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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 101. |
`2^x =5^y=10^(-z)`. find `(1/x+1/y+1/z)` |
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Answer» Let `2^x = 5^y = 10^-z = k` Taking log for each value, `xlog2 = logk => x = logk/log2 = log_2k => 1/x = log_k2` `ylog5 = log k => y = log_5k => 1/y = log_k5` `-zlog10 = logk => -z = log_10k => 1/z = log_k(1/10)` `:. 1/x+1/y+1/z = log_k2+ log_k5+log_k(1/10) = log_k(2**5**1/10) = log_k1 = 0` `:.1/x+1/y+1/z = = 0` |
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| 102. |
`a(y+z)=x, b(z+x)=y, c(x+y)=z` prove that `x^2/(a(1-bc))=y^2/(b(1-ca))=z^2/(c(1-ab))` |
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Answer» `a=x/(y+z), b=y/(z+x),c=z/(x+y)` `x^2/(a(1-bc))=x^2/(x/(y+z)(1-y/(z+x)xxz/(x+y))` `=(x(x+y)(y+z)(z+x))/(x(z+x+y))` `=((x+y)(y+z)(z+x))/(z+x+y)`equation1 `y^2/(b(1-ac))=y^2/(y/(z+x)(1-x/(y+z)xxz/(x+y))` `=(y(x-y)(y+z)(z+x))/(y(x+y+z))` `=((x+y)(y+z)(z+x))/(z+x+y)`equation2 `z^2/(c(1-ab))=(z^2)/((z/(x+y)(1-x/(y+z)xxy/x-z)` `=(z(x-y)(y-z)(z+x))/(z(y+x+z)` `=((x+y)(y+z)(z+x))/(z+x+y)`equation3 `then` `equation1=equation2=equation3` |
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| 103. |
Locate `sqrt10 and sqrt17` on number line. |
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Answer» We can write, `(sqrt10)^2 = 3^2+1^2` So, we can create a right angle triangle with these three values. On number line, at number `3`, we can create a perpendicular of `1` unit and then, from `0`, we can create an arc to top of perpendicular. Point that the arc will cut on number line will be `sqrt10`. `(sqrt17)^2 = 4^2+1^2` So, we can create a right angle triangle with these three values. On number line, at number `4`, we can create a perpendicular of `1` unit and then, from `0`, we can create an arc to top of perpendicular. Point that the arc will cut on number line will be `sqrt17`. Please refer to video to see the location on number line. |
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| 104. |
`sqrt(10).sqrt(15)` is equal toA. `6sqrt(5)`B. `5sqrt(6)`C. `sqrt(25)`D. `10sqrt(5)` |
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Answer» Correct Answer - B `sqrt(10). sqrt(15) = sqrt(2.5). sqrt(3.5) = sqrt(2) . sqrt(5) . sqrt(3) . sqrt(5)` `= (sqrt(2) . sqrt(3)) (sqrt(5). sqrt(5)) = sqrt(6).5 = 5sqrt(6)` |
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| 105. |
The number obtained on rationalising the denominator of `(1)/(sqrt(7) - 2)` isA. `(sqrt(7) + 2)/(3)`B. `(sqrt(2) - 2)/(3)`C. `(sqrt(7) + 2)/(5)`D. `(sqrt(7) + 2)/(45)` |
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Answer» Correct Answer - A `(1)/(sqrt(7) - 2) = (1)/(sqrt(7) - 2) . (sqrt(7) + 2)/(sqrt(7) + 2)` , [multiplying numerator and denominator by `sqrt(7) + 2`] `= (sqrt(7) + 2)/((sqrt(7))^(2) - (sqrt(2))^(2)) = (sqrt(7) +2)/(7-4) = (sqrt(7) +2)/(3)` [using identity `(a-b) (a+b) = a^(2) - b^(2)`] |
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| 106. |
If the area of a parallelogram ABCD is `30 cm^2`,then the length of altitude AQ is |
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Answer» Area of parallelogram ` A= 30cm^2` We can divide this parallelogram into two triangles of equal area by drawing diagonal `AC`. Please refer to video for the diagram. `:.` Area of `Delta ACD`+Area of `Delta ABC` `= 30` `=> 2**`Area of `Delta ABC= 30` `=>`Area of `Delta ABC= 15` `=>1/2**BC**AQ = 15` `=>6**AQ = 30` `=>AQ = 30/6 = 5cm` |
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| 107. |
Find the value of :`2sin^2[30^@]-3cos^2[30]^@+tan^2[60]^@+3sin^2[90]^@` |
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Answer» `2sin^2 30^@ - 3cos^2 30^@+tan^2 60^@+3sin^2 90^@` `=2(1/2)^2-3(sqrt3/2)^2 +(sqrt3)^2+ 3(1)^2` `=1/2-9/4+3+3` `=(2-9+12+12)/4 = 17/4` |
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| 108. |
Find a and b if both a and b are rational.`(sqrt(2) +sqrt(3))/(3sqrt(2) - 2sqrt(3))=a-bsqrt(6)` |
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Answer» `(sqrt2+sqrt3)/(3sqrt2-2sqrt3)` `=(sqrt2+sqrt3)/((sqrt2sqrt3)(sqrt3+sqrt2))` `=((sqrt2+sqrt3)sqrt6(sqrt3+sqrt2))/(sqrt6(sqrt3-sqrt2)sqrt6(sqrt3+sqrt2)` `=((sqrt3+sqrt2)^2sqrt6)/(sqrt6((sqrt3)^2-(sqrt2)^2)` `=(3+2+2sqrt6)/(6(3-2))` `=(5/6+(2sqrt6)/6)sqrt6` `=5/sqrt6+2` `=5/sqrt6+sqrt6/sqrt6 +2` `=5/6(sqrt6)+2` `=2+5/6(sqrt6)` `a=2` `b=-5/6` |
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| 109. |
Let `M=10^[10^[10^10]`.If M ends with N zeroes and N ends with P zeroes.The number of digits in P are: |
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Answer» `M =10^(10^(10^10))` We know, `10^a` will always have `a` zeroes. `:. N = (10^(10^10))` Similarly,`P = 10^10=10000000000` So, `P` has `11` digits. |
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| 110. |
A, B, C are three towns forming a triangle. A man has to walk from one town to the next town, then ride to the next, and then again drive towards his starting point. He can walk, ride, and drive a km in a, b, c minutes respectively. If he starts from B, he takes a + c - b hours, if he starts from C, he takes b + a - c hours, and if he starts from A, he takes c + b - a hours. The length of the triangle is : |
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Answer» Walking speed of the man `= 60/a**a = 60 ` km/h Riding speed of the man `= 60/b**a = 60a/b`km/h Driving speed of the man `= 60/c**a = 60a/c`km/h Now, let distance from `B` to `C` is `z` km, distance from `C` to `A` is `y` km, distance from `A` to `B` is `x` km. Then,`z/60+(by)/(60a)+(cx)/(60b) = a+c-b->(1)` `y/60+(bx)/(60a)+(cz)/(60b) = b+a-c->(2)` `x/60+(bz)/(60a)+(cy)/(60b) = c+b-a->(3)` Adding (1),(2) and (3), `(x+y+z)/60+b/(60a)(x+y+z) + c/(60a)(x+y+z) = a+b+c` `=>(x+y+z)/60(1+b/(a)+c/(a)) = a+b+c` `=>(x+y+z)/60((a+b+c)/a) = a+b+c` `=>(x+y+z) = 60a` So, length of triangle is `60a` km. |
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| 111. |
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius 20 feets and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point 10 vertical feet above the bottom? |
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Answer» `R=2pi(ratios)/(min)` `Costheta=(R/2)/(R)` `Costheta=1/2` `theta=pi/3 radian` `T=theta/(2theta)=(pi/3)/(2pi)` `T=1/6` minutes `T=1/6*60=10 seconds`. |
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| 112. |
A certain sum of money amounts to Rs. 678 in 2 years and to Rs. 736.50 in `3 1/2` years. The rate of interests is: |
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Answer» `A=P(1+(RT)/100)` `678=P(1+(R*2)/100)-(1)` `736.5=P(1+(R*(7/2))/100)-(2)` dividing equation 1 and 2 `678/736.5=(1+(2R)/100)/(1+(R)/100)` solving this R=6.5% option B is correct. |
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| 113. |
The area of a playground is `3650 m^2`.Find the cost of covering it with the gravel 1.2 cm deep, if the gravel costs rs 6.40 per cubic metre |
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Answer» Volume of the ground covered with gravel `V =` Area of ground * depth of gravel `V = 3650**1.2/100 = 43.8 m^3` Now, cost of gravel in rupees ` = 6.40` per `m^3` `:.` Cost of covering ground with gravel `= 43.8**6.40 = 280.32` Rs |
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| 114. |
If `p=3 and q=2` , then `(3p-4q)^(q-p)div(4p-3q)^(2q-p)` =_______A. 1B. 6C. `(1)/(6)`D. `(2)/(3)` |
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Answer» Correct Answer - C Substitute the values of p and q. |
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| 115. |
The cost of car and a flat together is Rs. 1,04,78,930. If the car cost Rs. 15,23,485. what is the cost of the flat. |
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Answer» Cost of both car and flat in rupees `= 10478930` Cost of car in rupees ` = 1523485` `:.` Cost of flat `= 10478930 - 1523485 = 8955445` Rs |
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| 116. |
The area of equilateral triangle measure `36sqrt(3) cm^2` Find the perimeter of triangle. |
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Answer» Area of equilateral triangle=`sqrt3/4a^2` `36sqrt3=sqrt3/4a^2` `a^2=36*4` `a=6*2=12cm` Perimeter=3a=3*12=36 cm. |
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| 117. |
In the given figure, O is the centre of the circle and BA = AC. If `angleABC=50^@`, find `angleBOC` and `angleBDC.` |
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Answer» BA=AC `/_ABC=/_ACB` `50^@=/_ACB` `In/_ABC` `/_A+/_B+/_C=180^@` `/_A+50+50=180` `/_A=80^@` `/_BOC=2/_A` `/_BOC=2*80=160^@` `/_BAC+/_BDC=180` `80+/_BDC=180` `/_BDC=100`. |
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| 118. |
find three rational numbers between (i) –1 and –2 (ii) 0.1 and 0.11(iii)5/7 and6/ 7 (iv)1/4 and1/ 5A. `-1 and -2`B. 0.1 and 0.11C. `5/7 and 6/7`D. `1/4 and 1/5` |
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Answer» Let y =-1 and x = -2 Here, ` x gt y` and we have to find three ratonal numbers, so n=3 `d= (y-x)/(n+1) = (-1 + 2)/(3+1) = 1/ 4 ` Since, the three, rational numbers x and y are x+ d, x+2d and x+ 3d now , ` x=d =-2 + 1/4 = (-8+1)/4 = -7/4 ` `x+ 2d =-2 + 2/ 4 = (-8 + 2)/4 = -6/4 = (-3)/2` and ` x+ 3d = 2-3/4 = (-8+3)/4 = (-5)/4 ` Hene, three rational numbers between -1 and -2 are `(-7)/4 , (-3) / 2 and (-5)/ 4 ` Alternate method let x=-1 and y =-2 we know, a rational number between x and y `= (x+y)/2 ` A rational number between -1 and -2 `= (-1-2)/2 = -3/2 ` and a rational number between -1 and `- 3/2 = (-1-3/2)/2 = (-2-3)/4 = 5/4 ` Similarly, `-7/4` is a rational number between - 1and -2 Hence, required solution `=3/2,-5/4,-7/4` (ii) let x= 0.1 and y= 0.11 Here, `x lt y ` and we have to find three rational numbers, so consider, n=3 `d= (y-x)/(n+1) = (0.11-0.1)/(3+1) = 0.01/4` Since, the three rational numbers between x and y are (x + d) , ( x + 2d) and ( x+ 3d) now, `x + d = 0.01 + (0.01)/4 = (0.4+0.01)/4 = (0.41)/4 = (0.41)/4 = 0.1025` ` x+ 2d = 0.1 + (0.02)/4 = (0.4+ 0.02)/4 = (0.42)/4 = 0.105` `and x + 3d = 0.1 + 0.03/4 = (0.4+0.03)/4 = (0.43)/4 = 0.1075` Hence , three rational numbers between, 0.1 and 0.11 and 0.1025, 0.105, 0.1075 Also, without using above formula the three rational numbers between 0.1 and 0.11 are 0.101, 0.102, 0.103. (iii) Let , `x= 5/7 and y=6/7` Here, ` x lt y ` Here, we have to find three rational numbers. consider, n=3, `d=(y-x)/(n+1)` `d=(6/7-5/7)/4 = (1/7)/4= 1/28` Since, the three rational number between x and y are ( x+d) , (x+2d) and (x+3d) now, `x+d=5/7+1/28=(20+1)/28= 21/28` `x+2d=5/7+2/28= (20+2)/28= 22/28` ` x+3d=5/7=3/28=(20+3)/28=23/28` Hence, three rational numbers between ` 5/7and 6/7 are 21/28, 22/28,23/28` Also, without using formula, the three rational numbers between `5/7 and 6/7 are 51/70 , 52/70, 53/70` Let ` x=1/5 and y1/4` Here, `xlty` Here, we have to find three rational numbers. consider, , n=3 `d=(y-x)/(n+1)= (1/4-1/5)/(3+1)=((5-4)/20)/4= 1/80` since, the three rational numbers between x and y are x+d , x+2d adn x+3d. `now x+d= 1/5+1/80=(16+1)/80 = 17/80` `x+2d=1/5+2/80=(16+2)/80 = 18/80=9/40` `x+3d = 1/5+3/80=(16+3)/80=19/80` Hence, three rational numbers between ` 1/4 and 1/5 are 17/80 , 9/40,19/80` Alternate method Let ` x=1/4 and y=1/5` so, a rational number between x and `y= (x + y)/2` A rational number between ` 1/4 and 1/5 = (1/4+1/5)/2=((5+4)/20)/2` `9/(2xx20)=9/40` Again, a rational number between `1/4 and 9/40= (1/4 + 9/40)/2= ((10+9)/40)/2=19/(2xx40)=19/80` Again , a rational number between `1/5 and 9/40= (1/5+9/40)/2= ((8+9)/40)/2=(17/40)/2=17/(40xx2)=17/80` Hence, three rational numbers between `1/4 and 1/5 are ,9/40, 19/80, 17/80` |
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| 119. |
Inset a rational number and an irrational number between the following (i) 2 and 3 , (ii) 0 and 0.1, (iii) `1/3 and 1/2` (iv) `(-2)/5 and 1/2 ` , (v) 0.15 and 0.16 , (iv) `sqrt6 and sqrt3` (vii) 2.357 and 3.121 , (vii) .0001 and .001 (ix) 3.623623 and 0.484848 , (x) 3.375289 and 6.375738 |
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Answer» We know that, there are infinitely many rational and irrational values between any two numbers. (i) A rational number between 2 and 3 is 2.1. To find an irrational number between 2 and 3. Find a number which is non-terminatiing non-recuring lying between them. Such number will be 2.040040004.... (ii) A rational number between 0 and 0.1 is 0.03 An irrational number between 0 and 0.1 is 0.007000700007..... (iii) A rational number between `1/3 and 1/2 is 5/12`. an irrational number between `1/3 and 1/2` i.e between `0.bar3and 0.5 is 0.141141114......` (iv) A rational number between `(-2)/5 and 1/2 is 0` An irrational number between `-2/5 and 1/2` i.e between -0.4 and 0.5 is 0.151151115..... (v) A rational number between 0.15 a nd 0.16 is 0.151. An irrational number between 0.15 and 0.16 is 0.1515515551..... (vi ) A rational number between `sqrt2 and sqrt3` i.e between 1.4142 ....... and 1.7320....... is 1.5. An irrational number between `sqrt2 and sqrt3` is 1.585585558....... (vii) A rational number between 2.357 and 3.121 is 3. An irrational number between 2.357 and 3.121 is 3.101101110...... (viii) A rational number between 0.0001 and 0.001 is 0.00011. An irrational number between 0.0001 and 0.001 is 0.0001131331333...... (ix) A rational number between 3.623623 and 0.484848 is 1. An irrational number between 3.623623 and 0.484848 is 1.909009000..... (X) A rational number between 6.375289 and 6.375738 is 6.3753. An irrational number between 6.375289 and 6.375738 is 6.3754141141111........ |
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| 120. |
The following are the steps involved in finding the least amont `sqrt(3),root(3)(4) and root(6)(15)` . Arrange them in sequential order. (A) `therefore root(6)(15)` is the smallest (B) `therefore 3^(1//2)=3^(3//6), 4^(1//3)=4^(2//6), 15^(1//6)=15^(1//6)` (C)The LCM fo the denominators of the exponents is 6 (D) `sqrt(3)=3^(1//2), root(3)(4)=4^(1//3), root(6)(15)=15^(1//6)` (E) `therefore sqrt(3)=root(6)(27), root(3)(4)=root(6)(16)root(6)(15)=root(6)(15)`A. DCABEB. DABEBC. DCBEAD. DBCAE |
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Answer» Correct Answer - C DCBEA is the required sequential order. |
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| 121. |
Give examples of two irrational no. , the product of which is1. A rational no.2. A irrational no. |
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Answer» `1)i)sqrt2 ii)3sqrt2` there product=`sqrt2*3sqrt2=3*2=6` Which is a rational number. `2)i)sqrt3 ii)sqrt5` There product=`sqrt3*sqrt5=sqrt15`. |
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| 122. |
Represent each of the following rational numbers on the number line : `(i) 5 (ii) -3 (iii)5/7` |
| Answer» video shows all the points in video. | |
| 123. |
`(3sqrt(2)-2sqrt(3))/(3sqrt(2)+2sqrt(3))+(sqrt(12))/(sqrt(3)-sqrt(2))` |
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Answer» `(3sqrt2-2sqrt3)/(3sqrt2+2sqrt3)*(3sqrt2-2sqrt3)/(3sqrt2-2sqrt3)+sqrt12/(sqrt3-sqrt2)*(sqrt3+sqrt2)/(sqrt3+sqrt2)` `(18+12-12sqrt6)/(18-12)+(2sqrt3(sqrt3+sqrt2))/(3-2)` `(30-12sqrt6)/6+(6+2sqrt6)/1` `5-2sqrt6+6+2sqrt6` `11. |
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| 124. |
`2[(16-15)^(-1)+25(13-8)^(-2)]^(-1)+(1024)^(0)=`_______.A. 2B. 3C. 1D. 5 |
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Answer» Correct Answer - A Simplify the expression. |
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| 125. |
If a = ` 6 - sqrt 35`, find the value of `a^2 + 1/a^2` |
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Answer» `a = 6-sqrt35` `:. a^2 = 36+35-12sqrt35 = 71-12sqrt35` `1/a^2 = 1/(71-12sqrt35)**(71+12sqrt35)/(71-12sqrt35) = (71+12sqrt35)/(71^2-(12sqrt35)^2)` `=>1/a^2 = (71+12sqrt35)/1` `:. a+1/a^2 = 71-12sqrt35+71+12sqrt35 = 142` |
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| 126. |
Write the conjugate of : (a) `sqrt(3)+sqrt(5)` (b) `5+root(3)(7)` |
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Answer» `sqrt(3)-sqrt(5)` is the conjugate of `sqrt(3)+sqrt(5) and sqrt(5)-sqrt(3)` is also the conjugate of `sqrt(3)+sqrt(5)` (b) `5-root(3)(7)` is conjugate of `5+root(3)(49)` |
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| 127. |
If x=2 and y=4 , then `((x)/(y))^(x-y)+((y)/(x))^(y-x)`=________.`A. 4B. 8C. 12D. 2 |
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Answer» Correct Answer - B Subsitute the values of x and y. |
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| 128. |
Which of the following is the greatest ?A. `7^(2)`B. `(49)^(3//2)`C. `((1)/(343))^(-1//3)`D. `(2401)^(-1//4)` |
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Answer» Correct Answer - B Convert all base into same number. |
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| 129. |
Multiply : `root(3)(5)` by `root(4)(2)`. |
| Answer» Correct Answer - `root(12)(5000)` | |
| 130. |
(a) Find the RF of `2^(1/3)+ 2^(-1/3)`(b) Find the RF of `5^(1/3)-5^(-1/3)` |
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Answer» (a) `2^(1//3)+2^(-1//3)` Let a`=2^(1//3) and b =2^(-1//3)` `a^(3)=(2^(1//3))^(3)=2` `b^(3)=(2^(-1//3))^(3)=2^(-1)=(1)/(2)` But `a^(3)+b^(3)=(a+b)(a^(2)-ab+b^(2))` `therefore a^(2)-ab+b^(2)=(2^(1//3))^(2)-(2^(1//3).2^(-1//3))+(2^(-1//3)+(2^(-1//3))^(2)=2^(2//3)-1+2^(-2//3)` `therefore ` RF of `2^(1//3)+2^(-1//3)` is `2^(2//3)-1+2^(-2//3)` (b) `5^(1//3)-5^(-1//3)` We have `a^(3)-b^(3)=(a-b)(a^(2)+ab+b^(2))` `therefore ` RF of `5^(1//3)-5^(-1//3)` is `[5^(1//3)]^(2)+[5^(1//3).5^(-1//3)]+[5^(-1//3)]^(2)` `=5^(2//3)+1+5^(-2//3)` |
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| 131. |
Let x and y be rational and irrational numbers , respectively. Is x+y necessarily an irrational number ? Give an example in suppot of your answer. |
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Answer» Yes ,(x+y) is necessarily an irrational number . e.g… Let `" " x=2, Y=sqrt3` Then, ` " " x+y =2+sqrt3` if possible, let x+y =2 `+ sqrt3` be a rational number. Consider , ` " " a=2,+sqrt3` On Squaring both sides, we get `a^(2)=(2+sqrt3)^(2) " " [ "unsing identity"(a+b)^(2)=a^(2) + b^(2) + 2ab]` ` implies " " a^(2) = 2^(2) + (sqrt3)^(2))+2(2)(sqrt(3))` `a^(2)=4 +3+4sqrt3 implies (a^(2)-7)/4=sqrt(3)` So, a is rational `Rightarrow (a^(2) -7)/4` is rational ` implies sqrt3` is rational. But, this contradicits the fact that `sqrt3` is an irrational number. thus, our assumption is wrong. Hence, x+ y is an irraional number. |
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| 132. |
If `y=3-sqrt(8)`, then `(y-(1)/(y))^(2)` =_______.A. 9B. 81C. 4D. 32 |
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Answer» Correct Answer - D Given that, `y=3-sqrt(8)` `(1)/(y)=(1)/(3-sqrt(8))=(3+sqrt(8))/((3-sqrt(8))(3+sqrt(8)))` `(3+sqrt(8))/(9-8)=3+sqrt(8)` Now , `(y-(1)/(y))^(2)=(3-sqrt(8)-3-sqrt(8))^(2)` `=(-2sqrt(8))^(2)=32` |
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| 133. |
Which is greater , `sqrt(2)` or `root(3)(3)`? |
| Answer» Correct Answer - `root(3)(3)` | |
| 134. |
Among `sqrt(7)-sqrt(3)andsqrt(11)-sqrt(7)` which is greater? |
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Answer» By rationalizing , `sqrt(7)-sqrt(3)=((sqrt(7)-sqrt(3))(sqrt(7)+sqrt(3)))/(sqrt(7)+sqrt(3))=(4)/(sqrt(7)+sqrt(3))` `sqrt(11)-sqrt(7)=((sqrt(11)-sqrt(7))(sqrt(11)+sqrt(7)))/(sqrt(11)+sqrt(7))=(4)/(sqrt(11)+sqrt(7))` The numberator of each of the irrational number is 4. But `sqrt(11)+sqrt(7) gt sqrt(7)+sqrt(3)` `therefore (4)/(sqrt(7)+sqrt(3)) gt (4)/(sqrt(11)+sqrt(7))` `sqrt(7)-sqrt(3) gt sqrt(11)-sqrt(7)` |
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| 135. |
In which of the following pairs of surds are the given two surds similar /A. `sqrt(5),7sqrt(5)`B. `root(3)(7),root(2)(7)`C. `sqrt(7), sqrt(28)`D. Both (a) and (c) |
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Answer» Correct Answer - D Recall the defination of similar surds. |
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| 136. |
The value of x in the equation: `(x-b-c)/a+(x-c-a)/b+(x-a-b)/c=3` |
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Answer» Taking LCM `(xbc-b^2c-bc^2+xac-ac^2-a^2c+xab-a^2b-ab^2)/(abc)` `x(bc+ac+ab)=b^2c+bc^2+ac^2+a^2c+a^2b+Ab^2+3abc` `x(bc+ab+ac)=bc(a+b+c)+ac(a+b+c)+ab(a+b+c)` `x(ab+bc+ca)=(a+b+c)(bc+ac+ab)` `x=a+b+c`. |
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| 137. |
`(x-y-z)^2-(x^2+y^2+z^2)=2(yz-zx-xy)` |
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Answer» `L.H.S. = (x-y-z)^2-(x^2+y^2+z^2)` `=(x^2+y^2+z^2-2xy-2zx+2yz) - (x^2+y^2+z^2)` `=2yz-2zx-2xy` `=2(yz-zx-xy) = R.H.S.` |
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| 138. |
If `x`is a rational number and `y`is an irrational number, thenboth `x + y a n d x y`arenecessarily rationalboth `x + y a n d x y`arenecessarily irrational`x y`isnecessarily irrational, but `x + y`can beeither rational or irrational`x + y`isnecessarily irrational, but `x y`can beeither rational or irrational |
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Answer» No. (xy) is necessarily an irrational only when `x ne O`. Let x be a non-zero rational and y be an irrational. Then , we have to show that xy be an irrational . If possible, let xy be a rational number. Since, quotient of two non- zero rational so , `((xy)/x)` is a rational number ` Rgihtarrow ` Y is a rational number . But, this contradicts the fact that y is an irrational number. thus, our supposition is wrong. Hence , xy is an irrational number. but, when x=0, then xy=0, rational number. |
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| 139. |
The following are the steps involved in finding the value of `a+b` from `(2+sqrt(3))/(2-sqrt(3))=a+bsqrt(3)` . Arrange them in sequential order. (A) `(2^(2)+(sqrt(3))^(2) + 2xx2xxsqrt(3))/(2^(2)-(sqrt(3))^(2))=a+bsqrt(3)` `a+b=7+4=11` (C) `((2+sqrt(3))(2+sqrt(3)))/((2-sqrt(3))(2+sqrt(3)))=a+bsqrt(3)` (D) `(7+4sqrt(3))/(4-3)=a+bsqrt(3)` (E)=`a=7 and b=4`A. CDAEBB. CAEBDC. CADEBD. CEDAB |
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Answer» Correct Answer - C CADEB is the required sequential order |
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| 140. |
Which of the two expressions , `sqrt(11)-sqrt(10)` and `sqrt(12)-sqrt(11)` is greater? |
| Answer» Correct Answer - `sqrt(11)-sqrt(10)` | |
| 141. |
Which of the following pairs is having two equal values ? (where `x in R`) ______.A. `9^(x//2),24^(x//3)`B. `(256)^(4//x),(4^(3))^(4//x)`C. `(343)^(x//3),(7^(4))^(x//12)`D. `(36^(2))^(2//7),(6^(3))^(2//7)` |
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Answer» Correct Answer - A Simplify the numbers given in options |
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| 142. |
express in rational number `0.bar(621)` |
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Answer» Let`x=0.overline621-(1)` multiply both side by thousand `1000x=621.overline621-(2)` Subtracting equation 1 from equation 2 `999x=621` `x=23/37`. |
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| 143. |
Which of the following are rational or irrational no ;- `(2+sqrt3)^2` , `(5-sqrt3)^2` |
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Answer» 1)`x=(2+sqrt3)^2` `x=4+3+4sqrt3` `x=7+4sqrt3` This is an irrational number 2)`x=(5-sqrt3)^2` `x=25+3-10sqrt3` `x=28-10sqrt3` This is an irrational number. |
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| 144. |
`sqrt(11sqrt(11sqrt(11...........4terms)))`A. `root(16)(11^(5))`B. `root(16)(11)`C. `root(16)(11^(14))`D. `root(16)(11^(15))` |
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Answer» Correct Answer - D `sqrt(asqrt(asqrt(a….)))n` terms `=a^(2^(n)-1)/(2^(n))` |
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| 145. |
The following are the steps involved in finding the value of x-y from`(sqrt(8)-sqrt(5))/(sqrt(8)+sqrt(5))=x-ysqrt(40)`. Arrange them in sequential order. (A) `(13-2sqrt(40))/(8-5)=x-ysqrt(40)` (B) `((sqrt(8))^(2)+(sqrt(5))^(2)-2(sqrt(8))(sqrt(5)))/((sqrt(8))^(2)-(sqrt(5))^(2))=x-ysqrt(40)` (C) `x-y=(11)/(3)` (D) `x=(13)/(3) and y=(2)/(3)` (E) `((sqrt(8)-sqrt(5))(sqrt(8)-sqrt(5)))/((sqrt(8)+sqrt(5))(sqrt(8)-sqrt(5)))=x-ysqrt(40)`A. EABDCB. EBADCC. ABDECD. DEBAC |
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Answer» Correct Answer - B EBADC is the required sequential order. |
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| 146. |
What is the value of `4^(2x-2)`, if `(16)^(2x+3)=(64)^(x+3)` ?A. 64B. 256C. 32D. 512 |
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Answer» Correct Answer - B Find the value of x. |
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| 147. |
Express each of the decimal in p/q form- |
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Answer» 1)Let`x=0.overline35-(1)` Multiply both side by 100 `100x=35.overline35-(2)` subtracting equation 1 from equation 2 `99x=35` `x=35/99` 2)similarly`x=123/999`. |
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| 148. |
Express `0.bar(34) + 0.3bar(4)` as a single decimal.A. `0.67bar(88)`B. `0.6bar(89)`C. `0.68bar(78)`D. `0.6bar(87)` |
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Answer» Correct Answer - D Express them in `(p)/(q)` form. |
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| 149. |
If `(5-sqrt(3))/(2+sqrt(3))=x+ysqrt(3)` , then (x,y) isA. `(13,-7)`B. `(-13,7)`C. `(-13,-7)`D. `(13,7)` |
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Answer» Correct Answer - A Rationalize the denominator of `((5-sqrt(3)))/((2+sqrt(3)))`. |
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| 150. |
if `x^4+1=1297` and `y^4-1=2400 `,then `y^2-x^2=`A. 10B. 25C. 13D. 43 |
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Answer» Correct Answer - C Find x and y |
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