List of practice Questions

Consider the experiment of throwing a die, if a multiple of 3 comes up throw the die again and if any other number comes toss a coin. Find the conditional probability of the event “the coin shows a tail”, given that “at least one die shows a 3''.

There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5%nitrogen and 10%phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14kg of phosphoric acid for her crop. If F1 costs Rs6/kg and F2 costs Rs5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs4 per unit food and F2 costs Rs6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimal nutritional requirements.

A merchant plans to sell two types of personal computers desktop model and a portable model that will cost Rs25000 and Rs40000 respectively. He estimates that the total monthly demand for computers will not exceed 250 units. Determine the number of units of each type of computer that the merchant should stock to get maximum profit if he does not want to invest more than Rs70lakhs and if his profit on the desktop model is Rs4500 and on portable model is Rs5000.

If P(A) = \(\frac 12\), P(B) = 0, then P(A|B) is :

One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
(i)E:'the card drawn is a spade'
F:'the card drawn is an ace'
(ii)E:'the card drawn is black'
F:'the card drawn is a king'
(iii)E:'the card drawn is a king or queen'
F:'the card drawn is a queen or jack'

If A and B are events such that \(P(A|B)=P(B|A)\), then:

Area of a rectangle having vertices \(A,B,C,and \space D\) with position vectors\( -\hat{i}+\frac{1}{2}\hat{j}+4\hat{k},\hat{i}+\frac{1}{2}\hat{j}+4\hat{k},\hat{i}-\frac{1}{2}\hat{j}+4\hat{k}\space and -\hat{i}-\frac{1}{2}\hat{j}+4\hat{k}\) respectively is

If \(P(A)=\frac 35\) and \(P(B)=\frac 15\), find \(P(A∩B\)) if A and B are independent events.

Find the equation of the plane passing through the line of intersection of the planes \(\vec r.(\hat i+\hat j+\hat k)=1 \) and \(\vec r.(2\hat i+3\hat j-\hat k)+4=0\) and parallel to x-axis.

Find the area of the triangle with vertices \( A(1,1,2),B(2,3,5),and \space C(1,5,5).\)

A girl walks \(4km\) towards west,then she walk \(3km\) in a direction \(30°\)east of north and stops.Determine the girls displacement from her initial point to departure.

If O be the origin and the coordinates of P be (1, 2, -3),then find the equation of the plane passing through P and perpendicular to OP.

If a unit vector \(\vec{a}\) makes an angles \(\frac{\pi}{3}\) with \( \hat{i}\),\(\frac{\pi}{4}\) with \(\hat{j}\) and an acute angle \(θ\) with \(\hat{k}\) then find \(θ\) and hence,the compounds of \(\vec{a}\).

Find a unit vector perpendicular to each of the vector \(\vec{a}+\vec{b} \space and\space \vec{a}-\vec{b}\),where \(\vec{a}=3\hat{i}+2\vec{j}+2\vec{k}\space and \space \vec{b}=\hat{i}+2\hat{j}-2\hat{k}.\)

A laboratory blood test is \(99\%\) defecting a certain disease when it is, infact present.However the test also yields a falls positive result for \(0.5\%\) of the healthy person tested(i.e.,if a healthy person is tested,then with probability 0.005,the test will imply he has the diesease).If \(0.1\%\) of the population actually has the disease,what is the probability that a person has the disease given that his test result is positive?
In answering a question on a multiple choice test a student either knows the answeror guesses.Let \(\frac{3}{4}\) be the probability that he knows the answer and \(\frac{1}{4}\) be the probability that he guesses.Assuming that the student who guessws the answer,will be correct with probability \(\frac{1}{4}\).What is the probability that a student knows the answer given that he answered it correctly?

If \(\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=2\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\),find a unit vector parallel to the vector \(2\vec{a}-\vec{b}+3\vec{c}.\)

Determine (E|F): Mother, Father and son line up at random for a family picture.
E: Son on one end
F: Father in middle

A cottage industry manufactures pedestal lamps and wooden shades,each requiring the use of a grinding/cutting machine and a sprayer.It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp.It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade.On any day the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours.The profit from the sale of a lamp is Rs5 and that from a shade is Rs3.Assuming that the manufacturer can sell all the lamps and shades that he produces,how should he schedule his daily production in order to maximize his profit?
Of the students in a college,it is known that \(60\%\) reside in hostel and \(40\%\) are day scholers.(not residing in hostel).Previous year result report that \(30\%\) of all students who reside in hostel attain A grade and \(20\%\) of day scholars attain A grade in their annual examination.At the end of the year,one student is choosen at random from the college and he has an A grade,What is the probability that the student is a hostler?

Determine P: A coin is tossed three times, where

  1. E: heads on third toss,F:heads on first two tosses.
  2. E: at least two heads,F:at most two heads.
  3. E: at most two tails,F:at least one tail.

Find a vector of magnitude 5units, and parallel to the resultant of the vectors \(\vec{a}=2\hat{i}+3\hat{j}-\hat{k}\space  and\space \vec{b}=\hat{i}-2\hat{j}+\hat{k}.\)

A factory makes tennis rackets and cricket bats.A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman's time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman's time.In a day,the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman,s time. (ii)What number of rackets and bats must be made if the factory is to work at full capacity? (iii)if the profit on a racket and on a bat is Rs20 and Rs10 respectively, find the maximum profit of the factory when it works at full capacity.

\(If \ P(A)=\frac {6}{11},\ P(B)=\frac {5}{11}\  and\  P(A∪B) =Find:\) 

  1. \(P(A∩B)\)
  2. \(P(A|B)\)
  3. \(P(B|A)\)