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maths ncert solution class eleveth chapter mathametical reasoning

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Aakash Educational Services Pvt. Ltd.
Regd. Office
: Aakash Tower, Plot No.-4, Sector-11, Dwarka, New Delhi-75 Ph.011-47623456
SECTION - A
School/Board Exam. Type QuestionsVery Short Answer Type Questions :
1.(i)
p
: Grass is green.(ii)
q
: There are 366 days in a leap year.(iii)
r
: Product of an even and odd number is an even number.2.(i)
p
: He is a chemistry graduate(ii)
q
: It
x
and
y
are two positive integer then
x
–
y
is always greater than zero.(iii)
r
: How much old he is3.Definition : If
p
is a statement, then the negation of
p
is also a statement and is denoted by ~
p
, and read asnot
p
.Example :
p
:
4 32
is an irrational number So,
~ :4 32
p
is not an irrational number 4.Component statements are
q
: Aeroplane flies in the air
r
: Ships sails on the water 5.(i)
p
is false. In
p
basic connective “and” is used, and one component statement is false, so
p
is false.(ii)
q
is true. In
q
basic connective “or” is used and in this case statement is true if any one of it’s componentstatements are true.6.(i)Inclusive OR(ii)Exclusive OR7.If a person cannot apply for post of P.O. then he has not a bachelor degree.8.If
x
is complex number then it is not real.9.Volume of a cone is one third of cylinder if and only if base radius and height of cone and cylinder are same.10.
n
= 2, is a prime number and even number also.
Solutions
Chapter
14
Mathematical Reasoning
36
Mathematical Reasoning(School/Board Exams.) Solutions
Aakash Educational Services Pvt. Ltd.
Regd. Office
: Aakash Tower, Plot No.-4, Sector-11, Dwarka, New Delhi-75 Ph.011-47623456
Short Answer Type Questions :
11.(i)
p
: There are 33 days in a month. We know that a month cannot have 33 days, so above sentence isfalse, so we can say this sentence is statement.(ii)
p
: The product of 5 and 6 is 30. This sentence is true so it is statement.(iii)
p
: Sum of two positive numbers is always positive. This sentence is always true, so it is statement.12.(i)Call the police. It is an order, so it is not a statement.(ii)If x and y are integers, then
x y
is an integer. This sentence is sometimes true, sometimes false, so itis not a statement.(iii)Chandigarh is far from here. This sentence does not specify particular place, so it is not a statement.13.(i)~
p
: 0 is not a natural number. (True)(ii)~
p
: 70 is not a multiple of 20. (True)(iii)~
p
: Diameter is not the longent chord of circle.14.Yes, they are negation of each other. Since negation of a statement
p
is also a statement, and we can formnegation of statement using phrases like “It is not the case” or “ It is false that” before
p
or, if possible byinserting in
p
the word “not”.15.~
p
: It is false that square of an odd integer is of the form 8
q
+ 1, for some positive integer
q
.or ~
p
: It is not the case that square of an odd positive integer is of the form of 8
q
+ 1, for some positive integer
k.
or ~
p
: square of an odd positive integer is not of the form 8
q
+ 1, for some positive integer
q
.16.Component statements of
p
are
q
: Taj Mahal is in India (True)
r
: Niagara falls is in U.S.A. (True)17.Component statements of p are
: 12
q
is a rational number (False)
: 12
r
is an irrational number (True)
: 12
s
is a complex number (False)18.Compound statement using connective AND “rational number follows commutative property for addition andmultiplication”.Compound statement using connective OR “rational number follows commutative property for addition or multiplication”.19.(i)By using basic connective “And”All sides and all angles of two congruent triangles are equal(ii)By using basic connective “OR”All sides or all angles of two congruent triangles are equal
37
(School/Board Exams.) SolutionsMathematical Reasoning
Aakash Educational Services Pvt. Ltd.
Regd. Office
: Aakash Tower, Plot No.-4, Sector-11, Dwarka, New Delhi-75 Ph.011-47623456
20.Component statements are
q
: Chord of a circle lies within the circle (True)
r
: Tangent of circle lies both outside and inside of circle (False)In compound statement
p
basic connective “And” is used and it is true only when all component statementare true.So, compound statement
p
is false.21.Component statements of
p
are
q
: From a point outside the circle we can draw two tangents to a circle (True)
r
: From a point inside the circle we can draw one tangent to circle (False)In compound statement p basic connective “OR” is used, so
p
will be true if at least one component statementis true, so compound statement is true.22.Compound statements of p are
q
: Solution of a quadratic equation
ax
2
+
bx
+
c
= 0 is real
r
: Solution of a quadratic equation
ax
2
+
bx
+
c
= 0 is complexWe know that when
q
is true
r
is false and when
r
is true
q
is false, both cannot be true at a time. So ExclusiveOR is used.23.(a)In
p
quantifier is “for every”.~
p
: For every natural number
x
, 7
x
is not greater than 7.(b)In
q
quantifier is “There exists”~
q
: There does not exist a tangent which is chord to the circle.24.(a)
p
: If two circle touch other then we can draw three common tangents to both circle.or If three common tangent can be drawn to two circles then they touch each other.(b)
q
: If chord of a circle subtends an angle 90º in alternate segment then it is diameter.25.A convex polygon is pentagon if and only if it has 5 diagonals.26.(i)If two chords are not diameter then they will not bisect each other.(ii)If corresponding angles of two triangles are not equal then they are not similar.27.(i)If radius of two circles are equal then they are congruent.(ii)If two polygons overlap each other then they are congruent.28.In order to prove that “
p
and
q
” is true, we have to follow following steps.Step-1 : Show that the statement
p
is true.Step-2 : Show that the statement
q
is true.In order to show that “
p
or
q
” is true, one must consider the followingCase-1: By assuming that p is false, show that
q
must be true.Case-2 : By assuming that q is false, show that
p
must be true.
38
Mathematical Reasoning(School/Board Exams.) Solutions
Aakash Educational Services Pvt. Ltd.
Regd. Office
: Aakash Tower, Plot No.-4, Sector-11, Dwarka, New Delhi-75 Ph.011-47623456
29.Numerical value of area of a circle and area of a triangle can be equal but they are not congruent.30.
x
3
– 6
x
2
+ 11
x
– 6 = 0
(
x
– 1) (
x
– 2) (
x
– 3) = 0So, given equation has three roots, so it will cut
x
-axis at three times.
Long Answer Type Questions :
31.Following are four types of sentences which are never considered as statementType-1 : Sentences involving variable time such as “Today”, tomorrow” or “Yesterday” are not statements,because it is not known what time is referred here.Example : Tomorrow is Friday.Type-2 : Sentences which are an order are never considered as statements.Example : Bring a glass of water Type-3 : Sentences which are exclamation are not considered as statements.Example : Too smooth.Type-4 : Sentences with pronouns unless a particular person is referred to.Example : He is a football player 32.(a)
p
contains pronoun “here” which refers to variable place, here it doesnot mention the place from whichwe are measuring the distance of Hisar.(b)
q
is true only on Sunday but not on other days.(c)
r
is exclamation so it is not a sentence33.(i)~
p
: It is false that everyone in Europe plays footballor ~
p
: It is not the case that everyone in Europe plays footballor ~
p
: Everyone in Europe does not play football.~
p
says that at least one person in Europe does not play football.(ii)~
q
: It is false that every quadratic equation has two real rootsor ~
q
: It is not the case that every quadratic equation has two real rootsor ~
q
: Every quadratic equation has not two real roots.~
q
Says that at least one quadratic equation exists whose roots are not real34.Definition : A compound statement is a statement which is made up of two or more statements. In this case,each statement is called a component statement.Examples are(i)Ganga is a river and Mansarover is a lake.(ii)8 is a natural number of integer.(iii)Product of a rational and an irrational number is real or irrational number.

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