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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.
| 51. |
An ant starts from the top rim of the cylinder section of a pipe and reaches a point directly below its starting point after making three complete revolutions around the pipe. The distance traveled by the ant if the height of the cylinder is 24 cm and radius of the pipe is cm `3/pi` is : |
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Answer» here the formed rectangle has length`= 2 pi r = 6` breadth is `= 24` `AB=CD=BC= 10` total distance= `10xx 3 = 30 cm` option A is correct Answer |
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| 52. |
If `a+b-c=14` then find the value of `2b^2c^2+2c^2a^2+2a^2b^2-a^4-b^4-c^4` |
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Answer» `=4a^2b^2-(a^4+b^4+c^4-2c^2a^2-2b^2c^2+2a^2b^2)` `=(2ab)^2-(a^2=b^2-c^2)^2` `=(2ab-(a^2+b^2-c^2))(2ab+a^2+b^2-c^2)` `=(2ab-a^2-b^2+c^2)(2ab+a^2+b^2-c^2)` `=(c^2-(a-b)^2)((a+b)^2-c^2)` `=(c-(a-b))(c+(a-b))((a+b)-c)(a+b+c)` `=(c-a+b)(c+a-b)(a+b-c)(a+b+c)` `a+b-c=14` `14(c-a+b)(c+a-b)(a+b+c)` `a=7,b=7,c=0`. |
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| 53. |
If three one rupee coins can be placed around a ten-paise coin such that each one rupee coin touches the ten paise coin as well as the other two one-rupee coins, the ratio of the radii of the one rupee and ten paise coin is : |
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Answer» Let `R` is the radius of 1-rupee coin and `r` is the radius of 10 paise coin. We can create a diagram with the given details. If we join centers of all 1-rupees coin, it will form an equilateral triangle `ABC`. Please refer to video for the diagram. If `O` is the center of 10 paise coin, then, `/_OAB = /_OBA = 60/2 = 30^@` Here, `AB = 2R and OA = r+R`. So, we can say that, `2(r+R)cos30^@ = 2R` `(r+R)/R = 2/sqrt3` `r/R + 1= 2/sqrt3=> r/R = 2/sqrt3-1` `r/R = (2-sqrt3)/sqrt3 = (2sqrt3-3)/sqrt3`, which is the required ratio. |
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| 54. |
If `(2.3)^x=(0.23)^y=1000` then find the value of `x` and `y`. |
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Answer» `(2.3)^x=1000` Take log on both side `log_10(2.3)^x=log_10(1000)` `xlog_10(2.3)=3log_10 10=3` `x=3/(log2.3)` `(0.23)^y=1000` Take log both side `ylog(0.23)=log1000` `y=3/(log(0.23))`. |
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| 55. |
`sin 2 theta = 2 sin theta cos theta` |
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Answer» `sin(A+B)=sinAcosB+cosAsinB` `A=theta,B=theta` `sin(2theta)=sintheta*costheta+costheta*sintheta` `=2sinthetacostheta`. |
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| 56. |
If `sin x+cosec x=2` then `sin^n x +cosec^n x=?` |
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Answer» `sinx+cosecx = 2` Taking square on both sides, `sin^2x+cosec^2x+2sinxcosecx = 4` `=>sin^2x+cosec^2x+2 = 4` (As `sinxcosecx = 1`) `=>sin^2x+cosec^2x = 2` Similarly, we can show that, `sin^4x+cosec^4x = 2` Now, `sin^3x+cosec^3x = (sinx+cosecx)(sin^2x+cosec^2x-sinxcosecx)` `=2(2-1) = 2` `:. sin^nx+cosec^nx = 2` |
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| 57. |
` ( sin theta tan theta)/( 1- cos theta) = 1 + sec theta` |
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Answer» LHS `(sin^2theta)/costheta*1/(1-costheta)*(1+costheta)/(1+costheta)` `sin^2theta/(costheta(1-cos^2theta))*(1+costheta)` `sin^2theta/sin^2theta*(1+costheta)/costheta` `1+sectheta` RHS. |
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| 58. |
`sin theta/cosec(theta)+cos(theta)/sec (theta)=1` |
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Answer» LHS=`sintheta/(cosectheta)+costheta/sectheta` `=sinthetasintheta+costhetacostheta` `=sin^2theta+cos^2theta=1=RHS` |
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| 59. |
If `theta` is an acute angle such that `sec^2 theta=3`, then the value of `(tan^2 theta-cosec^2 theta)/(tan^2 theta+cosec^2 theta)` is |
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Answer» `sec^2 theta = 3` `:. cos^2theta = 1/3 => cos theta = 1/sqrt3` `:. sin^2theta = 1- cos^2theta = 1-1/3 = 2/3` `:. tan^2theta = sin^2theta/cos^2theta = 2` `cosec^2theta = 1/sin^2theta = 3/2` `:. (tan^2theta-cosec^2theta)/(tan^2theta+cosec^2theta) =(2-3/2)/(2+3/2) = (1/2)/(7/2) = 1/7 ` |
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| 60. |
prove that : sin 48 sec 42 + cos 48 cosec 42 =2 |
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Answer» LHS=`sin48/cos42+cos48/sin42` `=(sin48sin42+cos48cos42)/(sin42cos42)` `=cos(48-42)/(sin42cos42)` `=(2cos6)/sin84` `=2cos6/cos6` `=2=RHS` hence proved. |
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| 61. |
A puts Rs 600 more in a business than B, but B has invested his capital for 5 months while A has invested his for 4 months. If the share of A is Rs.48 more than that of B out of the total profits of Rs.528, find the capital contributed by each? |
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Answer» Let `B` invests `x` rupees. Then, amount invested by `A` is `x+600` Rs. Total amount invested by `B` in `5` months in rupees ` = 5x` Total amount invested by `A` in `4` months in rupees ` = 4(x+600) = 4x+2400` Total amount invested by both `A` and `B` in rupees ` = 5x+4x+2400 = 9x+2400` Now, total profit is `528` Rs and share of `A` is Rs `48` more than that of `B`. `:. (4x+2400)/(9x+2400)**528 = (5x)/(9x+2400)**528 +48` `=> (4x+2400)/(9x+2400)**11 = 48((5x)/(9x+2400)**11 +1)` `=> (4x+2400)/(9x+2400)**11 = (5x)/(9x+2400)**11 +1` `=>44x+26400 = 55x+9x+2400` `=>20x = 24000` `=>x = 1200` So, amount invested by `B = 1200` Rs Amount invested by `A = 1200+600 = 1800` Rs. |
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| 62. |
If `x^2 + y^2=6xy` prove that `2log(x+y) = logx+ logy + 3log2` |
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Answer» `x^2+y^2 = 6xy` `=>(x+y)^2-2xy = 6xy` `=>(x+y)^2 = 8xy` Taking log both sides, `log(x+y)^2 = log(8xy)` `=>2log(x+y) = log8+logx+logy` `=>2log(x+y) = log2^3+logx+logy` `=>2log(x+y) = 3log2+logx+logy` |
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| 63. |
what will be the remainder when `5^97` is divided by `52` |
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Answer» `5^(97) = 5^96*5 = (5^4)^24*5 = (625)^24*5` `:. 5^97/52 = ((625)^24*5)/52` If we divide, `625` by `52`, then remainder is `1`. `:.` Remainder when `(625)^24*5` is divided by `52 = 1^24*5 = 5` |
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| 64. |
the zeroes of quedratic polynomiac p(x)= `6x^2+mx+2n` are -3/2 and 4/3 evaluate the value of m and n . |
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Answer» `p(x) = 6x^2+mx+2n` Here, sum of zeroes ` = -m/6` Product of zeroes ` = (2n)/6 = n/3` `:. -m/6 = -3/2+4/3` `=>-m/6 = (-9+8)/6 = -1/6` `=> m = 1` Also, `n/3 = -4/3**3/2` `=>n/3 = -2` `=> n = -6` `=>m = 1 and n = -6.` |
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| 65. |
A hollow sphere of internal and external radii 6 cm and 8 cm respectively, is melted and recast into small cones of base radius 2 cm and height 8 cm.Find the number of cones. |
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Answer» Volume of sphere=N*Volume of 1 cone `4/3pi(R^3-r^3)=N*1/3pir^2h` `4/3*22/7*(8^3-6^3)=N*1/3*22/7*2^2*8` `N=37`. |
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| 66. |
In Fig. 10.64, BC is a tangent to the circle with centre O.OE bisects AP. Prove that `Delta ABC ~ Delta AEO` |
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Answer» `/_AEO and /_PEO` OE=OE(same) CP=EA(given) OA=OD(radius) `/_AEo cong /_PEO` `/_AEO=/_PEO=90^0` `/_AEO and /_ABC` `/_A=/_A`(common) `/_AED=/_ABC=90^0` `/_AEO cong /_ABC (A A similarity)` |
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| 67. |
Triangle ABC is circumscribed touching a circle at P,Q,R, If AP = 4 cm, BP = 6 cm, AC = 12 cm and BC = x cm, find the value of x |
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Answer» We can create a diagram with the given details. Pleae refer to video for the diagram. we know, length of tangents from any common point outside a circle are always equal. `:. AP = AR = 4` cm `BP = BQ = 6` `CR = CQ` We are given,`AC = 12` cm `=>AR+CR = 12=> CR = 12-AR =>CR = 12-4 = 8` cm Now, `BC = BQ+QC = 6+CR =>x= 6+8 = 14` cm |
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| 68. |
A(6,1),B(8,2) and C(9,4) are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of `Delta` ADE. |
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Answer» P is MP of AC and BC P=`((6+9)/2,(1+4)/2)` `(15/2,5/2)=((x+8)/2,(y+2)/2)` `x+8=15` `x=7` `y+2=5` `y=3` D(7,3) Area=`1/2(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)` `=1/2(6(3-7/2)+7(7/2-1)+8(1-3))` Area`=3/4unit^2`. |
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| 69. |
The leastnumber of square tiles required to pave the ceiling of a room 15 m 17 cm longand 9 m 2 cm broad is(a)656 (b) 738 (c) 814 (d) 902 |
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Answer» Correct Answer - 814 Side of each square tile = HCF ( 1517 cm, 902 cm) = 41 cm. Required number of tiles = `((1517 xx 902)/(41 xx 41)) = 814` |
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| 70. |
Out of 7 consonants and 4 vowels. how many words of 3 consonant and 2 vowels can be formed? |
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Answer» 7C->3C 4V=2V Number of ways of selecting C and V=`7C_3*4C_2` `=(7*6*5)/(3*2)*(4*2)/(2*1)=210` Number of ways arraange words=`5!=120` Total words=120*210=25200. |
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| 71. |
solve for x: `9x^2-6b^2x-(a^4-b^4)=0` |
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Answer» `x=(6b^2pmsqrt(6b^2)^2-4*9(-a^4-b^4))/(2*9)` `x=(6b^2pm6a^2)/18` `x=(b^2pma^2)/3` `x=(b^2+a^2)/3,(b^2-a^2)/3`. |
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| 72. |
A train enters into a tunnel AB at A and exits at B. A jackal is sitting at 0 in another by-passing tunnel AOB, which is connected to AB at A and B, where OA is perpendicular to OB. A cat s sitting at P inside the tunnel AB making the shortest possible distance between O and P, such that AO=30 km and PB=32 km. when a train before entering into the tunnel AB makes a whistle (or siren’) somewhere before A ,the Jackal and cat run towards A. they meet with accident with the train) at the entrance A. The ratio of speeds of jackal and cat is |
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Answer» OP`_|_`AB AO=30km PB=32km `/_AOB=90^0` `/_OBA=theta` `tantheta=(OA)/(OB)=30/(OB)` `Sintheta=(OA)/(AB)=30/(AP+32)` `In/_APO` `Cos(90-theta)=(AP)/(AO)` `sintheta=(AB)/(30)-(2)` `(AP)/30=30/(AP+32)` `AP(AP+32)=30*30` `(AP)^2+32(AP)-900=0` `AP=(-32pmsqrt(3^3+3600))/2` `=(-32pm68)/2=36/2,-50` AP=18km. `OA=V_v*t` `PA=V_c*t` `V_c/V_v=(PA)/(AO)=18/30=3/5`. |
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| 73. |
A man went to market with Rs. 100 in hand. He bought 3 kgs potatoes, y kgs of tomatoes and 4 kgs of cucumber. The amount spent on tomatoes was double than that spent on cucumbers and also is same as one third of that spent on potatoes. After the shopping spree, the man was left with Rs. 10in his pocket. The sum of the rates of tomatoes, potatoes and cucumber is Rs. 27.5. |
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Answer» The man hasd `100` Rs in hand and he was left with `10` Rs after shopping. So, he spent `90` Rs. Let the rate of the potatoes is `P` Rs per kg, rate of tomatoes is `T` Rs per kg and rate of cucumber is `C` Rs per kg. `:. 3P+yT+4C = 90->(1)` Also, `yT = 2(4C) =>yT = 8C->(2)` Also, `yT = 1/3(3P) =>yT = P->(3)` From (1), (2) and (3), `3(8C)+8C+4C = 90` `=>36C = 90` `=>C = 2.5` `=>P = 8C = *(2.5) = 20` It is alos given that, `P+T+C = 27.5` `=>20+2.5+T = 27.5` So, ratio of rate of tomotoes and rate of tomoatoes `= 5/20 = 1/4` So, statement `(A)` is true. `=>T = 5` |
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| 74. |
In an examination, 51% candidate failed in physics and 43% failed in mathematics, 14% failed in both the subjects. If the total number of candidates who passed in physics alone are 580, then the total number of candidates is:(A). 2100 (B). 2000 (C). 2500 (D). None of the above |
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Answer» Number of candidates who failed in Maths `= 43%` Number of candidates who failed in both Physics and Maths ` = 14%` `:. `Number of candidates who failed only in Maths ` = 43-14 = 29%` It means the number of candidates who passed only in Physics will also be equal to `29%`. Let `x` is the number of totals candidates. Then, `x**29/100 = 580=> x = 20**100 = 2000` So, total number of candidates are `2000`. |
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| 75. |
25 trees are planted in a straight line 5 metre apart from each other. To water them the gardener must bring water for each tree separately from a well 10 metre from the first tree in line with the trees. The distance he will move in order to water all the trees beginning with the first if he starts from the well is : |
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Answer» Total distance that the gardener need to move to water all the trees will be given by, `S = 2(10+(10+5)+(10+10)+(10+15)+...(10+5(24)) -(10+5(24))` `S = 2(10+15+20+...+130)-130` Sum of AP, `S_m = (a+l)/2*n ` Here, `a =10, l = 130 and n = 25` `:. S = (2**(((10+130))/2**25))-130 = 3500-130 = 3370` m So, required distance is `3370 m`. |
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| 76. |
The number 3.14636363….isA. an integerB. a rational numberC. an irrational numberD. none of these |
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Answer» Correct Answer - B The number 3.24636363 … is a nonterminating repearing decimal. So, it a rational number. |
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| 77. |
2.13113111311113…. IsA. an integerB. a rational numberC. an irrational numberD. none of these |
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Answer» Correct Answer - C 2.131113111311113 …. Is a nonterminating nonrepeating decimal , so it is irrational |
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| 78. |
`2.overline(35) ` isA. an integerB. a rational numberC. an irrational numberD. none of these |
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Answer» Correct Answer - B ` 2.bar(35) = 2.353535`…., which is a repeating decimal. `2.bar(35)` is rational. |
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| 79. |
` pi ` isA. an integerB. a rational numberC. an irrational numberD. none of these |
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Answer» Correct Answer - C `pi` is an irrational number. |
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| 80. |
Which of the following is an irrational number?A. `22/7`B. 3.141141114C. `3.overline(1416)`D. 3.141141114… |
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Answer» Correct Answer - D 3.141141114 … is a nonterminating m norepeating decial, so, it is irrational. |
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| 81. |
A, B and C are collinear, and B is between A and C.The ratio of AB to AC is 2:5. If A is at (-6,9) and B is at (-2,3), what are the coordinates of point C? |
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Answer» Here, ratio of `AB` to `AC` is `2:5`. `:. B` divides `AC` into `2:3` ratio. Let `(x,y)` are the coordinates of `C`. Then, `(2(x)+3(-6))/(2+3) = -2 and (2(y)+3(9))/(2+3) = 3` `=>(2x+(-18))/5 = -2 and (2y+27)/5 = 3` `=>2x = -10+18 and y = 2y = 15-27` `=>x = 4 and y = -6` So, coordinates of `C` are `(4,-6)`. |
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| 82. |
find q and r for the following pairs of positive integers a and b satisfying` a = bq+r``(1)a =13 , b=13 ``(2)a=8 , b=20``(3)a=125 , b=5``(4)a=132 ,b=11` |
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Answer» given that, ` a= b*q +r ` in option 1 , given that `a= 13, b=13 ` `13= 13*q + r` `13*(1-q) = r` `if q= 0, r= -13` `if q=1, r= 0` `if q= 2 , r=-13` not possible `:.` 2 solutions can be possible. for option 2. given that `a= 8 , b=20` `8= 20*q+ r` `4(2-5*q)= r` `if q= 0 . r=8` `if q=1, r=-12 ` not possible so. 1 solution is possible. `a= 125 & b= 5` `125 = 5*q + r` `5*(25-q) = r` `if q=0, r=125` `if q=1, r=24*5` `if q=2, r= 23*5` `..` `..` `..` `..` `if q=25, r= 0` `if q= 26 , r=-ve` so, 25 solutions are possible.`a= 132, b=11` `132= 11*q + r` `132 - 11*q = r` `11*(12- q) = r` `q= 0 , r= 12*11` `q=1 , r= 11*11` `q=2 , r = 10*11` `..` `..` `q=12, r=0` `q= 13 , r= -ve` so, 12 solutions are possible. |
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| 83. |
If the point `P(3,4)` is equidistant from the points `A(a+b,b-a` and `B(a-b,a+b)`, then prove that `3b-4a=0` |
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Answer» `AP^2=(a+b-3)^2+(b-a-4)^2` `=(a+b)^2-6(a+b)+a+(b-a)^2-8(b-a)+16` `BP^2=(a-b-3)^2+(a+b-4)^2` `=(a-b)^2-6(a-b)+a+(a+b)^2-8(a+b)+16` `AP=BP` `AP^2=BP^2` `-6(a+b)-8(b-a)=-6(a-b)-8(a+b)` `8(a+b-b+a)+6(a-b-a-b)=0` `16a-12b=0` `3b-4a=0`. |
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| 84. |
Find the units digit of `8^25`. |
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Answer» Possibilites of units digy of `8^n` ar `8,4,2` and 6. the units digit of `8^n` gets repeated for every `4^th` power of 8. `therefore` Remainder obtained when 25 is divided by 4 is 1. `8^25 =8^(4xx6xx1)cong8^1`. `therefore` Units digit `8^25 is 8`. |
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| 85. |
Roshan wanted to type the first 190 whole numbers. Find the number of times he had to press the numbered keys. |
| Answer» Number of signle digit whole number from 0 and 9 i.e., number of times he has to press the keys `=10xx1 =10`. | |
| 86. |
Raj wanted to type the first `200` natural numbers,how many times does he have to press the keysA. 400B. 365C. 492D. 489 |
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Answer» Correct Answer - C (i) Count the number of digits appear from 1 to 200, when written at a stretch. (ii) For each single, double and triple digit number we have to press keys once, twice and thrice respectively. (iii) There are 9 single digit , 90 double digit and 101 triple digit number from 1 to 200. |
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| 87. |
There are 20 balls. The balls are numbered consecutively starting from anyone of the numbers from 1 to 20. For any case, the sum of the numbers on all the balls will be a/anA. Odd numberB. Even numberC. Prime numberD. Cannot say |
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Answer» Correct Answer - B Use the properties of even odd numbers. |
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| 88. |
Find the HCF and express as linear combination of them : 1288 and 575 |
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Answer» LCM(1288)=2*575+138 LCM(575)=138*4+23 HCF(1288,575)=23. |
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| 89. |
Solve the following quadratic equation by applying the quadratic formula`p^2x^2+(p^2-q^2)x-q^2=0` |
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Answer» `x = (-(p^2 - q^2) +- sqrt((p^2- q^2)^2 + 4p^2q^2))/(2p^2) ` `= (q^2 - p^2 +- sqrt(p^4 + q^4-2p^2q^2 + 4p^2q^2))/(2p^2)` `= (q^2 - p^2 +- sqrt(p^4 + q^2 + 2p^2q^2))/(2p^2)` `= (q^2 - p^2 +- (p^2 + q^2))/(2p^2)` `x = (q^2 - p^2 + p^2 + q^2)/(2p^2)` and `x= (q^2 - p^2 - p^2 - q^2)/(2p^2)` so, `x= q^2 /p^2 and x= -1` Answer |
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| 90. |
Which of the following numbers are divisible by 3? (a) `121212(b) 505550 (c) 4132 (d) 453052 (e)97621 (f) 182391 (g) 165651 (h)16 8681` |
| Answer» Correct Answer - (a),(f),(g),(h) | |
| 91. |
Given `l=28, S=144`, and there are total 9 terms. Find a |
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Answer» l=28 s=144 n=9 `S_n=n/2(a+l)` `144=9/2(a+28)` a=4. |
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| 92. |
Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be meaused an exact number of times, using any of the rods. |
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Answer» Correct Answer - 9.6 m Required length = LCM ( 64m, 80 m , 96 cm) = 960 cm , 9.6 m |
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| 93. |
Every rational number can be represented by some point on the number line, is this statement true or false? |
| Answer» Correct Answer - True | |
| 94. |
Obtain (i) one rational number and (ii) there rational numbers between `(1)/(3)and (1)/(2)`. |
| Answer» Correct Answer - N//A | |
| 95. |
How many rational numbers exist between any two rational numbers ? |
| Answer» Correct Answer - Infinite rational number exist between any two rational number. | |
| 96. |
Is `pi` a rational or an irrational numbers? |
| Answer» Correct Answer - Irrational number | |
| 97. |
`("Divided -Remainder")/("Divisor")=`_________ |
| Answer» Correct Answer - Quotient | |
| 98. |
How many prime number are there between 1 and 50 ? |
| Answer» Correct Answer - 15 | |
| 99. |
show that there is no value of n for which ` ( 2^(n) xx 5 ^(n))` ends in 5. |
| Answer» `(2^(n) xx5^(n)) = ( 2xx5)^(n) = 10^(n)` , which always ends in a zero. | |
| 100. |
`x^2+3x^2+3x+1` |
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Answer» `f(x)=x^3+3x^2+3x+1` `g(x)=x+1` dividing f(x) by g(x) f(x)/g(x)=`x^2+2x+1`. |
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