Higher Secondary Certificate Examination (Bangladesh) · Higher Mathematics

Matrices, Determinants, Vectors and Coordinate Geometry

35 practice questions, each with the answer and an explanation.

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All 35 questions

  1. What is the determinant of the 2x2 matrix [[4, -3], [2, 5]]?
  2. For what value of k is the matrix [[k, 6], [2, 3]] singular?
  3. If matrix A has dimension 3 x 4 and matrix B has dimension 4 x 2, what is the dimension of product matrix AB?
  4. If every element of an n x n square matrix A is multiplied by a non-zero scalar k, what is the determinant of kA?
  5. Which condition defines a symmetric matrix A?
  6. What is the trace of the 3x3 matrix with diagonal entries 5, -2, and 8?
  7. If A is an invertible matrix with det(A) = 5, what is the determinant of the inverse matrix A^(-1)?
  8. What is the value of the determinant of [[1, 2, 3], [0, 4, 5], [0, 0, 6]]?
  9. What is the scalar dot product of vectors u = 2i - 3j + 4k and v = i + 2j - k?
  10. If |u| = 6, |v| = 5, and the angle theta between u and v is 30 degrees, what is |u x v|?
  11. What is the unit vector in the direction of vector w = 3i - 4j?
  12. If two non-zero vectors a and b satisfy a . b = 0, what is the angle between them?
  13. What is the distance between points A(1, -2) and B(4, 2)?
  14. What are the coordinates of the midpoint of segment joining (-3, 5) and (7, -1)?
  15. What is the slope of the line passing through points (2, 3) and (6, 11)?
  16. What is the slope of a straight line perpendicular to the line 3x - 4y + 7 = 0?
  17. What is the perpendicular distance from the point (1, 2) to the straight line 3x + 4y - 1 = 0?
  18. What are the center and radius of the circle x^2 + y^2 - 6x + 8y = 0?
  19. What is the constant term c in the general circle equation x^2 + y^2 + 2gx + 2fy + c = 0 if the circle passes through the origin (0, 0)?
  20. For what value of c is the line y = 2x + c tangent to the circle x^2 + y^2 = 5?
  21. At what point do the lines 2x + y = 7 and x - y = 2 intersect?
  22. What is the area of the triangle with vertices (0, 0), (4, 0), and (0, 6)?
  23. What is the centroid of a triangle with vertices at (1, 2), (3, 6), and (5, 4)?
  24. If points (1, 2), (3, 4), and (k, 8) are collinear, what is the value of k?
  25. What is the acute angle between the straight lines y = x and y = root 3 x?
  26. What is the length of the tangent drawn from point (5, 0) to the circle x^2 + y^2 = 9?
  27. In a right-handed Cartesian coordinate system, what is the vector cross product i x j?
  28. Which relation correctly describes the algebraic symmetry of vector cross product?
  29. What is the scalar projection of vector a = 3i + 4j onto vector b = i?
  30. What is the equation of the straight line parallel to the x-axis passing through (4, -5)?
  31. If Q is an orthogonal matrix (Q^T Q = I), what are the only possible real values of det(Q)?
  32. What is the necessary and sufficient condition for three vectors u, v, and w to be coplanar?
  33. What are the x-intercept and y-intercept of the straight line 2x + 5y = 10?
  34. The two angle bisectors of two intersecting straight lines are always at what angle to each other?
  35. Under what condition does a system of n linear equations in n variables have a unique solution by Cramer's Rule?

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