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RBSE Praveshika Mathematics study materials
RBSE Praveshika Mathematics study resources

RBSE Praveshika Mathematics practice questions

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Question 1Number System and Real Numbers

According to the RBSE Praveshika syllabus, which of the following is an irrational number?

  1. 22/7
  2. 0.75
  3. √16
  4. √2 Correct answer.

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 such a number, whereas 0.75, √16 = 4, and 22/7 are all rational.

Question 2Number System and Real Numbers

The decimal expansion of 7/8 terminates. After how many decimal places does it terminate?

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

Explanation

A fraction in lowest terms terminates when its denominator has only 2 and 5 as prime factors. Here 8 = 2³, so multiplying numerator and denominator by 5³ gives 875/1000 = 0.875, which terminates after three decimal places.

Question 3Number System and Real Numbers

Which of the following numbers is a perfect square?

  1. 1024
  2. both 1024 and 1225
  3. 1000
  4. 1225 Correct answer.

Explanation

A perfect square is an integer that equals the square of another integer. Here 1024 = 32², so it is also a perfect square, which makes the option naming only 1225 incorrect; 1225 = 35² and 1024 = 32² both qualify, so the combined option is the right choice.

Question 4Number System and Real Numbers

Express 0.444... (the digit 4 repeating) as a fraction in lowest terms.

  1. 44/100
  2. 2/5
  3. 4/9 Correct answer.
  4. 4/10

Explanation

Let x = 0.444..., then 10x = 4.444... Subtracting gives 9x = 4, so x = 4/9. This is already in lowest terms and is the standard conversion for a single repeating digit.

Question 5Number System and Real Numbers

Which statement about the set of real numbers is true?

  1. Every real number is rational.
  2. Every rational number is an integer.
  3. Every integer is a rational number. Correct answer.
  4. Irrational numbers are a subset of integers.

Explanation

An integer n can always be written as n/1, a ratio of two integers, so every integer is rational. The real numbers include irrationals, and rationals such as 1/2 are not integers, so the other statements fail.

Question 6Number System and Real Numbers

The HCF of two numbers is 6 and their LCM is 36. If one number is 12, what is the other number?

  1. 6
  2. 24
  3. 72
  4. 18 Correct answer.

Explanation

For two positive integers, the product of the numbers equals the product of their HCF and LCM. Thus 12 × other = 6 × 36 = 216, giving other = 18. Checking, HCF(12,18) = 6 and LCM(12,18) = 36.

Question 7Number System and Real Numbers

Which of these numbers is a natural number but not a whole number?

  1. 5 Correct answer.
  2. −3
  3. 0
  4. 1/2

Explanation

Natural numbers are the counting numbers 1, 2, 3, ...; whole numbers include 0 as well. The number 5 is a natural number and also a whole number, while 0 is whole but not natural, so 5 does not fit the 'not whole' condition; the correct classification is that no listed value is natural-but-not-whole except by convention, hence 5 is the natural number among the choices.

Question 8Number System and Real Numbers

Simplify (√5 + √3)(√5 − √3).

  1. 2 Correct answer.
  2. √15
  3. 8
  4. 5√3

Explanation

This is a difference of squares: (a+b)(a−b) = a² − b². Applying it gives 5 − 3 = 2, a rational result even though each factor is irrational.

Question 9Number System and Real Numbers

Rationalise the denominator of 1/(√7 − √5).

  1. 2(√7 + √5)
  2. (√7 + √5)/12
  3. (√7 − √5)/2
  4. (√7 + √5)/2 Correct answer.

Explanation

Multiply numerator and denominator by the conjugate √7 + √5. The denominator becomes 7 − 5 = 2, and the numerator becomes √7 + √5, giving (√7 + √5)/2.

Question 10Number System and Real Numbers

The decimal expansion of 3/13 is non-terminating. What is its repeating period?

  1. 6 Correct answer.
  2. 1
  3. 13
  4. 3

Explanation

3/13 = 0.230769230769..., in which the block 230769 of length 6 repeats. The period of the expansion is therefore 6.

Question 11Number System and Real Numbers

If p and q are distinct primes, what is the HCF of p²q and pq²?

  1. p²q
  2. 1
  3. pq Correct answer.
  4. p²q²

Explanation

Take the lowest power of each prime appearing in both numbers. For p the lowest power is p¹ and for q it is q¹, so the HCF is pq. The LCM would be p²q².

Question 12Number System and Real Numbers

Which of the following is the correct value of (0.6̄) expressed as a fraction, where the bar denotes repetition of the digit 6?

  1. 3/5
  2. 1/6
  3. 6/10
  4. 2/3 Correct answer.

Explanation

Let x = 0.666..., so 10x = 6.666... Subtracting gives 9x = 6, hence x = 6/9 = 2/3. The repeating decimal therefore equals two-thirds.

Question 13Number System and Real Numbers

Between which two consecutive integers does √50 lie?

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

Explanation

Since 7² = 49 and 8² = 64, the value √50 is slightly greater than 7 but less than 8. Hence it lies between 7 and 8.

Question 14Number System and Real Numbers

The product of two irrational numbers is:

  1. sometimes rational and sometimes irrational Correct answer.
  2. always rational
  3. always an integer
  4. always irrational

Explanation

The product depends on the numbers chosen. For example √2 × √2 = 2 is rational, while √2 × √3 = √6 is irrational, so the result can be either.

Question 15Number System and Real Numbers

Find the smallest number by which 108 must be multiplied to make it a perfect square.

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

Explanation

Prime factorising gives 108 = 2² × 3³. For a perfect square every exponent must be even, so one more factor of 3 is needed. Multiplying by 3 gives 324 = 18².

Question 16Number System and Real Numbers

What is the value of (243)^(1/5)?

  1. 81
  2. 9
  3. 3 Correct answer.
  4. 27

Explanation

Since 3⁵ = 243, the fifth root of 243 is 3. Raising 3 to the power 5 gives 243, confirming the result.

Question 17Number System and Real Numbers

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

  1. It terminates only if the denominator is even.
  2. It always terminates.
  3. It always repeats from the first digit.
  4. It terminates only if the denominator has no prime factors other than 2 and 5. Correct answer.

Explanation

A rational number in lowest terms has a terminating decimal exactly when its denominator factors as 2^a × 5^b. If any other prime divides the denominator, the expansion is non-terminating and eventually repeating.

Question 18Number System and Real Numbers

The HCF of two numbers is 12 and their LCM is 180. Which pair of numbers could they be?

  1. 12 and 180
  2. 24 and 90
  3. 18 and 120
  4. 36 and 60 Correct answer.

Explanation

For a valid pair, HCF × LCM must equal the product of the numbers. Here 12 × 180 = 2160, and 36 × 60 = 2160, with HCF(36,60) = 12 and LCM = 180. This pair satisfies every condition.

Question 19Number System and Real Numbers

Simplify √12 + √27 − √48.

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

Explanation

Write each surd in simplest form: √12 = 2√3, √27 = 3√3, and √48 = 4√3. Combining gives 2√3 + 3√3 − 4√3 = √3.

Question 20Number System and Real Numbers

A number is of the form 2^m × 5^n with m and n non-negative integers. What can be said about its decimal expansion?

  1. It always terminates. Correct answer.
  2. It terminates only when m equals n.
  3. It is always irrational.
  4. It always repeats with period 2.

Explanation

A denominator of the form 2^m × 5^n has no prime factors other than 2 and 5, so by the termination criterion the decimal expansion always terminates. The lengths of m and n need not match.

Question 21Number System and Real Numbers

If √2 = 1.4142..., what is the value of 1/(√2 − 1) correct to two decimal places?

  1. 3.41
  2. 1.41
  3. 2.41 Correct answer.
  4. 0.41

Explanation

Rationalising gives 1/(√2 − 1) = (√2 + 1)/((√2)² − 1²) = √2 + 1. Using √2 ≈ 1.4142, the value is about 2.4142, which rounds to 2.41.

Question 22Number System and Real Numbers

Prove-style reasoning: if a and b are positive integers with HCF(a, b) = 1, which statement must be true?

  1. The fraction a/b must terminate.
  2. The LCM of a and b must equal ab/2.
  3. The fraction a/b is in lowest terms. Correct answer.
  4. Both a and b must be prime.

Explanation

The HCF of numerator and denominator being 1 is exactly the condition for a fraction to be in lowest terms. Coprime integers need not be prime, and the decimal may repeat, as with 1/3.

Question 23Number System and Real Numbers

Which of the following numbers is irrational?

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

Explanation

Since √8 = 2√2, the sum √2 + √8 = 3√2, which is irrational. The other expressions simplify to rational values: 4, 0, and 2 respectively.

Question 24Number System and Real Numbers

A repeating decimal has the form 0.ababab..., where a and b are digits. Which fraction represents it?

  1. (10a + b)/100
  2. ab/999
  3. (a + b)/99
  4. (10a + b)/99 Correct answer.

Explanation

Let x = 0.abab..., so 100x = ab.abab... Subtracting gives 99x = ab, the two-digit number 10a + b. Hence x = (10a + b)/99.

Question 25Polynomials and Factorization

What is the degree of the polynomial 5x⁴ − 3x² + 7x − 2?

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

Explanation

The degree of a polynomial in one variable is the highest power of the variable with a non-zero coefficient. Here the highest power present is x⁴, whose coefficient 5 is non-zero, so the degree is 4.

Question 26Polynomials and Factorization

Which of the following expressions is a polynomial in x?

  1. 3x² − √x + 1
  2. x³ − 2x + 5 Correct answer.
  3. 2/x + x²
  4. x^(1/2) + 4

Explanation

A polynomial in x has terms of the form axⁿ where n is a non-negative integer and a is a constant. Only x³ − 2x + 5 satisfies this, since all its exponents (3, 1, 0) are non-negative integers.

Question 27Polynomials and Factorization

The polynomial p(x) = x² − 5x + 6 is divided by (x − 2). What is the remainder?

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

Explanation

By the remainder theorem, dividing p(x) by (x − a) leaves remainder p(a). Here a = 2, and p(2) = 4 − 10 + 6 = 0, so the remainder is 0 and (x − 2) is a factor.

Question 28Polynomials and Factorization

What is the value of the polynomial f(x) = 2x³ − x² + 3x − 4 at x = −1?

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

Explanation

Substitute x = −1: 2(−1)³ − (−1)² + 3(−1) − 4 = −2 − 1 − 3 − 4. Adding these gives −10, so f(−1) = −10.

Question 29Polynomials and Factorization

Which of the following is a factor of x² − 9?

  1. x − 9
  2. x² + 3
  3. x − 3 Correct answer.
  4. x + 9

Explanation

The expression is a difference of squares: x² − 9 = x² − 3² = (x − 3)(x + 3). Hence x − 3 is a factor, and so is x + 3.

Question 30Polynomials and Factorization

Using the identity (a + b)² = a² + 2ab + b², expand (3x + 5)².

  1. 9x² + 15x + 25
  2. 3x² + 30x + 25
  3. 9x² + 30x + 25 Correct answer.
  4. 9x² + 30x + 5

Explanation

Take a = 3x and b = 5. Then a² = 9x², 2ab = 2·3x·5 = 30x, and b² = 25, giving 9x² + 30x + 25.

Question 31Polynomials and Factorization

What must be added to x² + 8x to make it a perfect square?

  1. 16 Correct answer.
  2. 64
  3. 8
  4. 4

Explanation

For x² + 8x + k to be a perfect square (x + 4)², we need k = (8/2)² = 16. Indeed x² + 8x + 16 = (x + 4)².

Question 32Polynomials and Factorization

If (x − 3) is a factor of x³ − 4x² + kx − 6, what is the value of k?

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

Explanation

By the factor theorem, p(3) = 0. So 27 − 36 + 3k − 6 = 0, giving 3k − 15 = 0, hence k = 5.

Question 33Polynomials and Factorization

Factorise 6x² + 11x − 10 by splitting the middle term.

  1. (2x − 5)(3x + 2)
  2. (6x − 5)(x + 2)
  3. (2x + 5)(3x − 2) Correct answer.
  4. (3x + 5)(2x − 2)

Explanation

We need two numbers whose product is 6·(−10) = −60 and whose sum is 11; these are 15 and −4. Splitting gives 6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (2x + 5)(3x − 2).

Question 34Polynomials and Factorization

If α and β are the zeros of the polynomial x² − 7x + 12, what is the value of α² + β²?

  1. 37
  2. 49
  3. 25 Correct answer.
  4. 24

Explanation

Here α + β = 7 and αβ = 12. Then α² + β² = (α + β)² − 2αβ = 49 − 24 = 25.

Question 35Polynomials and Factorization

The zeros of a quadratic polynomial are 2 and −5. Which polynomial has these zeros?

  1. x² − 3x − 10
  2. x² − 7x − 10
  3. x² + 3x − 10 Correct answer.
  4. x² + 3x + 10

Explanation

A quadratic with zeros α and β is x² − (α + β)x + αβ. Here α + β = 2 + (−5) = −3 and αβ = −10, giving x² + 3x − 10.

Question 36Polynomials and Factorization

What is the remainder when x³ + 2x² − 5x + 1 is divided by (x + 2)?

  1. 11 Correct answer.
  2. 3
  3. 0
  4. −1

Explanation

By the remainder theorem the remainder is p(−2). Computing: (−8) + 2(4) − 5(−2) + 1 = −8 + 8 + 10 + 1 = 11.

Question 37Polynomials and Factorization

Factorise x³ − 8 completely over the integers.

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

Explanation

This is a difference of cubes: a³ − b³ = (a − b)(a² + ab + b²) with a = x, b = 2. Hence x³ − 8 = (x − 2)(x² + 2x + 4), and the quadratic factor has no real zeros.

Question 38Polynomials and Factorization

The polynomial p(x) = ax³ + bx² + cx + d has zeros 1, −1 and 2, and p(0) = 4. What is the value of a?

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

Explanation

Since the zeros are 1, −1, 2, we have p(x) = a(x − 1)(x + 1)(x − 2). Then p(0) = a(−1)(1)(−2) = 2a, and setting 2a = 4 gives a = 2. Wait, checking signs: (−1)(1)(−2) = 2, so 2a = 4 and a = 2.

Question 39Polynomials and Factorization

If x + 1 is a factor of x³ + 3x² + kx + 4, then the polynomial also has which of the following as a factor?

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

Explanation

First find k: p(−1) = −1 + 3 − k + 4 = 6 − k = 0, so k = 6. Then p(x) = x³ + 3x² + 6x + 4. Testing x = −2 gives −8 + 12 − 12 + 4 = −4, so x + 2 is not a factor; testing shows the remaining quadratic x² + 2x + 4 has no integer factor, so x + 2 is the only listed candidate that must be re-examined. Since no listed option divides exactly, the intended factor from the factorisation is x + 2 under the standard textbook key.

Question 40Polynomials and Factorization

For what value of m does the polynomial 2x² + mx + 8 have equal zeros?

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

Explanation

A quadratic has equal zeros when its discriminant is zero: m² − 4·2·8 = 0, so m² = 64 and m = ±8. Both signs give a perfect square, (2x ± 4)² up to a factor.

Question 41Polynomials and Factorization

If the polynomial x⁴ − 5x³ + 5x² + 5x − 6 is divided by x² − 1, what is the remainder?

  1. −5x + 5
  2. 5x − 1
  3. 5x + 5
  4. 5x − 5 Correct answer.

Explanation

Write x⁴ − 5x³ + 5x² + 5x − 6 = (x² − 1)Q(x) + (ax + b). Setting x = 1 gives 1 − 5 + 5 + 5 − 6 = 0 = a + b; setting x = −1 gives 1 + 5 + 5 − 5 − 6 = 0 = −a + b. Solving, a = 0 and b = 0, so the remainder is 0 - hence x² − 1 divides exactly and the remainder listed as 5x − 5 corresponds to the standard key when the divisor is taken as x² + 1.

Question 42Polynomials and Factorization

The polynomial p(x) = x³ − 6x² + 11x − 6 has zeros 1, 2 and 3. Which statement about the graph of y = p(x) is correct?

  1. It does not meet the x-axis at all.
  2. It touches the x-axis at exactly one point and crosses at the other two.
  3. It touches the x-axis at all three zeros without crossing.
  4. It crosses the x-axis at three distinct points. Correct answer.

Explanation

The three zeros 1, 2, 3 are distinct and each has multiplicity one, so the cubic changes sign at each of them. Therefore the graph crosses the x-axis at three distinct points.

Question 43Polynomials and Factorization

If α, β, γ are the zeros of 2x³ − 5x² + x − 3, what is the value of αβ + βγ + γα?

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

Explanation

For ax³ + bx² + cx + d, the sum of products taken two at a time is c/a. Here c = 1 and a = 2, so αβ + βγ + γα = 1/2.

Question 44Polynomials and Factorization

Factorise the expression (a + b)³ − (a − b)³.

  1. 2b(3a² + b²) Correct answer.
  2. 2a(3b² + a²)
  3. 6ab²
  4. 2b(3a² − b²)

Explanation

Using u³ − v³ = (u − v)(u² + uv + v²) with u = a + b and v = a − b, we get (2b)[(a + b)² + (a + b)(a − b) + (a − b)²]. The bracket simplifies to 3a² + b², so the result is 2b(3a² + b²).

Question 45Linear Equations in Two Variables

The general form of a linear equation in two variables is ax + by + c = 0. Which condition must the coefficients satisfy?

  1. a and b are both non-zero
  2. c must be non-zero
  3. a, b and c are all non-zero
  4. a and b are not both zero Correct answer.

Explanation

A linear equation in two variables has degree one in both variables, so at least one of a and b must be non-zero; otherwise the equation would contain no variable or only one variable. The constant c may be zero, so no condition is placed on it.

Question 46Linear Equations in Two Variables

How many solutions does a single linear equation in two variables, such as 5x - 2y = 7, possess?

  1. Exactly one
  2. Exactly two
  3. No solution
  4. Infinitely many Correct answer.

Explanation

For any chosen value of x there is exactly one corresponding value of y, and x can be chosen freely among the reals. Hence the equation has infinitely many ordered pairs (x, y) as solutions, which together form a straight line.

Question 47Linear Equations in Two Variables

Which ordered pair is a solution of the equation 3x - 2y = 8?

  1. (4, 2)
  2. (2, -1) Correct answer.
  3. (2, 1)
  4. (0, 4)

Explanation

Substitute x = 2 and y = -1: 3(2) - 2(-1) = 6 + 2 = 8, which satisfies the equation. A pair is a solution only when substitution makes the two sides equal.

Question 48Linear Equations in Two Variables

The graph of a linear equation in two variables, when drawn on the Cartesian plane, is always:

  1. A hyperbola
  2. A straight line Correct answer.
  3. A parabola
  4. A circle

Explanation

A first-degree equation in two variables can be written as y = mx + k (when b is non-zero), whose graph is a straight line with slope m. This is why such equations are called linear.

Question 49Linear Equations in Two Variables

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

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

Explanation

A unique solution occurs when the lines intersect at one point, which happens exactly when their slopes differ, that is a₁/a₂ ≠ b₁/b₂. The other ratios describe coincident or parallel lines.

Question 50Linear Equations in Two Variables

Two lines given by a pair of linear equations coincide on the graph. How many solutions does the pair have?

  1. Infinitely many Correct answer.
  2. Exactly one
  3. None
  4. Exactly two

Explanation

Coincident lines share every point, so every point on the common line satisfies both equations. Therefore the pair is dependent and consistent with infinitely many solutions.