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151.	Which of the following statements is true ? (I).If x is a conjugate surd of y, then x can be a RF of y. (II).If x is RF of y, then x need not be the conjugate of y.A. Only IB. Only IIC. Both I and IID. Neither I nor II
Answer» Correct Answer - C Recall the definitions of RF and conjugate.

152.	Find the decimal expansions of `(10)/3,7/8`and `1/7`.
Answer» to find the decimal, we will divide `10/3 = 3.3333....` `7/8 =0.875 ` `1/7 = 0.1428571428571.... = 0. bar 142857`

153.	The simplified form of `sqrt(125)+sqrt(125)-sqrt(845)` isA. `sqrt(15)`B. `2sqrt(5)`C. `-sqrt(5)`D. `-2sqrt(5)`
Answer» Correct Answer - C Factorize each rational part in the compound surd.

154.	Find three consecutive numbers such that twice the first, three times the second and four times the third together make 191.
Answer» According to Question `2n+3(n+1)+4(n+2)=191` `2n+3n+3+4n+8=191` `9n+11=191` `n=20` numbers are 20,21,22 Option 3 is correct.

155.	If `sqrt(5^(n))` =125, then `5^(root(n)(64))` =_______A. 25B. `(1)/(125)`C. 625D. `(1)/(25)`
Answer» Correct Answer - A Find n.

156.	Find: (i) `64^(1/2)` (ii) `32^(1/5)` (iii) `125^(1/3)`
Answer» (i) `64^(1/2) = 8^(2(1/2)) = 8 ` (ii) `32 ^(1/5) = (2)^(5(1/5))` `= 2^(5/5) = 2` `= 125^(1/3) = 5^(3(1/3))` `= 5` answer

157.	Between two rational numbers(A) there is no rational number(B) there is exactly one rational number(C) there are infinitely many rational numbers(D) there are only rational numbers and no irrational numbersA. there is no rational numberB. there is exactly one rational numberC. there are infinitely many rational numberD. there are only rational numbers and no irrational numbers
Answer» Correct Answer - C Between the rational numbers, there are infiniitely many rational numbers. e.g., `3/5` and `4/5` are two rational numbers, then `31/50, 32/50, 33/50, 34/50, 35/50"….."` are inifinite rational numbers between them.

158.	Find five rational numbers between 1 and 2. We can approach this problem in at least two ways.
Answer» we can rewrite the givens as `1/1 = (16)/(16) = 6/6` `2/1 = (26)/(16)= 12/6` now rational no betwen them are `7/6 , 8/6 , 9/6 ,10/6, 11/6` answer

159.	Simplify `1/(sqrt(5)+sqrt(6)-sqrt(11))`
Answer» `1/(sqrt5+sqrt6-sqrt11) = 1/(sqrt5+sqrt6-sqrt11)**((sqrt5+sqrt6)+sqrt11)/((sqrt5+sqrt6)+sqrt11)` `= ((sqrt5+sqrt6)+sqrt11)/(((sqrt5+sqrt6)^2-sqrt11^2))` `=(sqrt5+sqrt6+sqrt11)/(5+6+2sqrt30 -11)` `=(sqrt5+sqrt6+sqrt11)/(2sqrt30)`, which is the required solution.

160.	Find : (i) `9^(3/2)` (ii) `32^(2/5)` (iii) `16^(3/4)` (iv) `125^((-1)/3)`
Answer» Here, we will use `(a^m)^n = a^(mn)` (i) `9^3/2 = (3^2)^(3/2) = 3^3 = 27` (ii)`32^(2/5) = (2^5)^(2/5) = 2^2 = 4` (iii) `16^(3/4) = (2^4)^(3/4) = 2^3 = 8` (iv)`125^(-1/3) = (5^3)^(-1/3) = 5^-1 = 1/5`

161.	State whether the following statements are true or false. Justify your answers, (i) Every irrational number is a real number.(ii) Every point on the number line is of the form `sqrt(m)`, where m is a natural number.(iii) Every real number is an irrational number.
Answer» (i) It is true as both rational and irrational numbers are real numbers. (ii) It is false as there may ne negative values for m and `sqrt(-m)` is not possible for natural numbers. (iii) It is false as both rational and irrational numbers are real numbers.Real numbers also contain rational numbers.

162.	Locate `sqrt(2)`on the number line.
Answer» `sqrt2 = sqrt(1+1)` `2 = 1+1` `(sqrt2)^2 = (1)^2 + (1)^2` `(sqrt2)^2` is taken as hypotenuse with centre as O, we can make a triangle and locate `sqrt2` OA & OC will be the radii of the circle

163.	Every rational number is an integer.It is true or falseA. a natural numberB. an integerC. a real numberD. a whole number
Answer» Correct Answer - C Since, real numbers are the combination of rational and irrational numbers. Hence, every rational number is a real number.

164.	Simplify : (i) `2^(2/3). 2^(1/5)` (ii) `(1/(3^3))^7` (iii) `(11^(1/2))/(11^(1/4))` (iv) `7^(1/2). 8^(1/2)`
Answer» (i) `2^(2/3).2^(1/5)` As, `a^m.a^n = a^(m+n)` `2^(2/3).2^(1/5)= 2^(2/3+1/5)=2^(13/15)` (ii) `(1/3^3)^7` As, `(a^m)^n)= a^(mn)` `(1/3^3)^7 = 1/3^21 = 3^-21` (iii) `(11^1/2)/(11^1/4)` As, `a^m/a^n = a^(m-n)` `(11^1/2)/(11^1/4)=11^(1/2-1/4) = 11^(1/4)` (iv)`7^(1/2).8^(1/2)` As, `a^m.b^m = (ab)^m` `7^(1/2).8^(1/2) = (7.8)^1/2 = 56^(1/2)`

165.	Simplify each of the following expressions: (i) `(3+sqrt(3))(2+sqrt(2))` (ii) `(3+sqrt(3))(3-sqrt(3))` (iii) `(sqrt(5)+sqrt(2))^2`(iv) `(sqrt(5)-sqrt(2))(sqrt(5)+sqrt(2))`
Answer» (i) `(3+ sqrt3)(2+ sqrt2)` `3(2+ sqrt2) + sqrt3(2+ sqrt2)` `= 6 + 3sqrt2 + 2sqrt3 + sqrt6` (ii) `(3+ sqrt3)(3- sqrt3)` `=3(3-sqrt 3) + sqrt3(3 - sqrt3)` `= 9 cancel( - 3sqrt3) + cancel(3sqrt3) - (sqrt 3)^2` `= 9 - 3 = 6` (iii) `(sqrt 5 + sqrt 2 )^2` `= (sqrt 5)^2 + (sqrt 2)^2 + 2(sqrt5)(sqrt2)` `=5+2+2sqrt10` `= 7 + 2sqrt10` (iv) `(sqrt5 - sqrt2)(sqrt5 + sqrt2)` `= (sqrt 5)^2 - (sqrt2)^2` using identity `(a+b)(a-b) = a^2 - b^2` `=5-2 =3`

166.	Are the following statements true or false? Give reasons for your answers.(i) Every whole number is a natural number.(ii) Every integer is a rational number.(iii) Every rational number is an integer.
Answer» (i) false as, whole no : 0,1,2,3,4..... natural no : 1,2,3,4.... 0 is not a natural no but it is whole no so false (ii) true 1,2, 3 are integer as well as rational no (iii) false 3/5 is a rational no but not integer

167.	Recall, `pi` is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, `pi=c/d`. This seems to contradict the fact the `pi`is irrational How will you resolve this contradiction?
Answer» When we measure circumference`(c)` and `diameter(d)` for any instrument, it is always irrational. So, `c/d` will also be irrational.As, `pi = c/d` , `pi` is also always irrational.

168.	The side BC of a `DeltaABC` is bisected at D. O is any point on AD. BO and CO are produced to meet AC and AB at E & F respectively. AD is produced to C1 so that D is the mid-point of OC1. Prove that FE\|\|BC.
Answer» `square BC_1CO` BC and `OC_1` are bisecting each other `square BC_1CO` is a parallelogram `BC_1\|\|OC and C C_1\|\|BO` `BC_1\|\|FC and C C_1\|\|BE` `C C_1\|\|OE` In`/_ABC_1` `OF\|\|BC_1`(proved) `(AF)/(FB)=(AD)/(OC_1)`(Proporsnality) In`/_AC C_1`(Proved) `(AE)/(EC)=(AF)/(FB)` FE\|\|BC (By P. Theorem).

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