Prove that locus of z is circle and find its centre and radius if is purely imaginary.
Concept: Introduction of Relations and Functions
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
Concept: Basic Concepts of Differential Equation
An urn contains 25 balls of which 10 balls are red and the remaining green. A ball is drawn at random from the urn, the colour is noted and the ball is replaced. If 6 balls are drawn in this way, find the probability that:
(i) All the balls are red.
(ii) Not more than 2 balls are green.
(iii) The number of red balls and green balls is equal.
Concept: Introduction of Probability
In a bolt factory, three machines A, B, and C manufacture 25%, 35% and 40% of the total production respectively. Of their respective outputs, 5%, 4% and 2% are defective. A bolt is drawn at random from the total production and it is found to be defective. Find the probability that it was manufactured by machine C.
Concept: Introduction of Probability
One dialing certain telephone numbers assume that on an average, one telephone number out of five is busy, Ten telephone numbers are randomly selected and dialed. Find the probability that at least three of them will be busy.
Concept: Introduction of Probability
A person borrows ₹ 68962 on the condition that he will repay the money with compound interest at 5% per annum in 4 equal annual installments, the first one being payable at the end of the first year. Find the value of each installment.
Concept: Application of Calculus in Commerce and Economics in the Cost Function
A firm has the cost function `C = x^3/3 - 7x^2 + 111x + 50` and demand function x = 100 – p.
(i) Write the total revenue function in terms of x.
(ii) Formulate the total profit function P in terms of x.
(iii) Find the profit-maximizing level of output x.
Concept: Application of Calculus in Commerce and Economics in the Cost Function
A bill of ₹ 5050 is drawn on 13th April 2013. It was discounted on 4th July 2013 at 5% per annum. If the banker’s gain on the transaction is ₹ 0.50, find the nominal date of the maturity of the bill.
Concept: Application of Calculus in Commerce and Economics in the Cost Function
The price of six different commodities for years 2009 and year 2011 are as follows:
Commodities | A | B | C | D | E | F |
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
Concept: Basic Concepts of Differential Equation
A machine costs ₹ 60,000 and its effective life is estimated to be 25 years. A sinking fund is to be created for replacing the machine at the end of its life when its scrap value is estimated as ₹ 5000. The price of the new machine is estimated to be 100% more than the price of the present one. Find the amount that should be set aside at the end of each year, out of the profits, for the sinking fund it accumulates at an interest of 6% per annum compounded annually.
Concept: Application of Calculus in Commerce and Economics in the Cost Function
A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.
Concept: Introduction of Probability
In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. On a given day, one of the three persons A, B, and C carries out this task. A has a 45% chance, B has a 35% chance and C has a 20% chance of doing the task.
The probability that A, B, and C will take more than the allotted time is `(1)/(6), (1)/(10), and (1)/(20)` respectively. If it is found that the time taken is more than the allotted time, what is the probability that A has done the task?
Concept: Introduction of Probability
A man borrows ₹ 20,000 at 12% per annum, compounded semi-annually and agrees to pay it in 10 equal semi-annual installments. Find the value of each installment, if the first payment is due at the end of two years.
Concept: Application of Calculus in Commerce and Economics in the Cost Function
A bill of ₹ 1,800 drawn on 10th September 2010 at 6 months was discounted for ₹ 1,782 at a bank. If the rate of interest was 5% per annum, on what date was the bill discounted?
Concept: Application of Calculus in Commerce and Economics in the Cost Function
Evaluate:
`int (1 + sin "x")/(1 - sin "x") "dx"`
Concept: Indefinite Integral
Evaluate:
`int "dx"/(sin "x" + sin 2"x")`
Concept: Indefinite Integral
If `bar"X"` = 40, `bar"Y"` = 6, σ_{x} = 10, σ_{y} = 1.5 and r = 0.9 for the two sets of data X and Y, then the regression line of X on Y will be:
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).
Concept: Area of the Region Bounded by a Curve and a Line
The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4
Concept: Types of Relations
Draw a rough sketch of the curve and find the area of the region bounded by curve y^{2} = 8x and the line x =2.
Concept: Area of the Region Bounded by a Curve and a Line