(i) Simplify :- (1^{3} + 2^{3} + 3^{3})^{1/2
(ii) }Simplify :-
(iii) Simplify :-
(iv) Simplify :-
(v) Simplify :-
(vi) Simplify :-
(vii) Simplify :-
(i) Answer. 6
Solution. (1^{3} + 2^{3} + 3^{3})^{1/2}
We know that
1^{3} = 1.1.1 = 1
2^{3} = 2.2.2 = 8
3^{3} = 3.3.3 = 27
Putting these values we get
(1^{3} + 2^{3} + 3^{3})^{1/2}
Hence the answer is 6
(ii) Answer.
Solution.
We know that
8 = 2.2.2 = 2^{3}
32 = 2.2.2.2.2 = 2^{5}
Hence the answer is
(iii) Answer. 9
Solution. Given
We know that
27 = 3.3.3 = 3^{3
}
= 3^{2} = 9
Hence the answer is 9
(iv) Answer. 5
Solution. Given
We know that
= 5^{1} = 5
Hence the answer is 5
(v) Answer.
Solution. We have
Now we know that
9 = 3.3 = 3^{2}
27 = 3.3.3 = 3^{3
}
Hence the answer is ^{}
(vi) Answer. – 3
Solution. We have ,
We know that 64 =4.4.4=4^{3
}
= – 3
Hence the answer is – 3
(vii) Answer. 16
Solution. Given ,
We know that
8 = 2.2.2 = 2^{3}
16 = 2.2.2.2 = 2^{4}
32 = 2.2.2.2.2 = 2^{5
and
}
Hence the answer is 16.
(i) Rationalize the denominator in each of the following and hence evaluate by taking and , upto three places of decimal :
(ii) Rationalize the denominator in each of the following and hence evaluate by taking and , upto three places of decimal :
(iii) Rationalize the denominator in each of the following and hence evaluate by taking and , upto three places of decimal :
(iv) Rationalize the denominator in each of the following and hence evaluate by taking and , upto three places of decimal :
(v) Rationalize the denominator in each of the following and hence evaluate by taking and , upto three places of decimal :
(i) Answer. 2.3093
Solution. Given:
Rationalising,
(Given that )
= 2.3093
Hence the answer is 2.3093
(ii) Answer. 2.449
Solution. Given:
Rationalising,
Putting the given values,
We get :
Hence the answer is 2.449
(iii) Answer. 0.462852
Solution. Given that
This can be written as
Now putting the given values,
We get :
= 0.462852
Hence the answer is 0.462852
(iv) Answer. 0.414
Solution. Given:
Rationalising,
Using (a – b) (a + b) = a^{2} – b^{2
}
Putting the given value of
We get
= 1.414 – 1
= 0.414
Hence the answer is 0.414
(v) Answer. 0.318
Solution. Given that
Rationalising,
Using (a – b) (a + b) = a^{2} – b^{2
}
Putting the given values,
We get,
= 1.732 – 1.414
= 0.318
Hence the answer is 0.318
(i) Find the values of a in each of the following :
(ii) Find the values of a in the following :
(iii) Find the values of b in the following :
(iv) Find the values of a and b in the following :
(i) Answer. a = 11
Solution. We have,
LHS =
Rationalising the denominator, we get:
{Using (a – b) (a + b) = a^{2} – b^{2}}
Now RHS
Hence a = 11 is the required answer
(ii)Answer.
Solution. Given that,
LHS =
Rationalising the denominator, we get:
LHS
{Using (a – b) (a + b) = a^{2} – b^{2}}
Now RHS
Comparing both , we get
Hence is the correct answer
(iii) Answer:
Solution:
Given:
LHS =
Rationalize
=
=
=
=
(iv) Answer. a = 0, b = 1
Solution. Given,
LHS
Using (a – b) (a + b) = a^{2} – b^{2}
(a + b)^{2} = a^{2} + b^{2} + 2ab
(a - b)^{2} = a^{2} + b^{2} - 2ab
RHS
Now LHS = RHS
a = 0, b = 1
Hence the answer is a = 0, b = 1
(i) Rationalise the denominator of the following :
(ii)Rationalise the denominator of the following :
(iii) Rationalise the denominator of the following :
(iv)Rationalise the denominator of the following :
(v) Rationalise the denominator of the following :
(vi) Rationalise the denominator of the following :
(vii) Rationalise the denominator of the following :
(viii) Rationalise the denominator of the following :
(ix) Rationalise the denominator of the following :
(i) Answer.
Solution. We have,
Rationalising the denominator, we get:
Hence the answer is
(ii) Answer.
Solution. We have ,
We know that, 40 = (2) (2) (10)
Rationalising the denominator, we get:
Hence the answer is:
(iii) Answer.
Solution. We have
Rationalising the denominator, we get:
Hence the answer is
(iv) Answer.
Solution. We have
Rationalising the denominator, we get:
Using the identity (a – b) (a + b) = a^{2} – b^{2}
We get:
Hence the answer is
(v) Answer.
Solution. We have,
Rationalising the denominator, we get:
Using (a – b) (a + b) = a^{2} – b^{2}
and (a + b)^{2} = a^{2} + b^{2} + 2ab
Hence the answer is
(vi)Answer.
Solution. We have,
Rationalising the denominator, we get:
Using the identity (a – b) (a + b) = a^{2} – b^{2}
We get:
Hence the answer
(vii) Answer.
Solution. We have,
Rationalising the denominator, we get:
Using (a – b) (a + b) = a^{2} – b^{2}
and (a + b)^{2} = a^{2} + b^{2} + 2ab
Hence the answer is
(viii)
Answer:
Solution:
We have
Rationalize
(ix) Answer:
Solution:
We have
Rationalize
View Full Answer(1)
Study 40% syllabus and score up to 100% marks in JEE
(i) Simplify the following :
(ii) Simplify the following :
(iii) Simplify the following :
(iv) Simplify the following :
(v) Simplify the following :
(vi) Simplify the following :
(vii) Simplify the following :
(viii) Simplify the following :
(ix) Simplify the following :
(i) Answer.
Solution.
We know that,
45 =
20 =
So we get
Hence the answer is
(ii) Answer.
Solution. We have,
We know that,
So we get
Taking LCM (3,4) = 12
(iii) Answer.
Solution. We have
We know that
12 =
6 =
So we get,
=
=
Hence the number is .
(iv)Answer.
Solution. We have,
We know that
28 =
So we can write,
=
=
Hence the answer is
(v) Answer.
We know that
27 =
So, =
(Rationalising the denominator)
(Taking common)
Now LCM (1,1,3) = 3
Hence the answer is 19.63
(vi) Answer.
Solution. Given,
We know that (a + b)^{2} = a^{2} – 2ab + b^{2}
Comparing the given equation with the identity, we get:
= 3 + 2 –
Hence the answer is
(vii) Answer. 0
Solution. We have,
We know that
81 =
216 =
32 =
225 =
So,
= 3 – 8 × 6 + 15 × 2 + 15
= 3 – 48 + 30 + 15
= – 45 + 45
= 0
Hence the answer is 0
(viii) Answer.
Solution. We have,
We know that, 8 =
So,
Hence the answer is
(ix) Answer.
Solution. We have,
LCM (3,6) = 6
Hence the answer is .
(i) Express the following in the form , where p and q are integers and . 0.2
(ii) Express the following in the form , where p and q are integers and .
0.888…….
(iii) Express the following in the form , where p and q are integers and .
(iv) Express the following in the form , where p and q are integers and .
(v) Express the following in the form , where p and q are integers and .0.2555……
(vii) Express the following in the form , where p and q are integers and ..00323232…..
(viii) Express the following in the form , where p and q are integers and ..404040……..
(i) Answer.
Solution. We know that
0.2 can be written as
Now,
Hence the answer is
(ii) Answer.
Solution. Let x = 0.888….. .…(i)
Multiply RHS and LHS by 10
10 x = 8.88……. …(ii)
Subtracting equation (i) from (ii)
We get
10x – x = 8.8 – 0.8
Hence answer is
(iii) Answer.
Solution. Let x = …eq. (1)
Multiply by 10 on both sides
10x = …eq (2)
Subtracting equation (1) from (2)
We get
10x – x = –
9x = 47
x =
Hence the answer is
(iv) Answer.
Solution. Let x = …. Eq. (1)
Multiply by 1000 on both sides
1000 x = …eq.(2)
Subtracting eq. (1) from (2)
We get
1000 x – x = –
999x = 1
x =
Hence the answer is
(v) Answer.
Solution. Let x = 0.2555 ….. …eq.(1)
Multiply by 10 on both sides
10x = 2.555… …eq.(2)
Multiply by 100 on both sides
100x = 25.55… …eq. (3)
Subtracting eq. (2) from (3),
We get
100x – 10x = 25.555… – 2.555…
90x = 23
x =
Hence the answer is
(vii) Answer.
Solution. Let x = 0.00323232….. …eq.(1)
Multiply with 100 on both sides.
We get
100x = 0.3232… …eq(2)
Multiply again with 100 on both sides,
10000x = 32.3232… …eq(3)
Equation (3) – (2)
we get,
10000 x – 100x = 32.3232… – 0.3232…
9900x = 32
x =
x =
Hence the answer is
(viii) Answer.
Solution. Let x = 0.404040……. …(1)
Multiplying by 100 on both sides
we get
100x = 40.40… …(2)
Subtracting equation (1) from (2)
we get
100x – x = 40.40 – 0.40
99x = 40
x =
Hence the answer is
(i) Represent geometrically the following numbers on the number line :
(ii) Represent geometrically the following numbers on the number line :
(iii) Presentation of on number line :
(iv) Presentation of on number line:
(i) Solution. AB = 4.5 units, BC = 1 unit
OC = OD = = 2.75 units
OD^{2} = OB^{2} + BD^{2}
So the length of BD will be the required one so mark an arc of length BD on number line, this will result in the required length.
(ii) Solution. Presentation of on number line.
Mark the distance 5.6 units from a fixed point A on a given line to obtain a point B such that AB = 5.6 units. From B mark a distance of 1 unit and mark a new point C. Find the mid point of AC and mark that point as O. Draw a semicircle with center O and radius OC. Draw a line
perpendicular to AC passing through B and intersecting the semicircle at O. Then BD =
(iii) Solution
Mark the distance 8.1 units from a fixed point A on a given line to obtain a point B such that AB = 8.1 units. From B mark a distance of 1 unit and mark the new point AB. Find the mid point of AC and mark a point as O. Draw a semi circle with point O and radius OC. Draw a line perpendicular to AC passing through B and intersecting
the semicircle at D. Then BD -
(iv) Solution
Mark the distance 2.3 unit from a fixed point A on a given line. To obtain a point B such that AB = 2.3 units. From B mark a distance of 1 unit and mark a new point as C. Find the mid point of AC and mark the point asO. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then BD =
View Full Answer(1)
(i)Insert a rational number and an irrational number between the following. 2 and 3
(ii)Insert a rational number and an irrational number between the following.0.1 and 0.1
(iii)Insert a rational number and an irrational number between the following.
(iv)Insert a rational number and an irrational number between the following.
(v)Insert a rational number and an irrational number between the following.0.15 and 0.16
(vi)Insert a rational number and an irrational number between the following. and
(vii)Insert a rational number and an irrational number between the following.2.357 and 3.121
(viii)Insert a rational number and an irrational number between the following..0001 and .001
(ix)Insert a rational number and an irrational number between the following.3.623623 and 0.484848
(x) Insert a rational number and an irrational number between the following.6.375289 and 6.375738
(i) Answer. Rational number:
Irrational number: 2.040040004 ……….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 2 and 3
Rational number: 2.5 =
and irrational number : 2.040040004
(ii) Answer. Rational number:
Irrational number 0.0105000500005 ……..
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0 and 0.1:
0.1 can be written as 0.10
Rational number: 0.019 =
and irrational number 0.0105000500005
(iii)Answer. Rational number
Irrational number : 0.414114111 ……
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions. They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
LCM of 3 and 2 is 6.
We can write as
and as
Also, = 0.333333….
And
So, rational number between and is
and irrational number : 0.414114111 ……
(iv)Answer. Rational number: 0
Irrational number: 0.151551555 …….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
and
Rational number between -0.4 and 0.5 is 0
And irrational number: 0.151551555 …….
(v) Answer. Rational number:
Irrational number: 0.151551555 ……….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0.15 and 0.16
Rational number : 0.151 =
and irrational number 0.151551555
(vi) Answer. Rational number:
Irrational number: 1.585585558 ………
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions. They are non-terminating non-recurring in nature.
Between
Rational number: 1.5 =
and irrational number: 1.585585558
(vii) Answer. Rational number: 3
Irrational number: 3.101101110………
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 2.357 and 3.121
Rational number: 3
Irrational number: 3.101101110………
(viii) Answer. Rational number:
Irrational number: 0.000113133133 ……….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0.0001 and 0.001
Rational number: 0.0002 =
Irrational number: 0.000113133133
(ix) Answer. Rational number: 1
Irrational number: 1.909009000 ……
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 3.623623 and 0.484848
A rational number between 3.623623 and 0.484848 is 1.
An irrational number between 3.623623 and 0.484848 is 1.909009000 ……
(x) Answer. A rational number is
An irrational number is 6.375414114111……..
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 6.375289 and 6.375738:
A rational number is 6.3753 =
An irrational number is 6.375414114111……..
Represent the following numbers on the number line. 7, 7.2,
Solution.
Firstly we draw a number line whose mid-point is O. Mark positive numbers on right hand side of O and negative numbers on left hand side of O.
(i) Number 7 is a positive number. So we mark a number 7 on the right
hand side of O, which is at a 7 units distance form zero.
(ii) Number 7.2 is a positive number. So, we mark a number 7.2 on the right hand side of O, which is 7.2 units distance from zero.
(iii) Number or – 1.5 is a negative number so, we mark a number 1.5 on the left hand side of zero, which is at 1.5 units distance from zero.
(iv) Number or –2.4 is a negative number. So, we mark a number 2.4 on the left hand side of zero, which is at 2.4 units distance from zero.
View Full Answer(1)If a = 2 + then find the value of
Answer.
Solution. Given that a =
We have
Rationalising,
Using (a – b) (a + b) = a^{2} – b^{2}
Now,
Hence the answer is
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The Solid State
Knowing our numbers
Integers
(i) Simplify :- (1power 3 + 2power 3 + 33) power 1 by2
(i)Find the values of a and b in each of the following : 5+2sqroot 3 by 7+4sqroot 3 = a-6 sqroot 3
Rationalise the denominator of the following : 2 by 3 sqroot 3
(i)Simplify the following : sqroot 45-3sqroot- 20+4sqroot 5
(i)Represent geometrically the following numbers on the number line : sqroot 4.5
(i)Insert a rational number and an irrational number between the following. 2 and 3
(i) Represent the following numbers on the number line. 7, 7.2, -3 by 2,-12 by 5