**Write the converse, inverse, and contrapositive of the following statement.**

"If it snows, then they do not drive the car"

Concept: Statement Patterns and Logical Equivalence

**Write the converse, inverse, and contrapositive of the following statement.**

"If it snows, then they do not drive the car"

Concept: Statement Patterns and Logical Equivalence

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? `("Given" sqrt(3/2) = 1.2247)`

Concept: Application of Differential Equations

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `at t = `pi/4`

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

**Examine whether the following statement pattern is a tautology, a contradiction or a contingency.**

(p ∧ ~ q) → (~ p ∧ ~ q)

Concept: Statement Patterns and Logical Equivalence

**Use quantifiers to convert the following open sentences defined on N, into a true statement.**

n^{2} ≥ 1

Concept: Quantifier and Quantified Statements in Logic

**Examine whether the following statement pattern is a tautology, a contradiction or a contingency.**

(p ∧ ~ q) → (~ p ∧ ~ q)

Concept: Statement Patterns and Logical Equivalence

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

**Choose the correct alternative:**

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?

Concept: Methods of Integration: Integration Using Partial Fractions

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

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