If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `.

Concept: Derivatives of Functions in Parametric Forms

A bill of Rs.7,500 was discounted for Rs. 7,290 at a bank on 28^{th} October 2006. If the rate of interest was 14% p.a., what is the legal due date ?

Concept: Commission, Brokerage and Discount

From the following table which relates to the number of animals of a certain

species at age x. complete the life table :

x |
0 |
1 |
2 |
3 |
4 |
5 |

l_{x} |
1000 | 850 | 760 | 360 | 25 | 0 |

Concept: Life Tables

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 = x^{3}-16x^{2 }+ 47x. For what values of x, with the average cost is decreasing'?

Concept: Derivatives of Functions in Parametric Forms

The price P for demand D is given as P = 183 + 120 D – 3D^{2}.

Find D for which the price is increasing.

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

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

For a bivariate data,

`bar x = 53 , bar y = 28 , "b"_"xy" = - 0.2` , `"b"_"yx" = -1.5` Find

Estimate of Y , When X = 50.

Concept: Regression Coefficient of X on Y and Y on X

The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo similar operations,

(a) None will recover

(b) Half of them will recover.

Concept: Probability Distribution of a Discrete Random Variable

The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class.

Also prepare marginal distributions :

(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )

Concept: Probability Distribution > Distribution Function of a Continuous Random Variable

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:

(a) the mean and

(b) P(X = 0). [Given: e^{-2} = 0.1353)

Concept: Probability Distribution > Distribution Function of a Continuous Random Variable

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