Q.1. Write the rationalising factor of the denominator in
Ans.
Here, the denominator i.e. 1 is a rational number. Thus, the rationalising factor of the denominator in
Q.2. Rationalise the denominator of each of the following.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Ans.
(i)
On multiplying the numerator and denominator of the given number by √7, we get:
(ii)
On multiplying the numerator and denominator of the given number by √3, we get:
(iii)
On multiplying the numerator and denominator of the given number by 2  √3, we get:
(iv)
On multiplying the numerator and denominator of the given number by √5 +2, we get:
(v)
On multiplying the numerator and denominator of the given number by 53√2, we get:
(vi)
Multiplying the numerator and denominator by √7+√6, we get
(vii)
Multiplying the numerator and denominator by √11+√7, we get
(viii)
Multiplying the numerator and denominator by 2+√2, we get
(ix)
Multiplying the numerator and denominator by 32√2, we get
Q.3. It being given that √2 = 1.414, √3 = 1.732, √5 = 2.236 and √10 = 3.162, find the value of three places of decimals, of each of the following.
(i)
(ii)
(iii)
Ans.
(i)
(ii)
(iii)
Q.4. Find rational numbers a and b such that
(i)
(ii)
(iii)
(iv)
Ans.
(i)
(ii)
(iii)
(iv)
Q.5. It being given that √3 = 1.732, √5 = 2.236, √6 = 2.449 and √10 = 3.162, find to three places of decimal, the value of each of the following.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Ans.
(i)
= 0.213
(ii)
=3 × (2.236 − 1.732)
= 1.512
(iii)
(iv)
(v)
= 16.660
(vi)
= 4.441
Q.6. Simplify by rationalising the denominator.
(i)
(ii)
Ans.
(i)
(ii)
Q.7. Simplify
(i)
(ii)
(iii)
(iv)
Ans.
(i)
(ii)
= 0
(iii)
= 16 − √3
(iv)
= 0
Q.8. Prove that
(i)
(ii)
Ans.
(i)
= 2/2
= 1
(ii)
Q.9. Find the values of a and b if
Ans.
Comparing with the given expression, we get
a = 0 and b = 1
Thus, the values of a and b are 0 and 1, respectively.
Q.10. Simplify
Ans.
Q.11. If x = 3 + 2√2, check whether is rational or irrational.
Ans.
x = 3 + 2√2 .....(1)
Adding (1) and (2), we get
which is a rational number
Thus, is rational.
Q.12.
If x = 2 − √3, find value of
Ans.
x = 2 − √3 .....(1)
Subtracting (2) from (1), we get
Thus, the value of
Q.13. If x = 9 − 4√5, find the value of
Ans.
x = 9 − 4√5 .....(1)
Adding (1) and (2), we get
Squaring on both sides, we get
Thus, the value of x^{2} + is 322.
Q.14. If x = find the value of
Ans.
Adding (1) and (2), we get
Thus, the value of x +is 5.
Q.15. If a = 3 − 2√2, find the value of a^{2 }
Ans.
a = 3−2√2
⇒ a^{2} = (3−2√2)^{2}
⇒ a^{2} = 9 + 8 − 12√2
⇒ a^{2 }= 17 − 12√2 .....(1)
Subtracting (2) from (1), we get
Thus, the value of a^{2} 
Q.16. If x = √13 + 2√3, find the value of x −
Ans.
Subtracting (2) from (1), we get
Thus, the value of x
Q.17.
If x = 2 + √3, find the value of
Ans.
Adding (1) and (2), we get
Cubing both sides, we get
Thus, the value of
Q.18. If andshow that
Ans.
Disclaimer: The question is incorrect.
The question is incorrect. Kindly check the question.
The question should have been to show that x − y =
Q.19. If a = and b = show that 3a^{2 }+ 4ab − 3b^{2 }=
Ans.
According to question,
Now,
3a^{2 }+ 4ab − 3b^{2}
= 3(a^{2 }− b^{2}) + 4ab
= 3 (a + b)(a − b) + 4ab
Hence, 3a^{2 }+ 4ab − 3b^{2 }=
Q.20.
If a = and b =find the value of a^{2} + b^{2} – 5ab.
Ans.
According to question,
Now,
Hence, the value of a^{2} + b^{2} – 5ab is 93.
Q.21.
If p = and q = find the value of p^{2} + q^{2}.
Ans.
According to question,
Now,
p^{2} + q^{2} = (p+q)^{2 }− 2pq
Hence, the value of p^{2} + q^{2} is 47.
Q.22. Rationalise the denominator of each of the following.
(i)
(ii)
(iii)
Ans.
(i)
Hence, the rationalised form is
(ii)
Hence, the rationalised form is
(iii)
Hence, the rationalised form is
Q.23. Given, √2 = 1.414 and √6 = 2.449, find the value of correct to 3 places of decimal.
Ans.
Hence, the value of correct to 3 places of decimal is −1.465.
Q.24. If x = find the value of x^{3} – 2x^{2} – 7x + 5.
Ans.
Now,
Also,
Now,
Hence, the value of x^{3 }– 2x^{2} – 7x + 5 is 3.
Q.25. Evaluate it being given that √5 = 2.236 and √10 = 3.162.
Hint
Ans.
Hence,
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