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451.

यदि `a!=b` है तो निम्न में से कौन सा कथन सत्य है?A. `(a+b)/2=sqrt(ab)`B. `(a+b)/2ltsqrt(ab)`C. `(a+b)/2 gtsqrt(ab)`D. All of the above

Answer» Correct Answer - C
Given that `a!=b,` Let `a=16, b=4`
`:.` by options
So `(a+b)/2=(16+4)/2=10`
and `sqrt(ab)=sqrt(16xx4)=8`
`:.(a+b)/2gtsqrt(ab)`
`:.` option (c) is correct.
Discussion
452.

यदि `x^(2)+1/5x+a^(2)` एक पूर्ण वर्ग है तो `a=?`A. `1/100`B. `+-1/10`C. `1/10`D. `-1/10`

Answer» Correct Answer - C
`x^(2)=1/5x+a^(2)`
`A^(2)+2xxAB+B^(2)=(A+B)^(2)`
`x^(2)+2xx1/10xx x+a^(2)=(x+1/10)^(2)`
`A=x`
`B=1/10`
`B=1/a=1/10`
Discussion
453.

`x` का मान ज्ञात करें। ‌जिसके लिए व्यंजक `2-3x-4x^(2)` का अधिकतम मान है?A. `41/16`B. `3/8`C. `-3/8`D. `41/16`

Answer» Correct Answer - D
`2-3x-4x^(2)=0`
`-4x^(2)-3x+2=0`
`ax^(2)+bx+c=0`
In quadratic equation
i when `agt0`
Minimum value `=(4ac-b^(2))/(4a)`
ii when `alt0`
Maximum value `=(4ac-b^(2))/(4a)`
`:.` In `-4x^(2)-3x-2`
`alt0`
`:.` Maximum value
`=` Maximum value
`(4x-4x2-(-3)^(2))/(4x-4)`
`=(-32-9)/(-16)=41/16`
Discussion
454.

यदि `x=(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))` और `y=(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))` है तो `x^(3)+y^(3)` का मान ज्ञात करें।A. 950B. 930C. 650D. 970

Answer» Correct Answer - D
`x=(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))`
`y=(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))`
`:. x=1/y`
`y=1/ximplies xy=1`
`:. x+y=(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))+(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))`
`implies ((sqrt(3)-sqrt(2))^(2)+(sqrt(3)+sqrt(2))^(2))/1`
`implies (3+2-2sqrt(6)+3+2+2sqrt(6))/1=10`
`implies (x+y)^(3)+x^(3)+y^(3)+3xy(x+y)`
`implies (10)^(3)=x^(3)+y^(3)+3xx1(10)`
`implies x^(3)+y^(3)=1000-30=970`
Discussion
455.

यदि `a=2.234, b=3.121` और `c=-5.355` है तो `a^(3)+b^(3)+c^(3)-3abc` का मान ज्ञात करें।A. -1B. 0C. 1D. 2

Answer» Correct Answer - B
`a=2.234`
`b=3.121`
`c=-5.355`
`:. a+b+c=0`
`a^(3)+b^(3)+c^(3)-3abc=(a+b+`
`c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)`
`=0`
Discussion
456.

`a`के किस मान के लिए `x+1/4 sqrt(x)+a^(2)` एक पूर्ण वर्ग होगा?A. `+-1/18`B. `1/8`C. `-1/5`D. `1/4`

Answer» Correct Answer - B
`x+1/4sqrt(x)+a^(2)`
`=(sqrt(x))^(2)+2xx1/8xxsqrt(x)+a^(2)`
`[(A^(2)+2BA+B^(2))=(A+B)^(2)]`
Here `, A=sqrt(x)` and `B=a`
`B=1/8:.a=1/8`
Discussion
457.

यदि व्यंजक `x^(2)+x+1,(x+1/2)^(2)+q^(2)`, के रूप में लिखा गया है तो `q` का संभावित मान क्या होगा?A. `+-1/3`B. `+-(sqrt(3))/2`C. `+-2/(sqrt(3))`D. `+-(1/2)`

Answer» Correct Answer - B
`x^(3)+x+1=0`………………i
`(x+1/2)^(2)+q^(2)=0`
`implies x^(2)+1/4+2xx1/2xx x+q^(2)`
`=x^(2)+x+(q^(2)+1/4)=0`…………….ii
Comparing constnat term of equation i and ii
`:. q^(2)+1/4=1`
`=q^(2)=1-1/4=3/4`
`q=+-sqrt(3/4)=+-(sqrt(3))/2`
Discussion
458.

`k` के किस मान के लिए व्यंजक `p+1/4+sqrt(p)+k^(2)` एक पूर्ण वर्ग होगा?

Answer» Correct Answer - A
`p+1/4+sqrt(p)+k^(2)`
`implies p+sqrt(p)+(k^(2)+1/4)`
`implies (sqrt(p))^(2)=2xx1/2xxsqrt(p)+(k^(2)+1/4)`
`implies A^(2)+2xxAxxB+B^(2)`
`implies A=sqrt(p)B^(2)=(k^(2)+1/4)`
`B=1/2`
`:. k^(2)+1/4=(1/2)^(2)`
`k^(2)+1/4=1/4`
`k^(2)=0`
`k=0`
Discussion
459.

यदि व्यंजक `(x^(2))/(y^(2))+tx+(y^(2))/4` एक पूर्ण वगै है तो `t` का मान होगा?A. `+-1`B. `+-2`C. `0`D. `+-3`

Answer» Correct Answer - A
`(x^(2))/(y^(2))+tx+(y^(2))/4` (given)
To make it a perfect square it should be in the form.
`A^(2) +-2AB+B^(2)=(A+-B)^(2)` ltbr `=(x/y)^(2)+-tx+(y/2)^(2)`
`= A^(2) +-2AB+B^(2)`
`A=x/y,B=y/2` & `2AB=tx`
So `tx=+-2xx x/y xx y/2`
`tx=+-x`
`t=+-1`
Discussion
460.

यदि `x=sqrt(5)+2` है तो `(2x^(2)-3x-2)/(3x^(2)-4x-3)` का मान ज्ञात करें।A. 0.1785B. 0.525C. 0.625D. 0.785

Answer» Correct Answer - C
`x=sqrt(5)+2` हर का परिमेयकरण
`implies 1/x=1/(sqrt(5)+2)xx(sqrt(5)-2)/(sqrt(5)-2)`
`implies (sqrt(5)-2)/(5-4)=sqrt(5)-2`
`implies x-1/x=sqrt(5)+2-sqrt(5)+2=4`
`:. (2x^(2)-3x-2)/(3x^(2)-4x-3)`
Multiply by `1/x` in numerator and denominator
`((2x^(2))/x-2/x-(3x)/x)/((3x^(2))/x-3/x-(4x)/x)implies(2x-2/x-3)/(3x-3/x-4)`
`=(2(x-1/x)-3)/(3(x-1/x)-4)implies (2xx4-3)/(3xx4-4)`
`=(8-3)/(12-4)=5/8=0.625`
Discussion
461.

In the given questions, two equations numbered l and II are given. You have to solve both the equations and mark the appropriate answerI. x2 – 9x + 20 = 0II. y2 – 16 = 01. x > y2. x ≥ y3. x < y4. x ≤ y5. x = y or the relationship cannot be established

Answer» Correct Answer - Option 2 : x ≥ y

Given:

I. x2 – 9x + 20 = 0

II. y2 – 16 = 0

Calculation:

I. x2 – 9x + 20 = 0

⇒ x2 – 5x – 4x + 20 = 0

⇒ x(x – 5) – 4(x – 5) = 0

⇒ (x – 5)(x – 4) = 0

Then, x = 5 or x = 4

Now,

II. y2 – 16 = 0

⇒ y2 = 16

⇒ y = 4 or y = – 4

Now,

Comparison between x and y (via Tabulation):

Value of X

Relation

Value of Y

5

x > y

  4

5

x > y

  -4

4

x = y

 4

4

x > y

– 4

∴ We can clearly see that x ≥ y
Discussion
462.

यदि `x=root(3)(2+sqrt(3))` है तो `x^(3)+1/(x^(3))` का मान ज्ञात करें।A. 8B. 9C. 2D. 4

Answer» Correct Answer - D
`x=root(3)(2+sqrt(3))`
`x^(3)+2+sqrt(3)`
`1/(x^(3))=1/(2+sqrt(3))xx(2-sqrt(3))/(2-sqrt(3))`
हर का परिमेयकरण
`implies (2-sqrt(3))/(4-3)=2-sqrt(3)`
`:. x^(3)+1/(x^(3))+2+sqrt(3)+2-sqrt(3)=4`
Discussion
463.

व्यंजक `x^(4)-2x^(2)+k` एक पूर्ण वर्ग होगा जब `k` का मान होगाA. 2B. 1C. -1D. -2

Answer» Correct Answer - B
`x^(4)-2x^(2)+k`
`implies (A+B)^(2)=A^(2)+2AB+B^(2)`
`(A-B)^(2)=A^(2)-2AB+B^(2)`
`implies (x^(2))^(2)-2xx x^(2)+k`
`(A)^(2)-2xx AB+B^(2)`
`:. A=x^(2),B=-1`
`B^(2)=K`
`(-1)^(2)=K`
`K=1`
Discussion
464.

In the given question, two equations numbered l and II are given. Solve both the equations and mark the appropriate answer.I. x2 + 9x + 14 = 0II. 3y2 + 4y + 1 = 01. x > y2. x < y3. x ≥ y4. x ≤ y5. x = y or relationship between x and y can't be established

Answer» Correct Answer - Option 2 : x < y

I. x2 + 9x + 14 = 0

⇒ x2 + 7x + 2x + 14 = 0

⇒ x (x + 7) + 2 (x + 7) = 0

⇒ (x + 2)(x + 7) = 0

⇒ x = - 2, -7

II. 3y2 + 4y + 1 = 0

⇒ 3y2 + 3y + y + 1 = 0

⇒ 3y(y + 1) + 1(y + 1) = 0

⇒ (3y + 1)(y + 1) = 0

⇒ \(y = - 1, - \frac{1}{3}\)

On comparing the values of x and y,

Value of x

Value of y

Relation

- 2

-1

x < y

- 2

 - (1/3)

x < y

 - 7

-1

x < y

 - 7

- (1/3)

x < y

 

∴ x < y 
Discussion
465.

यदि `a+b+c=0` तो `a^(3)+b^(3)+c^(3)` का मान ज्ञात कीजिए?A. `abc`B. `2abc`C. `3abc`D. `0`

Answer» Correct Answer - C
`a+b+c=0`
`a^(3)+b^(3)+c^(3)=T`
`a^(3)+b^(3)+c^(3)-3abc`
`=1/2(a+b+c)[(a-b)^(2)+`
`(b-c)^(2)+(c-a)^(2)]`
`a^(3)+b^(3)+c^(3)-3abc=0`
`a^(3)+b^(3)+c^(3)=3abc`
Discussion
466.

यदि `y:x=4:15` है तो `((x-y)/(x+y))` का मान ज्ञात करें।A. `11/9`B. `19/11`C. `4/11`D. `15/19`

Answer» Correct Answer - A
`y:x=4:15`
`:. y/x=4/15`
`:. (x-y)/(x+y)implies(x(1-y/x))/(x(1+y/x))` taking `x` common.
`implies (1-4/15)/(1+4/15)implies 11/15xx15/19=11/19`
Discussion
467.

यदि `x=sqrt(3)+sqrt(2)` है तो `(x+1/x)` का मान ज्ञात करें।A. `2sqrt(2)`B. `2sqrt(3)`C. `2`D. `3`

Answer» Correct Answer - B
`x=sqrt(3)+sqrt(2)`
`1/x=1/(sqrt(3)+sqrt(2))xx(sqrt(3)-sqrt(2))/(sqrt(3)-sqrt(2))=sqrt(3)-sqrt(2)`
`x=1/x=sqrt(3)+sqrt(2)+sqrt(3)-sqrt(2)=2sqrt(3)`
Discussion
468.

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

Answer» Correct Answer - D
`(x^(1/3)+x^((-1)/3))(x^(2/3)+x^((-2)/3)-1)`
We know that
`(a^(3)+b^(3))=(a+b)(a^(2)+b^(2)-ab)`
`(x^(1/3xx3)+x^(1/x)xx3)`
`(x+x^(-1))`
Alternative
`(x^(1/3)+1/(x^(1/3)))(x^(2/3)-1 1/(x^(2/3)))`
`(x^(1/3)+1/(x^(1/3)))(x^(2/3)+1/(x^(2/3))-1)`
`x^(1/3) . x^(2/3)+x^(1/3) . 1/(x^(2/3))-x^(1/3)+x^(2/3)`
`1/(x^(/3))+1/(x^(2/3)x^(1/3))-1/(x^(1/3))`
`x+x^((-1)/3)-x^(1/3)+x^(1/3)+1/x-x^((-1)/3)`
`x+x^(-1)`
Discussion
469.

यदि `1/p+1/q=1/(p+q)` तो `p^(3)-q^(3)` का मान क्या है?A. `p-q`B. `p q`C. `1`D. `0`

Answer» Correct Answer - D
`1/p+1/q=1/(p+q)`
`(p+q)/(pq)=1/(p+q)`
`(p+q)^(2)=1/(p+q)`
`(p+q)^(2)=pq`
`p^(2)+q^(2)+2pq-pq=0`
`(p^(2)+q^(2)+pq=0`
Multiply by `(p-q)`
both side
`(p-q)(p^(2)+q^(2)+pq)=(p-q)xx0`
`p^(3)-q^(3)=0`
Discussion
470.

यदि `n=7+4sqrt(3)` है तो `(sqrt(n)+1/(sqrt(n)))` का मान ज्ञात करें।A. `2sqrt(3)`B. `4`C. `-4`D. `-2sqrt(3)`

Answer» Correct Answer - B
`n=7+4sqrt(3)`
`n=4+3+4sqrt(3)`
`n=(2)^(2)+(sqrt(3))^(2)+2xx2xxsqrt(3)`
`n=(2+sqrt(3))^(2)`
`sqrt(n)=2+sqrt(3)`
`:. 1/(sqrt(n))=2-sqrt(3)`
`sqrt(n)+1/(sqrt(n))=2+sqrt(3)+2-sqrt(3)=4`
Discussion
471.

यदि `x=93, y=93, z=94` तो `x^(2)-y^(2)+10xz+10yz` का मान क्या है?A. 104784B. 147840C. 174840D. 184740

Answer» Correct Answer - C
`x^(2)-y^(2)+10xz+10yz`
`(x-y)(x+y)+10z(x+y)`
Put valueof `x,y` and `z`
`implies(93-93)(93+93)+10xx94`
`(93+93)`
`0+940xx186`
`=174840`
Discussion
472.

मान लिया कि `a=sqrt(6)-sqrt(5),b=sqrt(5)-2,c=2-sqrt(3)` है तो निम्न विकल्पों में से कौन सा विकल्प सही है?A. `b lt a lt c`B. `a lt c lt b`C. `b lt c lt a`D. `a lt b lt c`

Answer» Correct Answer - D
`a=sqrt(6)-sqrt(5)`
`b=sqrt(5)-2`
`c=2-sqrt(3)`
Rationalize all the terms
`a=(sqrt(6)-sqrt(5))/(sqrt(6)+sqrt(5))xxsqrt(6)+sqrt(5)`
`=1/(sqrt(6)+sqrt(5))`
`b=(sqrt(5)-2)/(sqrt(5)+2)xxsqrt(5)+2=1/(sqrt(5)+2)`
`=1/(sqrt(5)+sqrt(4))`
`c=(2-sqrt(3))/(2+sqrt(3))xx 2+sqrt(3)`
`=1/(2+sqrt(3))=1/(sqrt(4)+sqrt(3))`
NOw, `a=1/(sqrt(6)+sqrt(5))`
`b=1/(sqrt(5)+sqrt(4))`
`c=1/(sqrt(4)+sqrt(3))`
Now the term whose denominator is largest is the smallest term.
So `altbltc`.
Discussion
473.

यदि `a+b=1` और `a^(3)+b^(3)+3ab=k` है तो `k` का मान ज्ञात करें।A. 1B. 3C. 5D. 7

Answer» Correct Answer - A
`a+b=1`
By cubing
`a^(3)+b^(3)+3ab(a+b)=1^(3)`
`a^(3)+b^(3)+3ab=1(a+b=1)`
`a^(3)+b^(3)+3ab=k`
From above both equations
`k=1`
Discussion
474.

यदि `x=(4sqrt(15))/(sqrt(5)+sqrt(3))` है तो `(x+sqrt(20))/(x-sqrt(20))+(x+sqrt(12))/(x-sqrt(12))` का मान ज्ञात करें।A. 1B. 2C. `sqrt(3)`D. `sqrt(5)`

Answer» Correct Answer - B
`x=(4sqrt(15))/(sqrt(5)+sqrt(3))`
`(x+sqrt(20))/(x-sqrt(20))+(x+sqrt(12))/(x-sqrt(12))=?`
`:. x=(4xxsqrt(5)xxsqrt(3))/(sqrt(5)+sqrt(3))=(2xx(sqrt(5)xxsqrt(12)))/(sqrt(15)+sqrt(3))`
or `(2xxsqrt(20)xxsqrt(3))/(sqrt(5)+sqrt(3))`
`x/(sqrt(12))=(2sqrt(5))/(sqrt(5)+sqrt(3))`
by C-D rule
`(x+sqrt(12))/(x-sqrt(12))=(2sqrt(5)+sqrt(5)+sqrt(3))/(2sqrt(5)-sqrt(5)-sqrt(3))`
`=(3sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))`
`x/(sqrt(20))=(2sqrt(3))/(sqrt(5)+sqrt(3))`
`(x+sqrt(20))/(x-sqrt(20))=(2sqrt(3)+sqrt(5)+sqrt(3))/(2sqrt(3)-sqrt(5)-sqrt(3))`
`=(3sqrt(3)+sqrt(5))/(sqrt(3)-sqrt(5))`
`:. (x+sqrt(12))/(x-sqrt(12))+(x+sqrt(20))/(x-sqrt(20))`
`=(3sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))+(3sqrt(3)+sqrt(5))/(sqrt(3)-sqrt(5))`
`= (3sqrt(5)+sqrt(3)-3sqrt(3)-sqrt(5))/(sqrt(5)-sqrt(3))`
`=(2sqrt(5)-2sqrt(3))/(sqrt(5)-sqrt(3))=2`
Discussion
475.

यदि `ab=21` और `((a+b)^(2))/((a-b)^(2))=25/4` हो तो `a^(2)+b^(2)+3ab` का मान क्या होगा?A. 115B. 121C. 125D. 127

Answer» Correct Answer - B
`((a+b)^(2))/((a-b)^(2))=25/4`
`(a+b)/(a-b)=5/2`
by C & D
`(a+b+a-b)/(a+b-a+b)=(5+2)/(5-2)`
`(2a)/(2b)=7/3`
`a/b=7/3`
Now, the value of
`implies a^(2)+b^(2)+3ab`
`= 7^(2)+3^(2)+3xx7xx3`
`=49+9+63=58+63`
`=121`
Discussion
476.

यदि `x=5-sqrt(21)` है तो `(sqrt(x))/(sqrt(32-2x)-sqrt(21))` का मान क्या होगा?A. `1/(sqrt(2))(sqrt(3)-sqrt(7))`B. `1/(sqrt(2)(sqrt(7)-sqrt(3))`C. `1/(sqrt(2)(sqrt(7)+sqrt(3))`D. `1/(sqrt(2))(7+sqrt(3))`

Answer» Correct Answer - B
`x=5-sqrt(21)`
`2x=10-2sqrt(21)`…………….i
`implies2x=(sqrt(7))^(2)+(sqrt(3))^(2)-2(sqrt(7))(sqrt(3))`
`implies 2x=(sqrt(7)-sqrt(3))^(2)`
`implies x=1/2(sqrt(7)-sqrt(3))^(2)`
` implies sqrt(x)=1/(sqrt(2))sqrt((sqrt(7)-sqrt(3))^(2))` ltbgt `=1/(sqrt(2))(sqrt(7)-sqrt(3))`
`:. (sqrt(x))/(sqrt(32-2x)-sqrt(21))`
`(sqrt(7)-sqrt(3))/(sqrt(2)(sqrt(32-(10-2sqrt(21)))-sqrt(21)))`
`=(sqrt(7)-sqrt(3))/(sqrt(2)(sqrt(22+2sqrt(2))-sqrt(21)))`
`= (sqrt(7)-sqrt(3))/(sqrt(2)(sqrt((sqrt(21)+1)^(2))-sqrt(21)))`
`=(sqrt(7)-sqrt(3))/(sqrt(2)(sqrt(21a)+1-sqrt(21)))`
`=((sqrt(7)-sqrt(3)))/(sqrt(2))`
Discussion
477.

`(4x^(3)-x)/((2x+1)(6x-3))` का मान क्‍या है जब `x=9999` है?A. 1111B. 2222C. 3333D. 6666

Answer» Correct Answer - C
`x=9999`
`(4x^(3)-x)/((2x+1)(6x-3))=(x(4x^(2)-1))/(3(2x+1)(2x-1))`
`(x(4x^(2)-1))/(3(4x^(2)-1))=x/3`
`:. 9999/3=3333`
Discussion
478.

`(x^(a+b))^(a-b)(x!=0)` का मान क्‍या है?A. `1`B. `2`C. `-1`D. `0`

Answer» Correct Answer - A
`(x^(b+c))^(b-c)(x^(c+a))^(c-a)(x^(a+b))^(a-b)(x!=0)`
`x^(b^(2)c^(2)).x^(c(2)-a^(2)).x^(a^(2)-b^(2))`
`x^(b^(2)-c^(2)+c^(2)-a^(2)+a^(2)-b^(2)=x^(0)=1`
Discussion
479.

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

Answer» Correct Answer - C
`x/a=1/a=1/x`
`x/a-1/a=-1/x`
`((x-1)/a)=-1/x`
`(1-x)/a=1/x`
`x(1-x)=a`
`x-x^(2)=a`
Discussion
480.

यदि `x+1/x=99`, है तो `(100x)/(2x^(2)+102x+2)` का मान क्या होगा?A. `1/6`B. `1/2`C. `1/3`D. `1/4`

Answer» Correct Answer - C
`x+1/x=99`
`:. x^(2)+1=99x`
`2(x^(2)+1)=2xx99x`
`2x^(2)+2=198x`
`=(100x)/(2x^(2)+2+102x)`
`=(100x)/(198x+102x)=(100x)/(300x)=1/3`
Discussion
481.

य‌‌द‌ि `x=7-4-4sqrt(3)` है ते `(x+1/x)` का मान क्‍या है?A. `3sqrt(3)`B. `8sqrt(3)`C. `14+8sqrt(3)`D. `14`

Answer» Correct Answer - D
`x=7-4sqrt(3)`
`1/ximplies1/(7-4sqrt(3))`
By relationlisation
`1/x=1/(7-4sqrt(3))xx(7+4sqrt(3))/(7+4sqrt(3))`
`=(7+4sqrt(3))/(49-48)=7+4sqrt(3)`
`:. x+1/x=7-4sqrt(3)+7+4sqrt(3)`
`=14`
Discussion
482.

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

Answer» Correct Answer - B
`a^(2)+b^(2)+c^(3)+3=2(a+b+c)`
`a^(2)+b^(2)+c^(2)+3=2a+2b+2c`
`a^(2)-2a+1+b^(2)-2b+1+c^(2)-2c`
`+1=0`
`(a-1)^(2)+(b-1)^(2)+(c-1)^(2)=0`
`a=1`
`b=1`
`c=1`
`(a+b+c)implies 1+1+1=3`
Discussion
483.

यदि `x-1/x=5` है तो `x^(2)+1/(x^(2))=?`A. 2B. 25C. 27D. 23

Answer» Correct Answer - C
`x-1/x=5`
`x^(2)+1/(x^(2))-2=25`
`impliesx^(2)+1/(x^(2))=27`
Discussion
484.

If x = 3 + 2√2, then the value of \(x^2 + \frac{1}{x^2}\) is:1. 342. 303. 364. 32

Answer» Correct Answer - Option 1 : 34

Given:

x = 3 + 2√2

Formula used:

(a2 – b2) = (a – b)(a + b)

Calculation:

x = 3 + 2√2

Squaring on both side

x2= (3 + 2√2)2

x2 = 32 + 2 × 3 × 2√2 + (2√2)2

x2= 9 + 12√2 + 8

x2= 17 + 12√2      ----1

To find   \({x^2} + \frac{1}{{{x^2}}}\)

\({x^2} + \frac{1}{{{x^2}}}\; = \;17{\rm{\;}} + {\rm{\;}}12\sqrt 2 + {\rm{\;}}\frac{1}{{17{\rm{\;}} + {\rm{\;}}12\sqrt 2 }}{\rm{\;\;\;\;\;}}\)

By rationalisation method 

⇒ \(17{\rm{\;}} + {\rm{\;}}12\sqrt 2 {\rm{\;}} + {\rm{\;}}\frac{1}{{17{\rm{\;}} + {\rm{\;}}12\sqrt 2 }}\; \times \;\frac{{17{\rm{\;}} - {\rm{\;}}12\sqrt 2 }}{{17{\rm{\;}} - {\rm{\;}}12\sqrt 2 }}\)

Use (a2 – b2) = (a – b)(a + b)

⇒ \(17 + {\rm{\;}}12\sqrt {2\;} \; + \;\frac{{17{\rm{\;}} - {\rm{\;}}12\sqrt 2 }}{{{{17}^2}\; - \;{{(12\surd 2)}^2}}}\)

⇒ \(17{\rm{\;}} + {\rm{\;}}12\sqrt {2\;} + \;\frac{{17{\rm{\;}} - {\rm{\;}}12\sqrt {2\;} \;}}{{289\; - \;288}}\)

⇒ 17 + 12√2 + 17 – 12√2

⇒ 34

⇒  \({x^2} + \frac{1}{{{x^2}}} = 34\;\)

∴ The value of \({x^2} + \frac{1}{{{x^2}}}\) is 34
Discussion
485.

यदि `a=2+sqrt(3)` है तो `(a^(2)+1/(a^(2)))` का मान ज्ञात करें।A. 12B. 14C. 16D. 10

Answer» Correct Answer - B
`a=2+sqrt(3)`
`implies a^(2)=(2+sqrt(3))^(2)`
`implies 4+3+4sqrt(3)`
`implies 7+4sqrt(3)`
`implies 1/(a^(2))=1/(7+4sqrt(3))`
`=(7-4sqrt(3))/((7+4sqrt(3))(7-4sqrt(3)))=(7-4sqrt(3))/1`
`:. a^(2)+1/(a^(2))=7+4sqrt(3)+7-4sqrt(3)`
`=14`
Discussion
486.

यदि `x` एक परिमेय संख्या है और `((x+1)^(3)-(x-1)^(3))/((x+1)^(2)-(x-1)^(2))=2` है तो `x` के अंश तथा हर का योग ज्ञात करें।A. 3B. 4C. 5D. 7

Answer» Correct Answer - B
`((x+1)^(3)-(x-1)^(3))/((x+1)^(2)-(x-1)^(2))=2`
`implies A^(3)-B^(3)+(A-B)(A^(2)+AB+B^(2))`
`A^(2)-B^(2)=(A-B)(A+B)`
`implies ((x+1-x+1)((x+1)^(2)+(x-1)(x+1)+(x-1)^(2)))/((x+1-x+1)(x+1+x-1))`
`=2`
`implies ((x^(2)+1+3x+x^(2)-1+x^(2)+1-2x))/((2x))=2`
`implies (3x^(2)+1)/(2x)=`
`implies 3x^(2)+1=4x`
`3x^(2)-4x+1=0`
`3x^(2)-3x-x+1=0`
`3x(x-1)-1(x-1)=0`
`(3x-1)(x-1)=0`
`implies 3x-1=0`
`x=1/3`
`implies x-1=0`
for `x=1=1/1`
By adding numinator and denominator
`1+1=2`
No option is satisfied
`:. x=1/3`
`1+3=4`
Discussion
487.

यदि `a,b,c` धनात्मक है और `a+c+c=1` है तो `1/a+1/b+1/c` का मान ज्ञात करें।A. 9B. 5C. 3D. 1

Answer» Correct Answer - A
For minimum value of
`1/a+1/b+1/c`
`a=b=c`
`a+b+c=1` (given)
`:. a=b=c=1/3`
`1/a=1/b=1/c=3`
`:.` Minimum value of `1/a+1/b+1/c`
`=3+3+3=9`
Discussion
488.

`x+1/x` का व्‍यूत्‍क्रम क्‍या है?A. `x/(x^(2)+1)`B. `x/(x+1)`C. `x-1/x`D. `1/x+x`

Answer» Correct Answer - A
Reciprocal of `(x+1/x)`
`=1/((x+1/x))+x/(x^(2)+1)`
Discussion
489.

`x+y=2a` है `a/(x-a)+a/(y-a)` का मान है

Answer» Correct Answer - A
Given `x+y=2a` to
Find `a/((x-a))+1/((y-a))=?`
`implies ((x,+ y=,2a),(darr, darr, darr),(3, 1, 2))`
`implies` Let `x=3, y=1, a=2`
`impliesa/((x-a))+a/(y-a)`
`implies a/((3-2))+2/((1-2))`
`implies2/1+2/(-1)=0`
Discussion
490.

`(x-2)(x-9)` का न्‍यूनतम मान क्‍या है?A. `-11/4`B. `49/4`C. `0`D. `-49/4`

Answer» Correct Answer - D
`(x-2)(x-9)`
`=x^(2)-9x-2x+8`
`=x^(2)-11x+18`
`=ax^(2)+bx+c=0`
For minimum value
`(4ac-b^(2))/(4a)`
`=(4xx1xx18-(-11)^(2))/(4xx1)=(72-121)/4=(-49)/4`
Discussion
491.

If `(b-c)/a+(a+c)/b+(a-b)/c=1`and `a-b+c!=0` then which of the following is correct?A. `1/c=1/a+1/b`B. `1/a=1/b+1/c`C. `1/b=1/a-1/c`D. `1/b=1/a+1/c`

Answer» Correct Answer - B
`(b-c)/a+(a+c)/b+(a-b)/0=1`
`a-b+c!=0`
Let `b=c`
`:. (b-b)/a+(a+b)/b+(a-b)/b=1`
`implies 0+a/b+1+a/b=1=1`
`implies a/b+1/b=1`
`implies 1/b+1/b=1/a`
We take `b=c`
`:. 1/b+1/c=1/a`
Discussion
492.

`1/(x+y)` और `1/(x-y)` का योग हैA. `(2y)/(x^(2)-y^(2))`B. `(2x)/(x^(2)-y^(2))`C. `(-2y)/(x^(2)-y^(2))`D. `(2x)/(y^(2)-x^(2))`

Answer» Correct Answer - B
According to the question
`implies 1/(x+y)+1/(x-y)`
`implies (x-y+x+y)/(x^(2)-y^(2))implies(2x)/(x^(2)-y^(2))`
Discussion
493.

यदि `a^(2)-4a-1=0` है तो `a^(2)+1/(a^(2))+3a-3/a` का मान क्या होगा?A. 25B. 30C. 35D. 40

Answer» Correct Answer - B
`a^(2)-4a-1=0`
`a^(2)-1=4a`
`a-1/a=4`
Squaring both sides
`a^(2)+1/(a^(2))-2=16`
`a^(2)+1/(a^(2))=18`
`:. a^(2)+1/(a^(2))+3(a-1/a)`
`=18+3xx4=18+12=30`
Discussion
494.

`(x^(3)+y^(6))(x^(3)-y^(6))` किसके बराबर है?A. `x^(6)-y^(12)`B. `x^(9)-y^(16)`C. `x^(6)-y^(12)`D. `x^(9)+y^(36)`

Answer» Correct Answer - A
According to the question
`implies (x^(3)+y^(6))(x^(3)-y^(6))`
`implies x^(6)+x^(3)y^(6)-x^(3)^y^(6)-y^(12)`
`implies x^(6)-y^(12)`
Discussion
495.

यदि बिंदु `(K,2K-1)` का कोटिमान और भुजमान बराबर है तो `K` का मान क्या है?

Answer» Correct Answer - B
Here `k=2k-1`
Let `r=1`
`[(x=y),(darr darr),("abscissa"-"ordinate")]`
`k=2k-1`
`k=1`
Discussion
496.

यदि `x=root(3)(5)+2` है तो `x^(3)-6x^(2)+12x-13` का मान ज्ञात करें।A. `-1`B. `1`C. 2D. 0

Answer» Correct Answer - D
`x=root(3)(5)+2`
`impliesx-2=root(3)(5)`
Take cube on both sides
`implies (x-2)^(3)-(root(3)(5))^(3)`
`implies x^(3)-8-6x^(2)+12x=5`
`:. x^(3)-6x^(2)+12x-13=0`
Discussion
497.

यदि `x=5` है तो व्यंजक `x^(2)-2+1/(x^(2))` का मान है?A. `576/25`B. `24/25`C. `24/5`D. `625/24`

Answer» Correct Answer - A
According to the question
If `x=5`
`implies x^(2)-2+1/(x^(2))`
`implies (x-1/x)^(2)implies(5-1/5)^(2)`
`implies (24/5)^(2)implies576/25`
Discussion
498.

यदि `a+1/a=-1` है तो `(1-a+a^(2))(1+a-a^(2))` का मान है?A. 1B. 0C. -4D. 4

Answer» Correct Answer - D
According to the question
`a+1/a=-1` and `a^(2)+a=-1`
`a^(3)=1`
`(-a-a)(-a^(2)-a^(2))`
`(-2a)(-2a^(2))`
`4a^(3)=4xx1=4`
Discussion
499.

यदि `x=2,y=1` और `z=-3` है तो `x^(3)+y^(3)+z^(3)-3xyz` किसके बराबर है?A. 6B. 0C. 2D. 8

Answer» Correct Answer - B
According to the question
`x=2, y=1, z=-3`
`:. x^(3)+y^(3)+z^(3)-3xyz=?`
As we know that
`a+b+c=0` then `a^(3)+b^(3)+c^(3)`
`-3abc=0`
`:. 2+1-3=0`
`x^(3)+y^(3)+3^(3)-3xy=0`
Discussion
500.

यदि दो वास्तविक अचरों `a` और `b` के लिए व्यंजक `ax^(3)+3x^(2)-8x+b,(x+2)` और `(x-2)` से पूर्णतः विभाजित है तोA. `a=2, b=12`B. `a=12, b=2`C. `a=2,b=-12`D. `a=-2, b=12`

Answer» Correct Answer - C
`ax^(3)+3x^(2)-8x+b` is divisible
by `(x+2)` and `(x-2)`
`:. (x=2)` and `(x-2)` are factors
`:.x+2=0impliesx=-2`
`x-2-0impliesx=2`
Put `x=-2`
`:.a(-2)^(3)+3(-2)^(2)-8(-2)+b=0`
`=-8a+12+16+b=0`
`-8a+b+28=0`
`-8a+b=-28`.........i
and
put `x=2`
`implies a(2)^(3)+3(2)^(2)-8xx2+b=0`
`implies 8a+12-16+b=0`
`8a+b-4=0`

`8a+b=4`..........ii
From equation i & ii
`:. -8a+b=-28`
`ul(8a+b=4)`
`2b=-24`
`b=-12`
`a=2`
Discussion