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401.

`x/(2x^(2)+5x+2)=1/6` तो `(x+1/x)` का मान क्या होगा?A. `2`B. `1/2`C. `-1/2`D. `-2`

Answer» Correct Answer - B
`x/(2x^(2)+5x+2)=1/6`
`(x/x)/(2(x^(2))/x+(5x)/x+2/x)=1/6`
`implies 1/(2x+2/x+5)=1/6`
`implies 2(x+1/x)+5=6`
`=6-5=2(x+1/x)`
`implies1-2(x+1/x)`
`implies x+1/x=1/2`
Discussion
402.

यदि `(a+1/a)^(2)=3` हो तो `a^(18)+a^(12)+a^(6)+1` का मान ज्ञात कीजिये?A. 3B. 1C. 0D. 2

Answer» Correct Answer - C
`(a+1/a)^(2)=3`
`a+1/a=sqrt(3)`
`a^(3)+1/(a^(3))=0`
`a^(6)=-1`
`(a^(6)+1)=0`
so `a^(18)+a^(12)+a^(6)+1`
`implies a^(12)(a^(6)+1)+a^(6)+1`
`implies a^(12)xx0+0`
`implies 0`
Discussion
403.

यदि `x^(2)+5x+6=0` हो तो `(2x)/(x^(2)-7x+6)` का मान बताइए?A. `1/6`B. `1/3`C. `-1/6`D. `-1/3`

Answer» Correct Answer - C
If `x^(2)+5x+6=0`
then `x^(2)+6=-5x`
so, ltbgt `(2x)/(x^(2)-7x+6)=(2 x)/(x^(2)+6-7x)`
`=(2x)/(-5x-7x+6)=(2x)/(x^(2)+6-7x)`
`=(2x)/(-5x-7x)=(2x)/(-12x)=(-1)/6`
Discussion
404.

`(113^(2)+115^(2)+117^(2)-113xx115-115xx117-113xx117)` किसके बराबर है ?

Answer» Correct Answer - D
We know that
`(a^(2)+b^(2)+c^(2)-ab-bc-ca)=1/2`
`[(a-b)^(2)+(b-c)^(2)+(c-d)^(2)]`
`:. [(113^(2)+115^(2)+117^(2)-113xx115),(-115xx117-113xx117)]`
`=1/2[(113-115)^(2)+(115-117)^(2))`
`+(117-113)^(2)]`
`=1/2[4+4+16]=12`
Discussion
405.

दो वास्तविक संख्याओं को वर्गें का जोड़ 41 है और उनका अपना जोड़ 9 है तो इन संख्याओं के घनों का जोड़ क्या होगा?A. 169B. 209C. 189D. 198

Answer» Correct Answer - C
We no are `a,b`
`a^(2)+b^(2)=41`
`a+b=9`
`(a+b)^(2)=a^(2)+b^(2)+2ab`
`a^(2)=41+2ab`
`81=41+2ab`
`ab=20`
take `=a=5`
`b=4`
`a^(3)+b^(3)=5^(3)+4^(3)`
`=125+64=189`
Discussion
406.

यदि `x/(a+b)+1=x/(a-b)+(a-b)/(a+b)` है तो `x` किसके बराबर है ?A. `2a-b`B. `a+b`C. `a-b`D. `2a+b`

Answer» Correct Answer - C
`x/(a+b)+1=x/(a-b)+(a-b)/(a+b)`
`x/(a+b)-x/(a-b)=(a-b)/(a+b)-1`
`(ax-bx-ax-bx)/((a+b)(a-b))=(a-b-a-b)/(a+b)`
`(-2bx)/((a+b)(a-b))=(-2b)/((a+b))`
`x/((a+b)(a-b))=1/((a+b))`
`x=((a+b)(a-b))/((a+b))`
`x=(a-b)`
Discussion
407.

The difference between two number is 3 and the difference between their cubes is 999. Find the difference between their squares.1. 632. 363. 814. 18

Answer» Correct Answer - Option 1 : 63

Given-

Difference between two number = 3

Difference between their cubes is 999

Formula Used-

x3 - y3 = (x - y)3 + 3xy(x - y)

(x + y)2 = (x - y)2 + 4xy

x2 - y2 = (x + y)(x - y)

Calculation-

Let the two numbers be x and y

According to Question-

x - y = 3 -------(i)

x3 - y3 = 999  -------------(ii)

⇒ (x - y)3 + 3xy(x - y) = 999

⇒ 27 + 9xy = 999

⇒ xy = 108

Now,

(x + y)2 = (x - y)2 + 4xy

⇒ (x + y)2 = 9 + 432

⇒ x + y = 21

x2 - y2 = (x + y)(x - y)

⇒ x2 - y2 = 21 × 3

⇒ 63

∴ Difference between their squares = 63
Discussion
408.

यदि `x^(2)+y^(2)+2x+1=0` है तो `x^(31)+y^(35)` का मान क्या होगा?A. -1B. 0C. 1D. 2

Answer» Correct Answer - A
`x^(2)+y^(2)+2x+1=0`
`implies x^(2)+2x+1+y^(2)=0`
`(x+1)^(2)+y^(2)=0`
Hence both terms are squares and there addition is zero so, it can be possible only when both terms are zeros.
`:. x+1=0`
`impliesx=-1`
`y=0`
`:. x^(31)+y^(35)`
=`(-1)^(31)+(0)^(35)=-1`
Discussion
409.

यदि `x+1/x=c+1/c` तो `x` मान बताइए?A. `c,1//c`B. `c,c^(2)`C. `c,2c`D. `0,1`

Answer» Correct Answer - A
If `n+1/n=c+1/c`
go through the option
put `n=c, 1/c`
if `n=c`
`c+1/c=` satisfied
if `n=1/c`
then `n+1/n`
`1/c+1/(1//c)=1/c+c` satisfied
Discussion
410.

यदि `x+1/x=6` हो तो `x^(2)+1//x^(2)` का मान ज्ञात कीजिए?A. 23B. 16C. 34D. 32

Answer» Correct Answer - C
`x+1/x=6` on squaring both side
`x^(2)+1/(x^(2))=36-2`
`x^(2)+1/(x^(2))=34`
Discussion
411.

यदि `a-b=1` और `a^(3)-b^(3)=61` तो `ab` का मान क्या होगा?A. -20B. 20C. 30D. 60

Answer» Correct Answer - B
`a^(3)-b^(3)=61`
`(a-b)(a^(2)+ab+b^(2))=61`
(Given `a-b=1`)
`a^(2)+ab+b^(2)=61`……………i
Now
`a-b=1`
On squaring both side
`a^(2)+b^(2)-2ab=1`
`underbrace(a^(2)+b^(2)ab)-3ab=1`
from equ i
`61-3ab=1`
`3ab=60`
`ab=60/3`
`ab=20`
Discussion
412.

यदि `P=999` तो `P(P^(2)+3P+3)` का मान बताइए?A. 9999B. 999999C. 99999D. 999999

Answer» Correct Answer - B
`P=99`
`P(P^(2)+3P+3)`
`=P^(3)+3P^(2)+3P+1-1`
`=(P+1)^(3)-1=(99+1)^(3)-1`
`=(100)^(3)-1=(100)^(3)-1`
`=1000000-1=999999`
Discussion
413.

यदि `x^(3)+y^(3)=35` और `x+y=5` है तो `1/x+1/y` का मान क्या होगा?A. `1/3`B. `5/6`C. `6`D. `2/3`

Answer» Correct Answer - B
`x^(3)+y^(3)=35`
`implies x+y=5`
Take cube on both sides
`(x+y)^(3)=(5)^(3)`
`x^(3)+y^(3)+3xy(x+y)=125`
`35+3xy(5)=125`
`15xy=125-35`
`15xy=90`
`xy=6`
`:. 1/x+1/y=(x+y)/(xy)=5/6`
Discussion
414.

यदि `x=997, y=998` और `z=999` है तो `x^(2)+y^(2)+z^(2)-xy-zx` का मान ज्ञात करें।

Answer» Correct Answer - D
`x^(2)+y^(2)+z^(2)-xy-xz-yz=`
`1/2[(x-y)^(2)+(y-z)^(2)+(z-x)^(2)]`
`=1/2[(997-998)^(2)+(998-`
`(999)^(2)+(999-997)^(2)]`
`=1/2(1+1+4)=3`
Discussion
415.

यदि `p=99` तो `p(p^(2)+3p+3)` का मान क्या होगा?A. 999999B. 1000000C. 1000001D. 999998

Answer» Correct Answer - A
`P(p^(2)+3p+3)`
`=p^(3)+3p^(2)+3p+1-1`
(on adding and subtraction one both side) `=(p+1)^(2)-1`
`(99+1)^(3)-1=100^(3)-1`
`=1000000-1=999999`
Discussion
416.

यदि `2x+1/(4x)=1` हो तो `x^(2)+1/(64x^(2))` का काम क्या होगा?

Answer» Correct Answer - A
`2x+1/(4x)=1`
Dividing by 2 both side
`x+1/(8x)=1/2`
Squaring both side
`x^(2)+1/(64x^(2))+2xx x xx 1/(8x)=1/4`
`x^(2)+1/(64x^(2))=1/4-1/4=0`
Discussion
417.

यदि `a^(3)-b^(3)=56` और `a-b=2` है तो `a^(2)+b^(2)` का मान क्या होगा?A. 48B. 20C. 22D. 5

Answer» Correct Answer - B
`a^(3)-b^(3)=56`
`implies a-b=2`
`implies a^(3)-b^(3)-3ab(a-b)=(2)^(3)`
(By cubing)
`56-3abxx2=8`
`-6ab=8-56`
`6ab=48`
`implies ab=8`
`(a-b)=2`
`(a-b)^(2)=a^(2)+b^(2)-2ab=4`
`=a^(2)+b^(2)=4+2ab`
`a^(2)+b^(2)=4+2xx8=20`
Discussion
418.

यदि `x^(2)-3x+1=0` है तो `x^(2)+x+1/x+1/(x^(2))` का मान ज्ञात करें।A. 10B. 2C. 6D. 8

Answer» Correct Answer - A
`x^(2)-3x+1=0`
`implies x^(2)+1=3x`
`x+1/x=3`
Squaring both sides
`implies x^(2)+1/(x^(2))+2=9`
`x^(2)+1/(x^(2))=7`
`x^(2)+1/(x^(2))+x+1/x`
`=7+3=10`
Discussion
419.

यदि `x=(sqrt(3)+1)/(sqrt(3)-1)` & `y=(sqrt(3)-1)/(sqrt(3)+1)`, है तो `x^(2)+y^(2)` का मान क्या होगा?A. 14B. 13C. 15D. 10

Answer» Correct Answer - A
`(sqrt(3)+1)/(sqrt(3)-)` and `y=(sqrt(3)-1)/(sqrt(3)+1)`
`implies :. x=1/y`
`x=(sqrt(3)+1)/(sqrt(3)-1)xx(sqrt(3)+1)/(sqrt(3)+1)`
`=(sqrt(3)+1)^(2)/(3-1)`
`= (3+1+2sqrt(3))/2=(4+2sqrt(3))/2`
`=(2+sqrt(3))`
`x^(2)=(2+sqrt(3))^(2)=4+3+4sqrt(3)`
`=7+4sqrt(3)`
`y^(2)=1/(7+4sqrt(3))xx(7-4sqrt(3))/(7-4sqrt(3))`
`y^(2)=(7-4sqrt(3))/(49-48)=(7-4sqrt(3))/1=`
`7-4sqrt(3)`
`:. x^(2)+y^(2)=7+4sqrt(3)+7-4sqrt(3)`
Alternate:
`x^(2)+y^(2)=x^(2)+1/(x^(2))`
`=(x+1/x)^(2)-2( :. x=1/y)`
`=(sqrt(3)+1)/(sqrt(3)-1)+(sqrt(3)-1)/(sqrt(3)+1^(2))-2`
`=[((sqrt(3)+1)^(2)+(sqrt(3)-1)^(2))/(3-1)]^(2)-2`
`:. (a+b)^(2)+(a-b)^(2)=2(a^(2)+b^(2))`
`=[(2((sqrt(3))^(2)+(1)^(2)))]^(2)-2`
`=(3+1)^(2)-2=16-2=14`
Discussion
420.

यदि `p=999` है तो `root(3)(p(p^(2)+3p+3)+1)` का मान क्या होगा?A. 1000B. 999C. 998D. 1002

Answer» Correct Answer - A
`P=999`
`root(3)(p(p^(2)+3p+3)+1)`
`impliesroot(3)(p^(3)+3p^(2)+3p+1)`
`:. root(3)((p+1)^(3))`
`=root(3)((999+1)^(3))`
`=root(3)((1000)^(3))=1000`
Discussion
421.

यदि `sqrt(x)-sqrt(y)=1, sqrt(x)+sqrt(y)=17` हो तो `sqrt(xy)=?`A. `sqrt(72)`B. `72`C. `32`D. `24`

Answer» Correct Answer - B
`sqrt(x)-sqrt(y)=1`…………..i
`sqrt(x)+sqrt(y)=17`…………….ii
From eqn i and eqn ii
`sqrt(x)=9, sqrt(y)=8`
So `sqrt(xy)=9xx8=72`
Discussion
422.

`a^(3)-b^(3)=56` और `a-b=2` हो तो `a^(2)+b^(2)` का मान क्या होगा?A. 12B. 20C. 28D. 32

Answer» Correct Answer - B
`a^(3)-b^(3)=(a-b)(a^(2)+b^(2)+ab)`
`56=2(a^(2)+b^(2)+ab)`
`25=a^(2)+b^(2)+ab`…………i
`a-b=2`
`a^(2)+b^(2)-2ab=4`……………..ii
From equation i and eq ii
`a^(2)+b^(2)=20`
OR
put `a=4 , b-2 a-b=4-2`
`=2` (satisfy)
So `a^(2)+b^(2)=16+4=20`
Discussion
423.

यदि `x(3-2/x)=3/x` है तो `x^(2)+1/(x^(2))` का मान क्या होगा?A. `2 1/9`B. `2 4/9`C. `3 1/9`D. `3 4/9`

Answer» Correct Answer - B
`x(3-2/x)=3/x`
`implies 3x-2=3/x`
`implies 3x-3/x=2`
`implies 3(x-1/x)=2`
`implies x-1/x=2/3`
Squaring both sides
`implies x^(2)+1/(x^(2))-2=4/9`
`implies x^(2)1/(x^(2))=4/9+2=2 4/9`
Discussion
424.

यदि `(x+1/x)=4` है तो `x^(4)+1/(x^(4))` का मान ज्ञात करें।A. 64B. 194C. 81D. 124

Answer» Correct Answer - B
`x+1/x=4`
Squaring both sides
`x^(2)+1/(x^(2))+2=16`
`x^(2)+1/(x^(2))=14`
Squaring again
`x^(4)+1/(x^(4))=196-2=194`
Discussion
425.

यदि `x+9/x=6` है तो `(x^(2)+9/(x^(2)))` का मान ज्ञात करें?A. 8B. 9C. 10D. 12

Answer» Correct Answer - C
`x+9/x=6`
Take values of `x`
Let `x=3`
`3+9/3=6` (Proved)
So `x=3`
`:. x^(3)+9/(x^(2))=9+9/9=10`
Alternate:
`x+9/x=6`
On squaring
`(x=9/x)^(2)=36`
`x^(2)+81/(x^(2))+2xx9/x xx x=36`
`x^(2)+81/(x^(2))-18=0`
`(x-9/x)^(2)=0`
`x=9/x`
`x^(2)=9`
Hence `x^(2)+9/(x^(2))=9+9/9=10`
Discussion
426.

यदि `144/0.144=14.4/x` है तो `x` का मान ज्ञात करें?A. 144B. 14.4C. 1.44D. 0.0144

Answer» Correct Answer - D
`144/0.144=14.4/x`
`implies(144xx1000)/144=144/(x xx 10)`
`implies1000=144/(10x)`
`implies x=144/(000xx 10)`
`x=144/1000=0.0144`
Discussion
427.

यदि `A:B:C=2:3:4` है, तो `A/B:B/C:C/A` किसके बराबर है?A. `8:9:16`B. `8:9:12`C. `8:9:24`D. `4:9:16`

Answer» Correct Answer - C
`A:B:C`
`:. A/B:B/C:C/A`
(Multiply with ABC)
`:.(AxxABC)/B:(BxxABC)/C: (CxxABC)/A`
`A^(2)C:B^(2)A:BC^(2)`
`implies (2)^(2)xx4:(3)^(2)xx2:3xx(4)^(2)`
`implies 16:18:48`
`8:9:24`
Alternate :
`A/B:B/C:C/A=2/3:3/4:4/2`
`implies2/3xx12:3/4xx12:4/2xx12`
`implies 8:9:24`
Discussion
428.

यदि `x=3+2sqrt(2)`, है तो `(sqrt(x)1-1/(sqrt(x)))` का मान क्या होगा?A. 1B. 2C. `2sqrt(2)`D. `3sqrt(3)`

Answer» Correct Answer - B
`x=3+2sqrt(2)`
`x=2+1+2sqrt(2)=(sqrt(2)+1)^(2)`
`sqrt(x)=sqrt(2)+1`
`1/(sqrt(x))=1/(sqrt(2)+1)`
`implies1/(sqrt(x))=1/(sqrt(2)+1)xx(sqrt(2)-1)/(sqrt(2)-1)`
हर का ‌परिमेय करने पर
`=(sqrt(2)-1)/1=sqrt(2)-1`
`:. sqrt(x)-1/(sqrt(x))`
`=sqrt(2)+1-(sqrt(2)-1)`
`=sqrt(2)+1-sqrt(2)+1=2`
Discussion
429.

यदि `x=sqrt(3)+1/(sqrt(3))` हो तो `(x-(sqrt(126))/(sqrt(42))) (x-1/(x-(2sqrt(3))/3))` का मान क्या है?A. `5 (sqrt(3))/6`B. `(2sqrt(3))/3`C. `5/6`D. `2/3`

Answer» Correct Answer - C
`x=sqrt(3)+1/(sqrt(3))=(3+1)/(sqrt(3))=4/(sqrt(3))`
So `(x-(sqrt(126))/(sqrt(42)))(x-1/(x-(2sqrt(3))/3))`
`=(4/(sqrt(3))-(sqrt(126))/(sqrt(42)))(x-1/(4/(sqrt(3))-2/(sqrt(3))))`
`=((4sqrt(42)-sqrt(126)xxsqrt(3))/(sqrt(3)xxsqrt(142)))(x-1/(2//sqrt(3)))`
`=(4sqrt(42)-3sqrt(42))/(sqrt(3)xx sqrt(42))(x-(sqrt(3))/2)`
`=(4sqrt(42)-3sqrt(42))/(sqrt(3)xxsqrt(42))(4/(sqrt(3))-(sqrt(3))/2)`
`=((sqrt(42))/(sqrt(3)xxsqrt(42)))((8-3)/(2sqrt(3)))=5/6`
Discussion
430.

यदि `x=3+2sqrt(2)` है तो `x^(2)+1/(x^(2))` का मान क्या होगा?A. 36B. 30C. 32D. 34

Answer» Correct Answer - D
`x=3+2sqrt(2)`
`x^(2)=(3+2sqrt(2))^(2)`
(Squaring both sides)
`=9+8+12sqrt(2)=17+12sqrt(2)`
`1/(x^(2))=1/(17+12sqrt(2))xx(17-12sqrt(2))/(17-12sqrt(2))`
`= 17-12sqrt(2)`
`:. 1/(x^(2))+x^(2)`
`= 17+12sqrt(2)+17-12sqrt(2)=34`
Discussion
431.

`agtb` के लिए यदि `a+b=5` और `ab=6` है तो `(a^(2)-b^(2))` का मान ज्ञात करें?A. 1B. 3C. 5D. 7

Answer» Correct Answer - C
`agtb`
`a+b=5`
`axxb=6` Assume `a=3`
`a^(2)-b^(2)` Values `b=2`
`=9-4=5`
Alternate:
`a+b=5` given
`implies(a+b)^(2)=(5)^(2)`
`a^(2)+b^(2)+2ab=25`
`ab=6` given
`=a^(2)+b^(2)+2xx6=25`
`=a^(2)+b^(2)=13`
`implies a^(2)+b^(2)+2ab-2ab=13`
`a^(2)+b^(2)-2ab=13-2ab`
`(a-b)^(2)=13-2xx6`
`a-b-1`
`:. (a+b)(a-b)`
`a^(2)-b^(2)=5`
Discussion
432.

यदि `a/3=b/4=c/7` है तो `(a+b+c)/c` किसके बराबर है?

Answer» Correct Answer - C
`a/3=b/4=c/7=k`
`:. a=3k`
`b=4k`
`c=7k`
`:.(a+b+c)/c`
`=(3k+4k+7k)/(7k)=2`
Discussion
433.

यदि `x/(x^(2)-2x+1)=1/3` है तो `x^(3)+1/(x^(3))` का मान ज्ञात करें।A. 64B. 110C. 81D. 124

Answer» Correct Answer - B
`x/(x^(2)-2x+1)=1/3`
`implies 1/((x^(2))/x-(2x)/x+1/x)=1/3`
`implies 1/(x+1/x-2)=1/3`
`x+1/x-2=3`
`implies x+1/x=3+2=5`
Take cube on both sides
`implies (x+1/x)^(3)=(5)^(3)`
`implies x^(3)+1/(x^(3))+3xx5=125`
`implies x^(3)+1/(x^(3)=110`
Discussion
434.

`(x^(2)+x+1)/(x^(2)+x+1)=3/2` है `(x+1/x)` का मान क्‍या है?A. 5B. `-5`C. 6D. 8

Answer» Correct Answer - B
`(x^(2)-x+1)/(x^(2)+x+1)=3/2` (Given)
`implies({(x+1/x)-1})/(x{(x+1/x)+1})=3/2`
Let `(x=1/x=y)`
`implies ((x+1/x)-1)/((x+1/x)+1)=3/2`
`(y-1)/(y+1)=3/2`
`implies 2(y-1)=3(y+1)`
`2y-2=3y+3`
`y=-2-3=-5`
`:.x+1/x=-5`
Discussion
435.

यदि `x=5+2sqrt(6)`, है तो `(sqrt(x)+1/(sqrt(x)))` का मान क्या होगा?A. `2sqrt(2)`B. `3sqrt(2)`C. `2sqrt(3)`D. `sqrt(3)`

Answer» Correct Answer - C
`x=5+2sqrt(6)`
`x=3+2+2sqrt(3)xxsqrt(2)`
`x=(sqrt(3))^(2)+(sqrt(2))^(2)+2sqrt(3)xxsqrt(2)`
`x=(sqrt(3)+sqrt(2))^(2)`
`sqrt(x)=sqrt(3)+sqrt(2)`
Similarly
`implies1/(sqrt(x))=sqrt(3)-sqrt(2)`
`sqrt(x)+1/(sqrt(x))=sqrt(3)+sqrt(2)+sqrt(3)-sqrt(2)`
`=2sqrt(3)`
Discussion
436.

यदि `(2a-5b)/(3a+6b)=4/7` है तो `a,b` किसके बराबर है?A. `21:36`B. `2:59`C. `59:2`D. `36:21`

Answer» Correct Answer - C
`(2a-5b)/(3a+6b)=4/7`
`implies7(2a-5b)=4(3a+6b)`
`implies14a-35b=12a+24b`
`implies2a=59b`
`a/b=59/2`
`:. A:b=59:2`
Discussion
437.

यदि `(a+1/a)^(2)=3` है तो `a^(3)+1/(a^(3))=?`A. `2sqrt(3)`B. `2`C. `3sqrt(3)`D. 0

Answer» Correct Answer - D
`(a+1/a)^(2)=3`
`a+1/a=sqrt(3)`
Take cube on both sides
`(a+1/a)^(3)=(sqrt(3))^(3)`
`=a^(3)+1/(a^(3))+3axx1/a(a+1/a)=3sqrt(3)`
`=a^(3)+1/(a^(3))+3sqrt(3)=3sqrt(3)`
`=a^(3)+1/(a^(3))=0`
Discussion
438.

यदि `a^(2)+b^(2)+c^(2)=2(a-b-c)-3` है तो `4a-3b+5c` का मान ज्ञात करें।A. 2B. 3C. 5D. 6

Answer» Correct Answer - A
`a^(2)+b^(3)+c^(2)=2(a-b-c)-3`
`implies a^(2)+b^(2)_c^(2)_2a+2b+2c+1+1`
`+1=0`
`implies a^(2)-2a+1+b^(2)+2b+1+c^(2)+2c`
`+1=0`
`implies(a-1)^(2)+(b+1)^(2)+(c+1)^(2)=0`
`((darr,darr,darr),(x,y,z))`
यह तभी संभव है जब `x,y` तथा `z` का मान 0 होगा क्योंकि इनकी घातांक सम संख्या है जो कभी किसी भी संख्या को –ve नहीं आने देगी अतः `x,y` एवं `z` का मान 0 ही संभव हो।
`implies a-1=0, a=1`
`implies b=1=0 , b=-1`
`implies c+1=0, c=-1`
`=4xx1+3xx-1+5xx-1`
`=4+3-5=2`
Discussion
439.

यदि `x=3+sqrt(8)` है तो `x^(2)+1/(x^(2))` किसके बराबर होगा?A. 38B. 36C. 34D. 30

Answer» Correct Answer - C
`x=3+sqrt(8)`
`x^(2)=9+8+2xx3sqrt(8)`
`x^(2)=17+6sqrt(8)`
`1/(x^(2))=17-6sqrt(8)`
`x^(2)+1/(x^(2))=17+6sqrt(8)+17-6sqrt(8)=34`
Discussion
440.

`(x^(4)+3x^(3)+5x^(2)+3x+1)/(x^(4)+1)` का मान ज्ञात करें।A. `43/23`B. `47/21`C. `41/23`D. `45/21`

Answer» Correct Answer - A
`x+1/x=5, x^(2)+1/(x^(2))+2=25`
`x^(2)+1/(x^(2))=23`
Now
`(x^(4)+3x^(3)+5x^(2)+3x+1)/(x^(4)+1)`
divided by `x^(2)`
`((x^(4))/(x^(2))+(3x^(3))/(x^(2))+(5x^(2))/(x^(2))+(3x)/(x^(2))+1/(x^(2)))/((x^(1))/(x^(2))+1/(x^(2)))`
`=(x^(2)+3x+5+3x+1/(x^(2)))/(x^(2)+1/(x^(2))`
`=(x^(2)+1/(x^(2))+3(x+1/x)+5)/(x^(2)+1/(x^(2)))`
`=(23+3(5)+5)/23=43/23`
Discussion
441.

यदि `x:y=2:1` है तो `(5x^(2)-13xy+6y^(2))` किसके बराबर होगाA. `3/4`B. `4/3`C. `0`D. `55/4`

Answer» Correct Answer - C
`x:y`
`2:1`
then `5x^(2)-13xy+6y^(2)`
`implies5xx4-13xx2xx1+6xx1^(2)`
`implies20-26+6=0`
Discussion
442.

यदि `xgt1` और `x^(2)+1/(x^(2))=83` है तो `x^(3)-1/(x^(3))=?`A. 764B. 750C. 756D. 760

Answer» Correct Answer - C
`x^(2)+1/(x^(2))=83`
Subtracting 2 from both sides
`implies x^(2)+1/(x^(2))-2=83-2`
`implies (x+1/x)^(2)=81`
`implies x-1/x=9`
Take cube on both sides
`x^(3)-1/(x^(3))-3(x-1/x)=729`
`x^(3)-1/(x^(3))-3xx9=729`
Take cube on both sides
`x^(3)-1/(x^(3))-3(x-1/x)=729`
`x^(3)-1/(x^(3))-3xx9=729`
`x^(3)-1/(x^(3))=729+27=756`
Discussion
443.

यदि `x^(2)-4x-1=0` के लिए `x` एक वास्तविक संख्या है तो `x^(2)+1/(x^(2))` का मान क्या होगा?A. 4B. 10C. 12D. 18

Answer» Correct Answer - D
`x^(2)-4x-1=0`
`x^(2)-1=4x` (divide `x` both sides)
`x-1/x=4`
`x^(2)+1/x-2=16`
`x^(2)+1/(x^(2)=18`
Discussion
444.

यदि `(x-a)(x-b)=1` और `a-b+5=0` है तो `(x-a)^(3)1/((x-a)^(3))` का मान ज्ञात करें।A. -125B. 1C. 125D. 140

Answer» Correct Answer - D
`(x-a)(x-b)=1`
`(x-a)=1/((x-b))`
`implies a-b=-5` (given)
add and subtract `x`
`a-b+x-x=-5`
`(a-x)+(x-b)=-5`
`(x-b)-(x-a)=-5`
`(x-a)-(x-b)=+5`
`:. (x-a)=1/((x-b)=+5`
`:. (x-a)-1/((x-a))=+5`
Taking cube on both sides
`(x-a)^(3)-1/((x-a)^(3))-3(x-a)xx1/((x-a))`
`((x-a)-1/((x-a)))=(5)^(3)`
`implies (x-a)^(3)-1/((x-a)^(3))-3xx(5)=125`
`(x-a)^(3)-1/((x-a)^(3))=125+15=140`
Discussion
445.

यदि `(5x)/(2x^(2)+5x+1)=1/3, (x+1/(2x))` का मान ज्ञात करें।A. 15B. 10C. 20D. 5

Answer» Correct Answer - D
`(5x)/(2x^(2)+5x+1)=1/3`
`5/((2x^(2))/x+(5x)/x+1/x)=1/3`
`5/(2x+1/x+5)=1/3`
`(2x+1/x+5)=15`
`2x+1/x=10`
Divide by 2 both sides
`x+1/(2x)=10/2=5`
Discussion
446.

यदि `x=(sqrt(3))/2` है तो `(sqrt(1+x))/(1+sqrt(1+x))+(sqrt(1-x))/(1-sqrt(1-x)` किसके बराबर होगा?A. 1B. `2//sqrt(3)`C. `2-sqrt(3)`D. `2`

Answer» Correct Answer - B
`x=(sqrt(3))/2`
or `1+x=1+(sqrt(3))/2=(2+sqrt(3))/2`
`=(2(2+sqrt(3)))/(2xx2)`
(divides and multiply by 2)
`implies 1+x=(4+2sqrt(3))/4`
`=(1+3+2sqrt(3))/4`
`=((1)^(2)+(sqrt(3))^(2)+2xx1xxsqrt(3))/4`
`1+ximplies((1+sqrt(3))^(2))/4`
`:.sqrt(1+x)=(1+sqrt(3))/2`
Similarly,
`sqrt(1-x)implies(sqrt(3)-1)/2`
`:. (sqrt(1-x))/(1+sqrt(1+x))+(sqrt(1-x))/(1-sqrt(1-x))`
`=((1+sqrt(3))/2)/(1+(1+sqrt(3))/2)+((sqrt(3)-1)/2)/(1-(sqrt(3)-1)/2)`
`implies (1+sqrt(3))/(3+sqrt(3))+(sqrt(3)-1)/(3-sqrt(3))`
`= (1+sqrt(3))/(sqrt(3)(sqrt(3)+1))+(sqrt(3)-1)/(sqrt(3)(sqrt(3)-1))`
`=1/(sqrt(3))+1/(sqrt(3))=2/(sqrt(3))`
Discussion
447.

`(a^(2)+4b^(2)+4b-4ab-2a-8)` के गुणज है?A. `(a-2b-4)(a-2b+2)`B. `(a-b-2)(a+2b+2)`C. `(a+2b-4)(a+2b+2)`D. `(a+2b-4)(a-2b+2)`

Answer» Correct Answer - A
`a^(2)+4b^(2)+4b-4ab-2a-8`
`=a^(2)-4ab+4b^(2)-2a+4b-8`
`=(a-2b)^(2)-2(a-2b)-8`
put `t=a-2b`
`=t^(2)-2t-8`
`=t^(2)-4t+2t-8`
`=t(t-4)+2(t-4)`
`=(t+2)(t-4)`
`=(a-2b-4)(a-2b+2)`
(Put the value of assume `t`)
Discussion
448.

यदि `x,y` दो धनात्मक वास्तविक संख्याऐं हैं और `x^(1/3)=y^(1//4)` है तो निम्न से कौन सा संबंध सत्य है?A. `x^(3)=y^(4)`B. `x^(3)=y`C. `x=y^(4)`D. `x^(20)=y^(15)`

Answer» Correct Answer - D
`x^(1//3)=y^(1//4)`
`implies` LCM of ` 3,4=12`
`:. (x^(1//3))^(12)implies(y^(1//4))^(12)`
`x^(4)=y^(3)`
take power 5 on both sides
`implies(x^(4))^(5)=(y^(3))^(5)`
`implies x^(20)=y^(15)`
Discussion
449.

यदि `(x-1)` और `(x+3),x^(2)+k_(1)x+k_(2)` के गुणज है तोA. `k_(1)=-2,k_(2)=-3`B. `k_(1)=2,k_(2)=-3`C. `k_(1)=-2,k_(2)=3`D. `k_(1)=-2,k_(2)=3`

Answer» Correct Answer - B
If `(x-1)` and `(x+3)` are the factors of `x^(2)+k_(1)x+k_(2)`
`:.` when `(x-1)=0`
`x=1`
`x^(2)+k_(1)x+k_(2)=0`
`1+k_(1)+k_(2)=0`
`k_(1)+k_(2)=-1`………..i
When `(x+3)=0`
`x=-3`
`9-3k_(1)+k_(2)=0`
`-3k_(1)+k_(2)=-9`................ii
From equation i and ii
`k_(1)=2, k_(2)=-3`
Discussion
450.

यदि `x:y=3=3:4` है तो `(7x+3y): (7x-3y)` किसके बराबर हैA. `5:2`B. `4:3`C. `11:3`D. `37:19`

Answer» Correct Answer - C
`x:y=3:4`
`(7x+3y)/(7x-3y)=y/y((7 x/y +3)/(7 x/y -3))=(7xx 3/4+3)/(7xx 3/4-3)`
`(21/4+3)/(21/4-3)=((21+12)/4)/((21-12)/4)=11/3`
Discussion