Chapter I
Matrices and Determinants
Summary
• A matrix is a rectangular array of numbers arranged in rows and columns. A set of real or complex
numbers arranged in a rectangular array of ‘m’ rows and ‘n’ columns, of an order m x
•n (read as m by n) is called a matrix. Two matrices can be added only if they have same dimensions.
•Commutative property, Associative property, Distributive property is true for matrix addition.
• The matrix product AB is defined only when the number of columns in A is equal to the number of
rows in B.
•Commutative property is not true for matrix multiplication.
•Associative property ,Distributive property are true for matrix multiplication
•There exists additive identity, multiplicative identity, additive inverse and multiplicative inverse for a
matrix.
•A determinant is a square array of numbers.
•The 3 x 3 determinant values are evaluated by expansion by minors.
•System of equation can be solved using determinants with Cramer’s rule.
Self Assessment
1. If matrix A=
value of a21?
a. 3
b. 4
c. 2
d. 5
2. It is given that P=
and aij is the element of matrix A in ith row and jth column, then what is the
and Q =
.What is the value of x+y if P=Q?
a. 3
b. 5
c. 6
d. 8
3. A _______ is a rectangular array of numbers arranged in rows and columns.
a. Determinant
b. Matrix
c. Array
d. Transpose
4. Two matrices can be added only if they have __________.
a. same dimensions
b. different dimensions
c. plus sign
d. minus sign
5. When a matrix is multiplied by a number, then the process is called as _________.
a. matrix multiplication
b. scalar multiplication
c. square multiplication
d. rectangular multiplication
6. What type of matrix is A=
?
a. square
b. diagonal
c. null
d. identity
Chapter II
Mathematical Logic
Summary
•Mathematical Logic is a tool for providing a precise meaning to mathematical statements.
• A statement is a declarative sentence which is either true or false but not both.
• The truth value of a proposition is true (T) if it is of true proposition and false (F) if it is false
proposition.
• A table that gives the truth value of the compound statement in terms of its component part is called a
truth table. A compound statement is a combination of two or more statements.
• All declarative sentences to which it is possible to assign one and only of the two possible truth values
are called statements. It is possible to construct rather complicated statements from simpler statements
by using certain connecting words or expressions known as “sentential connectives”.
•The negation statement is generally formed by introducing the word “not” at a proper place in
statement with the phrase “It is not the case that” and read as “not P”. Let P and Q be two statements.
Then “If P, then Q” is the statement called an Implication or conditional statement, written P → Q.
Self Assessment:
1. “Canada is a country” .what kind of statement is it?
a. Primitive statement
b. Compound statement
c. Elementary statement
d. Primary statement
2. “A dog or a car is an animal”. What kind of statement it is?
a. Primitive statement
b. Compound statement
c. Elementary statement
d. Primary statement
3. What is the symbol for negation?
a. ∧
b. ∨
c. ∼
d. →
4. What is the symbol for conjunction?
a. ∧
b. ∨
c. ∼
d. →
5. What is the symbol for disjunction?
a. ∧
b. ∨
c. ∼
d. →
6. What is the symbol for implication?
a. ∧
b. ∨
c. ∼
d. →
7. If P is true and Q is False what is the value of P∨ Q?
a. T
b. F
c. Invalid
d. no value
8. If P is true and Q is False what is the value of P∧ Q?
a. T
b. F
c. Invalid
d. no value
9. If P is true and Q is False what is the value of P→ Q?
a. T
b. F
c. Invalid
d. no value
10. If P is true and Q is False what is the value of P↔ Q?
a. T
b. F
c. Invalid
d. no value
Chapter III
Set Theory
Summary
• A set is a collection of well defined object enclosed within curly brackets.
• The objects which form the set are called the elements or members of the set.
• In the tabular form, all elements of the set are enumerated or listed.
• Set P is a subset of a set Q, symbolised by P⊆ Q, if and only if all the elements of set P are also the
element of set Q.
• A set which contains no elements is called an empty set or a null set.
• Two sets are said to be disjoint sets if they do not have any common elements.
• Two sets are said to be overlapping sets, if they have some common elements.
• A universal set is the set that contains the element of all the sets under consideration.
• Two sets are said to be equal if they contain the same elements.
• The intersection of two sets is the set of elements common to both the given sets.
• The union of the two sets is the set containing the elements belonging to A and also the elements
belonging to B. A
• ∆ B = (A – B) ∪ (B – A) Demorgan’s Law (A∪ B) ’ = A’ ∩ B’
Self Assessment
1. A ____ is a collection of well defined object enclosed within curly brackets.
a. Set
b. Subset
c. Object
d. Element
2. A set which contains no elements is called _____ set.
a. equal
b. equivalent
c. empty
d. finite
3. Two sets are said to be __________ if they do not have any common elements.
a. disjoint sets
b. overlapping sets
c. intersection sets
d. equal sets
4. Two sets are said to be ______ if number of elements in one set is equal to the number of elements in
the other set.
a. disjoint set
b. equivalent set
c. overlapping set
d. intersection set
5. If A = {1,2,3,4,5} and B={4,5,6,7,8} ,what is A – B ?
a. {4,5}
b. {6,7,8}
c. {1,2,3}
d. {1,2,3,4,5}
6. Is P= {1, 3, 6} Q= {6, 1, 3} an equal set?
a. True
b. False
7. Is A= {1, 2, 3, 4} and B= {a, b, c, d} an equivalent set?
a. True
b. False
8. If A= {1,2,3,4} and U ={1,2,3,4,5,...,10},What is the value of A’?
a. {1,2,3,4,5}
b. {4,5,6,7}
c. {7,8,9,10}
d. {5,6,7,8,9,10}
9. If B= {2,4,6,8} and C= {3,4,5,6,8} what is the value of B∪ C?
a. {2,3,4,5,6,8}
b. {2,4,6,8}
c. {3,4,5,6,8}
d. {4,6,8}
10. If A= {1,2,3,4} and C= {3,4,5,6,8} what is the value of A∩ C?
a. {1,2,3,4}
b. {3,4}
c. {3,4,5,6,8}
d. {1,2,3,4}
Chapter IV
Progression
Self Assessment
1. ________ is a pattern of numbers wherein a quantity increases or decreases progressively by fixed
amount.
a. Progression
b. Mean
c. Geometric
d. Arithmetic
2. _______ . A succession of numbers formed and arranged in a definite order according to certain
definite rule, is called a
a. mean
b. arithmetic
c. geometric
d. progression
3. If quantities increase or decrease by common difference, then they are said to be in ________
progression.
a. geometric
b. arithmetic
c. harmonic
d. mean
4. The following series is in which progression? Series: 4,8,12,16,20,24…
a. geometric
b. arithmetic
c. harmonic
d. mean
5. The following series is in which progression? Series: 3,-6, 12,-24, 48,-96, 192…
a. geometric
b. arithmetic
c. harmonic
d. mean
6. The following series is in which progression? Series: 1/10, 3/20, 1/5, 1/4
if AM is 10, 20/3, 5, 4...
a. geometric
b. arithmetic
c. harmonic
d. mean
7. What is the sum of the series 5/2,4,11/2,7,….upto 21 terms?
a. 350
b. 360
c. 357.60
d. 367.60
8. What is the sum of the infinite series 1, 0.5, 0.25…?
a. 1
b. 2
c. 3
d. 4
9. What is the 74th term of the series 3, 7, 11, 15…?
a. 265
b. 285
c. 295
d. 300
10. What is the sum of first 6 terms of the series 3, 9, 27, 81 ….?
a. 1092
b. 1000
c. 1900
d. 1192
Chapter V
Probability
Self Assessment
1. __________ is the method to determine the chance of happening of a particular event in a random
experiment.
a. Probability
b. Experiment
c. Event
d. Deterministic
2. An operation which can produce some well defined outcomes is called an _________.
a. probability
b. experiment
c. event
d. deterministic
3. _________ experiment is the one which gives one or more results under identical conditions.
a. Deterministic
b. Random
c. Independent
d. Dependent
4. The set of all possible outcomes of a random experiment is known as the _________.
a. event
b. sample space
c. sample point
d. experiment
5. An event is a __________ of the sample space.
a. subset
b. set
c. proper subset
d. probability
6. Two or more events are said to be ___________ if the happening or non-happening of any event does
affect the happening of others.
a. independent
b. mutually exclusive
c. elementary
d. dependent
7. Two or more event are said to be _________ if these events cannot occur simultaneously.
a. independent
b. dependent
c. mutually exclusive
d. elementary
8. Two unbiased coins are tossed .What is the probability of getting at most one head?
a. 1/2
b. 1/4
c. 3/4
d. 5/2
9. An unbiased die is tossed .What is the probability of getting a multiple of 3?
a. 2/3
b. 5/6
c. 1/3
d. 1/6
10. A bag contains 6 white and 4 black balls. Two balls are drawn at random. What is the probability that
they are of the same colour?
a. 1/15
b. 4/15
c. 2/15
d. 7/15
Chapter VI
Permutations and Combinations
Self Assessment
1. ________ is the number of arrangements of given objects.
a. Permutation
b. Combination
c. Factorial
d. Selection
2. __________ is the number of selections out of given objects.
a. Permutation
b. Combination
c. Factorial
d. Selection
3. __________ is the method that helps us to determine the number of ways a group of objects can be
selected and then arranged.
a. Permutation
b. Combination
c. Factorial
d. Selection
4. __________ only tells us the number of ways the objects can be selected.
a. Permutation
b. Combination
c. Factorial
d. Selection
5. There are 3 varieties of mangoes(A,B,C),4 different varieties of Apples(D,E,F,G) and 7 different
varieties of grapes(H,I,J,L,M,N).In how many ways we can arrange 5 fruits on a table when G,M,N should
not be taken?
a. 22330
b. 33440
c. 44550
d. 55440
6. In how many ways they can be a part of a meeting of 7 People?
a. 5040
b. 3040
c. 2040
d. 1040
7. There are 11 players in a hockey team. In how many ways we can select 5 players for the board
meeting if captain and vice captain have to be there.
a. 81
b. 80
c. 83
d. 84
8. 7 people are there in a company. In how many ways they can be a part of a meeting of 7 People?
a. 7!
b. 7
c. 5040
d. 1
9. From 5 different green balls, four different blue balls and three different red balls, how many
combinations of balls can be chosen taking atleast one green and one blue ball?
a. 3120
b. 3720
c. 3520
d. 3420
10. From a pack of 52 cards in how many number of ways in which a king or a queen can be drawn?
a. 2
b. 1
c. 4
d. 3
Chapter VII
Interpolation
Summary
Algebraic expressions in which the variables concerned have only non-negative integral exponents are
called
•
polynomials The standard form of a polynomial in one variable is that in which the terms of the
polynomial are written in
•
the decreasing order of the exponents of the variable Interpolation provides us the missing quantity of a
series so that we can establish the while extrapolation are
•
the techniques of obtaining the most likely estimates of certain quantity under certain assumptions
Methods of Interpolation are Graphical method, Lagrange’s Method, Newton-Gauss Forward Method,
Newton’s•
Gauss Backward method. etc. Lagrange’s interpolating polynomial is another very good formula for
interpolation
•
This method has no restriction on the x-variable whether it should be equally spaced or not
•
Gauss Forward Formula is
•
fp=f0+pδ1/2 +G2 δ0 2+G3δ1/2 3 +....
Self Assessment
Graphical Interpolation method is_____.
1.
Simple
a. Algebrical
b. Fully reliable
c. Reliable
d. Interpolation is helpful in estimating:
2.
A series
a. An intermediary value of given argument
b. Entry of alternative values
c. A series of value
d. Lagrange Formula is useful for _____
3.
Interpolation
a. Arithmetic functions
b. Inverse range
c. Inverse extrapolation
d. Lagrange’s polynomials interpolation can be used even if;
4.
The given arguments are not equally spaced
a. Extrapolation is to be done
b. Inverse interpolation is to be done
c. Relation to be mapped
d. Interpolation formulae are based on the fundamental assumptions that the data can be expressed as;
5.
A linear function
a. A quadratic function
b. A polynomial function
c. A binomial function
d. The problems of interpolation are simpler than prediction because;
6.
Interpolation has fewer restrictions than prediction
a. Interpolation is based on more stringent restriction than prediction
b. There are no restriction than interpolation
c.
It is easier to find out
d. ________ is simplest method of Interpolation.
7.
Graphical method
a. Langrange’s method
b. Interpolation method
c. Inverse method
d.
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In which method the data is represented in graph?
8.
Langrange’s method
a. Interpolation method
b. Inverse method
c. Graphical method
d. The value to be interpolated is different from the ______ value.
9.
equivalent
a. equal
b. equidistant
c. equillibrium
d. _________ provides us the missing quantity of a series.
10.
Interpolation
a. Graphical method
b. Lagrange’s method
c. Newton’s method
d.
Chapter VIII
Consumer Arithmetic
Summary
The price at which the goods are sold is known as selling price (SP).
•
The price at which the goods are bought is known as cost price(CP).
• If selling price is more than the cost price, then it is said the seller has made a profit/gain.
•
If the selling price is less than the cost price ,then it is said the seller has made a loss.
•
The price at which the goods are intended to be sold is known as marked price.
•
When the Selling price is lesser than the marked price ,it is said that the seller has given a discount.
• Profit = selling price – cost price
•
Loss = cost price –selling price
•
Gain%=(Gain/cost price)* 100
•
Loss%=(Loss/cost price)*100
•
Interest is a fees paid by a borrower of assets to the owner of the assets as form of compensation for
the use of
•
assets. Simple interest is where the interest is paid to the investor/lender as and when it is due. Only
principle is
•
reinvested/renewed. S.I = (P*R*T)/100
• Recurring deposit is a deposit in which fixed sum of money is deposited at a regular interval for a
specified
•
period. Compound Interest is where the interest is not paid to the investor/lender as and when it is due
but is added to
•
the principle at the end of the time period agreed upon. Amount= P[1+R/100]
•
NT
C.I =Amount –Principle
Self Assessment
The price at which the goods are sold is known as _______.
1.
selling price
a. cost price
b. marked price
c. net price
d. The price at which the goods are bought is known as ______.
2.
selling price
a. cost price
b. marked price
c. net price
d. The price at which the goods are intended to be sold is known as ___________ .
3.
selling price
a. cost price
b. marked price
c. net price
d. When the Selling price is lesser than the marked price ,it is said that the seller has given a ______ .
4.
profit
a. discount
b. loss
c. gain
d. If the selling price is less than the cost price ,then it is said the seller has made a _____.
5.
profit
a. discount
b. loss
c. gain
d.
A man sells a pen at a profit of 20% .Had he bought it at 20% less and sold it for Rs 5 less, he would have
gained
6. 25% .What is the CP of the pen?
20
a. 22
b. 23
c. 25
d. If the SP of 10 articles is the same as the CP of 11 articles, what is the gain percent?
7. 7
a. 10
b. 9
c. 8
d.
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What will be the simple interest on Rs 625 at 6.5% per annum for 2.5 years?
8.
101.56
a. 102.56
b. 103.56
c. 104.56
d. What is the compound interest on Rs 4000 for 9 months at 6% per annum,the interest being reckoned
9. Quarterly? 181.70
a. 182
b. 182.71
c. 183
d. _________ is a fees paid by a borrower of assets to the owner of the assets as form of compensation
for the
10. use of assets. SI
a. CI
b. Recurring
c. Interest
d.
Chapter IX
Relations and Functions
Summary
Statistics helps in creating more efficiency in the decision making process
•
Statistics can be said as a collection of methods for planning experiments, obtaining data, and then
organising,
•
summarising, presenting, analysing, interpreting, and drawing conclusions The purpose of statistics is to
obtain some overall understanding of group characteristics
•
It is important to know how to understand statistics so that improper judgments are not made
•
Statistics does not deal with qualitative aspects like honesty, intelligence etc. It deals with only
quantitative
•
data. It does not study individual facts because individual items taken separately do not form a statistical
data.
• Exhaustive: Classification must be exhaustive. i.e. each and every item in the data must belong to one
of the
•
classes.
Homogeneity: The items included in each class must be homogeneous; else there should be further
classification
•
in to sub groups.
Flexibility: A good classification should be flexible. It should be adjustable. To the new and changed
situations
•
and conditions.
Stability: The basic principle of classification should be retained
Self Assessment
Statistics deals with which of the following?
1.
Qualitative data
a. Qualitative and quantitative data
b. Quantitative data
c. Productive data
d. Which of the following is not a characteristic of Statistics?
2.
Statistics are aggregates of facts.
a. Statistics are affected to a marked extent by multiplicity of causes.
b. It helps in bringing out the hidden relations between variables.
c. Statistics are collected in a systemic manner.
d. Common man cannot handle statistics properly, only
3.
can handle statistics properly.
technicians
a. statisticians
b. artisans
c. administrators
d.
“By statistics we mean quantitative data affected to a market extent by multiplicity of causes.” This
definition
4.
is defined by:
Yule and Kendall
a. Webster
b. Prof.A.L.Bowley
c. Tippet
d. Which among the following is not a function of statistics?
5.
It presents the facts in a definite for.
a. With the help of statistics, decision making process becomes easier.
b.
It simplifies the mass of data.
c. It helps in comparison of data of different kind.
d. Statistics can be referred as a subject that deals with
6.
facts and figures.
Alphanumerical
a. Alphabetical
b. Numerical
c. Quantitative
d.
“The classified facts respecting the condition of the people in a state especially those facts which can be
stated
7.
in numbers or in tables of numbers or in any tabular or classified arrangement.”This definition is given
by
__________. Tippet
a. Peter Drucker
b. Webstar
c. Oxford
d.
Quantitative Techniques
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_______of data leads to false conclusions.
8.
Misinterpretation
a. Understanding
b. Conceptualisation
c. Summarising
d. Which statement is true?
9.
Statistical analysis is based on certainty and not on probability.
a. Common man can handle statistics properly.
b. Statistical tools provide the best solution to problems under all circumstances.
c. Statistics does not study individual facts because individual items taken separately do not form a
statistical
d. data.
Discrete distribution is also called _______________ .
10.
Cumulative frequency distribution
a. Grouped frequency distribution
b. Ungrouped frequency distribution
c. Continuous distribution
d.