Derivative and Matrix Business Mathematics NEB 12
Business Math Mastery: 45 Solved Questions
Section 1: Stationary Points & Inflection
Hint: Set the 1st derivative to 0 for stationary points. Set the 2nd derivative to 0 for inflection points.
Q1: Find stationary points for y = x^2 - 4x + 3.
* Solution: Derivative is 2x - 4. Set to 0: 2x = 4, so x = 2. Plug into original: y = -1.
* Answer: (2, -1).
Q2: Find the point of inflection for y = x^3 - 3x^2 - 9.
* Solution: 1st derivative is 3x^2 - 6x. 2nd derivative is 6x - 6. Set to 0: 6x = 6, so x = 1.
* Answer: x = 1.
Q3: Find critical points for f(x) = x^3 - 3x + 4.
* Solution: Derivative is 3x^2 - 3. Set to 0: x^2 = 1.
* Answer: x = 1 and x = -1.
Q4: Does y = x^3 + 6x + 30 have stationary points?
* Solution: Derivative is 3x^2 + 6. Since x^2 is always positive, 3x^2 + 6 can never be 0.
* Answer: No stationary points.
Q5: Find the y-coordinate of the inflection point for y = x^3 - 3x^2 - 9x + 15.
* Solution: 2nd derivative is 6x - 6. Set to 0: x = 1. Plug x=1 into original: 1 - 3 - 9 + 15 = 4.
* Answer: (1, 4).
* Solution: Derivative is 2x - 4. Set to 0: 2x = 4, so x = 2. Plug into original: y = -1.
* Answer: (2, -1).
Q2: Find the point of inflection for y = x^3 - 3x^2 - 9.
* Solution: 1st derivative is 3x^2 - 6x. 2nd derivative is 6x - 6. Set to 0: 6x = 6, so x = 1.
* Answer: x = 1.
Q3: Find critical points for f(x) = x^3 - 3x + 4.
* Solution: Derivative is 3x^2 - 3. Set to 0: x^2 = 1.
* Answer: x = 1 and x = -1.
Q4: Does y = x^3 + 6x + 30 have stationary points?
* Solution: Derivative is 3x^2 + 6. Since x^2 is always positive, 3x^2 + 6 can never be 0.
* Answer: No stationary points.
Q5: Find the y-coordinate of the inflection point for y = x^3 - 3x^2 - 9x + 15.
* Solution: 2nd derivative is 6x - 6. Set to 0: x = 1. Plug x=1 into original: 1 - 3 - 9 + 15 = 4.
* Answer: (1, 4).
Section 2: Increasing & Decreasing Functions
Trick: If f'(x) is Positive (+), it is Increasing. If Negative (-), it is Decreasing.
Q6: Is f(x) = 3x^3 - 24x + 1 increasing at x = 4?
* Solution: f'(x) = 9x^2 - 24. f'(4) = 9(16) - 24 = 120 (Positive).
* Answer: Increasing.
Q7: Is f(x) = 2x^3 - x^2 + 4 decreasing at x = 1/4?
* Solution: f'(x) = 6x^2 - 2x. f'(1/4) = 6(1/16) - 2(1/4) = -1/8 (Negative).
* Answer: Decreasing.
Q8: Find the interval where f(x) = x^2 - 4x is increasing.
* Solution: Derivative is 2x - 4. Set > 0: 2x > 4, so x > 2.
* Answer: x > 2.
Q9: Check f(x) = x^3 + 2x^2 - 1 at x = -1.
* Solution: f'(x) = 3x^2 + 4x. f'(-1) = 3 - 4 = -1 (Negative).
* Answer: Decreasing.
Q10: Interval check for f(x) = 2x^3 - 15x^2 + 36x + 1 between x=2 and x=3.
* Solution: Testing x=2.5 in f'(x) = 6x^2 - 30x + 36 gives a negative result.
* Answer: Decreasing.
* Solution: f'(x) = 9x^2 - 24. f'(4) = 9(16) - 24 = 120 (Positive).
* Answer: Increasing.
Q7: Is f(x) = 2x^3 - x^2 + 4 decreasing at x = 1/4?
* Solution: f'(x) = 6x^2 - 2x. f'(1/4) = 6(1/16) - 2(1/4) = -1/8 (Negative).
* Answer: Decreasing.
Q8: Find the interval where f(x) = x^2 - 4x is increasing.
* Solution: Derivative is 2x - 4. Set > 0: 2x > 4, so x > 2.
* Answer: x > 2.
Q9: Check f(x) = x^3 + 2x^2 - 1 at x = -1.
* Solution: f'(x) = 3x^2 + 4x. f'(-1) = 3 - 4 = -1 (Negative).
* Answer: Decreasing.
Q10: Interval check for f(x) = 2x^3 - 15x^2 + 36x + 1 between x=2 and x=3.
* Solution: Testing x=2.5 in f'(x) = 6x^2 - 30x + 36 gives a negative result.
* Answer: Decreasing.
Section 3: Elasticity of Demand
Formula: Ed = - (P / Q) * (dQ / dP).
Q11: Find Ed for P = 8 - 5x^2.
* Solution: Ed = (8 - 5x^2) / (10x^2).
Q12: Calculate Ed for Q = 48 - 3P^2 at P = 3.
* Solution: Q = 21. dQ/dP = -18. Ed = - (3/21) * (-18) = 2.57.
* Answer: 2.57 (Elastic).
Q13: Find x for unitary elasticity (Ed=1) for P = 4 - 5x^2.
* Solution: Setting Ed = 1 gives x = 2 / Root(15).
Q14: If P = 40 - Q^2, find Ed at P = 20.
* Solution: Q^2 = 20. Formula gives Ed = 0.5.
* Answer: 0.5 (Inelastic).
Q15: Constant Elasticity check: Q = k / P^n.
* Hint: Elasticity is always equal to 'n' for this type of function.
* Answer: Ed = n.
* Solution: Ed = (8 - 5x^2) / (10x^2).
Q12: Calculate Ed for Q = 48 - 3P^2 at P = 3.
* Solution: Q = 21. dQ/dP = -18. Ed = - (3/21) * (-18) = 2.57.
* Answer: 2.57 (Elastic).
Q13: Find x for unitary elasticity (Ed=1) for P = 4 - 5x^2.
* Solution: Setting Ed = 1 gives x = 2 / Root(15).
Q14: If P = 40 - Q^2, find Ed at P = 20.
* Solution: Q^2 = 20. Formula gives Ed = 0.5.
* Answer: 0.5 (Inelastic).
Q15: Constant Elasticity check: Q = k / P^n.
* Hint: Elasticity is always equal to 'n' for this type of function.
* Answer: Ed = n.
Section 4: Revenue & Profit Maximization
Trick: Profit is max when Marginal Revenue (MR) = Marginal Cost (MC).
Q16: Find Marginal Revenue (MR) if P = 20 - Q.
* Solution: Revenue R = 20Q - Q^2. MR = dR/dQ = 20 - 2Q.
Q17: Find Q for max profit if R = 20Q - Q^2 and C = Q^2 + 8Q + 2.
* Solution: Profit = -2Q^2 + 12Q - 2. Derivative = -4Q + 12 = 0.
* Answer: Q = 3.
Q18: Marginal Cost (MC) if C = 1/3 Q^3 - 3Q^2 + 9Q.
* Solution: MC = dC/dQ = Q^2 - 6Q + 9.
Q19: Average Cost (AC) minimum condition for C = 1/3 Q^3 - 3Q^2 + 9Q.
* Solution: AC = 1/3 Q^2 - 3Q + 9. Derivative of AC set to 0 gives Q = 4.5.
* Answer: Q = 4.5.
Q20: Show max profit at Q = 8 for Market Price 500 and Cost C = 1/3 Q^3 - Q^2 + 452Q + 50.
* Solution: Derivative of Profit = -Q^2 + 2Q + 48 = 0. Factoring gives Q=8.
* Solution: Revenue R = 20Q - Q^2. MR = dR/dQ = 20 - 2Q.
Q17: Find Q for max profit if R = 20Q - Q^2 and C = Q^2 + 8Q + 2.
* Solution: Profit = -2Q^2 + 12Q - 2. Derivative = -4Q + 12 = 0.
* Answer: Q = 3.
Q18: Marginal Cost (MC) if C = 1/3 Q^3 - 3Q^2 + 9Q.
* Solution: MC = dC/dQ = Q^2 - 6Q + 9.
Q19: Average Cost (AC) minimum condition for C = 1/3 Q^3 - 3Q^2 + 9Q.
* Solution: AC = 1/3 Q^2 - 3Q + 9. Derivative of AC set to 0 gives Q = 4.5.
* Answer: Q = 4.5.
Q20: Show max profit at Q = 8 for Market Price 500 and Cost C = 1/3 Q^3 - Q^2 + 452Q + 50.
* Solution: Derivative of Profit = -Q^2 + 2Q + 48 = 0. Factoring gives Q=8.
Section 5: Applied Maxima (Area & Numbers)
Q21: Two numbers with sum 10 whose product is max.
* Solution: x + y = 10. Max product occurs when x = y = 5.
* Answer: 5 and 5.
Q22: Min sum of squares of two numbers with sum 10.
* Solution: x = 5, y = 5. Sum of squares = 50.
Q23: Max area of rectangle with perimeter 120m.
* Solution: Area is max when the rectangle is a square with side 30m.
* Answer: 900 sq. meters.
Q24: Max product of two numbers with sum 28.
* Answer: 14 and 14.
Q25: Min sum of two positive numbers with product 36.
* Solution: Numbers are equal (6 and 6) for minimum sum.
* Solution: x + y = 10. Max product occurs when x = y = 5.
* Answer: 5 and 5.
Q22: Min sum of squares of two numbers with sum 10.
* Solution: x = 5, y = 5. Sum of squares = 50.
Q23: Max area of rectangle with perimeter 120m.
* Solution: Area is max when the rectangle is a square with side 30m.
* Answer: 900 sq. meters.
Q24: Max product of two numbers with sum 28.
* Answer: 14 and 14.
Q25: Min sum of two positive numbers with product 36.
* Solution: Numbers are equal (6 and 6) for minimum sum.
Section 6: Local Maxima & Minima Values
Q26: Max value of f(x) = x^3 - 3x + 4.
* Answer: 6 at x = -1.
Q27: Min value of f(x) = x^3 - 3x + 4.
* Answer: 2 at x = 1.
Q28: Max value of f(x) = 2x^3 - 3x^2 - 36x + 10.
* Answer: 54 at x = -2.
Q29: Min value of f(x) = 2x^3 - 3x^2 - 36x + 10.
* Answer: -71 at x = 3.
Q30: Max value of f(x) = 1/2 x^4 - x^2 + 1.
* Answer: 1 at x = 0.
* Answer: 6 at x = -1.
Q27: Min value of f(x) = x^3 - 3x + 4.
* Answer: 2 at x = 1.
Q28: Max value of f(x) = 2x^3 - 3x^2 - 36x + 10.
* Answer: 54 at x = -2.
Q29: Min value of f(x) = 2x^3 - 3x^2 - 36x + 10.
* Answer: -71 at x = 3.
Q30: Max value of f(x) = 1/2 x^4 - x^2 + 1.
* Answer: 1 at x = 0.
Section 7: Equilibrium & Exponential Elasticity
Q31: Equilibrium P for P = 100 - Q^2 and Q = 2P - 10.
* Answer: P = 8.
Q32: Equilibrium Q for above case.
* Answer: Q = 6.
Q33: Elasticity expression for Q = 30 e^(-0.02P).
* Answer: Ed = 0.02P.
Q34: Percentage increase in demand if price drops 4% (Ed = 2.57).
* Answer: 10.28%.
Q35: Unity elasticity check for curve PQ = K.
* Hint: For PQ=K, elasticity is always 1.
* Answer: P = 8.
Q32: Equilibrium Q for above case.
* Answer: Q = 6.
Q33: Elasticity expression for Q = 30 e^(-0.02P).
* Answer: Ed = 0.02P.
Q34: Percentage increase in demand if price drops 4% (Ed = 2.57).
* Answer: 10.28%.
Q35: Unity elasticity check for curve PQ = K.
* Hint: For PQ=K, elasticity is always 1.
Section 8: Matrix Properties & Operations
Q36: Find order of Matrix A = [1 2; 3 4].
* Answer: 2 x 2.
Q37: Can you add a 2x3 matrix to a 3x2 matrix?
* Answer: No (Orders must be identical).
Q38: Product order of (2x3) and (3x1) matrices.
* Answer: 2 x 1.
Q39: Example of an Identity matrix of order 2.
* Answer: [1 0; 0 1].
Q40: Transpose of row matrix [1 2 3].
* Answer: Column matrix [1; 2; 3].
* Answer: 2 x 2.
Q37: Can you add a 2x3 matrix to a 3x2 matrix?
* Answer: No (Orders must be identical).
Q38: Product order of (2x3) and (3x1) matrices.
* Answer: 2 x 1.
Q39: Example of an Identity matrix of order 2.
* Answer: [1 0; 0 1].
Q40: Transpose of row matrix [1 2 3].
* Answer: Column matrix [1; 2; 3].
Section 9: Determinants
Q41: Value of determinant |1 2; 3 4|.
* Answer: (1*4) - (2*3) = -2.
Q42: Matrix is called "Singular" if its determinant is?
* Answer: Zero (0).
Q43: Inverse exists only if the matrix is?
* Answer: Non-singular (Determinant is not zero).
* Answer: (1*4) - (2*3) = -2.
Q42: Matrix is called "Singular" if its determinant is?
* Answer: Zero (0).
Q43: Inverse exists only if the matrix is?
* Answer: Non-singular (Determinant is not zero).
Section 10: Summary Conditions
Q44: Profit maximization first order condition.
* Answer: MR - MC = 0.
Q45: Benefit of second derivative test in profit.
* Answer: Ensures the point is a Maximum (if negative).
* Answer: MR - MC = 0.
Q45: Benefit of second derivative test in profit.
* Answer: Ensures the point is a Maximum (if negative).
Business Math: 45 Solved Questions on Matrices
Section 1: Matrix Basics & Notation
Hint: The order of a matrix is always (Rows x Columns).
Q1: Define a Row Matrix.
* Answer: A matrix having only one row is known as a row matrix.
Q2: Define a Column Matrix.
* Answer: A matrix having only one column is known as a column matrix.
Q3: What is a Square Matrix?
* Answer: A matrix where the number of rows equals the number of columns (m = n).
Q4: What is the order of a matrix with 3 rows and 2 columns?
* Answer: 3 x 2.
Q5: What is a Null Matrix?
* Answer: A matrix where every element is zero, denoted by 0.
* Answer: A matrix having only one row is known as a row matrix.
Q2: Define a Column Matrix.
* Answer: A matrix having only one column is known as a column matrix.
Q3: What is a Square Matrix?
* Answer: A matrix where the number of rows equals the number of columns (m = n).
Q4: What is the order of a matrix with 3 rows and 2 columns?
* Answer: 3 x 2.
Q5: What is a Null Matrix?
* Answer: A matrix where every element is zero, denoted by 0.
Section 2: Special Matrices
Q6: Define a Diagonal Matrix.
* Answer: A square matrix where all elements except those in the leading diagonal are zero.
Q7: What is a Scalar Matrix?
* Answer: A diagonal matrix where all diagonal elements are equal.
Q8: What is an Identity (Unit) Matrix?
* Answer: A diagonal matrix where all diagonal elements are equal to 1, denoted by I.
Q9: Define an Upper Triangular Matrix.
* Answer: A square matrix where all elements below the principal diagonal are zero.
Q10: Define a Lower Triangular Matrix.
* Answer: A square matrix where all elements above the principal diagonal are zero.
* Answer: A square matrix where all elements except those in the leading diagonal are zero.
Q7: What is a Scalar Matrix?
* Answer: A diagonal matrix where all diagonal elements are equal.
Q8: What is an Identity (Unit) Matrix?
* Answer: A diagonal matrix where all diagonal elements are equal to 1, denoted by I.
Q9: Define an Upper Triangular Matrix.
* Answer: A square matrix where all elements below the principal diagonal are zero.
Q10: Define a Lower Triangular Matrix.
* Answer: A square matrix where all elements above the principal diagonal are zero.
Section 3: Matrix Operations (Addition/Subtraction)
Trick: You can only add or subtract matrices if they have the exact same order.
Q11: Can you add a 2x2 matrix and a 2x3 matrix?
* Answer: No, orders must be identical.
Q12: If A = [1 2] and B = [3 4], what is A + B?
* Answer: [4 6] (Add corresponding elements).
Q13: Is Matrix Addition commutative?
* Answer: Yes, A + B = B + A.
Q14: What is Scalar Multiplication?
* Answer: Multiplying every element of a matrix by a constant number (k).
Q15: If A is a 3x3 matrix, what is the order of -A?
* Answer: 3 x 3.
* Answer: No, orders must be identical.
Q12: If A = [1 2] and B = [3 4], what is A + B?
* Answer: [4 6] (Add corresponding elements).
Q13: Is Matrix Addition commutative?
* Answer: Yes, A + B = B + A.
Q14: What is Scalar Multiplication?
* Answer: Multiplying every element of a matrix by a constant number (k).
Q15: If A is a 3x3 matrix, what is the order of -A?
* Answer: 3 x 3.
Section 4: Matrix Multiplication
Rule: (Rows of A) x (Columns of B) is only possible if Columns of A = Rows of B.
Q16: If A is 2x3 and B is 3x4, is AB defined?
* Answer: Yes (3 = 3).
Q17: In Q16, what will be the order of AB?
* Answer: 2 x 4.
Q18: Is Matrix Multiplication generally commutative?
* Answer: No, AB is usually not equal to BA.
Q19: What is the result of A times Identity Matrix (I)?
* Answer: A.
Q20: Define Equal Matrices.
* Answer: Two matrices of the same order where every corresponding element is identical.
* Answer: Yes (3 = 3).
Q17: In Q16, what will be the order of AB?
* Answer: 2 x 4.
Q18: Is Matrix Multiplication generally commutative?
* Answer: No, AB is usually not equal to BA.
Q19: What is the result of A times Identity Matrix (I)?
* Answer: A.
Q20: Define Equal Matrices.
* Answer: Two matrices of the same order where every corresponding element is identical.
Section 5: Transpose of a Matrix
Q21: What is the Transpose of a Matrix?
* Answer: A matrix formed by interchanging rows into columns.
Q22: If A is 2x5, what is the order of A-transpose?
* Answer: 5 x 2.
Q23: What is (A-transpose)-transpose?
* Answer: Matrix A.
Q24: What is a Symmetric Matrix?
* Answer: A square matrix where A-transpose = A.
Q25: What is a Skew-Symmetric Matrix?
* Answer: A square matrix where A-transpose = -A.
* Answer: A matrix formed by interchanging rows into columns.
Q22: If A is 2x5, what is the order of A-transpose?
* Answer: 5 x 2.
Q23: What is (A-transpose)-transpose?
* Answer: Matrix A.
Q24: What is a Symmetric Matrix?
* Answer: A square matrix where A-transpose = A.
Q25: What is a Skew-Symmetric Matrix?
* Answer: A square matrix where A-transpose = -A.
Section 6: Determinants (Order 2)
Formula: For |a b; c d|, value = (ad - bc).
Q26: Find the determinant of [2 3; 1 4].
* Answer: (2*4) - (3*1) = 5.
Q27: Define a Singular Matrix.
* Answer: A square matrix whose determinant is zero.
Q28: Define a Non-Singular Matrix.
* Answer: A square matrix whose determinant is NOT zero.
Q29: If det(A) = 5, find det(A-transpose).
* Answer: 5 (Determinant doesn't change with transpose).
Q30: What is the determinant of an Identity Matrix?
* Answer: 1.
* Answer: (2*4) - (3*1) = 5.
Q27: Define a Singular Matrix.
* Answer: A square matrix whose determinant is zero.
Q28: Define a Non-Singular Matrix.
* Answer: A square matrix whose determinant is NOT zero.
Q29: If det(A) = 5, find det(A-transpose).
* Answer: 5 (Determinant doesn't change with transpose).
Q30: What is the determinant of an Identity Matrix?
* Answer: 1.
Section 7: Determinants (Order 3)
Q31: How many terms are in the expansion of a 3x3 determinant?
* Answer: 6.
Q32: What is the Sarrus Rule?
* Answer: A shortcut method used to evaluate 3x3 determinants.
Q33: Define Minor of an element.
* Answer: The determinant obtained by deleting the row and column of that element.
Q34: Define Cofactor of an element.
* Answer: A minor with a proper positive or negative sign.
Q35: If two rows of a determinant are identical, what is its value?
* Answer: Zero.
* Answer: 6.
Q32: What is the Sarrus Rule?
* Answer: A shortcut method used to evaluate 3x3 determinants.
Q33: Define Minor of an element.
* Answer: The determinant obtained by deleting the row and column of that element.
Q34: Define Cofactor of an element.
* Answer: A minor with a proper positive or negative sign.
Q35: If two rows of a determinant are identical, what is its value?
* Answer: Zero.
Section 8: Matrix Inversion
Q36: What is the Adjoint of a Matrix?
* Answer: The transpose of the cofactor matrix.
Q37: What is the formula for Matrix Inverse (A-inverse)?
* Answer: Adjoint(A) divided by det(A).
Q38: When does an inverse NOT exist?
* Answer: When the matrix is singular (det = 0).
Q39: What is A times A-inverse?
* Answer: Identity Matrix (I).
Q40: What is the inverse of an Identity Matrix?
* Answer: Itself (I).
* Answer: The transpose of the cofactor matrix.
Q37: What is the formula for Matrix Inverse (A-inverse)?
* Answer: Adjoint(A) divided by det(A).
Q38: When does an inverse NOT exist?
* Answer: When the matrix is singular (det = 0).
Q39: What is A times A-inverse?
* Answer: Identity Matrix (I).
Q40: What is the inverse of an Identity Matrix?
* Answer: Itself (I).
Section 9: Cramer's Rule
Q41: What is Cramer's Rule used for?
* Answer: Solving systems of linear equations using determinants.
Q42: In Cramer's Rule, how do you find x?
* Answer: x = Dx / D.
Q43: What happens if D = 0 in Cramer's Rule?
* Answer: The rule cannot be applied; the system may have no solution or infinite solutions.
* Answer: Solving systems of linear equations using determinants.
Q42: In Cramer's Rule, how do you find x?
* Answer: x = Dx / D.
Q43: What happens if D = 0 in Cramer's Rule?
* Answer: The rule cannot be applied; the system may have no solution or infinite solutions.
Section 10: Business Applications
Q44: How can matrices help in production planning?
* Answer: By organizing raw material needs and labor hours for different products.
Q45: How is Total Cost calculated using matrices?
* Answer: By multiplying the Quantity Matrix by the Price Matrix.
* Answer: By organizing raw material needs and labor hours for different products.
Q45: How is Total Cost calculated using matrices?
* Answer: By multiplying the Quantity Matrix by the Price Matrix.
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