Math Test 1

8/8/2019 Math Test 1

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OR


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Let

Thus, is a symmetric matrix.


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Thus, is a skew symmetric matrix.

Thus, A is represented as the sum of a symmetric and a skew symmetric matrix.

Q uestion 24 ( 6.0 marks) Find the vector equation of the plane passing through the intersection of the planes,

, and the point (1, 3, 1).

Solution:

H ere,

Using the relation , we obtain

It is given that the plane passes through the point (1, 3, 1). Therefore, it must satisfy equation (ii).Therefore, we obtain


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S ubstituting the value of in (i), we obtain

This is the required vector equation of the plane.

Q uestion 25 ( 6.0 marks) Use the following information to answer the next question.

A painting is in the form of a rectangle with a semi-circle along one of its lengths. The total perimeter of the painting is 5 m.

Find the dimensions of the rectangular part of the painting such that the area covered by the paintingis maximum.

Solution:Let x and y be the length and breadth of the rectangle respectively. Then, radius of the semi-circle

will be .

It is given that the perimeter of the painting is 5 m.

Therefore, we obtain


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Area of the painting is given by,

Therefore, area of the painting is maximum when


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Thus, the dimensions of the rectangular part are

Q uestion 26 ( 6.0 marks)

Find .

Solution:

Let I =

The integrand can be written as

Comparing the coefficients of x 2 and the constant terms, we obtain

3 A + B = 0 B = 3 A

4 A + 2 B + C = 1

5 A + 2 C = 3

S olving these equations, we obtain

Thus, the integrand is given by,


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Put 3 x 2 + 4 x + 5 = t (6 x + 4) dx = dt

Put

From (2), (3), and (4), we obtain

Therefore, from (1), we obtain


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Q uestion 27 ( 6.0 marks) Find the area of the region bounded by the ellipse, x 2 + 9 y 2 = 81, and the line, x = 3.

Solution:

The given ellipse, x 2 + 9 y 2 = 81 can be written as

The given ellipse is symmetrical about both x and y axis. Thus, the required area of the region BCDEis given by,


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Q uestion 28 ( 6.0 marks) A factory manufactures two types of cloths, A and B. The factory has two machines, which can workfor at most 3 hours. It takes 5 minutes on first machine and 2 minutes on second machine to produce10 m of cloth A. It takes 3 minutes on first and 3 minutes on second machine to produce 10 m of cloth B. The manufacturer earns revenue of Rs 60 and Rs 50 on 10 m of cloth A and 10 m of cloth Brespectively. Determine the maximum revenue that the factory can earn.

Solution:Let x units and y units of cloths A and B be manufactured, where 1 unit = 10 m. Therefore, totalprofit (Rs) = 60 x + 50 y

Let z = 60 x + 50 y

The mathematical model for the given problem is as follows.

Maximize z = 60 x + 50 y (1)

S ubject to

The feasible region OABC determined by the inequalities (2), (3), (4) is represented as follows.


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The objective function Z at each of the corner point of the shaded region is evaluated as

C orner point Z = 60 x + 50 y

O (0, 0)

A (0, 60)

B (20, 40)

C (36, 0)

0

3000

3200

2160

Thus, the maximum value of Z is 3200 at B (20, 40).

Therefore, the factory should produce 20 units of cloth A and 40 units of cloth B to earn maximumrevenue of Rs 3200.

That is the factory should produce 200 m of cloth A and 400 m of cloth B to earn maximum revenueof Rs 3200.

Q uestion 29 ( 6.0 marks) A pair of unbiased dice is thrown. A random variable X is the difference between the numbers thatappear on the top face of the two dice. Find the expectation of X. Also, find the variance of X.

OR

A die is thrown 10 times. If getting a number greater than or equal to 3 is a success, then find theprobability of

a. at most 3 successes

b . at least 8 successes

Solution:The sample space of the experiment consists of 36 elements.


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The random variable X is the difference between the numbers on the two dice. Therefore, it can takethe values 0, 1, 2, 3, 4, or 5.

The probability distribution of X is

X 0 1 2 3 4 5

P ( X )

Thus, the expectation of X is .


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Variance

OR

The event X getting a number greater than or equal to 3 is success.

Probability of success

It can be observed that X has a binomial distribution with n = 10 and

(a) P (at most 3 successes) = P(X 3)


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(b) P (at least 8 successes) = P(X 8)

Math Test 1

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