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A bag contains 19 slips, numbered 1 to 19. A slip is drawn and then another slip is drawn without replacement. Find the probability that both slips will show even numbers.

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Verify Rolle’s Theorem for the following functions: f(x) = (x - 1)(x - 2)2 on [1, 2].

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(a) f(x) is a polynomial function, therefore the function is continuous on [1, 2].


(b) f'(x) = 1.(x - 2)2 + (x - 1).2(x - 2)
 = (x - 2)[x - 2 + 2x - 2]
= (x - 2)(3x - 4)

Thus, function is differentiable on (1, 2)


(c) f(1) = 0 = f(2).

All the three conditions of Rolle’s theorem are true

at least one point c (1, 2) s.t. f'(c) = 0

(c - 2)(3c - 4) = 0

c = 4/3 or c = 2
but 2 (1, 2), hence this choice is rejected and the value of c is 4/3.

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Evaluate .

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Find the equations of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

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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?

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Solve the following differential equation:

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In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4 . What is the probability that the student knows the answer given that he answered it correctly?

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Let E1= Student knows the answer,

E2= Student guesses the answer, and

A = answer is correct

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The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of the awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix methods, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

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Therefore, the number of awardees for honesty, cooperation and supervision is 3, 4 and 5 respectively.

Discipline should be another value, which should be added by the management of the colony.
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A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

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Suppose that a pair of fair dice are tossed and let the variable X denote the sum of points.
(i) Obtain the probability distribution for X.
(ii) Find mean of X.
(iii) Find variance and standard deviation of X.

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(i) The probability distribution for X

X

2

3

4

5

6

7

8

9

10

11

12

P(X)

1/36

2/36

3/36

4/36

5/36

6/36

5/36

4/36

3/36

2/36

1/36

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Find the area of the region {x, y : 0 y x2 + 1, 0 y y x +1, 0 x 2}.

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A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?

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ABCD is a parallelogram. The coordinates of A, B, C are (2, 3), (1, 4) and(0, -2) respectively. Find the coordinates of D.

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Find the vector equation of the line passing through the points (–1, 0, 2) and (3, 4, 6).

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Find the area of the region {x, y : 0 y x2 + 1, 0 y y x +1, 0 x 2}.

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For manufacturing an item, the fixed cost is 1900 and the cost of producing each unit is 15. What is the cost function?

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Total fixed cost = 1900

Variable cost of producing one unit = 15

Total cost, TC = (1900 + 15x), where x is the number of units produced.

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Two regression lines are represented by 2x + 3y – 20 = 0 and 4x + y – 15 = 0. Find the line of regression of y on x.

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Find the equation of the two lines of regression for the following observations:
(3, 6), (4, 5), (5, 4),(6, 2),(7, 3)
Find an estimate of y for x = 3.5.

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The constructed table is given below:

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Given the total cost function for x units of a
commodity as

(i) the marginal cost function

(ii) average cost function

(iii) slope of average cost function

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The fixed cost of a new product is `3600 and the variable cost per unit is ` 110. If the demand function is p(x) = 800 – 30x, find the breakeven values.

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Let x units of the product be produced and sold.

Variable cost of producing x units = ` 110x
As the fixed cost is ` 3600,

So, total cost of producing x units,
C(x) = `(3600 + 110x)
Given demand function, p(x) = 800 – 30x

i.e., the selling price per unit = `(800 – 30x)

So, total revenue on selling x units,

R(x) = (price per unit)(number of units sold)
= `(800 – 30x)x

At breakeven values,

C(x) = R(x)

3600 + 110x = (800 – 30x)x
3600 + 110x = 800x – 30x2

30x2 – 690x + 3600 = 0

Dividing by 30, we get
x2 – 23 +120 = 0
(x – 8)(x – 15) = 0
This implies that either x = 8 or x = 15

Hence, the breakeven values of x are 8 and 15.

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A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?

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