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Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) study materials
Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) study resources

Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) practice questions

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Question 1Mathematical Logic and Set Theory

Which of the following is a statement in the sense of mathematical logic?

  1. What is the capital of India?
  2. The number 7 is prime. Correct answer.
  3. Please close the window.
  4. Let x be an even number.

Explanation

A statement is a declarative sentence that is either true or false but not both. 'The number 7 is prime' declares a definite truth value (true), so it qualifies. Requests, questions and instructions containing variables do not have a fixed truth value.

Question 2Mathematical Logic and Set Theory

The truth value of the compound statement (p ∧ q) ∨ ~p, when p is false and q is true, is:

  1. Cannot be determined
  2. Both true and false
  3. False
  4. True Correct answer.

Explanation

With p false, ~p is true, so the disjunction (p ∧ q) ∨ ~p is automatically true because one disjunct is true. Substituting fully: (F ∧ T) ∨ T = F ∨ T = T. A disjunction is true whenever at least one component is true.

Question 3Mathematical Logic and Set Theory

If p → q is false, then the truth values of p and q respectively are:

  1. False, False
  2. True, True
  3. False, True
  4. True, False Correct answer.

Explanation

An implication p → q is false only in the single case where the antecedent p is true and the consequent q is false. In every other combination the implication is true. Hence p must be true and q must be false.

Question 4Mathematical Logic and Set Theory

The negation of the statement 'All students in the commerce batch play cricket' is:

  1. All students in the commerce batch do not play cricket.
  2. Some students in the commerce batch do not play cricket. Correct answer.
  3. Some students in the commerce batch play cricket.
  4. No student in the commerce batch plays cricket.

Explanation

The negation of a universal statement 'All A are B' is the existential statement 'Some A are not B'. So the correct negation asserts that at least one commerce student does not play cricket. The original statement is false exactly when such a counterexample exists.

Question 5Mathematical Logic and Set Theory

In set theory, two sets A and B are said to be equal when:

  1. They have at least one element in common.
  2. Every element of A is an element of B and every element of B is an element of A. Correct answer.
  3. They have the same number of elements.
  4. A is a subset of B only.

Explanation

Sets are equal precisely when they have exactly the same elements, which is expressed by mutual inclusion: A ⊆ B and B ⊆ A together give A = B. Counting elements alone does not fix membership.

Question 6Mathematical Logic and Set Theory

Which of the following is a tautology?

  1. p ↔ ~p
  2. p → ~p
  3. p ∨ ~p Correct answer.
  4. p ∧ ~p

Explanation

A tautology is a statement true for every assignment of truth values to its components. The law of excluded middle p ∨ ~p is true whether p is true or false. The other options all fail for at least one value of p.

Question 7Mathematical Logic and Set Theory

For the statements p: '2 is even' and q: '3 is odd', the truth value of the biconditional p ↔ q is:

  1. True, because at least one statement is true
  2. True, because both statements have the same truth value Correct answer.
  3. False, because both statements are true
  4. False, because the statements are unrelated

Explanation

A biconditional p ↔ q is true exactly when p and q have the same truth value. Here both '2 is even' and '3 is odd' are true, so the biconditional is true. Content or subject matter does not affect the truth value.

Question 8Mathematical Logic and Set Theory

The statement 'If a number is divisible by 4, then it is divisible by 2' has the contrapositive:

  1. If a number is divisible by 2, then it is not divisible by 4.
  2. If a number is not divisible by 4, then it is not divisible by 2.
  3. If a number is not divisible by 2, then it is not divisible by 4. Correct answer.
  4. If a number is divisible by 2, then it is divisible by 4.

Explanation

For an implication p → q, the contrapositive is ~q → ~p, obtained by negating and interchanging the parts. Here p is 'divisible by 4' and q is 'divisible by 2', so the contrapositive is 'not divisible by 2 → not divisible by 4'. The contrapositive is logically equivalent to the original.

Question 9Mathematical Logic and Set Theory

If A = {1, 2, 3, 4} and B = {3, 4, 5}, then the symmetric difference A Δ B is:

  1. {3, 4}
  2. {1, 2, 3, 4, 5}
  3. {1, 2, 5} Correct answer.
  4. {1, 2}

Explanation

The symmetric difference A Δ B consists of elements belonging to exactly one of A and B, that is (A ∪ B) − (A ∩ B). The common elements 3 and 4 are removed, leaving 1, 2 from A and 5 from B. Hence A Δ B = {1, 2, 5}.

Question 10Mathematical Logic and Set Theory

In a class of 60 students, 35 play football, 25 play hockey and 10 play both games. The number of students who play at least one of the two games is:

  1. 70
  2. 50 Correct answer.
  3. 45
  4. 60

Explanation

By the inclusion–exclusion principle, n(F ∪ H) = n(F) + n(H) − n(F ∩ H) = 35 + 25 − 10 = 50. The subtraction prevents double-counting the students who play both games.

Question 11Mathematical Logic and Set Theory

The dual of the statement (p ∨ q) ∧ r is:

  1. (p ∧ q) ∨ r Correct answer.
  2. (p ∨ q) ∨ r
  3. (~p ∨ ~q) ∧ r
  4. (p ∧ q) ∧ r

Explanation

The dual of a compound statement is obtained by interchanging ∧ with ∨ and replacing tautologies with contradictions and vice versa, while keeping the variables unchanged. Applying this to (p ∨ q) ∧ r gives (p ∧ q) ∨ r.

Question 12Mathematical Logic and Set Theory

If the statement p → q is true, which of the following must also be true?

  1. p ∧ ~q
  2. ~q → ~p Correct answer.
  3. q → p
  4. ~p → ~q

Explanation

The contrapositive ~q → ~p is logically equivalent to p → q, so whenever the implication is true its contrapositive is true. The converse and inverse are not equivalent to the original.

Question 13Mathematical Logic and Set Theory

The set {x ∈ N : x is a prime number less than 10} written in roster form is:

  1. {2, 4, 6, 8}
  2. {2, 3, 5, 7, 9}
  3. {1, 2, 3, 5, 7}
  4. {2, 3, 5, 7} Correct answer.

Explanation

A prime number has exactly two distinct positive divisors, 1 and itself. Among natural numbers below 10 these are 2, 3, 5 and 7. Note that 1 is not prime and 9 = 3 × 3 is composite.

Question 14Mathematical Logic and Set Theory

For any two sets A and B, the set (A − B) ∪ (B − A) is equal to:

  1. A Δ B Correct answer.
  2. A ∪ B
  3. (A ∪ B) − (A ∩ B) complement
  4. A ∩ B

Explanation

The elements in A − B belong only to A, and those in B − A belong only to B, so their union collects exactly the elements belonging to precisely one set. That is the definition of the symmetric difference A Δ B. It can also be written (A ∪ B) − (A ∩ B).

Question 15Mathematical Logic and Set Theory

The number of rows in the truth table of a compound statement containing four distinct component statements is:

  1. 16 Correct answer.
  2. 32
  3. 8
  4. 4

Explanation

Each distinct component statement can take two truth values, and the rows of a truth table list every possible combination. With n components the number of rows is 2ⁿ, so for n = 4 we get 2⁴ = 16 rows.

Question 16Mathematical Logic and Set Theory

Using the algebra of statements, the expression p ∧ (p ∨ q) simplifies to:

  1. q
  2. p ∧ q
  3. p Correct answer.
  4. p ∨ q

Explanation

This is the absorption law: p ∧ (p ∨ q) ≡ p. Intuitively, if p is true the conjunction holds regardless of q, and if p is false the conjunction fails. Hence q has no effect on the result.

Question 17Mathematical Logic and Set Theory

The negation of the compound statement p ↔ q is logically equivalent to:

  1. ~p ∧ q
  2. p ↔ ~q Correct answer.
  3. ~p ∨ ~q
  4. ~p ↔ ~q

Explanation

The biconditional p ↔ q is true exactly when p and q have the same truth value, so its negation is true exactly when they differ. That is expressed by p ↔ ~q (equivalently ~p ↔ q). Negating both parts leaves the truth values still matching and so does not negate the statement.

Question 18Mathematical Logic and Set Theory

In a survey of 100 people, 60 read newspaper A, 45 read newspaper B and 20 read both. The number of people who read neither newspaper is:

  1. 25
  2. 5
  3. 35
  4. 15 Correct answer.

Explanation

Those reading at least one paper number 60 + 45 − 20 = 85. The remaining people read neither, so the count is 100 − 85 = 15. The complement of the union within the universal set gives this figure.

Question 19Mathematical Logic and Set Theory

The statement (p → q) ∧ (q → p) is logically equivalent to:

  1. ~p ∧ ~q
  2. p ↔ q Correct answer.
  3. p ∨ q
  4. p ∧ q

Explanation

The conjunction of an implication and its converse is precisely the biconditional: p ↔ q means 'p if and only if q'. It is true exactly when p and q share the same truth value. Neither conjunction nor disjunction captures this two-way condition.

Question 20Mathematical Logic and Set Theory

For non-empty sets A and B, if A ⊂ B, then which of the following is necessarily true?

  1. A − B = B
  2. A ∩ B = B
  3. A ∪ B = A
  4. B − A is non-empty Correct answer.

Explanation

A proper subset A ⊂ B means every element of A lies in B and B contains at least one element not in A. Therefore B − A contains that extra element and cannot be empty. The other equalities each contradict proper inclusion.

Question 21Matrices and Determinants

If A is a square matrix of order 3 and |A| = 5, what is the value of |kA| where k is a scalar?

  1. 5k³ Correct answer.
  2. 5k²
  3. 5k
  4. 15k

Explanation

For a square matrix of order n, multiplying every element by k multiplies the determinant by kⁿ. Here n = 3, so |kA| = k³|A| = 5k³.

Question 22Matrices and Determinants

A matrix A of order 3×3 satisfies A² = A. Which statement about A must be true?

  1. A is symmetric
  2. A is the zero matrix
  3. A is the identity matrix
  4. A is singular or A equals the identity Correct answer.

Explanation

From A² = A we get A(A − I) = O. If A is invertible, multiplying by A⁻¹ gives A = I. If A is not invertible, |A| = 0, so A is singular.

Question 23Matrices and Determinants

For the matrix A = [[2, 3], [1, 4]], what is the value of the element in row 2, column 1 of A⁻¹?

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

Explanation

For a 2×2 matrix [[a, b], [c, d]], the inverse is (1/(ad − bc))[[d, −b], [−c, a]]. Here |A| = 8 − 3 = 5, so A⁻¹ = (1/5)[[4, −3], [−1, 2]] and the (2,1) entry is −1/5.

Question 24Matrices and Determinants

The cofactor of the element 2 in the matrix [[1, 2], [3, 4]] is:

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

Explanation

The cofactor of a₁₂ is (−1)^(1+2) times the minor obtained by deleting row 1 and column 2, which leaves the single element 3. So the cofactor is −3.

Question 25Matrices and Determinants

A system of three linear equations in three unknowns has coefficient determinant equal to 0, but at least one 2×2 minor of the coefficient matrix is non-zero. The system is:

  1. inconsistent
  2. consistent with a unique solution
  3. consistent either with no solution or with infinitely many solutions Correct answer.
  4. consistent with infinitely many solutions

Explanation

When the coefficient determinant is zero, the system has no unique solution. Consistency then depends on the augmented matrix: it may be inconsistent, or it may have infinitely many solutions, so further checking is required.

Question 26Matrices and Determinants

If A and B are square matrices of the same order such that AB is invertible, which statement is correct?

  1. Neither A nor B need be invertible
  2. Only B must be invertible
  3. Both A and B must be invertible Correct answer.
  4. Only A must be invertible

Explanation

Since |AB| = |A||B| and |AB| ≠ 0, neither |A| nor |B| can be zero. Hence both A and B are invertible, and B⁻¹A⁻¹ is the inverse of AB.

Question 27Matrices and Determinants

The value of the determinant [[1, 1, 1], [a, b, c], [a², b², c²]] is:

  1. (a + b + c)(ab + bc + ca)
  2. (a − b)(b − c)(a − c)
  3. (b − a)(c − a)(c − b) Correct answer.
  4. (a − b)(b − c)(c − a)

Explanation

This is a Vandermonde determinant. Factoring out row differences gives (b − a)(c − a)(c − b), which is the standard product form for this determinant.

Question 28Matrices and Determinants

If A is a 2×2 matrix with |A| = 3, what is |adj A|?

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

Explanation

For an n×n matrix, |adj A| = |A|^(n−1). With n = 2 and |A| = 3, we get |adj A| = 3¹ = 3.

Question 29Matrices and Determinants

A matrix A satisfies Aᵀ = −A and is of order 3. What is the value of |A|?

  1. −1
  2. It depends on the entries of A
  3. 1
  4. 0 Correct answer.

Explanation

Taking determinants, |Aᵀ| = |A| and |−A| = (−1)³|A| = −|A|. Hence |A| = −|A|, so |A| = 0 for any skew-symmetric matrix of odd order.

Question 30Matrices and Determinants

If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 5]], what is the element in row 1, column 2 of AB?

  1. 8
  2. 12
  3. 10 Correct answer.
  4. 14

Explanation

The (1,2) entry of AB is the dot product of row 1 of A with column 2 of B: 1×0 + 2×5 = 10.

Question 31Matrices and Determinants

The matrix A = [[2, 3], [4, 6]] is multiplied by a non-zero matrix X so that AX = O. The matrix X must have:

  1. only zero entries
  2. an inverse
  3. order 3×3
  4. at least one non-zero entry Correct answer.

Explanation

Here |A| = 12 − 12 = 0, so A is singular and there exist non-zero matrices X with AX = O. Such an X is a non-trivial solution of a homogeneous system with singular coefficient matrix.

Question 32Matrices and Determinants

In a determinant, if two rows are identical, the value of the determinant is:

  1. −1
  2. 0 Correct answer.
  3. 1
  4. equal to the sum of that row

Explanation

Interchanging two identical rows leaves the determinant unchanged in value but, by the row-swap property, changes its sign. A number equal to its own negative must be 0.

Question 33Matrices and Determinants

The determinant of the matrix [[2, 0, 0], [5, 3, 0], [1, 4, 7]] is:

  1. 42 Correct answer.
  2. 30
  3. 21
  4. 35

Explanation

For a lower triangular matrix the determinant equals the product of the diagonal entries. Here 2 × 3 × 7 = 42.

Question 34Matrices and Determinants

If A is a non-singular matrix of order 2 with |A| = 4, what is the value of |A⁻¹|?

  1. 1/4 Correct answer.
  2. −4
  3. 4
  4. 16

Explanation

Since AA⁻¹ = I, taking determinants gives |A||A⁻¹| = 1. Therefore |A⁻¹| = 1/|A| = 1/4.

Question 35Matrices and Determinants

Using Cramer's rule on the system x + y = 3, 2x − y = 0, the value of the determinant D (the coefficient determinant) is:

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

Explanation

The coefficient matrix is [[1, 1], [2, −1]], whose determinant is 1×(−1) − 1×2 = −3. Cramer's rule then uses this D in the denominators.

Question 36Matrices and Determinants

If A = [[3, −1], [2, 5]], then A + Aᵀ equals:

  1. [[0, 1], [−1, 0]]
  2. [[6, 1], [1, 10]] Correct answer.
  3. [[6, 1], [−1, 10]]
  4. [[6, −3], [1, 10]]

Explanation

Aᵀ = [[3, 2], [−1, 5]]. Adding entrywise gives [[6, 1], [1, 10]], which is symmetric, as A + Aᵀ always is.

Question 37Matrices and Determinants

A square matrix A satisfies A² − 5A + 6I = O. Which statement is necessarily true?

  1. A is symmetric
  2. A is singular
  3. A is invertible Correct answer.
  4. A = 2I or A = 3I

Explanation

Rearranging gives A(5I − A) = 6I, so A has inverse (5I − A)/6. Hence A is invertible, whatever its particular entries are.

Question 38Matrices and Determinants

The adjoint of the matrix [[1, 2], [3, 4]] is:

  1. [[4, 2], [3, 1]]
  2. [[1, −2], [−3, 4]]
  3. [[4, −3], [−2, 1]]
  4. [[4, −2], [−3, 1]] Correct answer.

Explanation

For A = [[a, b], [c, d]], adj A = [[d, −b], [−c, a]]. Substituting a = 1, b = 2, c = 3, d = 4 gives [[4, −2], [−3, 1]].

Question 39Matrices and Determinants

If A is a 3×3 matrix with |A| = 2, what is |3A|?

  1. 18
  2. 54 Correct answer.
  3. 24
  4. 6

Explanation

Scaling a 3×3 matrix by 3 multiplies its determinant by 3³ = 27. Hence |3A| = 27 × 2 = 54.

Question 40Matrices and Determinants

The system of equations x + 2y = 4 and 2x + 4y = 8 has:

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

Explanation

The second equation is twice the first, so the two equations represent the same line. The coefficient determinant is 1×4 − 2×2 = 0, and the equations are consistent, giving infinitely many solutions.

Question 41Differentiation

For the function f(x) = 7x³ − 4x² + 9x − 5, what is f′(x)?

  1. 21x² − 8x + 4
  2. 21x³ − 8x² + 9x
  3. 21x² − 8x + 9 Correct answer.
  4. 7x² − 4x + 9

Explanation

Differentiate term by term using the power rule: d/dx(7x³) = 21x², d/dx(−4x²) = −8x, d/dx(9x) = 9, and the constant −5 differentiates to 0. Hence f′(x) = 21x² − 8x + 9.

Question 42Differentiation

If y = (3x + 5)(x² − 2), what is dy/dx by the product rule?

  1. 3x² + 10x − 6
  2. 9x² + 10x − 6 Correct answer.
  3. 6x² + 5x − 2
  4. 9x² + 10x + 6

Explanation

The product rule gives dy/dx = (3)(x² − 2) + (3x + 5)(2x) = 3x² − 6 + 6x² + 10x = 9x² + 10x − 6. Expanding the product first and then differentiating gives the same result.

Question 43Differentiation

If y = (2x + 1)/(x − 3), what is dy/dx by the quotient rule?

  1. −5/(x − 3)²
  2. −7/(x − 3)
  3. 7/(x − 3)²
  4. −7/(x − 3)² Correct answer.

Explanation

The quotient rule gives dy/dx = [2(x − 3) − (2x + 1)(1)]/(x − 3)² = (2x − 6 − 2x − 1)/(x − 3)² = −7/(x − 3)². The numerator simplifies to the constant −7.

Question 44Differentiation

If y = sin(5x), what is dy/dx?

  1. −5 sin(5x)
  2. cos(5x)
  3. 5 cos(5x) Correct answer.
  4. −5 cos(5x)

Explanation

By the chain rule, d/dx sin(u) = cos(u)·du/dx with u = 5x. Since du/dx = 5, dy/dx = 5 cos(5x).

Question 45Differentiation

If y = e^(4x), what is dy/dx?

  1. 4e^(4x) Correct answer.
  2. 4x·e^(4x)
  3. e^(4x)
  4. e^(4x)/4

Explanation

For y = e^(u) with u = 4x, the chain rule gives dy/dx = e^(u)·du/dx = e^(4x)·4 = 4e^(4x).

Question 46Differentiation

What is the derivative of log x with respect to x, for x > 0?

  1. 1/x Correct answer.
  2. log x / x
  3. x
  4. 1/(x log 10)

Explanation

The natural logarithm log x (base e, as used in this syllabus) has derivative 1/x. This follows from the inverse relationship with e^x.

Question 47Differentiation

If y = √(3x + 4), what is dy/dx?

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

Explanation

Write y = (3x + 4)^(1/2). By the chain rule, dy/dx = (1/2)(3x + 4)^(−1/2)·3 = 3/(2√(3x + 4)).

Question 48Differentiation

For f(x) = x³ − 6x² + 9x, at which values of x does f′(x) = 0?

  1. x = −1 and x = −3
  2. x = 2 and x = 4
  3. x = 0 and x = 2
  4. x = 1 and x = 3 Correct answer.

Explanation

f′(x) = 3x² − 12x + 9 = 3(x² − 4x + 3) = 3(x − 1)(x − 3). Setting this to zero gives x = 1 or x = 3.

Question 49Differentiation

The total cost of producing x units of a commodity is C(x) = 3x² + 40x + 200 rupees. What is the marginal cost when x = 10?

  1. ₹100 Correct answer.
  2. ₹120
  3. ₹60
  4. ₹80

Explanation

Marginal cost is C′(x) = 6x + 40. Substituting x = 10 gives C′(10) = 60 + 40 = 100 rupees per unit.

Question 50Differentiation

The total revenue function of a firm is R(x) = 50x − x²/2. What is the marginal revenue when x = 20?

  1. ₹50
  2. ₹40
  3. ₹10
  4. ₹30 Correct answer.

Explanation

Marginal revenue is R′(x) = 50 − x. At x = 20, R′(20) = 50 − 20 = 30 rupees per unit.