Time : 3 hours TEST – XII ( MATHS ) M.M.: 100
SECTION – A
Q.1If A = and A + A = then find the value of such that 0 ..
Q.2Show that the matrix BAB is skew symmetric if A is skew symmetric.
Q.3Area of a triangle is 35 sq. units with vertices (2, - 6), (5, 4) and (k, 4). Find the value of k.
Q.4State with reason whether Mean Value Theorem is applicable or not for f(x) = x1/3 , [-1,1].
Q.5S = { 1, 2, 3 }. F is a function from S to S such that F = { (1,3), (3, 1), (2, 2)}. Find if it exists.
Q.6Evaluate:
Q.7Evaluate:
Q.8If , then find the angle between .
Q.9Write down the unit vector in XY – plane which makes angle of 30 with the positive direction of x – axis.
Q.10Find the distance of a point ( 2, 5, -3 ) from the plane
SECTION – B
Q.11Without expanding, prove that:
Q.12A man of height 180 cm is moving away from a lamp post at the rate of 1.2 m/s. If the height of the lamp post is 4.5 m, find the rate at which (i) his shadow is lengthening (ii) the tip of his shadow is moving.
Q.13Solve for x:
Q.14If then prove that
Q.15If f(x), defined by the following, is a continuous function then find the values of a and b.
Q.16Evaluate:
Q.17Evaluate:
Q.18Evaluate
Q.19Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Q.20Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XY – plane.
Q.21If then express in the form such that is parallel to and is perpendicular to
Q.22Let * be a binary operation on the set Q of rational numbers defined by a * b = (i) Find whether it is commutative or not. (ii) Find its identity element.
SECTION – C
Q.23Solve by matrix method: x + 2y – 3z = - 4 ; 2x + 3y + 2z = 2 and 3x – 3y – 4z = 11.
Q.24Find the coordinates of the foot of the perpendicular drawn from the point (1, 1, 1) to the plane 2x – 3y + 4z – 6 =0. Also find the equation of the perpendicular.
Q.25A manufacturer makes 2 types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:
Types of Toys / MachinesI / II / III
A / 12 / 18 / 6
B / 6 / 0 / 9
Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs 7.50 and that on type B is Rs 5, show that 15 toys of type A and 30 of type B should be manufactured for maximum profit.
Q.26Given three identical boxes I, II and III, each containing two coins. In box I, both coins are of gold. In box II, both are silver coins and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. (i) Find the probability that it is gold coin. (ii) If the coin taken out is of gold what is probability that the other coin in that box is of silver?
Q.27Show that the height of the cylinder of greatest volume which can be inscribed in a sphere of radius R is . Also find the maximum volume.
Q.28Find the area bounded by the curves using integrations.
Q.29Solve the differential equation:
1