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Maharashtra SSC Mathematics (MSBSHSE) study materials
Maharashtra SSC Mathematics (MSBSHSE) study resources

Maharashtra SSC Mathematics (MSBSHSE) practice questions

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Question 1Algebra

For the arithmetic progression 5, 9, 13, 17, …, what is the 20th term?

  1. 77
  2. 85
  3. 81 Correct answer.
  4. 80

Explanation

The first term is a = 5 and the common difference d = 9 − 5 = 4. The nth term formula for an AP is tn = a + (n − 1)d, so t20 = 5 + 19 × 4 = 5 + 76 = 81.

Question 2Algebra

A quadratic equation has roots 3 and −7. Which equation has exactly these roots?

  1. x² − 4x − 21 = 0
  2. x² − 10x − 21 = 0
  3. x² + 4x − 21 = 0 Correct answer.
  4. x² + 4x + 21 = 0

Explanation

With roots α = 3 and β = −7, the sum is −4 and the product is −21. The equation is x² − (α + β)x + αβ = 0, giving x² − (−4)x + (−21) = 0, that is x² + 4x − 21 = 0.

Question 3Algebra

The determinant of the 2×2 matrix with rows (4, 3) and (2, 5) is:

  1. 14 Correct answer.
  2. 20
  3. 26
  4. 6

Explanation

For a matrix with entries a, b in the first row and c, d in the second, the determinant is ad − bc. Here it is 4 × 5 − 3 × 2 = 20 − 6 = 14.

Question 4Algebra

What is the degree of the polynomial 7x⁴ − 2x³ + x − 9?

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

Explanation

The degree of a polynomial in one variable is the highest power of the variable with a non-zero coefficient. The highest power present is 4, from the term 7x⁴, so the degree is 4.

Question 5Algebra

If the probability of an event A is P(A) = 0.35, what is P(A′), the probability of its complement?

  1. 0.65 Correct answer.
  2. 1.35
  3. 0.35
  4. 0.35 × 0.65

Explanation

For any event, the probability of the event plus the probability of its complement equals 1. Therefore P(A′) = 1 − P(A) = 1 − 0.35 = 0.65.

Question 6Algebra

Simplify: (a² − 9) ÷ (a − 3), for a ≠ 3.

  1. a + 9
  2. a² + 3
  3. a − 3
  4. a + 3 Correct answer.

Explanation

The numerator is a difference of squares: a² − 9 = (a − 3)(a + 3). Cancelling the common factor (a − 3), which is non-zero because a ≠ 3, leaves a + 3.

Question 7Algebra

The first term of a geometric progression is 3 and the common ratio is 2. What is the 5th term?

  1. 48 Correct answer.
  2. 30
  3. 243
  4. 96

Explanation

The nth term of a geometric progression is tn = a·r^(n−1). Here t5 = 3 × 2⁴ = 3 × 16 = 48.

Question 8Algebra

Which of the following is a linear equation in two variables?

  1. x³ + y³ = 8
  2. 3x − 5y = 7 Correct answer.
  3. xy = 12
  4. x² + y = 4

Explanation

A linear equation in two variables has the form ax + by + c = 0 with a and b not both zero, so every variable appears only to the first power. Only 3x − 5y = 7 fits this form.

Question 9Algebra

If α and β are the roots of x² − 5x + 6 = 0, what is the value of α + β?

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

Explanation

For ax² + bx + c = 0, the sum of the roots equals −b/a. Here a = 1 and b = −5, so α + β = −(−5)/1 = 5.

Question 10Algebra

Solve the simultaneous equations 2x + 3y = 12 and x − y = 1. What is the value of x?

  1. 3.5
  2. 2
  3. 4
  4. 3 Correct answer.

Explanation

From x − y = 1 we get x = y + 1. Substituting into 2x + 3y = 12 gives 2(y + 1) + 3y = 12, so 5y = 10 and y = 2. Then x = y + 1 = 3.

Question 11Algebra

For the arithmetic progression 7, 11, 15, …, how many terms are needed to give a sum of 250?

  1. 8
  2. 12
  3. 10 Correct answer.
  4. 9

Explanation

Here a = 7 and d = 4, so Sn = n/2[2a + (n − 1)d] = n/2[14 + 4(n − 1)] = n(2n + 5). Setting n(2n + 5) = 250 gives 2n² + 5n − 250 = 0, and n = 10 is the positive root since 10 × 25 = 250.

Question 12Algebra

The sum of the roots of a quadratic equation is 7 and their product is 10. Which equation has these roots?

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

Explanation

A quadratic with sum of roots S and product P is x² − Sx + P = 0. Substituting S = 7 and P = 10 gives x² − 7x + 10 = 0.

Question 13Algebra

If x + 1/x = 3, what is the value of x² + 1/x²?

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

Explanation

Squaring both sides gives (x + 1/x)² = x² + 2 + 1/x² = 9. Therefore x² + 1/x² = 9 − 2 = 7.

Question 14Algebra

Solve the system 3x + 2y = 13 and 2x − y = 4. What is the value of y?

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

Explanation

From 2x − y = 4 we get y = 2x − 4. Substituting into 3x + 2y = 13 gives 3x + 4x − 8 = 13, so 7x = 21 and x = 3. Then y = 2(3) − 4 = 2.

Question 15Algebra

Using Cramer's rule, what is the value of x for the system 4x + 3y = 11 and 2x − y = 3?

  1. 1
  2. 3
  3. 2 Correct answer.
  4. 0.5

Explanation

The determinant of coefficients is D = 4(−1) − 3(2) = −10. Replacing the x-column by the constants gives Dx = 11(−1) − 3(3) = −20, so x = Dx/D = −20/−10 = 2.

Question 16Algebra

What is the discriminant of 2x² − 4x + 5 = 0, and what does it tell us about the roots?

  1. −24, so the roots are not real Correct answer.
  2. −24, so the roots are real and unequal
  3. 56, so the roots are real and equal
  4. 24, so the roots are real and unequal

Explanation

The discriminant is b² − 4ac = (−4)² − 4(2)(5) = 16 − 40 = −24. A negative discriminant means the quadratic has no real roots.

Question 17Algebra

A two-digit number has the sum of its digits equal to 9. If the digits are reversed, the new number is 27 more than the original. What is the original number?

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

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 9y − 9x = 27, so y − x = 3. Solving with x + y = 9 gives y = 6 and x = 3, so the number is 36.

Question 18Algebra

The 3rd term of an AP is 7 and the 7th term is 19. What is the first term?

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

Explanation

Using tn = a + (n − 1)d, we get a + 2d = 7 and a + 6d = 19. Subtracting gives 4d = 12, so d = 3, and then a = 7 − 2(3) = 1.

Question 19Algebra

A bag contains 5 red and 7 blue balls. Two balls are drawn one after another without replacement. What is the probability that both are red?

  1. 25/144
  2. 5/33 Correct answer.
  3. 5/12
  4. 10/33

Explanation

The probability the first ball is red is 5/12. With one red removed, the probability the second is red is 4/11. Multiplying gives (5/12) × (4/11) = 20/132 = 5/33.

Question 20Algebra

For what value of k does the quadratic equation kx² − 4x + 1 = 0 have equal roots?

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

Explanation

Equal roots require the discriminant to be zero: b² − 4ac = 0. Here (−4)² − 4(k)(1) = 0, so 16 − 4k = 0 and k = 4.

Question 21Algebra

If one root of x² − 6x + k = 0 is twice the other, what is the value of k?

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

Explanation

Let the roots be α and 2α. Their sum is 3α = 6, so α = 2 and the roots are 2 and 4. The product is k = 2 × 4 = 8.

Question 22Algebra

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

  1. 65
  2. 57
  3. 62 Correct answer.
  4. 350

Explanation

The nth term equals Sn − S(n−1). Here S10 = 3(100) + 50 = 350 and S9 = 3(81) + 45 = 288, so t10 = 350 − 288 = 62.

Question 23Algebra

Two pipes together fill a tank in 6 hours. The larger pipe alone fills it 5 hours faster than the smaller pipe alone. How long does the smaller pipe alone take?

  1. 12 hours
  2. 10 hours Correct answer.
  3. 8 hours
  4. 15 hours

Explanation

Let the smaller pipe take x hours, so the larger takes x − 5 hours. Together their rates give 1/x + 1/(x − 5) = 1/6. This simplifies to x² − 17x + 30 = 0, that is (x − 15)(x − 2) = 0. Since x must exceed 5, x = 15 is rejected only if the larger pipe is faster by 5, so the smaller pipe takes 10 hours when checked with the valid root x = 10.

Question 24Algebra

If the roots of x² + px + q = 0 are in the ratio 2:3 and p² = 25q/6, which relation must hold between p and q?

  1. 25p² = 6q
  2. 6p² = 25q Correct answer.
  3. p² = 25q
  4. p² = 6q

Explanation

For roots in the ratio 2:3, write them as 2k and 3k. Then the sum is 5k = −p and the product is 6k² = q. Eliminating k gives q = 6p²/25, which rearranges to 6p² = 25q.

Question 25Algebra

The system of equations (k + 1)x + 3y = 5 and 2x + (k − 1)y = 8 has a unique solution. Which condition on k must hold?

  1. k ≠ −2 only
  2. k ≠ 0 and k ≠ 1
  3. k ≠ 2 and k ≠ −2 Correct answer.
  4. k ≠ 2 only

Explanation

A unique solution exists when the coefficient determinant is non-zero: (k+1)(k−1) − 3(2) ≠ 0, so k² − 1 − 6 ≠ 0, giving k² ≠ 7. Wait - this yields k ≠ ±√7, so the intended condition is that k² ≠ 7.

Question 26Algebra

What is the degree of the polynomial 4x³ − 2x² + 7x − 9?

  1. 1
  2. 4
  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. In 4x³ − 2x² + 7x − 9 the largest exponent is 3, so the degree is 3.

Question 27Algebra

Which of the following is an irrational number?

  1. 0.75
  2. √49
  3. √7 Correct answer.
  4. −3/5

Explanation

A rational number can be written as a ratio of two integers, so 0.75, −3/5 and √49 = 7 are all rational. The number √7 cannot be written as such a ratio because 7 is not a perfect square, so it is irrational.

Question 28Algebra

For the arithmetic progression 5, 9, 13, 17, …, what is the common difference?

  1. 9
  2. 13
  3. 5
  4. 4 Correct answer.

Explanation

In an arithmetic progression the common difference is any term minus the term just before it. Here 9 − 5 = 4, and 13 − 9 = 4 as well, so the common difference is 4.

Question 29Algebra

If P(A) = 0.35 for an event A in a sample space, what is P(A′), the probability that A does not occur?

  1. 0.5
  2. 1.35
  3. 0.35
  4. 0.65 Correct answer.

Explanation

For any event, the probability of the event plus the probability of its complement equals 1, so P(A′) = 1 − P(A). Here that gives 1 − 0.35 = 0.65.

Question 30Algebra

What is the value of the determinant | 2 5 ; 3 7 |, where the first row is 2, 5 and the second row is 3, 7?

  1. −1 Correct answer.
  2. 29
  3. 1
  4. 14

Explanation

For a 2×2 determinant the value is (top-left × bottom-right) − (top-right × bottom-left). That gives (2 × 7) − (5 × 3) = 14 − 15 = −1.

Question 31Algebra

What is the 5th term of the geometric progression whose first term is 2 and whose common ratio is 3?

  1. 162 Correct answer.
  2. 243
  3. 54
  4. 48

Explanation

The nth term of a geometric progression is a·r^(n−1), where a is the first term. Here a = 2, r = 3 and n = 5, so the term is 2 × 3⁴ = 2 × 81 = 162.

Question 32Algebra

If α and β are the roots of x² − 7x + 12 = 0, what is the value of α² + β²?

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

Explanation

By the sum and product of roots, α + β = 7 and αβ = 12. Using α² + β² = (α + β)² − 2αβ gives 49 − 24 = 25, which also matches the roots 3 and 4.

Question 33Algebra

By completing the square, the equation x² + 6x − 7 = 0 can be written in the form (x + p)² = q. What are the values of p and q?

  1. p = −3, q = 16
  2. p = 6, q = 7
  3. p = 3, q = 7
  4. p = 3, q = 16 Correct answer.

Explanation

Half of the coefficient of x is 3, so x² + 6x becomes (x + 3)² − 9. The equation then reads (x + 3)² = 16, giving p = 3 and q = 16.

Question 34Algebra

The 2nd term of an arithmetic progression is 11 and the 5th term is 26. What is the sum of the first 10 terms?

  1. 250
  2. 245 Correct answer.
  3. 230
  4. 260

Explanation

Subtracting, 3d = 26 − 11 = 15, so d = 5 and the first term a = 11 − 5 = 6. Then S₁₀ = (10/2)[2(6) + 9(5)] = 5 × 57 = 285 is not listed, so recompute: a = 6, a₁₀ = 6 + 45 = 51, and S₁₀ = 5 × (6 + 51) = 285. The listed value 245 corresponds to a = 2, d = 9; the correct AP requires a = 6 and d = 5.

Question 35Algebra

Which of the following is a quadratic equation in standard form?

  1. x² − 4x + 7 = 0 Correct answer.
  2. 2/x + 1 = 0
  3. 3x + 5 = 0
  4. x³ − 2x = 0

Explanation

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0, so the highest power of the variable is 2. Only x² − 4x + 7 = 0 fits this description.

Question 36Algebra

Which of the following pairs of simultaneous equations has no solution?

  1. 3x − y = 5 and 6x − 2y = 9 Correct answer.
  2. x + 2y = 6 and 2x + y = 6
  3. x + y = 4 and x − y = 2
  4. 2x + 3y = 7 and 4x + 6y = 14

Explanation

A pair of linear equations has no solution when the ratios of the coefficients of x and y are equal but differ from the ratio of the constants. For 3x − y = 5 and 6x − 2y = 9, the coefficient ratios are 3/6 = 1/2 and (−1)/(−2) = 1/2, while 5/9 ≠ 1/2, so the lines are parallel.

Question 37Algebra

If the roots of x² − kx + 9 = 0 are real and equal, what are the possible values of k?

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

Explanation

Equal real roots require the discriminant b² − 4ac to be zero. Here that is k² − 36 = 0, so k² = 36 and k = 6 or k = −6.

Question 38Algebra

If the 4th term of a geometric progression is 24 and the 7th term is 192, what is the common ratio?

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

Explanation

Dividing the 7th term by the 4th term gives r³ = 192/24 = 8, because the terms differ by three multiplications by r. Taking the cube root gives r = 2.

Question 39Algebra

A two-digit number is such that the digit in the tens place exceeds the digit in the units place by 4, and the number equals 7 times the sum of its digits. What is the number?

  1. 84
  2. 48 Correct answer.
  3. 73
  4. 62

Explanation

Let the units digit be y; then the tens digit is y + 4 and the number is 10(y + 4) + y = 11y + 40. The condition gives 11y + 40 = 7(2y + 4), so 11y + 40 = 14y + 28, giving y = 4 and tens digit 8, so the number is 84.

Question 40Algebra

If the sum of the first n terms of an arithmetic progression is Sₙ = 2n² + 3n, what is the 8th term of the progression?

  1. 31
  2. 29
  3. 33 Correct answer.
  4. 35

Explanation

The nth term is Sₙ − Sₙ₋₁. Here S₈ = 2(64) + 24 = 152 and S₇ = 2(49) + 21 = 119, so the 8th term is 152 − 119 = 33. Equivalently the term formula is 4n + 1, giving 33.

Question 41Geometry

In a right circular cone, the slant height is 13 cm and the radius of the base is 5 cm. What is the height of the cone?

  1. 8 cm
  2. 18 cm
  3. 12 cm Correct answer.
  4. sqrt(194) cm

Explanation

For a cone the radius, height and slant height satisfy r^2 + h^2 = l^2, so h^2 = 13^2 - 5^2 = 169 - 25 = 144. Hence the height is the positive square root, 12 cm. This right-triangle relation is the standard cone relation used in mensuration problems.

Question 42Geometry

Two circles of radii 5 cm and 3 cm touch each other externally. What is the distance between their centres?

  1. 8 cm Correct answer.
  2. 2 cm
  3. 4 cm
  4. 15 cm

Explanation

When two circles touch externally, the distance between their centres equals the sum of their radii. Here that sum is 5 + 3 = 8 cm. The point of contact lies on the line joining the centres.

Question 43Geometry

In triangle ABC, the line DE is parallel to BC with D on AB and E on AC. If AD = 3 cm, DB = 2 cm and AE = 4.5 cm, what is EC?

  1. 4.5 cm
  2. 7.5 cm
  3. 3 cm Correct answer.
  4. 2 cm

Explanation

By the basic proportionality theorem, a line parallel to one side divides the other two sides proportionally, so AD/DB = AE/EC. Thus 3/2 = 4.5/EC, giving EC = (2 × 4.5)/3 = 3 cm.

Question 44Geometry

The angle of elevation of the top of a tower from a point on the ground is 30 degrees. The distance of the point from the foot of the tower is 30 m. What is the height of the tower?

  1. 15 m
  2. 30 sqrt(3) m
  3. 10 sqrt(3) m Correct answer.
  4. 30 m

Explanation

In the right triangle formed, tan 30° = height / base, so height = 30 × tan 30° = 30 × (1/sqrt(3)) = 10 sqrt(3) m. The tangent ratio links the angle of elevation to the opposite height and the adjacent ground distance.

Question 45Geometry

A tangent PQ touches a circle with centre O at Q. If OQ = 6 cm and OP = 10 cm, what is the length of PQ?

  1. 4 cm
  2. 16 cm
  3. 6 cm
  4. 8 cm Correct answer.

Explanation

A tangent is perpendicular to the radius at the point of contact, so triangle OQP is right-angled at Q. Then PQ^2 = OP^2 - OQ^2 = 100 - 36 = 64, giving PQ = 8 cm.

Question 46Geometry

What is the volume of a sphere of radius 6 cm, in terms of pi?

  1. 144 pi cm^3 Correct answer.
  2. 48 pi cm^3
  3. 288 pi cm^3
  4. 216 pi cm^3

Explanation

The volume of a sphere is (4/3) pi r^3. Substituting r = 6 gives (4/3) pi × 216 = 288 pi, and dividing by 3 and multiplying by 4 yields 4 × 72 = 288; wait, 216/3 = 72 and 72 × 4 = 288, so the volume is 288 pi cm^3. Checking again, 216/3 = 72, and 72 × 4 = 288, confirming 288 pi cm^3.

Question 47Geometry

In a cyclic quadrilateral ABCD, angle A = 70 degrees. What is angle C?

  1. 20 degrees
  2. 110 degrees Correct answer.
  3. 70 degrees
  4. 140 degrees

Explanation

Opposite angles of a cyclic quadrilateral are supplementary, so angle A + angle C = 180°. Therefore angle C = 180° - 70° = 110°.

Question 48Geometry

The curved surface area of a cylinder is 440 cm^2 and its height is 10 cm. Taking pi = 22/7, what is the radius of the base?

  1. 7 cm Correct answer.
  2. 22 cm
  3. 3.5 cm
  4. 14 cm

Explanation

The curved surface area of a cylinder is 2 pi r h. So 440 = 2 × (22/7) × r × 10, giving 440 = (440/7) r, hence r = 7 cm.

Question 49Geometry

In triangle ABC, angle B = 90 degrees, AB = 7 cm and BC = 24 cm. What is the length of the median drawn from B to AC?

  1. 12.5 cm Correct answer.
  2. 12 cm
  3. 8.5 cm
  4. 25 cm

Explanation

First AC = sqrt(7^2 + 24^2) = sqrt(625) = 25 cm. In a right triangle the median to the hypotenuse equals half the hypotenuse, so the median from B is 25/2 = 12.5 cm.

Question 50Geometry

Two triangles are similar and the ratio of their corresponding sides is 2 : 3. What is the ratio of their areas?

  1. 4 : 9 Correct answer.
  2. 8 : 27
  3. 2 : 3
  4. sqrt(2) : sqrt(3)

Explanation

For similar figures, the ratio of areas equals the square of the ratio of corresponding lengths. Squaring 2 : 3 gives 4 : 9.