Duration: 1 hour
Section A: Each question carries 1 mark
Question 1: The value of k such that (x-4)/1 = (y-4)/1 = (x-4)/1 lies in the plane 2x-4y+z=7 is
(c) no real value
Question 2: The value of m for which straight line 3x - 2y + z + 3 = 0 = 4x - 3y + 4z + 1 is parallel to the plane 2x - y + mz - 2 = 0 is
Question 3: The pair of lines whose direction cosines are given by the equation 3l + m +5n = 0 and 6mm – 2nl + 5lm = 0 are
(c) inclined at cos^(-1) (1/6)
(d) none of these
Question 4: We have two planes represented by the equations: 3x - 2y + 3z = 8 and 6x - 4y + 6z = 32. Find the distance between these two planes.
Section B: Each question carries 2 marks
Question 1: Given that M is a 3rd order skewed matrix, write a step wise proof for determinant value of M to be equal to 0, |M| = 0.
Question 2: Find out how fast is the surface area of a cube increasing when the length of an edge of the cube is 15 cm and its volume is increasing at a steady rate of 6 cu cm/ sec.
Question 3: The point P lies on the line joining the points A(2,-1,3) and B(1,4,-3). The x co ordinate of the point P is 7. Find the y coordinate of the point.
Section C: Each question carries 4 marks
Question 1: Use the theorems and properties of determinants that you have learned about to prove that the value of the following determinant is -(x-y)(y-z)(z-x)
Question 2: Prove the following statement: For an equilateral triangle, the centroid and the in-centre are the same point. Use the result to find the co-ordinates of the in-centre of the triangle having vertices P(6, 4, 6), Q(12, 4, 0) and R(4, 2, -2).
Section D: Each question carries 6 marks
Question 1: A radioactive body decays at a rate that is proportional to its mass at a particular instant of time. The mass of the body after decaying for a day is 100 gms. After two days the mass further reduces to 80 gms. What was the initial mass of the body before it started to decay?
Question 2: The line passing through the points (1,2,3) and (2, -3, 1) cuts the plane that contains the points (4, 2, -3), (3, -4, 5) and (0, 4, 3). Find the co-ordinates of the point of intersection of the line with the plane.