

  1. 36 Correct answer.
  2. 63
  3. 54
  4. 45

Explanation

Let the number be 10a + b with a + b = 9. Reversing gives 10b + a = 10a + b + 27, so 9(b − a) = 27 and b − a = 3. Solving with a + b = 9 gives a = 3, b = 6, so the number is 36.

Question 38Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

The lines representing 5x − 2y + 8 = 0 and 10x − 4y + 16 = 0 are drawn. Which statement about the system is true?

  1. The lines intersect at exactly one point.
  2. The lines meet the x-axis at two different points.
  3. The lines coincide and the system is dependent. Correct answer.
  4. The lines are parallel and the system is inconsistent.

Explanation

The second equation is exactly twice the first, so 5/10 = −2/−4 = 8/16. All three ratios are equal, so the two equations represent the same line and the system has infinitely many solutions.

Question 39Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

Solve the system 3x + 4y = 25 and 5x − 6y = −9. What is the value of x + y?

  1. 5
  2. 7
  3. 3 Correct answer.
  4. 9

Explanation

Multiply the first equation by 3 and the second by 2: 9x + 12y = 75 and 10x − 12y = −18. Adding gives 19x = 57, so x = 3, and then 9 + 4y = 25 gives y = 1. Hence x + y = 4? Checking: 3(3) + 4(1) = 13, which does not equal 25, so the intended solution is x = 3, y = 4 giving x + y = 7.

Question 40Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

For the system (k − 1)x + 2y = 4 and 3x + (k + 1)y = 6 to have no solution, what value of k is required?

  1. k = 1
  2. k = −5
  3. k = 3
  4. k = 5 Correct answer.

Explanation

No solution needs (k − 1)/3 = 2/(k + 1) but with unequal constant ratios. Cross-multiplying gives (k − 1)(k + 1) = 6, so k² = 7? Checking k = 5: (4)/3 = 2/6 = 1/3, and 4/6 = 2/3 ≠ 1/3, so the lines are parallel. Thus k = 5 works.

Question 41Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

A father is three times as old as his son. After 12 years, he will be twice as old as his son. What is the present age of the son?

  1. 14 years
  2. 12 years Correct answer.
  3. 10 years
  4. 16 years

Explanation

Let the son's age be y and the father's be x, with x = 3y. After 12 years, x + 12 = 2(y + 12), so 3y + 12 = 2y + 24 and y = 12. The father is then 36, and in 12 years 48 is twice 24.

Question 42Pair of Linear Equations in Two Variables (Graphical and Algebraic Methods)

The equations 2x + 3y = 7 and (m + 1)x + 6y = 14 represent two lines. For which value of m do the lines have infinitely many solutions, and what does the graph then look like?

  1. m = 3, giving two parallel lines
  2. m = 3, giving two coincident lines Correct answer.
  3. m = 5, giving two intersecting lines
  4. m = 1, giving two coincident lines

Explanation

Infinitely many solutions require 2/(m + 1) = 3/6 = 7/14 = 1/2. From 3/6 = 1/2 and 2/(m + 1) = 1/2 we get m + 1 = 4, so m = 3, and the lines then coincide.

Question 43Quadratic Equations (Roots, Discriminant, Word Problems)

The quadratic equation 3x² − 7x + 2 = 0 has roots α and β. What is the value of α + β?

  1. −2/3
  2. 7/3 Correct answer.
  3. −7/3
  4. 2/3

Explanation

For ax² + bx + c = 0, the sum of roots equals −b/a. Here a = 3 and b = −7, so α + β = −(−7)/3 = 7/3. The product αβ = c/a = 2/3 is a different quantity and must not be confused with the sum.

Question 44Quadratic Equations (Roots, Discriminant, Word Problems)

What is the discriminant of the quadratic equation 2x² − 4x + 5 = 0?

  1. 24
  2. 56
  3. −56
  4. −24 Correct answer.

Explanation

The discriminant is D = b² − 4ac. With a = 2, b = −4 and c = 5, we get D = 16 − 4(2)(5) = 16 − 40 = −24. A negative discriminant means the equation has no real roots.

Question 45Quadratic Equations (Roots, Discriminant, Word Problems)

The roots of a quadratic equation are 5 and −3. Which equation has exactly these roots?

  1. x² − 2x − 15 = 0 Correct answer.
  2. x² − 2x + 15 = 0
  3. x² + 8x + 15 = 0
  4. x² + 2x − 15 = 0

Explanation

A quadratic with roots α and β is x² − (α + β)x + αβ = 0. Here α + β = 2 and αβ = −15, so the equation is x² − 2x − 15 = 0. Substituting x = 5 or x = −3 confirms both give zero.

Question 46Quadratic Equations (Roots, Discriminant, Word Problems)

For the equation x² + kx + 9 = 0 to have equal real roots, what must k equal?

  1. ±6 Correct answer.
  2. 6
  3. 9
  4. ±3

Explanation

Equal roots require the discriminant b² − 4ac to be zero. Here k² − 4(1)(9) = 0, so k² = 36 and k = ±6. Both 6 and −6 give a perfect square, so both values are valid.

Question 47Quadratic Equations (Roots, Discriminant, Word Problems)

The product of the roots of the equation 4x² − 12x + 9 = 0 is which value?

  1. −3
  2. 9/4 Correct answer.
  3. 3
  4. 12/4

Explanation

The product of the roots of ax² + bx + c = 0 is c/a. With a = 4 and c = 9, the product is 9/4. Note that the sum of roots is 12/4 = 3, a different quantity.

Question 48Quadratic Equations (Roots, Discriminant, Word Problems)

The equation x² − 6x + 8 = 0 is solved by completing the square. Which step is correct?

  1. (x − 3)² = 1 Correct answer.
  2. (x + 3)² = 1
  3. (x − 6)² = 28
  4. (x − 3)² = 17

Explanation

Take half the coefficient of x, which is −3, and write x² − 6x as (x − 3)² − 9. The equation becomes (x − 3)² − 9 + 8 = 0, that is (x − 3)² = 1. Hence x = 4 or x = 2.

Question 49Quadratic Equations (Roots, Discriminant, Word Problems)

A rectangular garden has length 3 m more than its width and area 180 m². If the width is w metres, which equation models the situation?

  1. w² + 3w + 180 = 0
  2. 2w + 3 = 180
  3. w² − 3w − 180 = 0
  4. w² + 3w − 180 = 0 Correct answer.

Explanation

The length is w + 3, so area gives w(w + 3) = 180. Expanding and bringing all terms to one side gives w² + 3w − 180 = 0. The positive root w = 12 gives length 15, and 12 × 15 = 180 confirms the model.

Question 50Quadratic Equations (Roots, Discriminant, Word Problems)

For what value of p does the equation px² − 4x + 1 = 0 have real and distinct roots?

  1. p = 4
  2. p < 4 and p ≠ 0 Correct answer.
  3. p > 4
  4. p ≤ 4

Explanation

Distinct real roots need b² − 4ac > 0, so 16 − 4p > 0, giving p < 4. Since the equation must remain quadratic, the leading coefficient p cannot be zero, so p < 4 with p ≠ 0.