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A toy manufacturing company produces five types of toys. Each toy has to go through three machines A, B, C in the order ABC. The time required in hours for each process is given in the following table.
| Type | 1 | 2 | 3 | 4 | 5 |
| Machine A | 16 | 20 | 12 | 14 | 22 |
| Machine B | 10 | 12 | 4 | 6 | 8 |
| Machine C | 8 | 18 | 16 | 12 | 10 |
Solve the problem for minimizing the total elapsed time.
Concept: Types of Sequencing Problem
Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).
Concept: Random Variables and Its Probability Distributions
Solve the following problem :
Let the p. m. f. of the r. v. X be
`"P"(x) = {((3 - x)/(10)", ","for" x = -1", "0", "1", "2.),(0,"otherwise".):}`
Calculate E(X) and Var(X).
Concept: Probability Distribution of Discrete Random Variables
A random variable X has the following probability distribution:
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(x) | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
Find:
- k
- P(X < 3)
- P(X > 4)
Concept: Random Variables and Its Probability Distributions
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
Concept: Continuous Function of Point
If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `
Concept: Derivatives of Functions in Parametric Forms
The cost C of producing x articles is given as C = x3-16x2 + 47x. For what values of x, with the average cost is decreasing'?
Concept: Derivatives of Functions in Parametric Forms
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Concept: Logarithmic Differentiation
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Concept: Increasing and Decreasing Functions
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Concept: Increasing and Decreasing Functions
Cost of assembling x wallclocks is `( x^3/3 - 40x^2)` and labour charges are 500x. Find the number of wall clocks to be manufactured for which average cost and marginal cost attain their respective minimum.
Concept: Maxima and Minima
Evaluate : `int_3^9 [root(3)(12-x)]/[ root(3)(x) + root(3)(12 - x)]`
Concept: Applications of Definite Integrals
If f(x) = `("e"^(2"x") - 1)/"ax"` , for x < 0 , a ≠ 0
= 1 for x = 0
= `("log" (1 + 7"x"))/"bx"` , for x > 0 , b ≠ 0
is continuous at x = 0, then find a and b.
Concept: Applications of Definite Integrals
Find the volume of a solid obtained by the complete revolution of the ellipse `x^2/36 + y^2/25 = 1` about X-axis.
Concept: Applications of Definite Integrals
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
Concept: Properties of Definite Integrals
The ratio of number of boys and girls in a school is 3 : 2. If 20% of the boys and 30% of the girls are scholarship holders, find the percentage of students who are not scholarship holders.
Concept: Ratio, Proportion and Partnership
If the Crude Death Rate (CDR) for the following data is 13.4 per thousand, find x:
| Age groups (in years) |
Population | Number of deaths |
| 0 - 20 | 40,000 | 350 |
| 20 - 65 | 65,000 | 650 |
| 65 and above | 15,000 | x |
Concept: Commission, Brokerage and Discount
Mrs. Menon plans to save for her daughter's marriage. She wants to accumulate a sum of ₹ 4,00,000 at the end of 4 years. How much should she invest at the end of each year from now, if she can get interest compounded at 10% p.a. ?
[Given (1.1)4 = 1.4641]
Concept: Insurance and Annuity
If the rank correlation coefficient is 0.6 and the sum of squares of differences of ranks is 66, then find the number of pairs of observations.
Concept: Rank Correlation
The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.
Find:
(a) Correlation coefficient
(b) `sigma_x/sigma_y`
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
