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Seven different lecturers are to deliver lectures in seven periods of a class on a particular day. A, B and C are three of the lectures.
The number of ways in which a routine for the day can be made such that A delivers his lecture before B, and B before C, is a b c d None of these If 33! The number of zeros at the end of ! In a city no persons have identical set of teeth and there is no person without a tooth.
Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is a 2 3 2 c 2.
The number of ways in which a mixed double game can be arranged from amongst 9 married couples if no husband and wife play in the same game is a c In a college examination, a candidate is required to answer 6 out of 10 questions which are divided into two sections each containing 5 questions, further the candidate. The number of ways in which he can make up a choice of 6 questions is a b c d 50 The number of ways in which 9 identical balls can be placed in three identical boxes is 9!
The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is a 1 P2 2 5.
Each part has one or more than one correct answer s. The number of points x,y, z is space, whose each co-ordinate is a negative 6. The number of ways in which 30 coins of one rupee each be given to six persons so that none of them receives less than 4 rupees is a b c d 4. The number of ways to select 2 numbers from 0, 1, 2, 3, 4 such that the sum of the squares of the selected numbers is divisible by 5 are repitition of digits is allowed.
If n objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is a " - 2 n - 3 r a - 4. Answers Multiple 1. The middle term depends upon the value of n. So there is only one middle term i. Where [ ] denotes the greatest integral part. Binomial Theorem 5. The term independent of JC in the expansion Nio. The sum of the series 1 1 1 1! The greatest coefficient in the expansion of , , , -. The last two digits of the number 3 4 0 0 are a 39 b 29 c 01 d The coefficient of Xn J.
The number of distinct terms in the expansion. The value of the sum of the series 3. If maximum and minimum values of the determinant 2.
The last digit of 3 a 1 c 3. The number - 1 is divisible by a b c d Definition Determinant of order 2, 3 and 4 are written as an Minors and Cofactors If we delete the row and column passing through the element a y, thus obtained is called the minor of a,y and is usually denoted by Mjj and cofactors of a,y is - 1 M j j and it is denoted by Ay or C,y.
Properties of Determinants i The determinant remains unaltered if its rows and columns are interchanged. Note that we can also multiply rows by columns or columns by rows or columns by columns. Cramer's Rule: Differentiation of Determinant Function If.
If x, y, z are integers in A. In a third order determinant a,-, denotes the element in the ith row and the yth column 7. The digits A, B, C are such that the three digit numbers A88, 6B8, 86C are divisible by 72 then the determinant A 6 8 8 B 6 is divisible by 8 8 C a 72 b c d If all elements of a third order determinant are equal to 1 or - 1 , then the determinant itself is a an odd number b an even number c an imaginary number d a real number The largest value of a third order 8.
Definition The probability of an event to occur is the ratio of the number of cases in its favour to the total number of cases equally likely. If a is the number of cases favourable to the event , b is the number of cases favourable to t h e e v e n t ", t h e n odds in favour of a r e a: Type of Events: The given events are said to be equally likely, if none of them is expected to occur in preference to the other.
Two events are said to be independent if the occurrence of one does not depend upon the other. A set of events is said to be mutually exclusive if occurrence of one of them precludes the occurrence of any of the remaining events.
If a set of events 1, 2 En for mutually exclusive events. A set of events is said to be Exhaustive if the performance of the experiment results in the occurrence of at least one of them If a set of Events 1, 2, A set of events is said to be mutually exclusive and exhaustive if above two conditions are satisfied.
If a set of Events 1, 2 En then for mutually exclusive and exhaustive events P 1 u 2 u. It is denoted by P 1. Multinomial Theorem If a die has m faces marked 1, 2, 3, Then the probability that the sum of the numbers on the upper faces is equal to ris given by.
If 1 and 2 are two events, then i 1 u 2 stands for occurrence of at least one of 1, 2 ii 1 n 2 stands for the simultaneous occurrence of 1, 2.
Expectation If p be the probability of success of a person in any venture and M be the sum of money which he will receive in case of success, the sum of money denoted by pM is called expectation. The probabilities that a student will obtain grades A, B, C or D are 0. The probability that he will receive atleast C grade, is a 0.
If a student is absent twice, the probability that he will miss at least one test, is:. An inspector takes out one transistor at random, examines it for defects, and replaces it. After it has been replaced another inspector does the same thing, and then so does a third inspector. There are 5 duplicate and 10 original items in an automobile shop and 3 items are brought at random by a customer.
Three letters are written to three different persons and addresses on the three envelopes are also written. Without looking at the addresses, the letters are kept in these envelopes. A bag contains 7 red and 2 white balls and another bag contains 5 red and 4 white balls. Two balls are drawn, one from each bag. A bag contains 5 red, 3 white and 2 black balls. The probability that 3 out of 4 bulbs, bought by a customer will not be defective, is: Fifteen coupons are numbered 1, 2, 3, Seven coupons are selected at random one at a time with replacement.
The probability that a man aged x years will die in a year is p. The probability that out of n men M,,M If the letters are placed in the envelopes at random, the probability that atleast one letter is not placed in the right envelope, is a 1 c 1 -.
Three athletes A, B and C participate in a race. Both A and B have the same probability of winning the race and each is twice as likely to win as C. A number is chosen at random from among the first 30 natural numbers.
Out of 13 applicants for a job, there are 8 men and 5 women. It is desired to select 2 persons for the job. Two athletes A and B participate in a race along with other athletes. Three players A, B, C in this order, cut a pack of cards, and the whole pack is reshuffled after each cut. The probability that the mapping selected is bijective, is. Three letters are written to three different 2" and addresses on the three envelopes persons are also written.
An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled four times. The probability that the 13th day of a randomly chosen month is a second Saturdav. A box contains cards numbered 1 to A card is drawn at random from the box. An integer is chosen at random from the numbers 1,2,.. Two players A and B throw a die alternately for a prize of Rs. If A starts the game, their respective expectations are a Rs. A three-digit number is selected at random from the set of all three-digit numbers.
A student of this college is setected at random. The probability that this student who has failed in Mathematics would have failed in Physics too, is: A purse contains 4 copper and 3 silver coins, and a second purse contains 6 copper and 2 silver coins. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, is. The probabilities of two events A and B are and respectively.
Then the probability that neither A nor B occurs is a 0 1 0 b c d The probability that the mapping selected is one to one is given by. Two persons each makes a single throw with a pair of dice. A bag contains a white and b black balls.
Two players A and B alternately draw a ball from the bag, replacing the ball each time after the draw. A begins the game. If the probability of A winning that is drawing a white ball is twice the probability of B winning, then the ratio a: One ticket is selected at random from tickets numbered 00, 01, 02, Let X be a set containing n elements. Of the 25 questions in a unit, a student worked out only In a sessional test of unit, two questions were asked by teacher.
Probability The probability that at least one of the events A and B occur is An element a, b of their cartesian product A x B is chosen at random. Dialing a telephone number, a man forgot the last two digits and remembering only that they are different, dialled them at random. The probability that they will say the same thing while describing a single event is a 0: A three digit number, which is multiple of 11, is chosen at random.
A box contains tickets numbered 1 to Two numbers. The probability that. A fair die is thrown until a score of less than five points is obtained. Seven digits from the digits 1. A fair coin is tossed n times. A wire of length I is cut into three pieces. What is the probability that the three pieces form a triangle? A bag contains four tickets marked with , , , one ticket is drawn at random from the bag.
A bag contains four tickets numbered 00, 01, 10, The probability that the number satisfies the inequation x2- The adjoining Fig. A man walking on the road AB takes a turn at random to reach the road A jB,.
The chance that he moves on a straight line from the road AB to theAjB, is a b 0 04 c d None of these Two distinct numbers are selected at random from the first twelve natural numbers.
Given that x e [0, 1] and y e [0, 1]. The probability that out of 10 persons, all born in April, at least two have the same birthday is 30, 30, '10 '10 a b 1.
A pair of fair dice is rolled together till a sum of either 5 or 7 is obtained, the probability that 5 comes before 7 is a b c d C. A second order determinant is writeen down at random using the numbers 1, - 1 as elements. A five digit number is chosen at random. Five numbers out of these are picked up at random. The probability that the five numbers have x20 as the middle number is 2Q. A bag contains 14 balls of two colours, the number of balls of each colour being the same.
The ball in hand is returned to the bag before each new drawn. All the spades are taken out from a pack of cards. From these cards; cards are drawn one by one without replacement till the ace of spades comes. If two subsets A and B of X are picked at random, the probability that A and B have the same number of elements is.
A natural number x is chosen at random from the first one hundred natural numbers. Three six faced fair dice are thrown together. Each part has one or more than one correct A four digit number numered from to is said to be lucky if sum of its first two digits is equal to sum of its last two digits. If a four digit number is picked up at random, the probability t h a t it is lucky number is a 1 67 b c 0 d 2.
A number is chosen at random from the numbers 10 to By seing the number a man will laugh if product of the digits is If he chosen three numbers with replacement then the probability t h a t he will laugh at least once is , t43 o s3 31 s3 a 1 b 1 45 45 41 42 c 1 d 1 - 7c 45 43 3. Two numbers b and c are chosen at random with replacement from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9. The probability t h a t 2.
Suppose n boys and m girls take their seats randomly round a circle. The probabilities of different faces of a biased dice to appear are as follows: A card is selected at random from cards numbered as 00, 01, 02, An event is said to have occured.
If product of digits of the card number is If card is selected 5 times with replacement each time, then the probability t h a t the event occurs exactly three times is 97 ,3 3 a 5 C 3 3 97 3 97 c 5 C 3 3 2 d 20 0 03 ' Let A, B, C be three mutually independent events.
Consider the two statements S j and S 2. A and B u C are independent S 2: They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair.
Assuming t h a t all the players are of equal strength, the probability that exactly one of. The probabilities t h a t a student in Mathematics, Physics and Chemistry are a, p and y respectively. Which of the following relations are true? Answers Multiple Choice l. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d which ever is appropriate. The domain of the function V log0 5 x is a 1, oo b 0, oo c 0, 1] d , 1 7.
The number log 2 7 is a an integer b a rational number c an irrational number d a prime number 8. The value of 1 1 log 2 n.
A There are 5 parts in this question. The interval of. If two matrices A and B are of the same order, then only their addition and subtraction is possible and these matrices are said to be conformable for addition or subtraction. On the other hand if the matrices A and B are of different orders then their addition and subtraction is not possible and these matrices are called non-conformable for addition and subtraction. Every square matrix can be uniquely expressed as the sum of symmetric and skew symmetric matrix.
A square matrix A is called an orthogonal matrix if the product of the matrix A and its transpose A' is an identity matrix. Deductions of a: Deducation 1. Deduction 2. Inverse of a non-singular diagonal matrix: Non-homogeneous linear equations also solved by Cramer's rule this method has been discussed in the chapter on determinants. The rank of A is denoted by p A. Note 1. The rank of a zero matrix is zero and the rank of an identity matrix of order n is n.
Note 2. The rank of a matrix in echelon form is equal to the number of non-zero rows of the matrix. Note 3. Rank of A Rank of C. In matrices: The characteristic of an orthogonal matrix A is a A '.
With 1 co, co2 as cube roots of unity, inverse of which of the following matrices exists? If I is the identity matrix of order n, then s -1 U" a does not exist b I c 0 d n I 2 -2 -4 If A is a skew-symmetric matrix, then trace of A is a -5 b 0 c 24 d 9 1 1 is unitary, then The rank of. Let a, b, c be positive real numbers. The following system of equations in x, y and z 2. For all values of X, the rank of the matrix. Formulas for the Domain of a Function 1. Domain f x.
Extension of a function If a function f x is defined on the interval [0,a], it can be extend on [- a, a] so that f x is either even or odd function on the interval [- a, a], i Even Extension: Let g b e the odd extension, then.
The smallest value of Tis called the period of the function. Methods to Find Period of a Periodic Function i sin" x, c o s " x, s e c " x, cosec" x are periodic functions with period 2n and k according as n be odd or even.
V x e A and ye B Then g is said to be inverse of f. If A and B have s a m e number of elements say n. Greatest Integer Function [x] denotes the greatest integer less than or equal to x. It is also known as ceiling of x. Which of the following functions is periodic with period n? The domain of definition of. Let f-. Which of the following functions is even function a. Let fix be a function defined on 0, 11 such that. The period of e is [.
Then the value of fix , a x. Let f: The period of. For each question, write the letters a, h, c, d corresponding to the correct answer s. Which of the following function is periodic. Of the following functions defined from [-1, 1] to [-1, 1] select those which are not bijective 2 a sin sin x b - sin sin x 7C c Sgnx ln ex d x Sgnx If the function f: The domain of. Right hand and left hand Limits In the definition of the limit we say that I is the limit of f x i.
Frequently used Limits i. Provided f ' a and cf a are not both zero. For other indeterminate terms we have to convert to or and then apply L' Hospital's Rule. Indicate your choice of correct answer for each question by writing one of letters a, b, c, d whichever is appropriate: The value of Lim. The value of the limit Jx lWx a -a ,. The integer n for which j.
Lim 7 is non zero finite then n 24 24 6 x sin x d None of these c must be equal to 24 b 2 a 1 " 7. Lim equals 0 sin x denotes the greater integer where [ function a 0 b 1 c tan d oo 2 9.
Continuity of a Function Continuity of a function f x can be discussed in two ways 1 at a point 2 in an interval. In other words, the function f x is continuous in interval [a, b].
Derivability or Differentiability of a Function The function f x is said to be differentiable at equal, and their common value is called the derivative The function f x is said to be non differentiable at i both Rf' a and Lf' a exist but are not equal iii either or both Rf' a and Lf' a do not exist.
Indicate your choice of correct answer for each question by writing one of the letter a, b, c, d whichever is appropriate: RtR f. If and 2 , otherwise 2. The jump of the function at the point of the discontinuity of the function 1 -k.
The following functions are continuous on 0,71 a tan x b f t sin. The function. The points of discontinuity of the function In. Which of the following functions are 1. If f x is a continuous function V x e R and the range of fix is 2. Alternative Method: This is called derivative of logarithmic function or.
Some Standard Substitutions Expression Va. Lagrange's mean value Theorem If a function f x is defined on [a, fa] satisfying i f is continuous on [a, b] ii f is differentiable on a, b then c e a, b such that f c.
If j x be a polynomial function of the second degree. The diff. The third derivative of a function f i x varishes for all A. The distance between the origin and the Ix. If the tangent at any point on the curve 4.
The subtangent, ordinate and subnormal to 2. If the parametric equation of a curve given x 2. The value of 3. The tangent and normal at the point. Monotonocity A function f defined on an interval [a, b] said to be i Monotonically increasing function: Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whicheven is appropriate.
For each question, write the letters a, b, c, d corresponding to the correct answerfs. Every invertible function is a monotonic function b constant function. Monotonocity 8.
The tangent to the curve at these points are parallel to x-axis, i. Maximum or minimum values are also called local extremum values.
For the points of local extremum either f ' x - o or f x does not exist.
Two important tips: The point. The optical value of. For each question, write the letters a, b, c, 1 corresponding to the correct answer s. The most economical speed if the fixed charges i.
The maximum area of the rectangle that can be inscribed in a circle of radius r is. The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs Rs. The critical points of t h e function ,.
I x-2 I. If the total distance to be travelled by all s t u d e n t s is to be as small as possible, t h e n the school should be built at a townfl b 45 km from town A c town A d 45 km from town B. Methods of Integration i Integration by Substitution or change of independent variable: If u and vare the differentiable functions of xthen J u. How to c h o o s e 1st and llnd functions: S u c c e s s i v e integration by parts: Integrals of the type. Type II: Integrals of the type i j.
Rule for iv: Rule for i and ii: Integrals of the form Type VIII: Integrals of the type: Type IX: Integral of the form J sinmxcos"xdx Case i: Case ii: Case iv: Case v: If a particle is moving with velocity v? If the derivative of f i x w.
Each part has one or more than one correct n -n 1. The value of the integral dx..
If l' means log log log repeated. In definite integrals constant of integration is never present. Properties of Definite Integrals: If m is the least value and M is the greatest value of the function f x on the interval [a, b]. If f x and g x are integrable on the interval a, b , the Schwarz-Bunyakovsky inequality takes place: Leibniz's Rule: Given an integral b jr f x dx a.
Definite Integral a s the Limit of a Sum Let f x be a continuous function defined on the closed interval [a, b]. Then eb. Here we proceed as follows: The lower and upper limits of integration will be the values of - for the first and last term or the limit of these values respectively. Let fix be an odd function in the interval T T with a period T. The value of the integral J [2 sin x] dx is [.
Suppose for every. Then J. Let f': If [x] stands for the greatest integer function,. For each quesion, write the letters a, b, c, d corresponding to the correct answer s. The value of I [tan x] dx is o when [. The values of a which satisfy.
Given that n is odd and m is even integer. The number of positive continuous functions f x defined in [0, 1] for which a one b infinite c two d zero. UU If some part of curves lies below the x-axis, then its area is negative but area cannot be negative.
Therefore we take its modulus. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate: The area of the region bounded by the curve 4. If its area is 2, then the value of 'b' is a - 3 c - 1.
Each part 1. Area of t h e region bounded by the curve x. The area bounded by t h e curves 2. Variable Separable: If the differential Equation of the form. Method of Substitution: If the differential equation is not in the form of variable separable but after proper substitution the equation reduces in variable separable form in the new variable. H o m o g e n e o u s Differential Equations: Working Rule: To get the solution of a homogeneous differential equation, we follow the following procedure: Quite handy!
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I felt like there are infinite type of questions on these topic and every question I pick up is a new one. I used to take a lot of time in solving few questions of algebra compared to other topics. I wanted a book which will categorize separately, all the type of questions that are mostly made on this topic. This book did it for me. Everything in this book is systematically categorized and explained. There are so many solved examples which built my confidence to tackle any type of question given.
What I didn't like about this book?