Surds Must Solve Questions
Question 1.
If let us calculate simplified of i.
and ii.
Answer:
Formula used.
(a + b)2 = a2 + b2 + 2ab
(a + b)3 = a3 + b3 + 3ab(a + b)
Let a = m and b =
(a + b)2 = a2 + b2 + 2ab
(m + )2 = m2 +
2 + 2 × m ×
(√3)2 = m2 + + 2 × 1
3 = m2 + + 2
m2 + = 3 – 2 = 1
(a + b)3 = a3 + b3 + 3ab(a + b)
(m + )3 = m3 +
3 + 2 × m ×
× (m +
)
(√3)3 = m3 + + 2 × 1 × (√3)
3√3 = m3 + + 2√3
m3 + = 3√3 – 2√3 = √3[3 – 2]
= √ 3
Question 2.
Let us show that,
Answer:
= 2√15
Question 3.
Let us simplify
Answer:
]
]
]
=
=
Question 4.
Let us simplify
Answer:
Simplifying part 1
= √35 – √14
Simplifying part 2
= √35 – √10
Simplifying part 3
= √14 – √10
Putting values we get;
[√35 – √14] – [√35 – √10] + [√14 – √10]
√35 – √14 – √35 + √10 + √14 – √10
= 0
Question 5.
Let us simplify
Answer:
Simplifying 1st part by rationalizing the expression by multiplying and dividing by 2 + √2
= =
= 4√3 + 2√6
Simplifying 2nd part by rationalizing the expression by multiplying and dividing by 4√3 + √18
=
Simplifying 3rd part by rationalizing the expression by multiplying and dividing by 3 + √12
= = –
Putting all values we get;
4√3 + 2√6 – (4√3 – √18) + (3√2 + √24)
4√3 + √6 × 22 – (4√3 – √18) + (√2 × 32 + √24)
4√3 + √24 – (4√3 – √18) + (√18 + √24)
= 2√24 = 4√6Hence,
= 4√6
Question 6.
Let us simplify
Answer:
Simplifying 1st part
= -√6 + √12
Simplifying 2nd part
= √18-√6
Simplifying 3rd part
= -√12 + √18
Putting all values we get;
(-√6 + √12) – (√18 – √6) + (-√12 + √18)
-√6 + √12 – √18 + √6 – √12 + √18
= 0
Question 7.
If x = 2, y = 3 and z = 6, let us write the calculating the value of
Answer:
Putting value x = 2, y = 3, z = 6;
Simplifying 1st part
= -√6 + √12
Simplifying 2nd part
= √18-√6
Simplifying 3rd part
= -√12 + √18
Putting all values we get;
(-√6 + √12) – (√18 – √6) + (-√12 + √18)
-√6 + √12 – √18 + √6 – √12 + √18
= 0
Question 8.
If let us calculate simplified value of
Answer:
x = √7 + √6
then;
=
By simplifying
=
=
=
=
= √7 – √6
Hence;
x – = (√7 + √6) – (√7 – √6)
= 2√6
Question 9.
If let us calculate simplified value of
Answer:
x = √7 + √6
then;
=
By simplifying
=
=
=
=
= √7 – √6
Hence;
x + = (√7 + √6) + (√7 – √6)
= 2√7
Question 10.
If let us calculate simplified value of
Answer:
Formula used.
(a + b)2 = a2 + b2 + 2ab
x + = 2√7
(a + b)2 = a2 + b2 + 2ab
Put a = x and b =
(x + )2 = x2 +
2 + 2 × x ×
(2√7)2 = x2 + + 2
4 × 7 = x2 + + 2
28 = x2 + + 2
x2 + = 28 – 2 = 26
Question 11.
If let us calculate simplified value of
Answer:
Formula used.
(a + b)3 = a3 + b3 + 3ab(a + b)
If
x + = 2√7
(a + b)3 = a3 + b3 + 3ab(a + b)
Put a = x and b =
(x + )3 = x3 +
3 + 3 × x ×
× (x +
)
(2√7)3 = x3 + + 3 × (2√7)
56√7 = x3 + + 6√7
x3 + = 56√7 – 6√7
x3 + = √7 [56 – 6]
= 50√7
Question 12.
Let us simplify :
If the simplified value is 14, let us write by calculating the value of x.
Answer:
4x2 – 2
If 4x2 – 2 = 14
4x2 = 14 + 2 = 16
x2 = = 4
x = √4 = ±2
Question 13.
If and
let us calculate the followings :
Answer:
a + b =
a + b = =
=
= 3
a-b =
a-b = =
=
= √5
ab = = 1
=
=
=
Putting values we get;
=
Question 14.
If and
let us calculate the followings :
Answer:
a + b =
a + b = =
=
= 3
a-b =
a-b = =
=
= √5
=
=
Question 15.
If and
let us calculate the followings :
Answer:
a + b =
a + b = =
=
= 3
a-b =
a-b = =
=
= √5
ab = = 1
=
=
=
Putting values we get;
=
Question 16.
If and
let us calculate the followings :
Answer:
a + b =
a + b = =
=
= 3
a-b =
a-b = =
=
= √5
ab = = 1
(a + b)3 = a3 + b3 + 3ab(a + b)
a3 + b3 = (a + b)3 – 3ab(a + b)
= (3)3 – 3 × 1 × 3
= 27 – 9 = 18
(a-b)3 = a3-b3-3ab(a-b)
a3-b3 = (a-b)3 + 3ab(a-b)
= (√5)3 + 3 × 1 × (√5)
= 5√5 + 3√5
= √5 [5 + 3]
= 8√5
Question 17.
If let us calculate the simplified value of
Answer:
If x = 2 + √3
Then;
Simplifying it we get;
=
= 2-√3
x – = 2 + √3 – [2 – √3]
= 2√3
Question 18.
If let us calculate the simplified value of
Answer:
If y = 2-√3
Then;
Simplifying it we get;
=
= 2 + √3
y + = 2 – √3 + [2 + √3] = 4
(y + )2 = y2 + [
]2 + 2 × y ×
(4)2 = y2 + []2 + 2
y2 + []2 = 16 – 2 = 14
Question 19.
If let us calculate the simplified value of
Answer:
Formula used.
(a-b)3 = a3-b3-3ab(a-b)
If x = 2 + √3
Then;
Simplifying it we get;
=
= 2-√3
x – = 2 + √3 – [2 – √3]
= 2√3
(x – )3 = x3–
3-3 × x ×
× (x –
)
(x – )3 = x3–
-3 × 1 × (x –
)
(2√3)3 = x3– -3 × (2√3)
x3 – = 24√3 + 6√3
x3 – = 30√3
Question 20.
If let us calculate the simplified value of
Answer:
x = 2 + √3
y = 2 – √3
xy = (2 + √3) × ( 2-√3)
xy = (2)2 – (√3)2
= 4 – 3
= 1
=
= 1 + 1 = 2
Question 21.
If let us calculate the simplified value of
Answer:
Formula used.
(a – b)2 = a2 – 2ab + b2
3x2 – 5xy + 3y2
Add and subtract xy to the equation.
3x2 – 5xy + 3y2 [ + xy – xy]
3x2 – 6xy + 3y2 + xy
3[x2 – 2xy + y2] + xy
3[x-y]2 + xy
x = 2 + √3
y = 2 – √3
xy = (2 + √3) × ( 2-√3)
xy = (2)2 – (√3)2
= 4 – 3
= 1
x – y = 2 + √3 – [2-√3]
x – y = 2√3
Putting the values we get;
3[2√3]2 + 1
3[12] + 1
36 + 1 = 37
Question 22.
If and xy = 1, let us show that
Answer:
Formula used.
(a – b)2 = a2 – 2ab + b2
x =
xy = 1
y =
y = =
x + y = +
=
=
= = 5
x-y = –
=
=
= = √21
Add and Subtract xy both on numerator and denominator
=
Putting values we get;
=
Hence proved.
Question 23.
Let us write which one is greater of and
Answer:
Formula used.
(a – b)2 = a2 – 2ab + b2
1st value is √7 + 1
Its square is
(√7 + 1)2 = (√7)2 + 12 + 2 × 1 × √7 = 7 + 1 + 2√7 = 8 + 2√7
2nd value is √5 + √3
Its square is
(√5 + √3)2 = (√5)2 + (√3)2 + 2 × √3 × √5 = 5 + 3 + 2√15 = 8 + 2√15
If 7<15
Then √7 < √15
Then 8 + √7 < 8 + √15
Then √(8 + √7) < √(8 + √15)
∴ √7 + 1 < √5 + √3
Question 24.
If the value of
is
A. 2
B.
C. 4
D.
Answer:
If x = 2 + √3
Then;
Simplifying it we get;
=
= 2-√3
x + = 2 + √3 + [2 – √3]
= 4
Question 25.
If and
then the value of pq is
A. 2
B. 18
C. 9
D. 8
Answer:
p + q = √13
p = √13 – q
p–q = √5
(√13 – q) – q = √5
2q = √13 – √5
q =
p = √13 – q = √13 – =
pq = ×
=
=
= 2
Question 26.
If and
the value of (a2 + b2) is
A. 8
B. 4
C. 2
D. 1
Answer:
a + b = √5
a = √5 – b
a–b = √3
(√5 – b) – b = √3
2b = √5 – √3
b =
a = √5 – b = √5 – =
ab = ×
=
=
(a + b)2 = a2 + b2 + 2ab
(√5)2 = a2 + b2 + 2 ×
a2 + b2 = 5 – 1 = 4
Question 27.
If we subtract from
the value is
A.
B.
C.
D. None of this
Answer:
√125 = √ (5 × 5 × 5) = 5√5
√ 125 – √ 5
= 5√5 – √5
= √5 [5-1]
= 4√5
= √((4 × 4) × 5)
= √80
Question 28.
The product of is
A. 22
B. 44
C. 2
D. 11
Answer:
(5-√3)(√3-1)(5 + √3)(√3 + 1)
(5-√3)(5 + √3)(√3-1)(√3 + 1)
(52 – (√3)2)((√3)2 – 12)
(25 – 3)(3 – 1)
22 × 2 = 44
Question 29.
Let us write whether the following statements are true or false :
i. and
are similar surds
ii. is a quadratic surd.
Answer:
(i) True.
√75 = √(5 × 5 × 3) = 5√3
√147 = √(7 × 7 × 3) = 7√3
√3 is common on both surds
(ii) False
π itself is an irrational number
hence;
Square root of π is not a surd.
Question 30.
Let us fill up the blank:
i. a _________ number (rational/ irrational)
ii. Conjugate surd of is ________.
iii. If the product and sum of two quadratic surds is a rational number, then the surds are _________ surds.
Answer:
(a) Irrational
As √11 is irrational number
Multiplying it with 5
Also get irrational number
(b) √3 + 5
Conjugate surds are surds which are having same terms but having different symbol( + to- and – to + ) in between both the terms.
(c) Conjugate Surds
While product of conjugate surds
They get square to become rational number
While sum of conjugate surds
They get cancel to become rational number
Question 31.
If x = 3 + 2√2 let us write the value of .
Answer:
If x = 3 + 2√2
Then;
Simplifying it we get;
=
= 3-2√2
x + = 3 + 2√2 + [3 – 2√2]
= 6
Question 32.
Let us write which one is greater of √15 + √3 and √10 + √8
Answer:
1st value is √15 + √3
Its square is
(√15 + √3)2 = (√15)2 + (√3)2 + 2 × √15 × √3 = 15 + 3 + 2√45 = 18 + 2√45
2nd value is √10 + √8
Its square is
(√10 + √8)2 = (√10)2 + (√8)2 + 2 × √10 × √8 = 10 + 8 + 2√80 = 18 + 2√80
If 45<80
Then √45 < √80
Then 18 + √45 < 18 + √80
Then √(18 + √45) < √(18 + √80)
∴ √15 + √3 < √10 + √8
Question 33.
Let us write two mixed quadratic surds of which product is a rational number.
Answer:
Two mixed surds are
5√6 and 7√6
Multiplying both
5√6 × 7√6
35 × 6 = 210
Which is a rational number.
Question 34.
Let us write what should be subtracted from √72 to get √32
Answer:
Let the number be x
√72 – x = √32
6√2 – x = 4√2
x = 6√2 – 4√2
= √2 [6 – 4]
= 2√2
Question 35.
Let us write simplified value of
Answer:
Simplifying part 1
= √2-1
Simplifying part 2
= √3-√2
Simplifying part 3
= √4-√3
Adding all we get;
√2-1 + √3-√2 + √4-√3
= √4 – 1
= 2 – 1 = 1
