Class 12th math notes

Q10/10: The Cartesian equations of a line are 3x + 1 = 6y - 2 = 1 - z. Find its direction ratios.

Q9/10: If the lines (x-1)/-3 = (y-2)/2k = (z-3)/2 and (x-1)/3k = (y-1)/1 = (z-6)/-5 are perpendicular, find the value of k.

Q8/10: Assertion (A): The lines x/1 = y/2 = z/3 and (x-1)/-2 = (y-2)/-4 = (z-3)/-6 are parallel. Reason (R): Two lines are parallel if their direction ratios are proportional.

Q7/10: What is the condition for two lines r = a1 + λ b1 and r = a2 + μ b2 to be coplanar?

Q6/10: The vector equation of a line passing through the point with position vector a and parallel to vector b is:

Q5/10: If a line makes angles 90, 60 and 30 degrees with the positive direction of x, y, and z axes respectively, find its direction cosines.

Q4/10: The direction ratios of the line joining points (2, -1, 4) and (1, 1, 2) are:

Q3/10: What are the direction cosines of the x-axis?

Q2/10: To find if a system of three linear equations representing intersecting planes in 3D space has a unique point of intersection, we check if the determinant of the coefficient matrix is:

Q1/10: A company produces three products: X, Y, and Z. The sales in two markets are given as matrices. Matrix multiplication is used to find total revenue. The operation requires:

Q10/10: If A is a 2x2 matrix and A^2 = 0 (Null matrix), then |A| is:

Q9/10: If A = [2, 4; 1, 2], find the cofactor of element 4.

Q8/10: Consider matrix A of order 3x3. If A is both symmetric and skew-symmetric, then A is:

Q7/10: Assertion (A): If all elements of a row of a determinant are multiplied by k, the value of the determinant is multiplied by k. Reason (R): |kA| = k|A| for any square matrix A.

Q6/10: Two matrices A and B can be added only if:

Q5/10: If A is a square matrix of order 3 and |A| = 5, what is the value of |A * adj(A)|?

Q4/10: If the inverse of matrix A is [1, -1; 2, 3], then the inverse of A^T is: