758 + Interview Questions in GENERAL QA IN ALGEBRA Page 3 InterviewSolution

101.	यदि `x+(1//x)=2` है तो `x^(7)+(1//x^(5))` का मान ज्ञात करें।A. `2^(5)`B. `2^(12)`C. `2`D. `2^(7)`
Answer» Correct Answer - C `x+1/x=2` Find `x^(7)+1/(x^(5))=2` `x+1/x=2` `implies ((darr,darr),(1,1))implies "Let" x=1` `implies x^(7)+1/(x^(5))implies 1^(7)+1/(1^(5))` `implies 1+1=2`

Discussion

102.	`(4x^(2)+8x)` में क्या जोड़ा जाए कि परिणाम पूर्ण वर्ग हो जाए।A. `2x`B. 2C. 1D. 4
Answer» Correct Answer - D Given expression `implies 4x^(2)+8x` Let `P` should added `implies 4x^(2)+8x+p` `implies (2x)^(2)+2xx(2x)xx2` `[(a+b)^(2)=a^(2)b^(2)+2ab]` Term that shold be added `=2^(2)=4` `P=4`

Discussion

103.	यदि `x=1/(2+sqrt(3)),y=1/(2-sqrt(3))` है तो `8xy(x^(2)+y^(2))` का मान क्या होगा?A. 112B. 194C. 290D. 196
Answer» Correct Answer - A According to the question `implies x=1/(2+sqrt(3)),y=1/(2-sqrt(3))` `implies x=1/(2+sqrt(3))xx(2-sqrt(3))/(2-sqrt(3))` `y=1/(2-sqrt(3))xx(2+sqrt(3))/(2+sqrt(3))` `implies x=2-sqrt(3),y=2+sqrt(3)` `8xy(x^(2)+y^(2))` `=8(2-sqrt(3))(2+sqrt(3))[(2-sqrt(3))^(2)+(2+sqrt(3))^(2)]` `implies 8xx1[7-2sqrt(3)+7+2sqrt(3)]=112`

Discussion

104.	यद‌ि `a/b=7/9,b/c=3/5` है तो `a:b:c` मान क्या होगा?A. `7:9:15`B. `7:9:5`C. `21:35:45`D. `7:3:15`
Answer» Correct Answer - A `a/b=7/9,b/c=3/5` `:. a:b:c` `7:9` `3_(x3):5_(x3)` `7:9:15`

Discussion

105.	यद‌ि `x=-2k` and `y=1-3k` है तो `k` के ‌ किस मान के लिए `x=y` है?
Answer» Correct Answer - B `y=1-3k` and `x=-2k` (given) `:.` for `x=y` `-2k=1-3k` `k=1`

Discussion

106.	यद‌ि `1/4xx2/6xx3/8xx5/12xx……..xx31/64=1/(2^(x))` है तो `x` का मान क्या होगा?A. 31B. 32C. 36D. 37
Answer» Correct Answer - C `1/4xx2/6xx3/8xx4/10xx5/12xx……xx` `31/64=1/(2^(x))` `implies(1/2)^(30)xx(1/2)^(6)` `=(1/2)^(30+6)=1/(2^(x))` or `1/(2^(36))=1/(2^(x))` `:.x=36`

Discussion

107.	यद‌ि `a^(2)+a+1=0` है तो `a^(9)` का मान क्‍या है?A. 2B. 3C. 1D. 0
Answer» Correct Answer - C `a^(2)+a+1=0` `[(a^(3)+1^(3)=(a+1)(a^(2)-a+1))/(a^(3)-1=(a-1)(a^(2)+1+a))]` `:. (a^(3)-1)=(a-1)xx0` `a^(3)-1=0` `a^(3)=1` `(a^(3))^(3)=1^(3)` `a^(9)=1`

Discussion

108.	यद‌ि `x+2/x=1` है तो ` (x^(2)+x+2)/(x^(2)(1-x))` का मान ज्ञात करें।A. 1B. -1C. 2D. -2
Answer» Correct Answer - A `x+2/x=1` `x^(2)+2=x` `x^(2)-x=-2` `x-x^(2)=2` `:.(x^(2)+x+2)/(x^(2)(1-x))=` divide & multiply by `x` `implies((x^(2))/x+x/x+1/x)/((x^(2))/x(1-x))=(x+2/x+1)/(x(1-x))` `implies x+(2/x+1)/(x-x^(2))=(1+1)/2=1`

Discussion

109.	यद‌ि `x-1/x=1` है तो `(x^(4)-1/(x^(2)))/(3x^(2)+5x-3)` का मान ज्ञात करें।A. `1/4`B. `1/2`C. `3/4`D. 0
Answer» Correct Answer - B `x-1/x=1` `implies (x^(4)-1/(x^(2)))/(3x^(2)-5x-3)` Divide and multiply by `x` `((x^(4))/x-1/(x^(3)))/((3x^(2))/x+(5x)/x-3/x)` `=(x^(3)-1/(x^(3)))/(3x-3/x+5)implies(x^(3)-1/(x^(3)))/(3(x-1/x)+5)` `implies x-1/x=1` Take cube on both sides `(x-1/x)^(3)=(1)^(3)` `implies x^(3)-1/(x^(3))-3(x-1/x)=1` `x^(3)-1/(x^(3)-3)=1` `implies x^(3)-1/(x^(3))=4` `= (x^(3)-1/(x^(3)))/(3(x-1/x)+5)=4/(3xx1+5)=4/8=1/2`

Discussion

110.	यद‌ि `x^(2)+1/(x^(2))=66` है तो `(x^(2)-1+2x)/x=?`A. `+-8`B. `10,-6`C. `6,-10`D. `+-4`
Answer» Correct Answer - B `x^(2)+1/(x^(2))=66` `implies x^(2)+1/(x^(2))-2=66-2` `(x-1/x)^(2)=64` `(x-1/x)^(2)=(8)^(2)` `implies x-1/x=+-8` `implies (x^(2)-1+2x)/x=((x^(2))/x-1/x+(2x)/x)/(x/x)` `implies (x-1/x+2)/1` When `x-1/x=+8` Then `(x-1/x)+2=8+2=10` When `x-1/x=-8` `-8+2=-6` `:. (10,-6)`

Discussion

111.	यद‌ि `x+y=15` है तो `(x-10)^(3)+(y-5)^(3)` का मान क्‍या है?A. 25B. 125C. 625D. 0
Answer» Correct Answer - D `x+y=15` `implies x-10=5-y` `x-10=-(y-5)` Take cube on both sides `implies (x-10)^(3)=(-(y-5))^(3)` `implies (x-10)^(3)+(y-5)^(3)-0`

Discussion

112.	यद‌ि `a+1/b=b+1/c=c+1/a(a!=b!=c)` है तो `abc` का मान क्‍या है?A. `+-1`B. `+-2`C. `0`D. `+-1/2`
Answer» Correct Answer - A `a+1/b=b+1/c=c+1/a` To save your time assume values of `a,b,c` according toequation. Let `a=2, b=-1` & `c=1/2` `2+1/(-1)=-1+1/(1//2)=1/2+1/2` `=1=1=1` `:. abc=2xx-1xx 1/2=-1`

Discussion

113.	`x^(2)+3Qx-2Q` का गुणज `(x-2)` है तो `Q` का मान क्‍या है?A. 2B. -2C. 1D. -1
Answer» Correct Answer - D `(x-2)` is a factor of `x^(2)+3Qx-2Q` for `(x-2)=0` `x^(2)+3Qx-2Q=0` `implies 4+3xxQxx2-2xxQ=0` `implies 4+6Q-2Q=0` `4Q=-4` `Q=-1`

Discussion

114.	यदि `a^(3)+b^(3)=9` और `a+b=3` है तो `1/a+1/b`का मान ज्ञात करें।A. `1/2`B. `3/2`C. `5/2`D. `-1`
Answer» Correct Answer - B `a^(3)+b^(3)=9` `a+b=3` Assume values `a=2, b=1` `:. (2)^(3)+1=9` `2+1=3` `:. 1/a+1/b=1/2+1=3/2`

Discussion

115.	यदि `xgt1` और `x+1/x=2 1/12` है तो `x^(4)-1/(x^(4))` का मान ज्ञात करें।A. `58975/20736`B. `59825/20736`C. `57985/20736`D. `57895/20736`
Answer» Correct Answer - A `x+1/x=2 1/12=25/12` `x^(2)+1/(x^(2))+2=675/144` `x^(2)+1/(x^(2))=625/144=2` `x^(2)+1/(x^(2))=(625-288)/144` `x^(2)+1/(x^(2))=337/144-2` `(x-1/x)^(2)=(337-288)/144=49/144` `x-1/x=7/12` `:. (x+1/x)(x-1/x)=25/12xx7/12=175/144` `:. (x^(2)-1/(x^(2))=175/144` `:. x^(4)=(x^(2)+1/(x^(2)))(x^(2)-1/(x^(2)))` `=175/144xx337/144=58975/20736`

Discussion

116.	यदि `xy=x^(2)+y^(2)-xy`, है तो `911` का मान क्या होगा?A. 93B. 103C. 113D. 121
Answer» Correct Answer - B `xy=y=x^(2)+y^(2)-xy` `911=(9)^(2)+(11)^(2)-11xx9` `81+121-99=103`

Discussion

117.	यद‌ि `5^(5x+5)=1` है, तो `x` किसके बराबर है?A. `0`B. `-1`C. `1`D. `-4/5`
Answer» Correct Answer - B `5^(5x+5)=1` `5^(5x+5)=5^(@)(a^(@)=1)` (If bases are same then power are equal) `5x+5=0` `5x+5=0` `5x=-5` `x=-1`

Discussion

118.	यदि `a` और `b` धनात्मक पूर्णांक इस प्रकार है `a^(2)-b^(2)=19` है तो `a` का मान ज्ञात करें।A. 19B. 20C. 9D. 10
Answer» Correct Answer - D `a^(2)-b^(2)=19` `(a-b)(a+b)=19xx1` `a+b=19to` ………….i `a-b-1to`……………….ii `2a=20` `a=10` (Adding equation i and ii)

Discussion

119.	यदि `2^(2x-y=)12` और `2^(x+y)=32` है `xy` का मान ज्ञात करें?A. 2B. 4C. 6D. 8
Answer» Correct Answer - C `2^(2x-y)=16` (given) `implies2^(2x-y)=2^(4)` `:. 2x-y=4` ………..i `2^(x+y)=32` (given) `2^(x+y)=2^(5)` and `x+y=5` ………….ii by equation i and ii `2x-y=4` `(x+y=5)/(3x=9)` `x=3` `:. y=2, xy=3xx2=6`

Discussion

120.	यदि `x=6+1/x` है तो `x^(4)+1/(x^(4))+2` का मान ज्ञात करें।A. 1448B. 1442C. 1444D. 1446
Answer» Correct Answer - C `x=6+1/x` `implies x=1/x=6` Taking square on both sides `x^(2)+1/(x^(2))-2=36` `implies x^(2)+1/(x^(2))=38` Again taking square on both sides `x^(4)+1/(x^(4))+2=(38)^(2)` `implies x^(4)+1/(x^(4))+2=1444` `x^(4)+1/(x^(4)+2)=1444`

Discussion

121.	यदि `x+1/x=3` है तो `(3x^(2)-4x+3)/(x^(2)-x+1)` का मान ज्ञात करें।A. `4/3`B. `3/2`C. `5/2`D. `5/3`
Answer» Correct Answer - C `k+1/x=3` `(3x^(2)+3-4x)/(x^(2)+1-x)=((3x^(2))/x+3/x-4)/((x^(2))/x+1/x-x/x)=(3(x+1/x)-4)/((x+1/x)-1)` `=(3xx3-4)/(3-1)=(9-4)/(3-1)=5/2`

Discussion

122.	ग्राफ `2x+1=0` तथा `3y-9=0` का प्रतिछेदी बिन्दु होगा?A. `(-1/2,-3)`B. `(-1/2,3)`C. `(1/2,-3)`D. none of these
Answer» Correct Answer - B `2x+1=0, x=(-1)/2` `3y-9=0, y=+9/3=3` `:. (x,y)=((-1)/2,3)` Point of intersection

Discussion

123.	यदि `x^(2)+9y^(2)=6xy` हो तो `x:y` होगाA. `1:3`B. `3:2`C. `3:1`D. `2:3`
Answer» Correct Answer - C `x^(2)+9y^(2)=6xy` `implies x^(2)+9y^(2)-6xy=0` `implies x^(2)+(3y)^(2)-2xx 3y xx x=0` `implies(x-3y)^(2)=0` `x=3y=0` `implies x=3y` `:. x:y` `3:1`

Discussion

124.	यदि `x=y=333` और `z=34` है तो `x^(3)+y^(3)+z^(3)-3xyz` का मान क्या होगा?
Answer» Correct Answer - C `x=y=333,z=334` `implies x^(3)+y^(3)+z^(3)-3xyz=1/2` `(x+y+z)((x-y)^(2)+(y-z)^(2)+(z-x)^(2))` `implies 1/2(333+333+334)[(333-333)^(2)+(333-334)^(2)+(334-333)^(2)]` `implies 1/2(1000)(0+1+1)=1000`

Discussion

125.	यदि `C+1/C=sqrt(3)` है तो `C^(3)+1/(C^(3))` का मान किसके बराबर है?
Answer» Correct Answer - A `C+1/C=sqrt(3)` On cubing both side `(C+1/C)^(3)=3sqrt(3)` `C^(3)+1/(C^(3))+3xxCxx1/C(C+1/C)=3sqrt(3)` `C^(3)+1/(C^(3))+3sqrt(3)=3sqrt(3)` `C^(3)+1/(C^(3))=3sqrt(3)-3sqrt(3)=0`

Discussion

126.	यदि `a+b+c=0` है तो `a^(3)+b^(3)+c^(3)` किसके बराबर है?A. `a+b+c`B. `abc`C. `2abc`D. `3abc`
Answer» Correct Answer - D If `a+b+c=0` then `a^(3)+b^(3)+c^(3)-3abc=0` `a^(3)+b^(3)+c^(3)=3abc`

Discussion

127.	यदि `x+1/x=2,x!=0` है तो `x^(2)+1/(x^(3))` का मान किसके बराबर है?A. 1B. 2C. 3D. 4
Answer» Correct Answer - B `x+1/x=2 x!=0` put `x=1` `1+1-1` `:. x^(2)+1/(x^(3))=1+1=2`

Discussion

128.	यदि `x=0.5` और `y=0.2` है तो `sqrt(0.6)xx(3y)^(x)` का मान किसके बराबर होगा?A. `1.0`B. `0.5`C. `0.6`D. `1.1`
Answer» Correct Answer - C `x=0.5` `y=0.2` `sqrt(0.6)xx(3y)^(x)=sqrt(0.6)xx(3xx0.2)^(0.5)` `= sqrt(0.6)xxsqrt(0.6)=0.6`

Discussion

129.	यदि `a-b=3` और `a^(3)-b^(3)=27` है तो `\|a+b\|` किसके बराबर है?A. 3B. 5C. 7D. 9
Answer» Correct Answer - A `a-b=3` `a^(3)-b^(3)=27` For Exams assume the values of `agtb` Let `a=3,b=0` `a-b=3-0=3` and `(3)^(3)-0=27` Hence `a+b=3+0=3`

Discussion

130.	निम्न में से कौन सा संबंध सही नही है:A. `(x+3)^(2)=x^(2)+6x+9`B. `(x+2y)^(3)=x^(3)+8y^(3)+6xy(x+2y)`C. `(x+2)^(2)=x^(2)+2x+4`D. `(x+3)(x-3)=x^(2)-9`
Answer» Correct Answer - C From option (c) LHD `(x+2)^(2)=x^(2)+4x+4` RHS `=x^(2)+2x+4` `:.` LHS `!=` RHS

Discussion

131.	यदि `a+b+c+d=4` है तो `1/((1-a)(a-b)(a-c))+1/((a-b)(a-c)(a-d))+1/((a-c)(1-d)(1-a))+1/((1-d)(1-a)(1-b))` का मान ज्ञात करें।
Answer» Correct Answer - A `a+b+c+d=4` `1/((1-a)(a-b)(1-c))+1/((1-b)(1-c)(1-d))` `+1/((1-c)(1-d)(1-a))+1/((1-d)(1-a)(1-b))` Put `a=0,b=0` and `c=2` and `d=2` `a+b+c+d=0+0+2+2=4=4` (satisfy) `1/((1-0)(1-0)(1-2))+1/((1-0)(1-2)(1-2))` `+1/((1-2)(1-2)(1-0))+1/((1-2)(1-0)(1-0))` `implies 1/(-1)+(1/(+1))+1/(-1xx-1)+1/(-1)` `implies -1+1+1-1=0`

Discussion

132.	यदि `a^(1//3)+b^(1/3)+c^(1/3)=0` है तो निम्न में से `a,b,c` के बीच कौन सा संबंध सही है?A. `a+b+c=0`B. `(a+b+b+c)^(3)=0`C. `a+b+c=3abc`D. `a^(3)+b^(3)+c^(3)=0`
Answer» Correct Answer - B `a^(1/3)+b^(1/3)+c^(1/3)=0` `implies a^(1/3)+b^(1/3)=-c^(1/3)` Take cube on both sides `(a^(1/3)+b^(1/3))^(3)=(-c^(1/3))^(3)` `implies a+b+3a^(1/3)b^(1/3)(a^(1/3)+b^(1/3))=-c` `implies a+b+3a^(1/3)b^(1/3)xx(-c^((+1)/3))=-c` `implies a+b+c=3a^(1/3)b^(1/3)c^(1/3)` again taking cube `(a+b+c)^(3)=27abc`

Discussion

133.	`121a^(2)+64b^(2)` में कौन सा पद जोड़ा जाए कि परिणाम पूर्ण वर्ग हो जाए।A. `176 ab`B. `276 a^(2)b`C. `178 ab`D. `188 b^(2)a`
Answer» Correct Answer - A `121a^(2)+64b^(2)` `=(11a)^(2)+(8b)^(2)+2xx11axx8b` `=(11a+8b)^(2)` `:.` So term added to make perfect squarre `=176ab`

Discussion

134.	यदि `P(x+y)^(2)=5` और `q(x-y)^(2)=3` तो `p^(2)(x+y)^(2)+4pq xy –q^(2)(x-y)^(2)` का सरलीकृत मान क्या होगा?A. `2(p+q)`B. `-(p+q)`C. `-2(p+q)`D. `p+q`
Answer» Correct Answer - A `p(x+y)^(2)=5` and `q(x-y)^(2)=3` Put the value of `x=2` and `y=1` `p(2+1)^(2)=` and `q(2-1)^(2)= 3` `p=5/9, q=3` `top^(2)(x+y)^(2)+4pqxy-q^(2)(x-y)^(2)` `=(5/9)^(2)(2+1)^(2)+4xx5/9xx3xx2xx` `1-(3)^(2)(2-1)^(2)` `=25/81xx9+40/3-9` `=25/9+40/3-9` `=25/9+40/3-9` `=(25+120-80)/9=64/9` Put the value of `p` and `q` option a Option `Ato2(p+q)` `=2(5/9+3)=2xx32/9=64/9` OPtion `A` is satisfied So `2(p+q)` is answer.

Discussion

135.	यदि `x+1/x=3` तो `x^(5)+1/(x^(5))` का मान बताइए-A. 123B. 110C. 132D. 122
Answer» Correct Answer - A::C `x+1/x=3` (Cube both sides) Formula: `x^(5)+1/(x^(5))=(x^(3)+1/(x^(3)))(x^(2)+1/(x^(2))-(x+1/x)` `x^(3)+1/(x^(3))+3x xx1/x(x+1/x)=27` `x^(3)+1/(x^(3))=18` `x+1/x=3` (squaring both sides) `x^(2)+1/(x^(2))+2xx x xx1/x=9` `x^(2)+1/(x^(2))=7` Put in formula, `x^(5)+1/(x^(5))=18xx7-3=123`

Discussion

136.	यदि `x^(2)=x=5` है तो `(x+3)^(3)+1/((x+30^(3))`A. 140B. 110C. 130D. 120
Answer» Correct Answer - B According to the question `=[(x+3)^(3)+1/((x+3)^(3))]` Let `t=x+3,x=t-3` `x^(2)+x=5` (given) `(x+1)x=5` `(t-3+1)(t-3)=5` `(t-2)(t-3)=5` `t^(2)-3t-2t+6=5` `t^(2)-5t=-1` `t+1/t=5` `(x+3)^(3)+1/((x+3)^(3))` `=t^(3)+1/(t^(3))` `=(t+1/t)^(3)-3xx txx 1/t(t+1/t)` `=5^(3)-3xx5=125-15` `=110=(x+3)^(3)+1/((x+3)^(3))=110`

Discussion

137.	यदि `x=z=225` और `y=226` है तो `x^(3)+y^(3)+z^(3)-3xyz` का मान ज्ञात करें।A. 765B. 676C. 674D. 576
Answer» Correct Answer - B According to the question `:, x=z=225` `y=226` `implies x^(3)+y^(3)+z^(3)-3xyz=?` `x^(3)+y^(3)+z^(3)-3xyz=1/2(x+y+z)` `[(x-y)^(2)+(y-z)^(2)+(z-x)^(2)]` `=1/2[225+225+226][(225-226)^(2)+` `(+226-225)^(2)+(225-225)^(2)]` `=676/2xx[1+1+0]=676`

Discussion

138.	यदि `x=222,y=223,z=225` हो तो `x^(3)+y^(3)+z^(3)-3xyz` का मान क्या है?A. 4590B. 4690C. 4950D. 4960
Answer» Correct Answer - B `x^(3)+y^(3)+z^(3)-3xyz=1/2(x+y+` `z)[(x-y)^(2)+(y-z)^(2)+(z-x)^(2)]` `=1/2(222+223+225) (222-223)^(2)` `+(223-225)^(2)+(225-222)^(2)]` `=1/2(970)[1+4+9]` `=1/2(670)[1+4+9]` `=1/2xx670xx14=4690`

Discussion

139.	यदि `x=a-b,y=b-c,z-c-a` है तो बीजगणितीय व्यंजक `x^(3)+y^(3)+z^(3)-3xyz` का आंकिक मान ज्ञात करें।A. `a+b+c`B. `0`C. `4(a+b+c)`D. `3 abc`
Answer» Correct Answer - B `x=a-b` `y=b-c` `z=c-a` `x+y+z=a-b+b-c+c=a=0` `:. x^(3)+y^(3)+z^(3)-3xyz=(x+y+z)` `(x^(2)+y^(2)+z^(2)-xy-yz-zx)=0`

Discussion

140.	यदि `x=1-sqrt(2),` है तो `(x-1/x)^(3)` का मान क्या होगा?A. -84B. 8C. `2sqrt(2)`D. 1
Answer» Correct Answer - B `x=1-sqrt(2)` `1/x=1/(1-sqrt(2))xx(1+sqrt(2))/(1+sqrt(2))` हर का परिमेयकरण `=(1+sqrt(2))/(1-2)=(1+sqrt(2))/(-1)=-(sqrt(2)+1)` `:. x-1/x=1-sqrt(2)+sqrt(2)+1=2` Take cube `(x-1/x)^(3)=(2)^(3)implies(x-1/x)^(3)=8`

Discussion

141.	यदि `x+y+z=6` और `x^(2)+y^(2)+z^(2)=20` है तो `x^(3)+y^(3)+z^(3)-3xyz` का मान ज्ञात करें।A. 64B. 70C. 72D. 76
Answer» Correct Answer - C `x+y+z=6` `x^(2)+y^(2)+z^(2)=20` `implies (x+y+z)^(2)=(6)^(2)` `implies x^(2)+y^(2)+z^(2)+2(xy+yz+zx)=36` `implies 20+2(xy+yz+zx)=36` `implies 2(xy+yz+zx)=16` `xy+yz+zx=8` `implies x^(3)+y^(3)+z^(3)-3xyz=` `(x+y+z)(x^(2)+y^(2)+z^(2)-xy-zx-yz)` `implies x^(3)+y^(3)+z^(3)-3xyz=6(20-8)` `=6xx12=72`

Discussion

142.	यदि `x=(sqrt(5)-sqrt(3))/(sqrt(5)+sqrt(3))` और `y=(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))` है तो `(x^(2)+xy+y^(2))/(x^(2)-xy+y^(2))=?`A. `65/63`B. `67/65`C. `69/67`D. `63/61`
Answer» Correct Answer - D Given `x=(sqrt(5)-sqrt(3))/(sqrt(5)+sqrt(3)),y=(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))` Find: `(x^(2)+y^(2)+xy)/(x^(2)+y^(2)-xy)=?` `implies (x^(2)+y^(2)+2xy-xy)/(x^(2)+y^(2)-2xy+xy)` `((x+y)^(2)-xy)/((x-y)^(2)+xy)=?` Now `x+y=((sqrt(5)-sqrt(3)))/((sqrt(5)+sqrt(3)))+((sqrt(5)-sqrt(3)))/((sqrt(5)-sqrt(3)))` `implies x+y=(2(sqrt(5)^(2)+sqrt(3)^(2)))/2` `implies x+y=8`...........i Again `x=y=((sqrt(5)-sqrt(3))(sqrt(5)+sqrt(3)))/((sqrt(5)+sqrt(3))(sqrt(5)-sqrt(3)))` `=(4xxsqrt(5)xxsqrt(3))/2` `implies (x-y)=2sqrt(15)`.............ii And `xy=((sqrt(5)-sqrt(3))/((sqrt(5)+sqrt(3)))xx(sqrt(5)+sqrt(3)))/((sqrt(5)-sqrt(3)))` `xy=1` Substitutes values in the question. `implies ((x+y)^(2)-xy)/((x=y)^(2)+xy)` `=(8^(2)-1)/((2sqrt(15))^(2)+1)implies 63/61`

Discussion

143.	If x is a real number, then \(\sqrt {log_e\frac{{4x - {x^2}}}{3}} \)is a real number if and only if1. - 3 ≤ x ≤ 32. 1 ≤ x ≤ 23. -1 ≤ x ≤ 34. 1 ≤ x ≤ 3
Answer» Correct Answer - Option 4 : 1 ≤ x ≤ 3 Calculation: According to the question, √(log_e(4x – x²)/3) to be a real number, because x is a real number Then, it's root should be greater than 0 ⇒ √(loge(4x – x2)/3) ≥ 0 If, log_e = 2.7(Approx) It means log_B A ≥ 0 If A ≥ 1 It's true then, B > 1 ⇒ (4x – x2)/3) ≥ 1 ⇒ 4x – x2 ≥ 3 ⇒ x² – 4x + 3 ≤ 0 ⇒ x² – 3x – x + 3 ≤ 0 ⇒ (x – 1)(x – 3) ≤ 0 ⇒ x belongs to (1, 3) ∴ x lies between 1 ≤ x ≤ 3.

143.

If x is a real number, then \(\sqrt {log_e\frac{{4x - {x^2}}}{3}} \)is a real number if and only if1. - 3 ≤ x ≤ 32. 1 ≤ x ≤ 23. -1 ≤ x ≤ 34. 1 ≤ x ≤ 3

Answer» Correct Answer - Option 4 : 1 ≤ x ≤ 3

Calculation:

According to the question,

√(log_e(4x – x²)/3) to be a real number, because x is a real number

Then, it's root should be greater than 0

⇒ √(loge(4x – x2)/3) ≥ 0

If, log_e = 2.7(Approx)

It means log_B A ≥ 0

If A ≥ 1

It's true then, B > 1

⇒ (4x – x2)/3) ≥ 1

⇒ 4x – x2 ≥ 3

⇒ x² – 4x + 3 ≤ 0

⇒ x² – 3x – x + 3 ≤ 0

⇒ (x – 1)(x – 3) ≤ 0

⇒ x belongs to (1, 3)

∴ x lies between 1 ≤ x ≤ 3.

Discussion

144.	If 5x - 3y = 13438 and 5x-1 + 3y+1 = 9686, then x + y equals
Answer» Calculation: 5^x – 3^y = 13438 ----(i) 5^{(x – 1)} + 3^{(y + 1)} = 968 (5^x/5) + 3 × 3^y = 9686 ----(ii) Multiplying equation 1 by 3 and adding, we get ⇒ 5^x = 15625 ⇒ 5^x = 5⁶ So, x = 6 Putting x in equation 1, we get ⇒ 3^y = 2187 ⇒ 3^y = 3⁷ So, y = 7 So, x + y = 7 + 6 = 13 ∴ The value of x + y is 13.

144.

If 5x - 3y = 13438 and 5x-1 + 3y+1 = 9686, then x + y equals

Answer»

Calculation:

5^x – 3^y = 13438 ----(i)

5^{(x – 1)} + 3^{(y + 1)} = 968

(5^x/5) + 3 × 3^y = 9686 ----(ii)

Multiplying equation 1 by 3 and adding, we get

⇒ 5^x = 15625

⇒ 5^x = 5⁶

So, x = 6

Putting x in equation 1, we get

⇒ 3^y = 2187

⇒ 3^y = 3⁷

So, y = 7

So, x + y = 7 + 6 = 13

∴ The value of x + y is 13.

Discussion

145.	If 9x + 4y = 35 and x + 5 = 8, then what is the value of y?1. 72. 63. 14. 2
Answer» Correct Answer - Option 4 : 2 Given 9x + 4y = 35 x + 5 = 8 Calculation x + 5 = 8 x = 8 - 5 x = 3 Put x = 3 in equation 9x + 4y = 35 9(3) + 4y = 35 4y = 35 - 27 4y = 8 y = 2

145.

If 9x + 4y = 35 and x + 5 = 8, then what is the value of y?1. 72. 63. 14. 2

Answer» Correct Answer - Option 4 : 2

Given

9x + 4y = 35

x + 5 = 8

Calculation

x + 5 = 8

x = 8 - 5

x = 3

Put x = 3 in equation 9x + 4y = 35

9(3) + 4y = 35

4y = 35 - 27

4y = 8

y = 2

Discussion

146.	If a2 + b2 = 2b + 4a – 5, then what is the value of 2(a + b)?1. 102. 123. 84. 6
Answer» Correct Answer - Option 4 : 6 Given: a2 + b2 = 2b + 4a – 5 Concept used: If (x – a)² + (y – b)² = 0 Then x – a = 0, y – b = 0 Formula used: (x – y)² = x² + y² – 2xy Calculations: a2 + b2 = 2b + 4a – 5 ⇒ a² – 4a + b² – 2b + 5 = 0 ⇒ a² – 4a + 4 + b² – 2b + 1 = 0 ⇒ (a – 2)² + (b – 1)² = 0 ⇒ a – 2 = 0 and b – 1 = 0 ⇒ a = 2 and b = 1 Thus 2(a + b) = 2(2 + 1) = 2 × 3 = 6 ∴ The value of 2(a + b) = 6.

146.

If a2 + b2 = 2b + 4a – 5, then what is the value of 2(a + b)?1. 102. 123. 84. 6

Answer» Correct Answer - Option 4 : 6

Given:

a2 + b2 = 2b + 4a – 5

Concept used:

If (x – a)² + (y – b)² = 0

Then x – a = 0, y – b = 0

Formula used:

(x – y)² = x² + y² – 2xy

Calculations:

a2 + b2 = 2b + 4a – 5

⇒ a² – 4a + b² – 2b + 5 = 0

⇒ a² – 4a + 4 + b² – 2b + 1 = 0

⇒ (a – 2)² + (b – 1)² = 0

⇒ a – 2 = 0 and b – 1 = 0

⇒ a = 2 and b = 1

Thus 2(a + b) = 2(2 + 1) = 2 × 3 = 6

∴ The value of 2(a + b) = 6.

Discussion

147.	If x/4 = y/5 = z/9, find (x – y + z)/(x + z).1. 8/132. 13/83. - 8/134. - 13/8
Answer» Correct Answer - Option 1 : 8/13 Calculation: Let assume that x/4 = y/5 = z/9 =k x = 4k y = 5k z = 9k Now, (4k – 5k + 9k)/(4k + 9k) ⇒ (8k)/(13k) ⇒ 8/13 ∴ (x – y + z)/(x + z) = 8/13 The correct option is 1 i.e. 8/13

147.

If x/4 = y/5 = z/9, find (x – y + z)/(x + z).1. 8/132. 13/83. - 8/134. - 13/8

Answer» Correct Answer - Option 1 : 8/13

Calculation:

Let assume that x/4 = y/5 = z/9 =k

x = 4k

y = 5k

z = 9k

Now,

(4k – 5k + 9k)/(4k + 9k)

⇒ (8k)/(13k)

⇒ 8/13

∴ (x – y + z)/(x + z) = 8/13

The correct option is 1 i.e. 8/13

Discussion

148.	If \(\sqrt {\rm{x}} + \frac{1}{{\sqrt {\rm{x}} }} = {\rm{\;}}3\) then what will be the value of \(\frac{{\left( {x + \frac{1}{x}} \right)\; + \left( {{x^2} + \frac{1}{{{x^2}}}} \right)}}{2}\)1. 252. 273. 264. 23
Answer» Correct Answer - Option 2 : 27 Given: \(\sqrt {\rm{x}} + \frac{1}{{\sqrt {\rm{x}} }} = {\rm{\;}}3\) Formula used: \({{\rm{a}}^2} + \frac{1}{{{{\rm{a}}^2}}} = {{\rm{c}}^2} - {\rm{\;}}2\) Calculation: \(\sqrt {\rm{x}} + \frac{1}{{\sqrt {\rm{x}} }} = {\rm{\;}}3\) ⇒ \({\rm{x}} + \frac{1}{{{\rm{x}}}} = {\rm{\;}}3^2 - 2\) ⇒ \({\rm{x}} + \frac{1}{{{\rm{x}}}} = {\rm{\;}}9 - 2\) ⇒ \({\rm{x}} + \frac{1}{{{\rm{x}}}} = {\rm{\;}}7\) ⇒ \({\rm{x^2}} + \frac{1}{{{\rm{x^2}}}} = {\rm{\;}}7^2 - 2\) ⇒ \({\rm{x^2}} + \frac{1}{{{\rm{x^2}}}} = {\rm{\;}}49 - 2\) ⇒ \({\rm{x^2}} + \frac{1}{{{\rm{x^2}}}} = {\rm{\;}}47\) \(\frac{{\left( {x + \frac{1}{x}} \right)\; + \left( {{x^2} + \frac{1}{{{x^2}}}} \right)}}{2}\) ⇒ (7 + 47)/2 ⇒ 54/2 ⇒ 27 ∴ Required value is 27

148.

If \(\sqrt {\rm{x}} + \frac{1}{{\sqrt {\rm{x}} }} = {\rm{\;}}3\) then what will be the value of \(\frac{{\left( {x + \frac{1}{x}} \right)\; + \left( {{x^2} + \frac{1}{{{x^2}}}} \right)}}{2}\)1. 252. 273. 264. 23

Answer» Correct Answer - Option 2 : 27

Given:

\(\sqrt {\rm{x}} + \frac{1}{{\sqrt {\rm{x}} }} = {\rm{\;}}3\)

Formula used:

\({{\rm{a}}^2} + \frac{1}{{{{\rm{a}}^2}}} = {{\rm{c}}^2} - {\rm{\;}}2\)

Calculation:

\(\sqrt {\rm{x}} + \frac{1}{{\sqrt {\rm{x}} }} = {\rm{\;}}3\)

⇒ \({\rm{x}} + \frac{1}{{{\rm{x}}}} = {\rm{\;}}3^2 - 2\)

⇒ \({\rm{x}} + \frac{1}{{{\rm{x}}}} = {\rm{\;}}9 - 2\)

⇒ \({\rm{x}} + \frac{1}{{{\rm{x}}}} = {\rm{\;}}7\)

⇒ \({\rm{x^2}} + \frac{1}{{{\rm{x^2}}}} = {\rm{\;}}7^2 - 2\)

⇒ \({\rm{x^2}} + \frac{1}{{{\rm{x^2}}}} = {\rm{\;}}49 - 2\)

⇒ \({\rm{x^2}} + \frac{1}{{{\rm{x^2}}}} = {\rm{\;}}47\)

\(\frac{{\left( {x + \frac{1}{x}} \right)\; + \left( {{x^2} + \frac{1}{{{x^2}}}} \right)}}{2}\)

⇒ (7 + 47)/2

⇒ 54/2

⇒ 27

∴ Required value is 27

Discussion

149.

In the given question, two equations numbered l and II are given. Solve both the equations and mark the appropriate answer.I. x2 - 37x + 330 = 0II. y2 - 28y + 195 = 01. x > y2. x < y3. x ≥ y4. x ≤ y5. x = y or relationship between x and y cannot be established.

Answer» Correct Answer - Option 3 : x ≥ y

Calculation∶

I. x² - 37x + 330 = 0

⇒ x² - 15x - 22x + 330 = 0

⇒ x (x - 15) - 22 (x - 15) = 0

⇒ (x - 15) (x - 22) = 0

⇒ x = 15, 22

II. y² - 28y + 195 = 0

⇒ y² - 13y - 15y + 195 = 0

⇒ y (y - 13) - 15 (y - 13) = 0

⇒ (y - 13) (y - 15) = 0

⇒ y = 13, 15

Value of x	Value of y	Relation
15	13	x > y
15	15	x = y
22	13	x > y
22	15	x > y

∴ x ≥ y

Discussion

150.

Find a positive value of x for which the given equation \(\frac{x^2 - 9}{5 + x^2} = -\frac{5}{9}\) is satisfied.1. 32. 23. 14. 4

Answer» Correct Answer - Option 2 : 2

Calculations:

\(\frac{x^2 - 9}{5 + x^2} = -\frac{5}{9}\)

⇒ 9x² – 81 = -25 – 5x²

⇒ 14x² = 56

⇒ x² = 4

⇒ x = 2

∴ The value of x is 2

Discussion

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If x is a real number, then \(\sqrt {log_e\frac{{4x - {x^2}}}{3}} \)is a real number if and only if1. - 3 ≤ x ≤ 32. 1 ≤ x ≤ 23. -1 ≤ x ≤ 34. 1 ≤ x ≤ 3

If 5x - 3y = 13438 and 5x-1 + 3y+1 = 9686, then x + y equals

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If a2 + b2 = 2b + 4a – 5, then what is the value of 2(a + b)?1. 102. 123. 84. 6

If x/4 = y/5 = z/9, find (x – y + z)/(x + z).1. 8/132. 13/83. - 8/134. - 13/8

If \(\sqrt {\rm{x}} + \frac{1}{{\sqrt {\rm{x}} }} = {\rm{\;}}3\) then what will be the value of \(\frac{{\left( {x + \frac{1}{x}} \right)\; + \left( {{x^2} + \frac{1}{{{x^2}}}} \right)}}{2}\)1. 252. 273. 264. 23

In the given question, two equations numbered l and II are given. Solve both the equations and mark the appropriate answer.I. x2 - 37x + 330 = 0II. y2 - 28y + 195 = 01. x > y2. x < y3. x ≥ y4. x ≤ y5. x = y or relationship between x and y cannot be established.

Find a positive value of x for which the given equation \(\frac{x^2 - 9}{5 + x^2} = -\frac{5}{9}\) is satisfied.1. 32. 23. 14. 4

यदि `xy=x^(2)+y^(2)-xy`, है तो `911` का मान क्या होगा?A. 93B. 103C. 113D. 121