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RBSE Secondary Mathematics study materials
RBSE Secondary Mathematics study resources

RBSE Secondary Mathematics practice questions

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Question 1Real Numbers

Which of the following is an irrational number?

  1. √49
  2. 0.125
  3. √2 Correct answer.
  4. 22/7

Explanation

An irrational number cannot be written as a ratio of two integers, and its decimal expansion is non-terminating and non-repeating. The square root of 2 is not a perfect square, so it is irrational. The other three values are all rational.

Question 2Real Numbers

The decimal expansion of 7/8 terminates. Which statement explains why?

  1. The numerator and denominator are coprime.
  2. The fraction is less than 1.
  3. The numerator 7 is a prime number.
  4. The denominator 8 has only 2 as a prime factor. Correct answer.

Explanation

A fraction in lowest terms has a terminating decimal exactly when its denominator factors into 2s and 5s only. Since 8 = 2³, the decimal expansion terminates, giving 0.875.

Question 3Real Numbers

What is the value of the digit string 0.333... (the 3 repeating forever) expressed as a fraction in lowest terms?

  1. 1/3 Correct answer.
  2. 33/100
  3. 1/30
  4. 3/10

Explanation

Let x = 0.333..., then 10x = 3.333..., and subtracting gives 9x = 3, so x = 3/9 = 1/3. The repeating decimal therefore equals one third exactly.

Question 4Real Numbers

Which list shows the numbers √9, π, −4, and 2/5 arranged in ascending order?

  1. −4, √9, 2/5, π
  2. 2/5, −4, √9, π
  3. −4, 2/5, √9, π Correct answer.
  4. −4, 2/5, π, √9

Explanation

Evaluating each: √9 = 3, π ≈ 3.14159, −4 is negative, and 2/5 = 0.4. Ascending order places the negative first, then 0.4, then 3, then π.

Question 5Real Numbers

Which statement about the set of real numbers is true?

  1. Every irrational number is a real number. Correct answer.
  2. Every integer is a whole number.
  3. Every rational number is an integer.
  4. Every real number is rational.

Explanation

The real numbers are the union of the rational and irrational numbers, so each irrational number belongs to the reals. Rational numbers include non-integers such as 1/2, and whole numbers exclude negatives.

Question 6Real Numbers

Express 0.474747... (the block 47 repeating) as a fraction in lowest terms.

  1. 47/90
  2. 47/100
  3. 47/99 Correct answer.
  4. 94/99

Explanation

Let x = 0.474747..., so 100x = 47.474747.... Subtracting gives 99x = 47, hence x = 47/99. The two-digit repeating block corresponds to a denominator of 99.

Question 7Real Numbers

The prime factorisation of the denominator of a fraction in lowest terms is 2² × 5³. The decimal expansion of this fraction is:

  1. non-terminating and non-repeating
  2. non-terminating and repeating
  3. terminating after three decimal places Correct answer.
  4. terminating after two decimal places

Explanation

A fraction in lowest terms terminates when the denominator has only 2 and 5 as prime factors. The number of decimal places equals the larger exponent, which is 3 here. So the expansion terminates after three places.

Question 8Real Numbers

What is the value of (√5 + √2)(√5 − √2)?

  1. √7
  2. 2√10
  3. 3 Correct answer.
  4. 7

Explanation

This is a difference of squares: (√5)² − (√2)² = 5 − 2 = 3. The cross terms cancel, leaving an integer.

Question 9Real Numbers

Which of these numbers is rational?

  1. √2 × √3
  2. (√3)² Correct answer.
  3. √12
  4. π + 1

Explanation

Squaring √3 gives exactly 3, an integer, so the value is rational. The others involve surds or π in forms that cannot be reduced to a ratio of integers.

Question 10Real Numbers

Between which two consecutive integers does √50 lie?

  1. 5 and 6
  2. 8 and 9
  3. 6 and 7
  4. 7 and 8 Correct answer.

Explanation

Since 7² = 49 and 8² = 64, and 49 < 50 < 64, it follows that 7 < √50 < 8. So √50 lies between 7 and 8.

Question 11Real Numbers

The number 0.101001000100001... (with the count of zeros growing each time) is:

  1. rational because the pattern is regular
  2. irrational because it is non-terminating and non-repeating Correct answer.
  3. an integer
  4. rational because it is less than 1

Explanation

A rational number's decimal expansion must terminate or repeat a fixed block forever. This expansion never settles into a repeating block, so the number is irrational.

Question 12Real Numbers

If p and q are distinct primes, which statement about √(pq) is always true?

  1. It is equal to √p × q.
  2. It is irrational. Correct answer.
  3. It is an integer.
  4. It is rational but not an integer.

Explanation

A square root of a positive integer is rational only when the integer is a perfect square. The product of two distinct primes has each prime to the first power, so it is not a perfect square and √(pq) is irrational.

Question 13Real Numbers

Simplify (3 + √7)/(3 − √7) by rationalising the denominator.

  1. (8 + 3√7)/1
  2. (8 + 3√7)/2 Correct answer.
  3. (16 + 6√7)/2
  4. (3 + √7)²/2

Explanation

Multiply numerator and denominator by the conjugate 3 + √7. The denominator becomes 9 − 7 = 2, and the numerator becomes (3 + √7)² = 16 + 6√7, giving (16 + 6√7)/2 = (8 + 3√7)/2.

Question 14Real Numbers

A repeating decimal has a repeating block of length 6. When it is converted to a fraction in lowest terms, which denominator is guaranteed to divide the denominator of that fraction?

  1. 6
  2. 999999 Correct answer.
  3. 60
  4. 10⁶

Explanation

For a block of length 6, multiplying by 10⁶ and subtracting produces an equation whose coefficient is 10⁶ − 1 = 999999. The resulting fraction has denominator dividing 999999, so that value is guaranteed to be a multiple of the reduced denominator.

Question 15Real Numbers

Let a = √2 + √3. Which statement is true about a²?

  1. a² is irrational, and a² = 5 + √6
  2. a² is irrational, and a² = 5 + 2√6 Correct answer.
  3. a² is an integer equal to 6
  4. a² is rational and equals 5

Explanation

Squaring gives a² = 2 + 3 + 2√6 = 5 + 2√6. Since √6 is irrational, 5 + 2√6 is also irrational, so a² is irrational.

Question 16Real Numbers

Suppose x is a positive real number such that x + 1/x = 3. What is the value of x² + 1/x²?

  1. 9
  2. 6
  3. 11
  4. 7 Correct answer.

Explanation

Squaring both sides gives x² + 2 + 1/x² = 9, since the cross term 2·x·(1/x) equals 2. Subtracting 2 yields x² + 1/x² = 7.

Question 17Polynomials

For the polynomial p(x) = 3x⁴ − 2x³ + 7x − 5, what is the degree of p(x)?

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

Explanation

The degree of a polynomial is the highest power of the variable with a non-zero coefficient. In 3x⁴ − 2x³ + 7x − 5 the highest power present is x⁴, so the degree equals 4. The constant −5 is the constant term, not the degree.

Question 18Polynomials

Which of the following expressions is a polynomial in one variable?

  1. x⁻² + 3x
  2. √x + 4
  3. 3/(x + 1)
  4. 2x² − 5x + 1 Correct answer.

Explanation

A polynomial in one variable is a sum of terms of the form axⁿ where n is a non-negative integer and a is a constant. Only 2x² − 5x + 1 fits, since all exponents (2, 1, 0) are non-negative integers. The other expressions involve negative or fractional exponents, or division by a variable.

Question 19Polynomials

If p(x) = x² − 4x + 3, what is the value of p(2)?

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

Explanation

Substitute x = 2 into p(x): p(2) = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1. Evaluate the squared term first, then the linear term, then add the constant.

Question 20Polynomials

The zeroes of the polynomial p(x) = x² − 7x + 12 are:

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

Explanation

Factor x² − 7x + 12 as (x − 3)(x − 4). Setting each factor to zero gives x = 3 and x = 4. The sum is 7 and the product is 12, which matches the coefficients.

Question 21Polynomials

If α and β are the zeroes of 2x² − 5x + 3, what is the value of α + β?

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

Explanation

For ax² + bx + c, the sum of zeroes α + β = −b/a. Here a = 2, b = −5, c = 3, so α + β = −(−5)/2 = 5/2. The product would be c/a = 3/2.

Question 22Polynomials

When the polynomial 2x³ + 3x² − 11x − 6 is divided by (x − 2), what is the remainder?

  1. 4
  2. 0 Correct answer.
  3. 12
  4. −4

Explanation

By the Remainder Theorem the remainder is p(2). Compute p(2) = 2(8) + 3(4) − 11(2) − 6 = 16 + 12 − 22 − 6 = 0. A remainder of 0 means (x − 2) is a factor of the polynomial.

Question 23Polynomials

Which of the following is a factor of the polynomial x³ − 6x² + 11x − 6?

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

Explanation

Test the candidate factors by substituting the corresponding root. For (x − 3), p(3) = 27 − 54 + 33 − 6 = 0, so (x − 3) is a factor. The polynomial factors completely as (x − 1)(x − 2)(x − 3).

Question 24Polynomials

If a polynomial p(x) is divided by (2x − 3), the remainder equals:

  1. p(−3/2)
  2. p(3)
  3. p(2/3)
  4. p(3/2) Correct answer.

Explanation

By the Remainder Theorem, dividing p(x) by (x − k) leaves remainder p(k). Rewrite 2x − 3 as 2(x − 3/2), so the root is x = 3/2 and the remainder is p(3/2).

Question 25Polynomials

The polynomial p(x) = x⁴ − 5x² + 4 has zeroes at:

  1. ±2 and ±4
  2. ±1 and ±4
  3. only ±1
  4. ±1 and ±2 Correct answer.

Explanation

Let y = x², giving y² − 5y + 4 = 0, which factors as (y − 1)(y − 4) = 0. Thus x² = 1 or x² = 4, giving x = ±1 and x = ±2. A quartic can have up to four real zeroes, and here all four are real.

Question 26Polynomials

A quadratic polynomial has zeroes 3 and −5. Which polynomial is a valid choice?

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

Explanation

A polynomial with zeroes α and β is k(x − α)(x − β). Using k = 1: (x − 3)(x + 5) = x² + 5x − 3x − 15 = x² + 2x − 15. Check: sum of zeroes = −2 = −2/1 and product = −15, both consistent.

Question 27Polynomials

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

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

Explanation

We know α + β = 5 and αβ = 6. Using the identity α² + β² = (α + β)² − 2αβ = 25 − 12 = 13. Directly, the zeroes are 2 and 3, and 4 + 9 = 13.

Question 28Polynomials

The polynomial 2x³ − 9x² + x + 12 has 3 as one of its zeroes. What is the sum of its remaining two zeroes?

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

Explanation

For ax³ + bx² + cx + d, the sum of all three zeroes is −b/a = 9/2. If one zero is 3, the other two sum to 9/2 − 3 = 3/2. Dividing by (x − 3) gives 2x² − 3x − 4, whose zeroes sum to 3/2.

Question 29Polynomials

If (x − 2) is a factor of x³ − kx² + 4x − 8, what is the value of k?

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

Explanation

By the Factor Theorem, p(2) must equal 0. Compute p(2) = 8 − 4k + 8 − 8 = 8 − 4k, set equal to 0, giving k = 2. Substituting back verifies that x³ − 2x² + 4x − 8 factors as (x − 2)(x² + 4).

Question 30Polynomials

The polynomial p(x) = x³ + ax² + bx + 6 has zeroes that are all positive integers. If one zero is 1, which set gives all three zeroes?

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

Explanation

The product of the zeroes equals −d/a = −6, so with one zero equal to 1 the other two must multiply to 6 and sum to −a (an integer). The distinct positive integer pairs multiplying to 6 are (2, 3) and (1, 6). The unique set with three distinct positive integers is 1, 2, 3, giving p(x) = (x − 1)(x − 2)(x − 3).

Question 31Pair of Linear Equations in Two Variables

For the pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, which condition guarantees a unique solution?

  1. a₁/a₂ = b₁/b₂ ≠ c₁/c₂
  2. a₁/a₂ = b₁/b₂ = c₁/c₂
  3. a₁/a₂ ≠ b₁/b₂ Correct answer.
  4. a₁a₂ + b₁b₂ = 0

Explanation

The graphs of the two equations are straight lines, and a unique solution means the lines intersect at exactly one point. That happens when their slopes differ, which in coefficient form is a₁/a₂ ≠ b₁/b₂. The other ratio conditions describe coincident or parallel lines.

Question 32Pair of Linear Equations in Two Variables

The pair of equations 2x + 3y = 7 and 4x + 6y = 10 has:

  1. exactly two solutions
  2. infinitely many solutions
  3. exactly one solution
  4. no solution Correct answer.

Explanation

Compare the ratios: 2/4 = 3/6 = 1/2, but 7/10 ≠ 1/2. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and distinct, so the system is inconsistent and has no solution.

Question 33Pair of Linear Equations in Two Variables

Solve the system x + y = 10 and x − y = 4.

  1. x = 6, y = 4
  2. x = 7, y = 3 Correct answer.
  3. x = 3, y = 7
  4. x = 5, y = 5

Explanation

Adding the two equations eliminates y: 2x = 14, so x = 7. Substituting back into x + y = 10 gives y = 3. Checking in the second equation, 7 − 3 = 4, which is consistent.

Question 34Pair of Linear Equations in Two Variables

In the substitution method applied to y = 2x + 1 and 3x + y = 11, which equation is obtained after substituting for y?

  1. 3x + 2x − 1 = 11
  2. 3(2x + 1) + y = 11
  3. 3x − 2x + 1 = 11
  4. 3x + 2x + 1 = 11 Correct answer.

Explanation

Substitution means replacing y everywhere it appears in the second equation by the expression 2x + 1. This gives 3x + (2x + 1) = 11, that is 3x + 2x + 1 = 11, an equation in x alone.

Question 35Pair of Linear Equations in Two Variables

The graph of a pair of linear equations consists of two straight lines drawn on the same axes. If the lines coincide, the number of solutions of the pair is:

  1. one
  2. infinitely many Correct answer.
  3. zero
  4. two

Explanation

Coincident lines share every point, and each common point is a solution of the pair. Hence the system has infinitely many solutions, and the equations are dependent and consistent.

Question 36Pair of Linear Equations in Two Variables

Find the value of k for which the system kx + 3y = k − 3 and 12x + ky = k has no solution.

  1. k = 6
  2. k = −6 Correct answer.
  3. k = 0
  4. k = 3

Explanation

No solution requires a₁/a₂ = b₁/b₂ ≠ c₁/c₂, so k/12 = 3/k, giving k² = 36 and k = ±6. Testing k = 6 makes the constant ratio equal, giving infinitely many solutions, so only k = −6 makes the lines parallel and distinct.

Question 37Pair of Linear Equations in Two Variables

A two-digit number has digits whose sum is 9. Reversing the digits increases the number by 27. The number is:

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

Explanation

Let the tens digit be x and the units digit be y, so x + y = 9 and (10y + x) − (10x + y) = 27, which simplifies to y − x = 3. Solving gives y = 6 and x = 3, so the number is 36. Reversing gives 63, which is indeed 27 more.

Question 38Pair of Linear Equations in Two Variables

The sum of the ages of a father and his son is 50 years. Five years ago the father was seven times as old as the son. The present age of the son is:

  1. 10 years Correct answer.
  2. 12 years
  3. 8 years
  4. 5 years

Explanation

Let the father's age be f and the son's be s, so f + s = 50. Five years ago, f − 5 = 7(s − 5), that is f = 7s − 30. Substituting gives 7s − 30 + s = 50, so 8s = 80 and s = 10, with f = 40.

Question 39Pair of Linear Equations in Two Variables

Two numbers differ by 5 and twice the smaller exceeds the larger by 3. The smaller number is:

  1. 6
  2. 10
  3. 8 Correct answer.
  4. 4

Explanation

Let the smaller be x and the larger be y, so y − x = 5 and 2x − y = 3. Adding the equations gives x = 8, and then y = 13. Checking, twice 8 is 16, which exceeds 13 by 3 as required.

Question 40Pair of Linear Equations in Two Variables

The pair of equations 3x − 5y = 7 and 6x − 10y = 14 has:

  1. no solution
  2. infinitely many solutions Correct answer.
  3. a unique solution
  4. exactly two solutions

Explanation

Here 3/6 = 5/10 = 7/14 = 1/2, so all three ratios are equal. The second equation is just twice the first, so the two lines coincide and every point on the line is a common solution.

Question 41Pair of Linear Equations in Two Variables

A boat covers 30 km downstream in 2 hours and the same distance upstream in 3 hours. The speed of the boat in still water is:

  1. 15 km/h
  2. 10 km/h
  3. 12 km/h
  4. 12.5 km/h Correct answer.

Explanation

Let the boat's speed be b and the stream's speed be s. Then b + s = 30/2 = 15 and b − s = 30/3 = 10. Adding gives 2b = 25, so b = 12.5 km/h, and the stream speed is 2.5 km/h.

Question 42Pair of Linear Equations in Two Variables

Using the elimination method on 4x + 3y = 25 and 5x − 2y = 8, which multiplier pair clears y in one step?

  1. Multiply the first by 5 and the second by 4
  2. Multiply the first by 3 and the second by 2
  3. Multiply the first by 4 and the second by 5
  4. Multiply the first by 2 and the second by 3 Correct answer.

Explanation

The coefficients of y are 3 and −2, whose magnitudes have LCM 6. Multiplying the first equation by 2 and the second by 3 gives 6y and −6y, which cancel when the equations are added.

Question 43Pair of Linear Equations in Two Variables

The lines representing the equations x + 2y = 5 and 3x + 6y = 15 are:

  1. coincident Correct answer.
  2. parallel and distinct
  3. intersecting at exactly one point
  4. perpendicular to each other

Explanation

Dividing the second equation by 3 gives x + 2y = 5, which is exactly the first equation. Since one equation is a constant multiple of the other, the two lines coincide and the system has infinitely many solutions.

Question 44Pair of Linear Equations in Two Variables

For what value of k does the system 2x + (k − 1)y = 5 and 4x + 6y = 10 have infinitely many solutions?

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

Explanation

Infinitely many solutions need a₁/a₂ = b₁/b₂ = c₁/c₂, and here 2/4 = 5/10 = 1/2 already. So (k − 1)/6 must equal 1/2, giving k − 1 = 3 and hence k = 4.

Question 45Pair of Linear Equations in Two Variables

A fraction becomes 9/11 when 2 is added to both numerator and denominator, and becomes 5/6 when 3 is added to both. The original fraction is:

  1. 11/13
  2. 7/9 Correct answer.
  3. 5/7
  4. 9/11

Explanation

Let the fraction be x/y. Then 11(x + 2) = 9(y + 2) and 6(x + 3) = 5(y + 3), giving 11x − 9y = −4 and 6x − 5y = −3. Solving, x = 7 and y = 9, and indeed 9/11 and 10/12 = 5/6 check out.

Question 46Pair of Linear Equations in Two Variables

The system ax + by = c and dx + ey = f has a unique solution. If the constant terms c and f are both doubled while a, b, d, e stay the same, the new system has:

  1. a unique solution that is generally different Correct answer.
  2. the same unique solution
  3. no solution
  4. infinitely many solutions

Explanation

Doubling only the constants changes the right-hand side, so the two lines keep their slopes but shift, and they still intersect in exactly one point because a/d ≠ b/e is unchanged. That intersection point is generally a different point from the original one, as solving the new system shows.

Question 47Quadratic Equations

The quadratic equation 3x² - 12x + 9 = 0 has roots α and β. What is the value of α + β?

  1. 4 Correct answer.
  2. 3
  3. -3
  4. -4

Explanation

For ax² + bx + c = 0, the sum of roots equals -b/a. Here a = 3 and b = -12, so α + β = -(-12)/3 = 4.

Question 48Quadratic Equations

What is the discriminant of the equation 2x² + 5x - 3 = 0?

  1. 1
  2. 49 Correct answer.
  3. -49
  4. 31

Explanation

The discriminant is b² - 4ac. With a = 2, b = 5 and c = -3, we get 25 - 4(2)(-3) = 25 + 24 = 49.

Question 49Quadratic Equations

Which of the following is a quadratic equation in the variable x?

  1. x² - 7x + 12 = 0 Correct answer.
  2. 5/x + x = 3
  3. x³ - 8 = 0
  4. 2x + 5 = 11

Explanation

A quadratic equation in one variable has the form ax² + bx + c = 0 with a ≠ 0, so the highest power of x is 2. Only x² - 7x + 12 = 0 fits this standard form.

Question 50Quadratic Equations

The roots of a quadratic equation are 5 and -2. Which equation has exactly these roots?

  1. x² + 3x - 10 = 0
  2. x² - 7x + 10 = 0
  3. x² + 7x + 10 = 0
  4. x² - 3x - 10 = 0 Correct answer.

Explanation

If the roots are p and q, the equation is x² - (p + q)x + pq = 0. Here p + q = 3 and pq = -10, giving x² - 3x - 10 = 0.