Indian Statistical Institute : Solved Problems

TOMATO (TEST OF MATHEMATICS AT THE 10+2 LEVELS) 

OBJECTIVE PROBLEM SOLVING

 

ISI's Test of Mathematics at 10 + 2 Level is a rigorous book for mathematics enthusiasts and for Class-12 students applying for an admission into the Bachelor of Mathematics at one of the Indian Statistical Institutes across the nation. The book provides advanced problems that are commonly found in the ISI entrance examinations and also prepares students for a study involving higher mathematics.

About the Indian Statistical Institute

ISI is an academic institute in India catering to studies in statistics. It was established in 1931 to improve the application of statistical methods across the nation. The institute provides academic programs at the undergraduate and postgraduate level in mathematics, statistics and numerical methods.

PART -1 QUESTION 1 TO QUESTION 28

Permutations - RD Sharma Solved Problems.

 A permutation is an arrangement of objects in a definite order. The members or elements of sets are arranged here in a sequence or linear order. For example, the permutation of set A={1,6} is 2, such as {1,6}, {6,1}. As you can see, there are no other ways to arrange the elements of set A.

In permutation, the elements should be arranged in a particular order whereas in combination the order of elements does not matter. Also, read: Permutation And Combination

When we look at the schedules of trains, buses and the flights we really wonder how they are scheduled according to the public’s convenience. Of course, the permutation is very much helpful to prepare the schedules on departure and arrival of these. Also, when we come across licence plates of vehicles which consists of few alphabets and digits. We can easily prepare these codes using permutations.

A permutation is defined as an arrangement in a definite order of a number of objects taken, some or all at a time. Counting permutations are merely counting the number of ways in which some or all objects at a time are rearranged. The convenient expression to denote permutation is defined as “ nPr ”.

The permutation formula is given by,

Pr = n!/(n-r)! ; 0 ≤ r ≤ n

Where the symbol “!” denotes the factorial which means that the product of all the integers is less than or equal to n but it should be greater than or equal to 1.

Permutation When all the Objects are Distinct

There are some theorems involved in finding the permutations when all the objects are distinct. They are :

Theorem 1: If the number of permutations of n different objects taken r at a time, it will satisfy the condition 0 < r ≤ n and the objects which do not repeat is n ( n – 1) ( n – 2)……( n – r + 1), then the notation to denote the permutation is given by “ n Pr”

Theorem 2: The number of permutations of different objects “n” taken r at a time, where repetition is allowed and is given by nr .

Permutation When all the Objects are not Distinct Objects

Theorem 3: To find the number of permutations of the objects ‘n’, and ‘p’s are of the objects of the same kind and rest is all different is given as n! / p!

Theorem 4: The number of permutations of n objects, where p1 are the objects of one kind, p2 are of the second kind, …, pk is of the kth kind and the rest, if any, are of a different kind, then the permutation is given by n! / ( p1!p2!…Pk!)

Functions Class XI - RD Sharma Solved Problems

 Functions

A relation ‘f’ is said to be a function, if every element of a non-empty set X, has only one image or range to a non-empty set Y.

Or

If ‘f’ is the function from X to Y and (x,y) ∊ f, then f(x) = y, where y is the image of x, under function f and x is the preimage of y, under ‘f’. It is denoted as;

f: X → Y.

Example: N be the set of Natural numbers and the relation R be defined as;

R = {(a,b) : b=a2, a,b ∈ N}. State whether R is a relation function or not.

Solution: From the relation R = {(a,b) : b=a2, a,b ∈ N}, we can see for every value of natural number, their is only one image. For example, if a=1 then b =1, if a=2 then b=4 and so on.

Therefore, R is a relation function here.

Real-Valued Function
A function f : A → B is called a real-valued function if B is a subset of R (set of all real numbers). If A and B both are subsets of R, then f is called a real function.

Some Specific Types of Functions
Identity function: The function f : R → R defined by f(x) = x for each x ∈ R is called identity function.
Domain of f = R; Range of f = R

Constant function: The function f : R → R defined by f(x) = C, x ∈ R, where C is a constant ∈ R, is called a constant function.
Domain of f = R; Range of f = C

Polynomial function: A real valued function f : R → R defined by f(x) = a0 + a1x + a2x2+…+ anxn, where n ∈ N and a0, a1, a2,…….. an ∈ R for each x ∈ R, is called polynomial function.

Signum function: The real function f : R → R defined
by f(x) = |x|x, x ≠ 0 and 0, if x = 0
or

is called the signum function.
Domain of f = R; Range of f = {-1, 0, 1}

Greatest integer function: The real function f : R → R defined by f (x) = {x}, x ∈ R assumes that the values of the greatest integer less than or equal to x, is called the greatest integer function.
Domain of f = R; Range of f = Integer

Fractional part function: The real function f : R → R defined by f(x) = {x}, x ∈ R is called the fractional part function.
f(x) = {x} = x – [x] for all x ∈R
Domain of f = R; Range of f = [0, 1)

Algebra of Real Functions
Addition of two real functions: Let f : X → R and g : X → R be any two real functions, where X ∈ R. Then, we define (f + g) : X → R by
{f + g) (x) = f(x) + g(x), for all x ∈ X.

Subtraction of a real function from another: Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, we define (f – g) : X → R by (f – g) (x) = f (x) – g(x), for all x ∈ X.

Multiplication by a scalar: Let f : X → R be a real function and K be any scalar belonging to R. Then, the product of Kf is function from X to R defined by (Kf)(x) = Kf(x) for all x ∈ X.

Multiplication of two real functions: Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, product of these two functions i.e. f.g : X → R is defined by (fg) x = f(x) . g(x) ∀ x ∈ X.

Quotient of two real functions: Let f and g be two real functions defined from X → R. The quotient of f by g denoted by fg is a function defined from X → R as

Relations Class XI - RD Sharma Solved Problems

Cartesian Product of Sets

Suppose there are two non-empty sets A and B. So, the cartesian product of A and B is the set of all ordered pairs of elements from A and B.

A × B = {(a,b) : a ∊ A, b ∊ B}

Let A = {a1, a2, a3, a4} and B = {b1, b2}

Then, The cartesian product of A and B will be;

A × B = {(a1, b1), (a2, b1), (a3, b1), (a4, b1), (a1, b2), (a2, b2), (a3, b2), (a4, b2)}

Example: Let us say, X = {a,b,c} and Y = { 1,2,3}

Therefore, X × Y = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}.

This set has 9 ordered pairs. We can also represent it as in a tabular form.

Note: Two ordered pair X and Y are equal, if and only if the corresponding first elements and second elements are equal.

Relations

Definition: A relation R is the subset of the cartesian product of X x Y, where X and Y are two non-empty elements. It is derived by stating the relationship between the first element and second element of the ordered pair of X × Y. The set of all primary elements of the ordered pairs is called a domain of R and the set of all second elements of the ordered pairs is called a range of R.

For two sets X = {a, b, c} and Y = {apple, ball, cat}, the cartesian product have 9 ordered pairs, which can be written as;

X × Y = {(a, apple), (a, ball), (a, cat), (b, apple), (b, ball), (b, cat), (c, apple), (c, ball), (c, cat)}

With this we can obtain a subset of X x Y by introducing a relation R, between the elements of X and Y as;

R = {(a,b) : a is the first letter of word b, a ∊ X, b ∊ Y}

Therefore, the relation between X and Y can be represented as;

R = {(a,apple),(b,ball),(c,cat)}

Example: Let X={a,b} and Y = {c,d}. Find the number of relations from X to Y.

Solution: X × Y = {(a,c),(a,d),(b,c),(b,d)}

Number of subsets, n (X × Y) = 24 . Therefore, the number of relations from X to Y is 24.

Solved Problems from the book of RD Sharma - Class XI CBSE

Surds Practice Problem Set

In Mathematics, surds are the values in square root that cannot be further simplified into whole numbers or integers. Surds are irrational numbers. The examples of surds are √2, √3, √5, etc., as these values cannot be further simplified. If we further simply them, we get decimal values, such as:

√2  = 1.4142135…

√3 = 1.7320508…

√5 = 2.2360679…

Surds Definition

Surds are the square roots  (√) of numbers that cannot be simplified into a whole or rational number. It cannot be accurately represented in a fraction. In other words, a surd is a root of the whole number that has an irrational value. Consider an example, √2 ≈ 1.414213. It is more accurate if we leave it as a surd √2.

Surds Worksheet

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Surd-Practice-I

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Surd-Practice-II

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Surd-Practice-III

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Trigonometry: Solved Problems

Solutions of Triangle is an important topic in the JEE Main and JEE Advanced and in class XI for CBSE and other state boards in India. This topic comprises various formulae and rules like the sine rule, cosine rule, tangent rule etc. Questions based on the application of these formulas are often asked in exams. Revising these problems will help students to remember them and easily solve other questions of similar type and also apply their learning to new problems:

Solved Problems from the Book of S.L Loney Plane Trigonometry: CLICK HERE

More Problems to follow soon :) 

WORKSHEET RELATION AND FUNCTIONS: CLASS XI

 In relations and functions Class 11, you will learn the cartesian product of sets along with relations and functions. In our day-to-day life, we have known the term relations in a pattern such as the relation between brothers and sisters, husband and wife or teacher and student. In Maths, the term relation is used to relate the numbers, symbols, variables, sets, group of sets, etc. For example, A is a subset of B denotes the relation of A and B. A function is a kind of relation which is operated between two quantities to yield output.

In this WORKSHEET, we will provide you with the relations and functions class 11 PROBLEMS, so that it would be easier for you to learn and understand the concepts.

If you have any query please comment in the comment box.

Assessment and Solved Problems: SET THEORY -XI Science

Please go through the slides one by one and solve the questions that would appear on the slides.

Solved Problems: Set Theory - 11 Science

Assessment on set theory. Answer all questions. Each question carries 1 mark and has only one correct option. Please enter a valid e-mail address. If you have any query, leave your comment in the comment box. 

Class Test: 

Total time: 40 mins  ( 6 pm - 6:40 pm )                                            Full Marks: 15

Thanks for participating, those who have not given the test should clarify for the same:

Scores:

11/18/2020 18:13:53	saya****@gmail.com	7 / 15	Sayan Saha 04
11/18/2020 18:20:28	deb***ari23@gmail.com	6 / 15	Debjit Kushari roll:- 07
11/18/2020 18:20:47	sa**as6767@gmail.com	9 / 15	Saima das 14 A
11/18/2020 18:21:09	di***g147@gmail.com	5 / 15	Disha Ghosh 16
11/18/2020 18:25:25	dha***nk@gmail.com	8 / 15	Subhadeep Dhar - 05
11/18/2020 18:26:54	ta****nog@gmail.com	5 / 15	Tamaghno Ghosh roll–3
11/18/2020 18:35:01	ag****dhar@gmail.com	10 / 15	Ritam Dhar.ROLL NO A-2.
11/18/2020 18:36:07	mistr***885@gmail.com	5 / 15	Jayanti Mistri 15A
11/18/2020 18:39:12	shre*****47@gmail.com	5 / 15	Shreya majumdar.18
11/18/2020 18:39:39	kau****74@gmail.com	10 / 15	Kaushik kundu...Roll no.-6A
11/18/2020 18:40:22	tap***572@gmail.com	9 / 15	Meghasree Mandal -11
11/18/2020 18:41:43	tap***etro@gmail.com	4 / 15	Tanmesh Mondal/A-17

Delayed Submission due to some technical issue:

11/18/2020 19:04:33

jai***s106@gmail.com

11 / 15

Jaita Das

Test conducted on the request of the student:

11/19/2020 18:36:35

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13 / 15

Mourya Vedajna, 20

You can't lie to your soul.