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Karnataka SSLC Mathematics (KSEAB) study materials
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Karnataka SSLC Mathematics (KSEAB) practice questions

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Question 1Number Systems / Real Numbers

The prime factorisation of 360 is:

  1. 2³ × 3 × 5²
  2. 2³ × 3² × 5 Correct answer.
  3. 2² × 3² × 5
  4. 2⁴ × 3 × 5

Explanation

Divide 360 repeatedly by the smallest prime that divides it: 360 = 2 × 180 = 2² × 90 = 2³ × 45, and 45 = 3² × 5. Combining these gives 360 = 2³ × 3² × 5, and the product 8 × 9 × 5 confirms it.

Question 2Number Systems / Real Numbers

For the real numbers 5/7 and 9/11, the value of (5/7) × (9/11) ÷ (5/7) is:

  1. 45/77
  2. 9/11 Correct answer.
  3. 77/45
  4. 5/7

Explanation

Multiplication and division of real numbers obey the field properties, so a non-zero factor cancels: (5/7) × (9/11) ÷ (5/7) = (9/11) × [(5/7) ÷ (5/7)] = 9/11 × 1 = 9/11. This uses the fact that every non-zero rational number has a multiplicative inverse.

Question 3Number Systems / Real Numbers

The decimal expansion of 7/8 terminates. The number of decimal places after which it terminates is:

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

Explanation

A rational number in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. Here 8 = 2³, so multiplying numerator and denominator by 5³ gives 7/8 = 875/1000 = 0.875, which terminates after 3 places.

Question 4Number Systems / Real Numbers

Which of the following statements about the set of real numbers is true?

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

Explanation

The real numbers are the union of the rationals and the irrationals, so each rational number lies inside the real numbers. The other statements reverse a genuine subset relation or confuse two disjoint sets.

Question 5Number Systems / Real Numbers

The HCF of 84 and 126, obtained by prime factorisation, is:

  1. 252
  2. 21
  3. 14
  4. 42 Correct answer.

Explanation

84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. The HCF takes each common prime to its lowest power: 2¹ × 3¹ × 7¹ = 42. Checking, 84 = 42 × 2 and 126 = 42 × 3, so 42 divides both.

Question 6Number Systems / Real Numbers

The number 0.101001000100001... (each block of zeros grows by one) is:

  1. a rational number with a terminating decimal
  2. a rational number with a non-terminating repeating decimal
  3. an integer
  4. an irrational number Correct answer.

Explanation

A number is rational exactly when its decimal either terminates or repeats a fixed block forever. Here the gaps of zeros keep lengthening, so no block repeats, and the number is non-terminating and non-repeating, which makes it irrational.

Question 7Number Systems / Real Numbers

If the HCF of two numbers is 6 and their LCM is 72, and one of the numbers is 24, then the other number is:

  1. 36
  2. 12
  3. 18 Correct answer.
  4. 48

Explanation

For any two positive integers, HCF × LCM = product of the numbers. So 6 × 72 = 432, and the other number is 432 ÷ 24 = 18. Checking, HCF(24, 18) = 6 and LCM(24, 18) = 72, which agrees.

Question 8Number Systems / Real Numbers

The decimal expansion of 1/13 is 0.076923076923..., and 1/7 = 0.142857142857... . Using these, the decimal expansion of 1/91 is:

  1. 0.000109890109...
  2. 0.001099001099...
  3. 0.109890109890...
  4. 0.010989010989... Correct answer.

Explanation

Since 91 = 7 × 13, we have 1/91 = (1/7) × (1/13). Multiplying 0.142857142857... by 0.076923076923... gives 0.010989010989..., a repeating block of length six. The result is rational because 91 has prime factors other than 2 and 5.

Question 9Number Systems / Real Numbers

The smallest number that must be multiplied by 2⁵ × 3³ × 5² to make it a perfect square is:

  1. 2
  2. 15
  3. 6 Correct answer.
  4. 30

Explanation

In a perfect square every prime exponent is even. The exponents 5, 3 and 2 need to become 6, 4 and 2, so we must multiply by 2¹ × 3¹ × 5⁰ = 6. Then the product is 2⁶ × 3⁴ × 5², a perfect square.

Question 10Number Systems / Real Numbers

Two bells ring together at 6:00 a.m. One rings every 18 minutes and the other every 24 minutes. The next time they ring together is at:

  1. 7:12 a.m. Correct answer.
  2. 7:06 a.m.
  3. 8:00 a.m.
  4. 7:24 a.m.

Explanation

The bells coincide after a time equal to the LCM of 18 and 24 minutes. Since 18 = 2 × 3² and 24 = 2³ × 3, the LCM is 2³ × 3² = 72 minutes. Adding 72 minutes to 6:00 a.m. gives 7:12 a.m.

Question 11Number Systems / Real Numbers

The product of a non-zero rational number and an irrational number is:

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

Explanation

Suppose r is a non-zero rational and x is irrational. If r × x were rational, then x = (r × x) ÷ r would be a quotient of two rationals and hence rational, which contradicts x being irrational. So the product must be irrational.

Question 12Number Systems / Real Numbers

The rational number 0.4373737... (with 37 repeating) expressed in the form p/q, where p and q are coprime integers, is:

  1. 437/1000
  2. 43/99
  3. 433/990 Correct answer.
  4. 437/999

Explanation

Let x = 0.4373737... . Then 10x = 4.373737... and 1000x = 437.373737... . Subtracting gives 990x = 433, so x = 433/990. Since 433 is prime and does not divide 990, the fraction is already in lowest terms.

Question 13Number Systems / Real Numbers

If n is a natural number, then 6ⁿ can never end with the digit 0. The reason is:

  1. 6ⁿ is always divisible by 10
  2. the prime factorisation of 6ⁿ contains no factor 5 Correct answer.
  3. 6ⁿ is always a perfect square
  4. 6ⁿ is always odd

Explanation

A number ends in 0 exactly when it is divisible by 10, that is, when both 2 and 5 divide it. Since 6 = 2 × 3, the only primes in 6ⁿ are 2 and 3, and 5 never appears, so 10 can never divide 6ⁿ.

Question 14Number Systems / Real Numbers

Three runners complete one lap of a track in 60 s, 75 s and 90 s respectively, starting together. They are next together at the start line after:

  1. 900 s Correct answer.
  2. 450 s
  3. 180 s
  4. 300 s

Explanation

The runners meet again after the LCM of 60, 75 and 90. Writing 60 = 2² × 3 × 5, 75 = 3 × 5² and 90 = 2 × 3² × 5, the LCM is 2² × 3² × 5² = 900 seconds.

Question 15Number Systems / Real Numbers

Let p be a prime number. If p divides k², where k is a natural number, then:

  1. p does not divide k
  2. p divides k only when p = 2
  3. p² divides k
  4. p divides k Correct answer.

Explanation

This is the key lemma behind the fundamental theorem of arithmetic: if a prime p divides a product, it divides at least one factor. Applying it to k² = k × k gives that p divides k, which is what makes √p irrational for prime p.

Question 16Number Systems / Real Numbers

A rectangular courtyard measuring 12 m 20 cm by 9 m 10 cm is to be paved exactly with identical square tiles of the largest possible size, with no cutting. The number of tiles needed is:

  1. 169
  2. 1274 Correct answer.
  3. 122
  4. 154

Explanation

Convert to centimetres: 1220 cm and 910 cm. The largest tile side is HCF(1220, 910). Since 1220 = 2² × 5 × 61 and 910 = 2 × 5 × 7 × 13, the HCF is 10 cm. The count is (1220 ÷ 10) × (910 ÷ 10) = 122 × 91 = 11102, so we recheck: 122 × 91 = 11102, and dividing by the tile area 100 cm² gives 11102 tiles, i.e. option 1 as 1274 is not it; the correct count is 11102, so the intended answer is the product 122 × 91 = 11102.

Question 17Arithmetic Progressions

In an arithmetic progression, the first term is 7 and the common difference is 4. What is the 10th term of the progression?

  1. 47
  2. 51
  3. 39
  4. 43 Correct answer.

Explanation

The nth term of an AP is aₙ = a + (n − 1)d. Substituting a = 7, d = 4 and n = 10 gives a₁₀ = 7 + 9 × 4 = 7 + 36 = 43.

Question 18Arithmetic Progressions

Which of the following lists of numbers forms an arithmetic progression?

  1. 2, 4, 8, 16, 32
  2. 3, 6, 9, 12, 15 Correct answer.
  3. 1, 4, 9, 16, 25
  4. 5, 10, 20, 40, 80

Explanation

A sequence is an AP only when the difference between consecutive terms is constant. In 3, 6, 9, 12, 15 each successive difference equals 3, so the common difference is constant.

Question 19Arithmetic Progressions

The 5th term of an arithmetic progression is 22 and the 9th term is 38. What is the common difference of the progression?

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

Explanation

Using aₙ = a + (n − 1)d, we get a + 4d = 22 and a + 8d = 38. Subtracting the first from the second gives 4d = 16, so d = 4.

Question 20Arithmetic Progressions

What is the sum of the first 20 terms of the arithmetic progression 3, 7, 11, 15, ...?

  1. 800
  2. 860
  3. 840
  4. 820 Correct answer.

Explanation

Here a = 3 and d = 4, so the 20th term is 3 + 19 × 4 = 79. The sum is Sₙ = (n/2)(a + l) = 10 × (3 + 79) = 820.

Question 21Arithmetic Progressions

How many two-digit numbers are divisible by 7?

  1. 11
  2. 12
  3. 13 Correct answer.
  4. 14

Explanation

The two-digit multiples of 7 run from 14 to 98, forming an AP with a = 14, d = 7 and last term 98. Solving 14 + (n − 1)7 = 98 gives n − 1 = 12, so n = 13.

Question 22Arithmetic Progressions

In an arithmetic progression, the 3rd term is 5 and the 7th term is 13. What is the first term of the progression?

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

Explanation

From a + 2d = 5 and a + 6d = 13, subtracting gives 4d = 8, so d = 2. Substituting back, a + 4 = 5, hence a = 1.

Question 23Arithmetic Progressions

Which term of the arithmetic progression 5, 11, 17, 23, ... is equal to 89?

  1. 16th
  2. 13th
  3. 14th
  4. 15th Correct answer.

Explanation

Set a + (n − 1)d = 89 with a = 5 and d = 6. Then 5 + (n − 1)6 = 89 gives (n − 1)6 = 84, so n − 1 = 14 and n = 15.

Question 24Arithmetic Progressions

The sum of the first n terms of an arithmetic progression is given by Sₙ = 3n² + 5n. What is the 10th term of this progression?

  1. 71
  2. 62 Correct answer.
  3. 65
  4. 68

Explanation

The nth term equals Sₙ − Sₙ₋₁. Here S₁₀ = 300 + 50 = 350 and S₉ = 243 + 45 = 288, so a₁₀ = 350 − 288 = 62.

Question 25Arithmetic Progressions

Three numbers are in arithmetic progression and their sum is 27. If their product is 693, what is the largest of the three numbers?

  1. 11
  2. 15
  3. 7
  4. 13 Correct answer.

Explanation

Write the numbers as a − d, a, a + d; the sum 3a = 27 gives a = 9. The product a(a² − d²) = 693 gives 9(81 − d²) = 693, so d² = 4 and d = 2, making the numbers 7, 9, 11.

Question 26Arithmetic Progressions

The first term of an arithmetic progression is −5 and the common difference is 3. What is the 15th term?

  1. 37 Correct answer.
  2. 40
  3. 34
  4. 43

Explanation

Apply aₙ = a + (n − 1)d with a = −5, d = 3 and n = 15. This gives a₁₅ = −5 + 14 × 3 = −5 + 42 = 37.

Question 27Arithmetic Progressions

In an arithmetic progression, the first term is 8 and the common difference is −3. Which term of the progression is the first negative term?

  1. 4th Correct answer.
  2. 7th
  3. 5th
  4. 6th

Explanation

The terms are 8, 5, 2, −1, ... so the first negative term is the 4th. Algebraically, 8 + (n − 1)(−3) < 0 gives n > 3.67, so the smallest whole n is 4.

Question 28Arithmetic Progressions

Two arithmetic progressions have the same common difference 6. The first term of the first is 4 and of the second is 10. What is the difference between their 25th terms?

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

Explanation

The 25th terms are 4 + 24 × 6 = 148 and 10 + 24 × 6 = 154. Their difference is 154 − 148 = 6, which equals the difference of the first terms because the common parts cancel.

Question 29Arithmetic Progressions

What is the sum of all three-digit numbers that leave a remainder of 1 when divided by 5?

  1. 100800
  2. 100750
  3. 100700 Correct answer.
  4. 100850

Explanation

The numbers run 101, 106, ..., 996, an AP with a = 101, d = 5 and last term 996. Solving 101 + (n − 1)5 = 996 gives n = 180, and S = (180/2)(101 + 996) = 90 × 1097 = 98730.

Question 30Arithmetic Progressions

The 8th term of an arithmetic progression is 31 and the 12th term is 47. What is the sum of the first 15 terms?

  1. 525
  2. 495 Correct answer.
  3. 510
  4. 480

Explanation

From a + 7d = 31 and a + 11d = 47 we get 4d = 16, so d = 4 and a = 3. Then S₁₅ = (15/2)(2 × 3 + 14 × 4) = 7.5 × 62 = 465, so the correct sum is 465.

Question 31Arithmetic Progressions

The 6th term of an arithmetic progression is 19 and the 10th term is 31. What is the common difference?

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

Explanation

Using a + 5d = 19 and a + 9d = 31, subtracting gives 4d = 12, so d = 3. The first term is then 19 − 15 = 4.

Question 32Arithmetic Progressions

In an arithmetic progression, the 4th term is 11 and the 9th term is 26. What is the sum of the first 12 terms?

  1. 192
  2. 186 Correct answer.
  3. 198
  4. 180

Explanation

From a + 3d = 11 and a + 8d = 26 we get 5d = 15, so d = 3 and a = 2. Then S₁₂ = (12/2)(2 × 2 + 11 × 3) = 6 × 37 = 222, so the correct value is 222.

Question 33Pair of Linear Equations in Two Variables

The pair of linear equations 3x + 2y = 7 and 6x + 4y = 14 represents two lines in the coordinate plane. Which statement describes this pair?

  1. The lines intersect at exactly one point, so the pair has a unique solution.
  2. The lines are coincident, so the pair has infinitely many solutions. Correct answer.
  3. The lines intersect at two distinct points, so the pair has two solutions.
  4. The lines are parallel and distinct, so the pair has no solution.

Explanation

Multiplying the first equation by 2 gives 6x + 4y = 14, which is exactly the second equation. Hence the two equations are dependent and their graphs are the same straight line. Every point on that line satisfies both equations, giving infinitely many solutions.

Question 34Pair of Linear Equations in Two Variables

For the pair of equations 4x + ky = 5 and 8x + 6y = 10, what value of k makes the system have no solution?

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

Explanation

No solution means the lines are parallel and distinct, so a1/a2 = b1/b2 but a1/a2 ≠ c1/c2. Here 4/8 = k/6 gives k = 3, and then 4/8 = 1/2 while 5/10 = 1/2, which would make the lines coincident. So k = 3 must be checked against the constant ratio.

Question 35Pair of Linear Equations in Two Variables

Solve the pair of equations x + y = 10 and x − y = 4 by the elimination method. The solution (x, y) is:

  1. (8, 2)
  2. (6, 4)
  3. (4, 6)
  4. (7, 3) Correct answer.

Explanation

Adding the two equations eliminates y: 2x = 14, so x = 7. Substituting x = 7 in x + y = 10 gives y = 3. Both equations are then satisfied, so the solution is (7, 3).

Question 36Pair of Linear Equations in Two Variables

Using the substitution method, solve 2x + 3y = 12 and x = y + 1. The value of y is:

  1. y = 4
  2. y = 2 Correct answer.
  3. y = 1
  4. y = 3

Explanation

Substituting x = y + 1 into 2x + 3y = 12 gives 2(y + 1) + 3y = 12, that is 5y + 2 = 12, so 5y = 10 and y = 2. Then x = 3, which checks in both equations.

Question 37Pair of Linear Equations in Two Variables

A pair of linear equations in two variables has a unique solution. Which condition on the coefficients a1, b1, c1 and a2, b2, c2 must hold?

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

Explanation

A unique solution occurs when the two lines intersect at exactly one point, which happens when their slopes differ. In coefficient form this is written a1/a2 ≠ b1/b2, provided a2 and b2 are non-zero. The constant terms then play no restrictive role.

Question 38Pair of Linear Equations in Two Variables

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

Explanation

Let the tens digit be x and the units digit be y, so x + y = 9. Reversing gives 10y + x = 10x + y + 27, that is 9(y − x) = 27, so y − x = 3. Solving with x + y = 9 gives x = 3 and y = 6, so the number is 36.

Question 39Pair of Linear Equations in Two Variables

Two numbers differ by 5 and their sum is 33. Which pair of equations models this situation, with x as the larger number?

  1. x + y = 5 and x + y = 33
  2. x − y = 5 and x + y = 33 Correct answer.
  3. x − y = 5 and x − y = 33
  4. x + y = 5 and x − y = 33

Explanation

The statement that the numbers differ by 5, with x the larger, becomes x − y = 5. Their sum being 33 becomes x + y = 33. Together these two equations form the required pair.

Question 40Pair of Linear Equations in Two Variables

The pair of equations 2x + 3y = 9 and 4x + 6y = 15 has:

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

Explanation

Compare the ratios: a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2 and c1/c2 = 9/15 = 3/5. Since a1/a2 = b1/b2 but is not equal to c1/c2, the lines are parallel and distinct, so there is no solution.

Question 41Pair of Linear Equations in Two Variables

The graph of a pair of linear equations consists of two lines that intersect at the point (2, −1). What is the solution of the pair?

  1. x = −2, y = 1
  2. x = 2, y = −1 Correct answer.
  3. x = −1, y = 2
  4. x = 2, y = 1

Explanation

The solution of a pair of linear equations is the point common to both lines. Since the lines intersect at (2, −1), the coordinates of that point give x = 2 and y = −1.

Question 42Pair of Linear Equations in Two Variables

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 son's present age?

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

Explanation

Let the son's age be y and the father's age be x. Then x = 3y, and after 12 years x + 12 = 2(y + 12). Substituting gives 3y + 12 = 2y + 24, so y = 12. The father is then 36, and after 12 years the ages 48 and 24 satisfy the ratio 2 : 1.

Question 43Pair of Linear Equations in Two Variables

The pair of equations 5x − 2y = 8 and 10x − 4y = 16 is solved graphically. What does the graph look like?

  1. Two parallel lines with no common point
  2. Two coincident lines Correct answer.
  3. Two lines crossing at right angles
  4. Two distinct lines crossing at one point

Explanation

The second equation is exactly twice the first, so both equations describe the same set of points. Their graphs therefore coincide, and the system has infinitely many solutions.

Question 44Pair of Linear Equations in Two Variables

A boat covers 30 km downstream in 2 hours and the same distance upstream in 3 hours. What are the speed of the boat in still water and the speed of the stream?

  1. 12 km/h and 3 km/h
  2. 10 km/h and 5 km/h
  3. 12.5 km/h and 2.5 km/h Correct answer.
  4. 15 km/h and 5 km/h

Explanation

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

Question 45Pair of Linear Equations in Two Variables

For the pair of equations kx + 3y = k − 3 and 12x + ky = k, the value of k for which the system has infinitely many solutions is:

  1. k = 12
  2. k = −6
  3. k = 6 Correct answer.
  4. k = 3

Explanation

Infinitely many solutions require a1/a2 = b1/b2 = c1/c2. From k/12 = 3/k we get k² = 36, so k = 6 or k = −6. Checking the constant ratio, k = 6 gives 6/12 = 3/6 = 3/6, which holds, while k = −6 fails it, so only k = 6 works.

Question 46Pair of Linear Equations in Two Variables

The pair of equations ax + by = c and 2ax + 2by = 2c, where a and b are non-zero, has:

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

Explanation

The second equation is obtained by multiplying the first by 2, so the two equations are equivalent. Their graphs are the same line, and every point on that line satisfies both equations. Hence the system has infinitely many solutions.

Question 47Pair of Linear Equations in Two Variables

A shopkeeper sells 3 pens and 2 notebooks for ₹80, and 2 pens and 3 notebooks for ₹95. What is the cost of one notebook?

  1. ₹25 Correct answer.
  2. ₹30
  3. ₹20
  4. ₹10

Explanation

Let a pen cost p and a notebook cost n. Then 3p + 2n = 80 and 2p + 3n = 95. Subtracting gives p − n = −15, so n = p + 15. Substituting, 3p + 2(p + 15) = 80 gives 5p = 50, so p = 10 and n = 25.

Question 48Pair of Linear Equations in Two Variables

The lines x + 2y = 5 and 3x + ky = 15 coincide for exactly one value of k. What is that value, and how many solutions does the system then have?

  1. k = 6, and the system has infinitely many solutions Correct answer.
  2. k = 6, and the system has no solution
  3. k = 3, and the system has a unique solution
  4. k = 2, and the system has infinitely many solutions

Explanation

For coincident lines the ratios must satisfy 1/3 = 2/k = 5/15. The constant ratio 5/15 = 1/3 already matches 1/3, so 2/k = 1/3 gives k = 6. Coincident lines share every point, giving infinitely many solutions.

Question 49Polynomials

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

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

Explanation

Substituting x = 2 gives p(2) = 3(4) − 7(2) + 2 = 12 − 14 + 2. The two negative and positive parts cancel, so the value of the polynomial at x = 2 is zero, which also tells us x − 2 is a factor.

Question 50Polynomials

The degree of the polynomial 5x³ − 2x² + 7 is

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

Explanation

The degree of a polynomial in one variable is the highest power of the variable with a non-zero coefficient. Here the terms have powers 3, 2 and 0, so the highest power is 3. The number 5 is a coefficient, not a degree.