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Tamil Nadu HSE Second Year Business Mathematics and Statistics study materials
Tamil Nadu HSE Second Year Business Mathematics and Statistics study resources

Tamil Nadu HSE Second Year Business Mathematics and Statistics practice questions

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Question 1Matrices and Determinants

If A is a square matrix of order 3 with |A| = 5, what is the value of |3A|?

  1. 135 Correct answer.
  2. 15
  3. 405
  4. 45

Explanation

For a square matrix of order n, |kA| = k^n |A|. Here n = 3 and k = 3, so |3A| = 3^3 × 5 = 27 × 5 = 135.

Question 2Matrices and Determinants

A matrix A of order 3 × 3 has all entries in its second row equal to zero. What is |A|?

  1. 0 Correct answer.
  2. Cannot be determined from the given information
  3. Equal to the sum of the cofactors of the second row
  4. 1

Explanation

If every element of one row (or one column) of a determinant is zero, the determinant expands to zero. Each term in the expansion of that row contains a zero factor.

Question 3Matrices and Determinants

For matrices A and B of order 2 × 2, which of the following statements is always true?

  1. (A + B)^2 = A^2 + 2AB + B^2
  2. A^2 is symmetric
  3. |AB| = |A||B| Correct answer.
  4. AB = BA

Explanation

The determinant of a product of two square matrices equals the product of their determinants, so |AB| = |A||B| holds for all such matrices. Matrix multiplication is not commutative in general, and the binomial expansion fails unless AB = BA.

Question 4Matrices and Determinants

What is the value of the determinant with first row 4, 7 and second row 2, 3?

  1. 2
  2. -26
  3. -2 Correct answer.
  4. 26

Explanation

For a 2 × 2 determinant, the value is (4)(3) − (7)(2) = 12 − 14 = −2. Subtract the product of the secondary diagonal from the product of the main diagonal.

Question 5Matrices and Determinants

If A is a 2 × 3 matrix and B is a 3 × 4 matrix, what is the order of AB?

  1. 2 × 4 Correct answer.
  2. The product AB is not defined
  3. 3 × 3
  4. 4 × 2

Explanation

The product AB is defined when the number of columns of A equals the number of rows of B, which is 3 here. The resulting matrix has the rows of A and the columns of B, so its order is 2 × 4.

Question 6Matrices and Determinants

A square matrix A satisfies A^T = A. Such a matrix is called

  1. singular
  2. symmetric Correct answer.
  3. orthogonal
  4. skew-symmetric

Explanation

A matrix equal to its own transpose is symmetric, meaning a_ij = a_ji for all entries. Skew-symmetric requires A^T = −A, and singular refers to a zero determinant.

Question 7Matrices and Determinants

If A is a square matrix with |A| = 4, what is |adj A| when A has order 3?

  1. 4
  2. 64
  3. 12
  4. 16 Correct answer.

Explanation

For a square matrix of order n, |adj A| = |A|^(n−1). With n = 3 and |A| = 4, this gives 4^2 = 16.

Question 8Matrices and Determinants

A matrix of order m × n has how many elements in total?

  1. m − n
  2. m + n
  3. mn Correct answer.
  4. m^n

Explanation

A matrix with m rows and n columns has one entry for each row-column pair, giving m × n entries. For example, a 2 × 3 matrix has six elements.

Question 9Matrices and Determinants

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

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

Explanation

Scaling a matrix of order n by k multiplies its determinant by k^n. Here |2A| = 2^3 × 2 = 8 × 2 = 16.

Question 10Matrices and Determinants

If A is a non-singular square matrix, which expression equals A^(−1)?

  1. adj A × |A|
  2. |A| adj A
  3. adj A / |A| Correct answer.
  4. A / |A|

Explanation

The inverse of a non-singular matrix is given by A^(−1) = adj A / |A|. This requires |A| ≠ 0, which is exactly the non-singular condition.

Question 11Matrices and Determinants

A 2 × 2 matrix A has |A| = 0. What can be said about A?

  1. A is the zero matrix
  2. A is singular Correct answer.
  3. A is invertible
  4. A is non-singular

Explanation

A square matrix whose determinant is zero is called singular and has no inverse. Singularity does not require all entries to be zero, only that the rows are linearly dependent.

Question 12Matrices and Determinants

Two matrices A and B satisfy AB = BA = I, where I is the identity matrix. What is B?

  1. |A|
  2. A^(−1) Correct answer.
  3. adj A
  4. A^T

Explanation

The matrix B that satisfies AB = BA = I is by definition the inverse of A, written A^(−1). The transpose and adjoint equal the inverse only in special cases.

Question 13Matrices and Determinants

If A is a 3 × 3 matrix with |A| = −3, what is |adj(adj A)|?

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

Explanation

For order n, |adj(adj A)| = |A|^((n−1)^2). With n = 3 the exponent is 4, so the value is (−3)^4 = 81. However, using the direct rule |adj(adj A)| = |A|^(n−1)^2 with n = 3 gives 81, so check the exponent: (3−1)^2 = 4 and (−3)^4 = 81. The stated options do not include 81, so the intended reading is |adj(adj A)| = |A| when n = 3 is misapplied; the correct value is 81.

Question 14Matrices and Determinants

A system of three linear equations in three unknowns has coefficient matrix A with |A| = 0, but the system is consistent. What does this imply?

  1. The system has a unique solution
  2. The coefficient matrix is invertible
  3. The system has exactly two solutions
  4. The system has infinitely many solutions or no solution Correct answer.

Explanation

When |A| = 0 the coefficient matrix is singular, so Cramer's rule fails and a unique solution is impossible. A consistent singular system has infinitely many solutions, while an inconsistent one has none.

Question 15Matrices and Determinants

If A is a square matrix such that A^2 = A, what is the value of (I + A)^3 − 7A?

  1. A
  2. I + A
  3. I Correct answer.
  4. I − A

Explanation

Since A^2 = A, higher powers collapse: A^3 = A. Expanding (I + A)^3 = I + 3A + 3A^2 + A^3 = I + 3A + 3A + A = I + 7A. Subtracting 7A leaves I.

Question 16Matrices and Determinants

The determinant of a 3 × 3 matrix A is 6. What is the determinant of the matrix obtained by interchanging two of its rows?

  1. −6 Correct answer.
  2. 6
  3. 12
  4. 0

Explanation

Interchanging any two rows of a determinant changes its sign. Since the original determinant is 6, the new determinant is −6.

Question 17Matrices and Determinants

If A is a 2 × 2 matrix with |A| = 7, what is |A^T|?

  1. −7
  2. 49
  3. 1/7
  4. 7 Correct answer.

Explanation

A matrix and its transpose have equal determinants, so |A^T| = |A| = 7. Transposing swaps rows and columns without changing the value.

Question 18Matrices and Determinants

A matrix A of order 3 × 3 has |A| = 4. What is |A · adj A|?

  1. 16
  2. 256
  3. 64 Correct answer.
  4. 4

Explanation

A · adj A = |A| · I for any square matrix. Taking determinants gives |A|^n = 4^3 = 64, since the order is 3.

Question 19Matrices and Determinants

Which of the following is a scalar matrix?

  1. A matrix with a single row
  2. A diagonal matrix with unequal diagonal entries
  3. A matrix with all entries equal to 1
  4. A matrix with all diagonal entries equal and all off-diagonal entries zero Correct answer.

Explanation

A scalar matrix is a diagonal matrix in which every diagonal entry is the same non-zero scalar. It is a special case of a diagonal matrix, obtained by multiplying the identity by a scalar.

Question 20Matrices and Determinants

Two matrices A and B are such that AB = O, where O is the zero matrix, but neither A nor B is the zero matrix. What can be concluded?

  1. A = B
  2. AB = BA
  3. Both A and B must be invertible
  4. At least one of A or B is singular Correct answer.

Explanation

Taking determinants of AB = O gives |A||B| = 0, so at least one determinant is zero and that matrix is singular. The zero product property for numbers does not carry over to matrices directly.

Question 21Matrices and Determinants

If A is a 3 × 3 matrix with |A| = 2, what is |A^(−1)|?

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

Explanation

The determinant of the inverse is the reciprocal of the determinant, so |A^(−1)| = 1/|A| = 1/2. This follows from |A||A^(−1)| = |I| = 1.

Question 22Matrices and Determinants

A 3 × 3 determinant has two identical columns. What is its value?

  1. Equal to the product of the diagonal entries
  2. Zero Correct answer.
  3. Equal to the common column sum
  4. Undefined

Explanation

If two columns of a determinant are identical, the determinant is zero because the columns are linearly dependent. Interchanging them would change the sign while leaving the determinant unchanged, forcing it to be zero.

Question 23Matrices and Determinants

If A is a square matrix of order 3 with |A| = 5, what is |adj A|?

  1. 25 Correct answer.
  2. 5
  3. 1/5
  4. 125

Explanation

The rule |adj A| = |A|^(n−1) with n = 3 gives 5^2 = 25. This follows from A · adj A = |A|I and taking determinants.

Question 24Matrices and Determinants

Matrices A and B satisfy A + B = A. What is B?

  1. The zero matrix Correct answer.
  2. A^T
  3. A
  4. The identity matrix

Explanation

Subtracting A from both sides of A + B = A gives B = O, the zero matrix of the same order. Matrix addition is associative and each matrix has an additive inverse.

Question 25Matrices and Determinants

The system x + y = 3 and 2x + 2y = 7 has coefficient determinant zero. What is the nature of this system?

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

Explanation

The second equation is twice the first on the left but not on the right, since 2 × 3 = 6 ≠ 7. The equations are inconsistent, so the system has no solution even though the coefficient determinant is zero.

Question 26Matrices and Determinants

If A is a square matrix of order 3 and |A| = 5, what is the value of |adj(adj A)|?

  1. 25
  2. 625
  3. 5
  4. 125 Correct answer.

Explanation

For a square matrix of order n, the identity |adj(adj A)| = |A|^((n−1)^2) holds. Here n = 3, so the exponent is (3−1)^2 = 4, giving |adj(adj A)| = 5^4? No - the exponent for adj(adj A) is (n−1)^2 = 4 only for |adj A| itself raised; the correct chain is |adj A| = |A|^(n−1) = 5^2 = 25, then |adj(adj A)| = |adj A|^(n−1) = 25^2 = 625. Wait, applying the formula directly: |adj(adj A)| = |A|^((n−1)^2) = 5^4 = 625.

Question 27Matrices and Determinants

A matrix has 12 elements and 3 rows. How many columns does it have, and what is its order?

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

Explanation

A matrix of order m × n contains m × n elements, where m is the number of rows and n the number of columns. With 3 rows and 12 elements, the number of columns is 12 ÷ 3 = 4, so the order is 3 × 4.

Question 28Matrices and Determinants

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

  1. 7
  2. 10 Correct answer.
  3. 9
  4. 12

Explanation

The (2,1) entry of AB is the dot product of the second row of A with the first column of B. That gives (4)(1) + (3)(2) = 4 + 6 = 10, so the required element is 10.

Question 29Matrices and Determinants

The system of equations x + 2y = 5 and 3x + 6y = 15 has coefficient determinant zero. What is the nature of this system?

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

Explanation

The coefficient determinant is (1)(6) − (2)(3) = 0, so the system is not uniquely solvable. The second equation is exactly three times the first, so both equations represent the same line and every point on it satisfies the system, giving infinitely many solutions.

Question 30Matrices and Determinants

If A is a square matrix of order 2 with |A| = 3, what is the value of |A · adj A|?

  1. 3
  2. 27
  3. 9 Correct answer.
  4. 6

Explanation

For any square matrix A, the product A · adj A equals |A| · I, where I is the identity of the same order. Taking determinants, |A · adj A| = |A|^n with n = 2, so the value is 3^2 = 9.

Question 31Analytical Geometry

The slope of the straight line joining the points (2, 3) and (6, 11) is

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

Explanation

The slope of the line through two points is m = (y2 − y1)/(x2 − x1). Here m = (11 − 3)/(6 − 2) = 8/4 = 2, so the line rises two units for every unit moved to the right.

Question 32Analytical Geometry

The equation of the straight line with slope 3 and y-intercept −5 is

  1. y = 3x − 5 Correct answer.
  2. y = −5x + 3
  3. y − 5 = 3x
  4. 3x + y + 5 = 0

Explanation

In slope-intercept form y = mx + c, the coefficient m is the slope and c is the y-intercept. Substituting m = 3 and c = −5 gives y = 3x − 5 directly.

Question 33Analytical Geometry

The distance between the points (1, 2) and (4, 6) is

  1. 7
  2. √7
  3. 25
  4. 5 Correct answer.

Explanation

Distance is found from d = √[(x2 − x1)² + (y2 − y1)²]. Here d = √(3² + 4²) = √25 = 5, the classic 3-4-5 right triangle.

Question 34Analytical Geometry

The midpoint of the line segment joining (−3, 4) and (7, −2) is

  1. (2, 1) Correct answer.
  2. (2, −1)
  3. (5, −3)
  4. (4, 2)

Explanation

The midpoint formula averages the coordinates: ((x1 + x2)/2, (y1 + y2)/2). This gives ((−3 + 7)/2, (4 − 2)/2) = (2, 1).

Question 35Analytical Geometry

The slope of any line parallel to 4x − 2y + 7 = 0 is

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

Explanation

Rearranging gives 2y = 4x + 7, so y = 2x + 7/2 and the slope is 2. Parallel lines share the same slope, so any parallel line also has slope 2.

Question 36Analytical Geometry

The equation x² + y² − 6x + 8y − 11 = 0 represents a circle whose radius is

  1. 6 Correct answer.
  2. 6√2
  3. 11
  4. √36

Explanation

Comparing with x² + y² + 2gx + 2fy + c = 0 gives g = −3, f = 4, c = −11. The radius is √(g² + f² − c) = √(9 + 16 + 11) = √36 = 6.

Question 37Analytical Geometry

The point of intersection of the lines x + y = 5 and x − y = 1 is

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

Explanation

Adding the two equations eliminates y and gives 2x = 6, so x = 3. Substituting into x + y = 5 gives y = 2, hence the intersection point is (3, 2).

Question 38Analytical Geometry

The angle between the lines y = x and y = −x is

  1. 60°
  2. 45°
  3. 90° Correct answer.
  4. 30°

Explanation

The slopes are m1 = 1 and m2 = −1, and their product m1·m2 = −1. Two lines whose slopes multiply to −1 are perpendicular, so the angle between them is 90°.

Question 39Analytical Geometry

The line x/4 + y/3 = 1 cuts the x-axis and y-axis respectively at

  1. (4, 0) and (0, 3) Correct answer.
  2. (3, 0) and (0, 4)
  3. (0, 4) and (3, 0)
  4. (1/4, 0) and (0, 1/3)

Explanation

In intercept form x/a + y/b = 1, the line meets the x-axis at (a, 0) and the y-axis at (0, b). With a = 4 and b = 3, the intercepts are (4, 0) and (0, 3).

Question 40Analytical Geometry

The slope of a line perpendicular to the line joining (1, 2) and (5, 10) is

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

Explanation

The slope of the joining line is (10 − 2)/(5 − 1) = 2. A perpendicular line has slope equal to the negative reciprocal, that is −1/2.

Question 41Analytical Geometry

The distance of the point (3, −4) from the origin is

  1. 5 Correct answer.
  2. 7
  3. 25
  4. 1

Explanation

Distance from the origin is √(x² + y²), giving √(9 + 16) = √25 = 5. The negative sign of the y-coordinate disappears on squaring.

Question 42Analytical Geometry

If the line kx + 3y − 6 = 0 passes through the point (2, 0), then the value of k is

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

Explanation

A point lies on a line when its coordinates satisfy the equation. Substituting x = 2 and y = 0 gives 2k − 6 = 0, so k = 3.

Question 43Analytical Geometry

The equation of the circle with centre (2, −3) and radius 4 is

  1. x² + y² − 4x + 6y − 3 = 0
  2. x² + y² + 4x − 6y + 9 = 0
  3. x² + y² − 2x + 3y + 9 = 0
  4. x² + y² − 4x + 6y + 9 = 0 Correct answer.

Explanation

The circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². Expanding (x − 2)² + (y + 3)² = 16 gives x² + y² − 4x + 6y − 3 = 0.

Question 44Analytical Geometry

The area of the triangle with vertices (0, 0), (4, 0) and (0, 3) is

  1. 6 square units Correct answer.
  2. 12 square units
  3. 7 square units
  4. 24 square units

Explanation

The legs along the axes have lengths 4 and 3, so the area is ½ × 4 × 3 = 6 square units. The determinant formula ½|x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)| gives the same result.

Question 45Analytical Geometry

The lines 2x + 3y = 7 and 4x + 6y = 15 are

  1. intersecting at exactly one point
  2. parallel and distinct Correct answer.
  3. perpendicular
  4. coincident

Explanation

Comparing coefficients, 2/4 = 3/6 = 1/2 but 7/15 ≠ 1/2. Equal slope ratios with unequal constant ratios mean the lines are parallel and do not coincide, so they never meet.

Question 46Analytical Geometry

The foot of the perpendicular from the origin to the line 3x + 4y = 25 is

  1. (5, 0)
  2. (25/3, 0)
  3. (3, 4) Correct answer.
  4. (0, 25/4)

Explanation

The perpendicular from the origin has slope 4/3, so its equation is 4x − 3y = 0. Solving with 3x + 4y = 25 gives x = 3, y = 4, and this point lies on the line since 9 + 16 = 25.

Question 47Analytical Geometry

The length of the perpendicular from the point (1, 1) to the line 3x + 4y − 12 = 0 is

  1. 12/5
  2. 2
  3. 1 Correct answer.
  4. 5

Explanation

The perpendicular distance is |ax1 + by1 + c| / √(a² + b²). This gives |3 + 4 − 12| / √(9 + 16) = 5/5 = 1.

Question 48Analytical Geometry

The circle x² + y² − 4x + 2y − 4 = 0 has its centre at

  1. (−2, 1)
  2. (2, −1) Correct answer.
  3. (4, −2)
  4. (2, 1)

Explanation

For x² + y² + 2gx + 2fy + c = 0 the centre is (−g, −f). Here 2g = −4 and 2f = 2, so g = −2 and f = 1, giving centre (2, −1).

Question 49Analytical Geometry

The value of p for which the lines px + 3y − 5 = 0 and 2x + 6y + 7 = 0 are parallel is

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

Explanation

Parallel lines satisfy a1/a2 = b1/b2, so p/2 = 3/6 = 1/2. Solving gives p = 1, and the constant ratio 5/7 differs, so the lines are distinct and parallel.

Question 50Analytical Geometry

The equation of the line passing through (2, 3) and perpendicular to the line x + 2y = 5 is

  1. x − 2y = −4
  2. 2x + y = 7
  3. x + 2y = 8
  4. 2x − y = 1 Correct answer.

Explanation

The given line has slope −1/2, so the perpendicular slope is 2. Using the point-slope form with (2, 3): y − 3 = 2(x − 2), which simplifies to 2x − y = 1.