Simplify each of the following :
(i) (ii)
(i)
=
=
(ii)
=
=
Simplify the following expressions :
(i) (4+) (3+)
(ii) (3+) (5 -)
(iii) ( -2)
(i)
=
=
(ii)
=
=
=
(iii)
=
=
=
Simplify the following expressions :
(i) (11+) (11-)
(ii) (5+) (5-)
(iii)
(iv) (3+) (3-)
(v)
(i)
Because
= 121 - 11 = 110
(ii)
= 25 – 7 = 18
(iii)
= 8 – 2 = 6
(iv)
= 9 – 3 = 6
(v)
= 5 – 2 = 3
Simplify the following expressions:
(i) (ii) (iii)
(i)
Because:
=
=
(ii)
(iii)
=
=
=
=
Rationalise the denominator of each of the following (i-vii) :
(i) (ii) (iii) (iv) (v) (vi) (vii)
(i) As there is √5 in the denominator and we know that √5 x √5 = 5
So, multiply numerator and denominator by √5,
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Find the value to three places of decimals of each of the following. It is given that
= 1.414, = 1.732, =2.236 and = 3.162.
(i) (ii) (iii) (iv)
(v) (vi)
(i) Given that √2 = 1.414, √3=1.732, √5 = 2.236 and √10= 3.162
So we have,
Rationalising factor of denominator is √3
=1.15466667 = 1.54
(ii) we have rationalisation factor of denominator is √10
(iii) we have rationalisation factor of denominator is √2
(iv) we have rationalisation factor of denominator is √2
=
=
(v) We have
=
(vi) We have rationalising factor of denominator is √5
Express each one of the following with rational denominator:
(i) (ii) (iii)
(iv) (v) (vi) (vii) (viii) (ix)
(i) we have rationalizing factor of the denominator is 3-√2
(ii) we have rationalizing factor of the denominator is
=
(iii) we have rationalizing factor of the denominator is
=
=
=
(iv) we have to rationalize factor of
=
=
=
=
(v) we have to rationalize factor of
(vi) we have to rationalize factor of
(vii) we have to rationalize factor of
=
Because;
(viii) we have to rationalize factor of
(ix) we have to rationalize factor of
Rationalies the denominator and simplify :
(i) (ii) (iii)
(iv) (v)
(vi)
i)
ii)
iii)
iv)
v)
vi)
Simplify :
(i) (ii)
(iii)
(iv)
(v)
i)
ii)
iii) rationalising factors of denominators are
(iv)
Rationalising factor for
For and
For
(v)
Rationalising factors for denominators are,
For
For and
For
In each of the following determine rational numbers a and b.
(i) = a-b(ii)= a-
(iii) = a+b (iv) =a+b
(v) = a-b
(vi) =a+b
(i)
Given,
Rationalising factor for denominator is
On equating rational and irrational parts,
We get a = 2 and b = 1
(ii) rationalising factor for the denominator is
We have
On equating rational and irrational parts we get,
a=3 and b=2
(iii)
Rationalising factor for the denominator is
On equating rational and irrational parts we get,
(iv) given,
Rationalising factor for denominator is
We have
On equating rational and irrational parts we get,
a= -1 and b=1
(v) given,
We have
On equating rational and irrational parts we get
(vi) given,
We have
On equating rational and irrational parts we have,
If x = 2+, find the value of x3 + .
Given and given to find the value of
We have
We know that ,
By putting we get
The value of is 52.
If x = 3+, find the value of x2+.
Given that
And given to find the value of
We have
The rationalising factor for denominator is
=
=
Also,
We know that,
= 36 - 2 = 34
The value of is 34.
Find the value of , it being given that = 1.732 and =2.236
Rationalising factor for the denominator is
We have
= 3(3.968)
= 11.904
Find the values of each of the following correct to three places of decimals, it being given that = 1.414, = 1.732, =2.236, = 2.4495 and = 3.162.
(i) (ii)
(i) We have rationalising factor for denominator is
=
We have
(ii) by putting the value of in the equation we get,
If x =, find the value of 4x3+2x2-8x+7.
Given and given to find the value of
Squaring on both the sides we get,
Now take
The value of is 10.
is equal to
A. 5
B. 6
C.
D.
√10 × √15 =( √5×√2) × ( √5×√3)
= 5 (√6)
Write the value of (2+) (2-).
(2+√3) (2-√3)
= (2)2 - (√3)2 [(a+b) (a-b) = a2 –b2 ]
= 4 – 3 = 1.
is equal to
A.
B.
C.
D.
5√6 × 5√6 = (6)1/5 × (6)1/5 = (36)1/5
= 5√36
Write the reciprocal of 5 +.
Reciprocal of 5 + √2 = 1/ (5 + √2)
=
Write the rationalisation factor of 7-3.
Rationalizing factor of 7- 3√5
=
The rationalisation factor of is
A. -
B.
C. 2
D. -2
Rationalisation factor of √3 = 1/√3
The rationalisation factor of 2+ is
A. 2-
B. 2+
C. -3
D. -2
Rationalisation factor of 2+√3 = 1/2+√3 = 2-√3
If = x+y, find the values of x and y.
Given,
=
So, x= 2 , y = -1
If x = -1, then write the value of .
Given . x = √2-1
If x = +2, then x - equals
A. 2
B. 4
C. 2
D.
Given x = √5+2
−2
= √5 + 2 – ( √5 – 2) = 4
Simplify .
Consider,
As we know,
(a+b)2 = a2 + b2 + 2ab
If = a – b, then
A. a = 2, b = 1
B. a = 2, b = -1
C. a = -2, b = 1
D. a = b = 1
Given
=
So, a = 2 , b = 1.
The simplest rationalising of is
A.
B.
C.
D. none of these
3√(125×4) = 5×3√4
Simplify .
√(3-2√2) = √ (√2)2 + (1)2 - 2× √2×1 = √ (√2 – 1 )2 = √2- 1.
If a = +1, then find the value of a -.
Given , a = √2 +1
=
=
The simplest rationalising factor of +is
A. -5
B. 3-
C.
D.
Simplest rationalizing factor of √3 + √5
1/ (√3+√5) = √3-√5
The simplest rationalising factor of 2√5 - √3 is
A. 2+3
B. 2+
C.
D.
Simplest rationalizing factor of 2√5 - √3
= 1/(2√5 - √3)
= 2√5 + √3
If x = 2+, find the value of x +.
Given, x = 2+√3
=
=
Write the rationalisation factor of -2.
Rationalizing factor of √5 – 2
=
If x = , then (x-3)2 =
A. 1
B. 3
C. 6
D. 7
Given X = 2/(3+√7)
=
= (x -3 )2 = ( 3 - √7 -3 )2 = √72 = 7
If x = 3 + 2, then find the value of .
Given x = 3+ 2√2
)2
=
So, √x – 1 /√x = √2+1 – ( √2- 1)
= 1 + 1 = 2.
If x = 7+4 and xy=1, then =
A. 64
B. 134
C. 194
D. 1/49
Given . x = 7+4√3 , xy = 1
Y = 1/x = 1/7+ 4√3 = 7-4√3
Y2 = 1/x2 = 49 + 48 - 56√3 = 97 - 56√3
Similarly, x = 1/y
= x2 = 1/y2 = ( 7 +4√3)2 = 49 + 48 + 56√3 = 97+ 56√3
So, 1/x2 + 1/y2 = 97 + 56√3 + 97 – 56√3 = 194
If x + = 4, then x+ =
A. 2
B. 4
C. 8
D. 1
Given x +√15 = 4
X = 4 - √15
1/x = 1/(4 - √15) = (4 + √15) / 16 -15 = 4 + √15
So, x + 1/x = 4 - √15 + 4+√15 = 8
If x = , then x3 + =
A. 2
B. 4
C. 8
D. 9
Given x =
= x3 = 2 + √3
Similarly, 1/x3 = 2 - √3
X3 + 1/x3 = 2+ √3 + 2 - √3 = 4.
If x = and y =, then x + y + xy =
A. 9
B. 5
C. 17
D. 7
Given x = √5 +√3 / √5 - √3 , y =√5 - √3 / √5 + √3
X=
Y =
Xy = 42 - √152 = 16 – 15 = 1
So,
X + y+ xy = 4+√15 + 4 - √15 + 1 = 9.
If x = and y =, then x2 + xy + y2 =
A. 101
B. 99
C. 98
D. 102
Given x = , y =
X= 5 – 2√6
X2 = (5 – 2√6)2 = 25 +24 -20√6) = 49 – 20√6
Similarily y = = 5 + 2√6
Y2 = (5 + 2√6 )2 = 49 + 20√6
Xy = (5- 2√6) ( 5 + 2√6) = 25 – 24 = 1
So, x2+ xy+y2 = 49 – 20√6 +1 + 49 + 20√6 = 99.
The value of is
A. -1
B. +1
C. -
D. +
( try to break the terms in form of (a+b)2 or (a – b )2 )
√(√2)2+ 12 – 2 ×√2×1) = √(√2-1)2 = √2 – 1 .
The value of is
A. -
B. +
C. +
D. none of these
( try to break the terms in form of (a+b)2 or (a – b )2 )
√(√2)2+ 12 – 2 ×√2×1) = √(√2-1)2 = √2 – 1 .
If =1.4142, then is equal to
A. 0.1718
B. 5.8282
C. 0.4142
D. 2.4142
Given √2 = 1.4142
√(√2-1)/√2+1) = √(√2-1)2 = √2 – 1 = 1.4142 – 1 = 0 .4142
If =1.414, then the value of upto three place of decimal is
A. 0.235
B. 0.707
C. 1.414
D. 0.471
Given , √2 = 1.414
√6 - √3
= √2 × √3 - √3
= √3(√2 - 1)
= 1.732 (1.414 – 1)
= 1.732 × 0.414
= 0.707
The positive square of 7 + is
A. 7 + 2
B. 7+
C. 2+
D. 3+
7 + √48
= 7 + √(16×3) = 7 + 4√3 ( try to break it in form of (a+b)2)
= (2)2 + (√3)2 + 2×2×√3 = (2+√3)2 = (2+√3) (2+√3).
is equal to
A. 3 + 2
B.
C. 3-2
D. -
1/ √9-√8
= 1/(√9 - √8) × (√9 + √8) / (√9+√8)
= √9 + √8 = 3 + 2√2
The value of is
A.
B. 4
C. 3
D.
√48 + √32 / √27 + √18
= 4√3 + 4√2 / 3√3 + 3√2 = (4√3+4√2)/(3√3+3√2) × (3√3- 3√2)/ (3√3-3√2)
=( 36 + 12√6 – 12√6 -24) / (27-18) = 12/9 = 4/3
If x = , then x2 + -2 =
A. 2
B. 2
C. 24
D. 20
Given x= √6 + √5
X2 = 11 + 2√11
1/x2 = 11- 2√11
So, x2 + 1/x2 – 2 = 11+ 2√11 + 11 – 2√11 -2 = 22-2 = 20.
If , then a =
A. -5
B. -6
C. -4
D. -2
√(13-a√10) =√8 +√5
Squaring both side,..
= 13 – a√10 = 8 + 5 + 2×√8×√5
= 13 – a√10 = 13 + 2√40
= - a√10 = 4√10
= a = -4
If == x+y, then
A. x = 13, y = -7
B. x = -13, y = 7
C. x = -13, y = -7
D. x = 13, y = 7
5 - √3/ 2+√3 = x + y√3
= ( 5-√3) / (2+ √3) × (2 -√3) / (2 -√3)
= (10 – 5√3 – 2√3 +3)/ (4-3)
= 10 – 7√3 +3
= 13 – 7√3 = x + y√3
So , x= 13 , y = -7