Section-A Short Questions Question1: Define Problem? : A Problem is defined as a cultural artifact, which is especially visible in a society s economic and industrial decision making process. Those managers that make effective decision concerning a known problem are good administrators. Question 2: What is set and what are various notations to define set? : A collection of well defined objects is called a set. The objects are called elements. The elements are definite and distinct. Notations: Sets are usually denoted by capital letters A, B, C, D.. And their elements are denoted by corresponding small letters a, b, c, d If a is an element of set A, then this fact is denoted by the symbol a A and read as a belongs to A. If a is not an element of A, then we write a A and read it as a does not belongs to A. Question 3: Differentiate between singleton Set and Empty Set. : Empty Set: A set which does not contain any element is called the empty set or the null set or the void set. Let A = {x: 1 < x < 2, x is a natural number}. Then A is the empty set, because there is no natural number between 1 and 2. Singleton Set: If a set A has only one element, we call it a singleton set. Thus, {a} is a singleton set. Question 4: Differentiate between Subset Set and Proper Subset. Subset: A set A is said to be a subset of a set B if every element of A is also an element of B. i.e. A B if whenever a A, then a B. we can write the definition of subset as follows: A B if a A for all a B Proper Subset: Let A and B be two sets. If A B and A B, then A is called a proper subset of B and B is called superset of A. For example, A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}. Question 5: Differentiate between Equal Set and Equivalent Set. Equal Set: Two sets A and B are said to be equal if they have exactly the same elements and write A = B. Otherwise, the sets are said to be unequal and we write A B. For Examples Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B. Equivalent Set: Two sets are called equivalent set, if and only if there is one to one correspondence between their elements. If A= {a, b, c} and B= {1, 2, 3}, then correspondence in the elements of A and B is one to one, A is equivalent to B, and we write it as A ~ B.
Question 6: What is Set of Sets? : If the elements of a set ate sets, then the set is called a set of sets. Example: {{a}, {a, b}, {a, b, c}} Question 7: Define Finite and Infinite Set. Finite Set: A set is said to be finite set if in counting its different elements, the counting process comes to an end. Thus a set with finite number of elements is a finite set. E.g.: The set of vowels = {a, e, i, o, u}. Infinite Set: A set, which is neither set nor a finite set, is called infinite set. The counting process can never come to an end in counting the elements of this set. E.g.: The set of natural number= {1, 2, 3, 4 } Question 8: Prove that Every set is subset of itself. Proof: Let A be any set and be the empty set. It is clear that has no element of A. Thus is a subset of A. Question 9: What is Venn diagram? Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of sets (groups of things). Venn diagrams were invented around 1880 by John Venn. They are used in many fields, including set theory, probability, logic, statistics, and computer science. Question 10: Define Log. : The logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number. For example, the logarithm of 1000 to the base 10 is 3, because 3 is how many 10s you must multiply to get 1000: thus 10 10 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 2 2 2 2 = 32. Question 11: What is use of log in real life? Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely related functions. In the equation b n = x, b can be determined with radicals, n with logarithms, and x with exponentials. Logarithms can be used to replace difficult operations on numbers by easier operations on their logs
Question 12: Define Arithmetic Progression. : A series of quantities from an arithmetic progression if each subsequent term is obtained by adding t previous term a constant amount, which is called the common difference. A arithmetic progression is always has the form: a, a+d, a+2d, a+3d,. a+ (n-1)d Here a is the first term, d is the common difference and n is the number of terms. Question 13: What is Geometric Progression? : A series quantities from a geometric progression if each term is obtained by multiplying the previous term a constant, which is called the common ratio. A G. P. always has the form: a, ar 2, ar 3 ar n Where a is the first term, r is the common ratio and n is the number of terms. Question 14: What will be the sum of a G.P. series, whose first term is a and common ration is r i. Containing n terms in series ii. Containing infinite terms in series (I) (II) Series containing n terms: S = a rn 1 r 1 Series containing infinite terms: S = a/(1 r) Where S is the summation of series, which has a as first term and r as common ratio Question 15: Define Linear Equation. : Linear Equation may be defined as an equation where the power of the variable(s) is one, and no cross or product terms are present. The general expressions of these linear equations look like the following: Ax + B = 0 Question 16: What is a Quadratic Equation? : A quadratic equation is a polynomial equation of the second degree. The general form is Ax 2 + bx + c = 0 Where a 0 (If a = 0, the equation becomes a linear equation.) The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x 2, the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term or constant term.
Question 17: Write Shri Dharacharya Formula to find the roots of quadratic equation. x = b± b2 4ac 2a Where quadratic equation is Ax 2 + bx + c = 0 Question 18: What are the roots of any Quadratic Equation? : The solutions to the equation are called the roots of the equation. Question 19: What is Matrix Algebra? : A matrix is a collection of numbers ordered by rows and columns. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces. Or A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. For example, X = Question 20: What is a Vector? A vector is a special type of matrix that has only one row (called a row vector) or one column (called a column vector). Below, a is a column vector while b is a row vector. For Example: [1 2 3] Question 21: What are determinants? In algebra, a determinant is a function depending on n that associates a scalar, det(a), to an n n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multi linear algebra. For a fixed nonnegative integer n, there is a unique determinant function for the n n matrices over any commutative ring R. In particular, this function exists when R is the field of real or complex numbers. Question 22: What is Order of a Matrix?
: A matrix having m rows and n columns is called a matrix of order m n or simply m n. For Example: Order of a Matrix containing 3 rows and 2 columns, will be 3 2. Question 23: State the Principle of Mathematical Induction. : Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one. According to Mathematical Induction, corresponding to each natural number n, there exists a natural number n+1 Mathematical Induction attempts to prove that a given statement is true for all the values of an infinite sequence through proving two sub-statements. A. Prove that the given statement is true for the first element of the infinite series, and then B. Prove that if the statement true for any other element then it is also true for the next element of the sequence. Question 24: Find the 10th term of the AP: 2, 7, 12,... Here, a = 2, d = 7 2 = 5 and n = 10. We have an = a + (n 1) d So, a10 = 2 + (10 1) 5 = 2 + 45 = 47 Question 25: Which term of the AP: 21, 18, 15... is 81? Also, is any term 0? Give reason for your answer. Here, a = 21, d = 18 21 = 3 and an = 81, and we have to find n. As an = a + ( n 1) d, we have 81 = 21 + (n 1)( 3) 81 = 24 3n 105 = 3n So, n = 35 Therefore, the 35th term of the given AP is 81. Next, we want to know if there is any n for which an = 0. If such an n is there, then 21 + (n 1) ( 3) = 0, i.e., 3(n 1) = 21 i.e., n = 8 So, the eighth term is 0. Question 26: Determine the AP whose 3rd term is 5 and the 7th term is 9. We have a3 = a + (3 1) d = a + 2d = 5 EQ (1) so a7 = a + (7 1) d = a + 6d = 9 EQ (2) Solving the pair of linear equations (1) and (2), we get a = 3, d = 1 Hence, the required AP is 3, 4, 5, 6, 7,... Question 27: Check whether 301 is a term of the list of numbers 5, 11, 17, 23,...
We have: a2 a1 = 11 5 = 6, a3 a2 = 17 11 = 6, a4 a3 = 23 17 = 6 As ak + 1 ak is the same for k = 1, 2, 3, etc., the given list of numbers is an AP. Now, a = 5 and d = 6. Let 301 be a term, say, the nth term of this AP. We know that an = a + (n 1) d So, 301 = 5 + (n 1) 6 i.e., 301 = 6n 1 So, n =302/6 = 151/3 But n should be a positive integer (Why?). So, 301 is not a term of the given list of numbers. Question 28: How many two-digit numbers are divisible by 3? The list of two-digit numbers divisible by 3 is : 12, 15, 18... 99 Is this an AP? Yes it is. Here, a = 12, d = 3, an = 99. As an = a + (n 1) d, We have 99 = 12 + (n 1) 3 i.e., 87 = (n 1) 3 i.e., n 1 = 87/3 = 29 i.e., n = 29 + 1 = 30 So, there are 30 two-digit numbers divisible by 3. Question 29: A sum of Rs 1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these interests form an AP? If so, find the interest at the end of 30 years making use of this fact. We know that the formula to calculate simple interest is given by Simple Interest =P R T/100 So, the interest at the end of the 1st year =Rs (1000 8 1)/100 = Rs 80 The interest at the end of the 2nd year = Rs (1000 8 2)/100 = Rs 160 The interest at the end of the 3rd year = Rs (1000 8 3)/100 = Rs 240 Similarly, we can obtain the interest at the end of the 4th year, 5th year, and so on. So, the interest (in Rs) at the end of the 1st, 2nd, 3rd,... years, respectively are 80, 160, 240.. It is an AP as the difference between the consecutive terms in the list is 80, i.e. d = 80. Also, a = 80. So, to find the interest at the end of 30 years, we shall find a30. Now, a30 = a + (30 1) d = 80 + 29 80 = 2400 So, the interest at the end of 30 years will be Rs 2400. Question 30: How many terms of the AP : 24, 21, 18,... must be taken so that their sum is 78? Here, a = 24, d = 21 24 = 3, Sn = 78. We need to find n.
We know that Sn = ( 1 )n 2a + n 1 d 2 So, 78 = 1 n 2 24 + n 1 3 2 So, 78 = 1 n 48 3n + 3 2 So, 156 = 51n 3n 2 So, n 2 17n + 52 = 0 So n 4 n 13 = 0 Both values of n are admissible. So, the number of terms is either 4 or 13. Questions 31: What will be the sum of series of A.P. containing n terms and whose first term is a and common difference is d? : S may represent the sum of n terms of an A. P. Where a is the first term of the series and d is common difference and n is total number of terms. Sn = ( 1 )n 2a + n 1 d 2 Question 32: Sets A and B has 3 and 6 elements respectively. What is the least number of elements in A B? : The number of elements in the set A B is least when A B. Then all the 3 elements of A are in B. Since B has 6 elements. Therefore, the least number of elements is 6.It is clear from the fact that A B So A B n (B) = 6 Question 33: If A B and B A, then prove that A = B. Proof: Given A B. If x A so x B EQ (1) Also B A. If x B so x A EQ (2) From EQ (1) and EQ (2),: x A and x B Therefore A = B (Hence Proved) Question 34: If A B and B C, Then A C. Proof: A B So x A and x B EQ (1)
Again B C EQ (2) So, x B and x C From (1) and (2), it is clear So, x A and x C Therefore A C (Hence Proved) Question 35: Given a set A {2, {}, a, I}. Find n (P (A)). Solution As the number of elements in given set A are 4, so total n (P (A)) = 16. P (A) ={2,, a, I, 2,, 2, a, 2, I,, a,, I, a, I, 2,, a, 2, a, I, (2,, I,), ({}, a, I),φ} Question 36: Find the value of X and Y for the following linear equations: 3X + 4Y = 8 5X + 7Y =15 3X + 4Y = 8 EQ (1) 5X + 7Y =15 EQ (2) By multiplying EQ (1) by 5 and EQ (2) by 3: 15X + 20Y = 40 15X + 21Y =45 Now by subtracting EQ (1) by EQ (2): 15X + 20Y 15X 21Y = 40 45 Y = 5 Or Y = 5 By putting the value of Y in EQ (1): 3X + 20 =8 Or X = 12 3 Or X = 4 So X = 4 and Y = 5
() Question 37: Which term of the A. P. 3, 8, 13, is 248. For the given A.P.:- a= 3, d= 5 and l = 248 We know that l = a + n 1 d So 248 = 3 + n 1 5 n 1 = 245 5 n 1 = 49 So n = 50 () Question 38: Find the value of log 10 1000 10 1 = X log 10 1000 10 1 = X Or log 10 1000 1 10 = X Or log 10 100 = X Or log 10 10 2 = X Or 2 log 10 10 = X X=2 () Question 39: Find the value of log 125 X = 1 6 log 125 X = 1 6 Or 125 1 6 = X Or X = 5 3 1 6 Or X = 5 1 2 Or X = 5
() Question 40: Solve the following matrices operations: a. A + B +C b. 2A 3B C A = 3 6 11 17 B = 7 8 8 7 C = 4 5 7 6 a. A + B + C = 3 + 7 + 4 11 8 + 7 6 + 8 + 5 17 7 + 6 A + B + C = 14 10 19 16 b. 2A 3B C = 2 3 3 7 4 2 11 + 3 8 7 2 6 3 8 5 2 17 + 3 7 6 () 2A 3B C = 2A 3B C = 19 39 17 49 6 21 4 22 + 24 7 12 24 5 34 + 21 6 Question 41: How many three digit numbers exist, which are divisible by 7? Three digit number starts from 100 and ends at 999. But first three digits number which is divisible by 7 is 105 and the last three digit number is 994. So required series is 105, 112 994. And this series is an A. P. So a = 105, d = 7 and l = 994 As l = a + (n 1) d 994 = 105 + n 1 7 n 1 7 = 889 n 1 = 127 n = 128 So there are 128 terms in this series, hence we can that there are 128 three digit numbers exists, which are divisible by 7.
Question 42: Question: Find the compound interest on Rs. 10,000 for 3 years at the rate of 10% per annum? P = 10,000, n = 3 years, r = 10% per annum I = P 1 + r 100 n I = 10,000 1 + 10 100 I = 10000 110 100 I = 13310 () 3 3 Question 43: Find the sum of series 2 + 4 + 6 + 8 +.. up to 25 terms? For given A. P.: a = 2, d = 2 and n = 25 S = n 2 [2a + n 1 d] S = 25 2 S = 25 2 2 2 + 25 1 2 4 + 48 S = 25 26 S = 650 () Question 44: What is depreciation? Depreciation is a specialized subset of amortization. Amortization simply means the spreading out of a cost over a period of time. It is a generic term used for any type of item that is being prorated over a time. Depreciation comes from the word deprecate indicating a breaking down or physical depreciation and wearing out. Question 45: Solve the following equations by Cramer s Rule method. 2X + 3Y = 13 5X 2Y = 4
2X + 3Y = 13 5X 2Y = 4 D = 2 3 5 2 A = 13 3 4 2 B = 2 13 5 4 = 2 2 5 3 = 4 15 = 19 = 13 2 4 3 = 26 12 = 38 = 4 2 5 13 = 8 65 = 57 X = A D So X = 38 19 X = 2 And Y = B D So Y = 57 19 Y = 3 So finally X = 2 and Y = 3 ()