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• Question 3

State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

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• Question 4

For a 2 × 2 matrix, A = [aij] whose elements are given by , write the value of a12.

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• Question 5

For what value of x, the matrix is singular?

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• Question 8

For what value of ‘a’ the vectors and are collinear?

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• Question 9

Write the direction cosines of the vector

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• Question 10

Write the intercept cut off by the plane 2x + y z = 5 on x-axis.

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• Question 11

Using properties of determinants, prove the following:

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• Question 12

Find the values of a and b such that the following function f (x) is a continuous function:

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• Question 13

Solve the following differential equation:

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• Question 14

If two vectors are such that then find the value of

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• Question 15

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min. {a, b}. Write the operation table of the operation *.

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• Question 16

Prove the following:

OR

Find the value of

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• Question 17
• Question 18

Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

OR

Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to x-axis.

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• Question 20

Solve the following differential equation:

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• Question 21

Find the angle between the following pair of lines:

and check whether the lines are parallel or perpendicular.

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• Question 22

Probabilities of solving a specific problem independently by A and B are and respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

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• Question 23

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six.

Find the probability that it is actually a six.

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• Question 24

Show that of all the rectangles of given area, the square has the smallest perimeter.

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• Question 25

Using matrix method, solve the following system of equations:

OR

Using elementary transformations, find the inverse of the matrix

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• Question 26

Using integration find the area of the triangular region whose sides have equations y = 2x + 1, y = 3x + 1 and x = 4.

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• Question 27
• Question 28

Find the equation of the plane which contains the line of intersection of the planes and which is perpendicular to the plane

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• Question 29

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve graphically.

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