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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.
| 1. |
If a, b, c are positive integers, then `((a^(2)+b^(2)+c^(2))/(a+b+c))^(a+b+c)gta^(x)b^(y)c^(z)`, thenA. `x=a, y=b, z=c`B. `x=b, y=a, z=c`C. `x=(1)/(a),y=(1)/(b),z=(1)/( c )`D. `x=y=z=1` |
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Answer» Correct Answer - A We know that Weighted `A.M.gt"Weighted G.M."` `implies" "(a.a+b.b+c.c)/(a+b+c)gt(a^(a)b^(b)c^(c))^((1)/(a+b+c))` `implies((a^(2)+b^(2)+c^(2))/(a+b+c))^(a+b+c)gta^(a)b^(b)c^(c)" "...(i)` Hence, x=a, y=b, z=c. |
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| 2. |
For all positive values of x and y, the value of `((1+x+x^(2))(1+y+y^(2)))/(xy)`,isA. `le9`B. `lt9`C. `ge9`D. `gt9` |
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Answer» Correct Answer - C Using `A.M.geG.M.`, we have `(x+(1)/(x)+1)/(3)ge(x xx(1)/(x)xx1)^(1//3)" and "(y+(1)/(y)+1)/(3)ge(yxx(1)/(y)xx1)^(1//3)` `implies" "(1+x+x^(2))/(3x)ge1" and "(1+y+y^(2))/(3y)ge1` `implies" "(1+x+x^(2))/(x)xx(1+y+y^(2))/(y)ge9` `implies" "((1+x+x^(2))(1+y+y^(2)))/(xy)ge9` |
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| 3. |
If `agt1,bgt1,cgt1,dgt1` then the minimum value of `log_(b)a+log_(a)b+log_(d)c+log_(c)d`, isA. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - D | |
| 4. |
If `x_(n)gt1` for all `n in N`, then the minimum value of the expression `log_(x_(2))x_(1)+log_(x_(3))x_(2)+...+log_(x_(n))x_(n-1)+log_(x_(1))x_(n)` is |
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Answer» Correct Answer - D Using `A.M.geG.M.`, we have `(log_(x_(2))x_(1)+log_(x_(3))x_(2)+...+log_(x_(n))x_(n-1)+log_(x_(1))x_(n))/(n)` `ge(log_(x_(2))x_(1).log_(x_(3))x_(2)....log_(x_(1))x_(n))^(1//n)` `implies" "log_(x_(2))x_(1)+log_(x_(3))x_(2)+...+log_(x_(1))x_(n)gen` Hence, the minimum value of the given expression is n and it attains this value when `x_(1)=x_(2)=......=x_(n)`. |
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| 5. |
Number of integers, which satisfy the inequality `((16)^(1/ x))/((2^(x+3))) gt1,` is equal to |
| Answer» Correct Answer - D | |
| 6. |
The number of solution of the equation `16(x^(2)+1)+pi^(2)=|tanx|+8pix` is equal toA. 2B. 4C. 1D. infinite |
| Answer» Correct Answer - A | |
| 7. |
The numbers of integral solutions of the equations `y^(2)(5x^(2)+1)=25(2x^(2)+13)" where "x, y inI`, isA. 2B. 4C. 8D. infinite |
| Answer» Correct Answer - B | |
| 8. |
The least value of `2^(sinx)+2^(cosx)`, isA. `2^(1//sqrt(2))`B. `2^(1-2^(-1//2))`C. `2^(1+2^(-1//2))`D. `2^(1-sqrt(2))` |
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Answer» Correct Answer - B Using `A.M.geG.M.,` we have `(2^(sinx)+2^(cosx))/(2)ge(2^(sinx)xx2^(cosx))^(1//2)` `implies" "2^(sinx)+2^(cosx)ge2{2^((1)/(2)(sinx+cosx))}` `implies" "2^(sinx)+2^(cosx)ge2^(1+(1)/(2)(sinx+cosx))" "...(i)` Now, `-sqrt(2)lesinx+cosxlesqrt(2)" for all "x` `implies" "-(1)/(sqrt(2))le(1)/(2)(sinx+cosx)le(1)/(sqrt(2))` `implies" "1-(1)/(sqrt(2))le1+(1)/(2)(sinx+cosx)le1+(1)/(sqrt(2))` `implies" "2^(1-(1)/(sqrt(2)))le2^(1+(1)/(2)(sinx+cosx))le2^(1+(1)/(sqrt(2)))" "...(ii)` From (i) and (ii), we get `2^(sinx)+2^(cosx)ge2^(1-(1)/(sqrt(2)))" for all "x in R.` Hence, the least value of `2^(sinx)+2^(cosx)" is "2^(1-(1)/(sqrt(2)))` |
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| 9. |
If `0ltxlt(pi)/(2)`, then the minimum value of `(sinx+cosx+cosec2x)`,isA. 27B. `(27)/(2)`C. `(25)/(4)`D. none of these |
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Answer» Correct Answer - D Using `A.M.geG.M.,` we have `(sinx+cosx+cosec2x)/(3)ge(sinx+cosx+cosec2x)^(1//3)` `sinx+cosx+cosec2ge3((1)/(3))^(1//3)=3xx1.2599=3.779` |
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| 10. |
Number of solutions of `|(1)/(|x|-1)|=x+sinx,` isA. 1B. 2C. 3D. none of these |
| Answer» Correct Answer - A | |
| 11. |
The least value of `5^(sinx-1)+5^(-sinx-1)`, isA. 10B. `(5)/(2)`C. `(2)/(5)`D. `(1)/(5)` |
| Answer» Correct Answer - C | |
| 12. |
If `a_(i)gt0` for `i u=1, 2, 3, … ,n` and `a_(1)a_(2)…a_(n)=1,` then the minimum value of `(1+a_(1))(1+a_(2))…(1+a_(n))`, isA. `2^(n//2)`B. `2^(n)`C. `2^(2n)`D. 1 |
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Answer» Correct Answer - B Using `A.MgeG.M`, we have `(1+a_(1))gesqrt(1xxa_(1))implies1+a_(1)ge2sqrt(a_(1))` `(1+a_(2))/(2)gesqrt(1xxa_(2))implies1+a_(2)ge2sqrt(a_(2))` `vdots" "vdots" "vdots` `(1+a_(n))/(2)lesqrt(1xxa_(n))implies1+a_(n)ge2sqrt(a_(n))` `:." "(1+a_(1))(1+a_(2))...(1+a_(n))ge2^(n)sqrt(a_(1)a_(2)...a_(n))` `implies" "(1+a_(1))(1+a_(2))...(1+a_(n))ge2^(n)` |
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| 13. |
If roots of the equation `x^(4)-8x^(3)+bx^(2)+cx+16=0` are positive, thenA. `b=8=c`B. `b=-24,c=-32`C. `b=24, c=-32`D. `b=24, c=32` |
| Answer» Correct Answer - C | |
| 14. |
The minimum value of the sum of the lengths of diagonals of a cyclic quadrilateral of area `a^(2)` square units, isA. `sqrt(2)a`B. `2sqrt(2)a`C. 2aD. none of these |
| Answer» Correct Answer - B | |
| 15. |
If `a_(1)gt0` for all `i=1,2,…..,n`. Then, the least value of `(a_(1)+a_(2)+……+a_(n))((1)/(a_(1))+(1)/(a_(2))+…+(1)/(a_(n)))`, isA. `n^(2)`B. 2nC. nD. `(1)/(n)` |
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Answer» Correct Answer - B Using `A.M.geG.M.,` we have `(a_(1)+a_(2)+...+a_(n))/(n)ge(a_(1)a_(2)...a_(n))^(1//n)" "...(i)` `implies(a_(1)+a_(2)+...+a_(n))gen(a_(1)a_(2)...a_(n))^(1//n)` Again, using `A.M.geG.M.,` we have `((1)/(a_(1))+(1)/(a_(2))+...+(1)/(a_(n)))/(n)ge((1)/(a_(1))xx(1)/(a_(2))xx...xx(1)/(a_(n)))^(1//n)` `implies((1)/(a_(1))+(1)/(a_(2))+...+(1)/(a_(n)))gen((1)/(a_(1)a_(2)......a_(n)))^(1//n)" "...(ii)` From (i) and (ii), we get `(a_(1)+a_(2)+...+a_(n))((1)/(a_(1))+(1)/(a_(2))+...+(1)/(a_(n)))` `gen|a_(1)a_(2)......a_(n)|^(1//n)xx{(n)/((a_(1)a_(2)....a_(n)))}^(1//n)` `implies(a_(1)+a_(2)+....+a_(n))((1)/(a_(1))+(1)/(a_(2))+....+(1)/(a_(n)))gen^(2)` |
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| 16. |
The least value of`n` such that` (n -2)x^2 + 8x+ n + 4 gt0`, where n e N,A. 2B. 3C. 4D. 5 |
| Answer» Correct Answer - D | |
| 17. |
The least perimeter of a cyclic quadrilateral of given area A square units, isA. `sqrt(A)`B. `2sqrt(A)`C. `3sqrt(A)`D. `4sqrt(A)` |
| Answer» Correct Answer - D | |
| 18. |
If `xyz=abc,` then the least value of `bcx+cay+abz,` isA. 3abcB. 6abcC. abcD. 4abc |
| Answer» Correct Answer - A | |
| 19. |
If `a+b+c=1`, the greatest value of `(ab)/(a+b)+(bc)/(b+c)+(ca)/(c+a)`, isA. `(1)/(3)`B. `(1)/(2sqrt(2))`C. `(1)/(2)`D. `(1)/(4)` |
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Answer» Correct Answer - C Using `A.MgeH.M.,` we have `(a+b)/(2)ge(2ab)/(a+b),(b+c)/(2)ge(2bc)/(b+c)" and "(c+a)/(2)ge(2ca)/(c+a)` `implies" "a+b+cge2((ab)/(a+b)+(bc)/(b+c)+(ca)/(c+a))` `implies" "(1)/(2)ge(ab)/(a+b)+(bc)/(b+c)+(ca)/(c+a)` |
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| 20. |
If a, b, c are positive real numbers such that `a+b+c=1`, then the greatest value of `(1-a)(1-b)(1-c), isA. `(1)/(27)`B. `(8)/(27)`C. `(4)/(27)`D. 9 |
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Answer» Correct Answer - B Using `G.M.leA.M.,` we have `{(1-a)(1-b)(1-c)}^(1//3)le(1-a+1-b+1-c)/(3)` `implies" "(1-a)(1-b)(1-c)le(8)/(27)` Hence, the greatest value is `(8)/(27)` |
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| 21. |
If `a,bgt0` then the maximum value of `(a^(3)b)/((a+b)^(4)),` isA. `(81)/(512)`B. `(27)/(256)`C. `(27)/(512)`D. `(81)/(256)` |
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Answer» Correct Answer - B Using `AMgeGM` for a, a, a, 3b, we get `(a+a+a+3b)/(4)ge(axxaxxaxx3b)^(1//4)` `implies" "(3(a+b))/(4)ge(3a^(3)b)^(1//4)` `implies" "(3^(4))/(4^(4))(a+b)^(4)ge3a^(3)b` `implies" "(27)/(256)ge(a^(3)b)/((a+b)^(4))` `implies" "(a^(3)b)/((a+b)^(4))le(27)/(256)" for all "a,bgt0.` |
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| 22. |
If `a, bgt0,a+b=1`, then the least value of `(1+(1)/(a))(1+(1)/(b))`, isA. 3B. 6C. 9D. 12 |
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Answer» Correct Answer - C We have, `(1+(1)/(a))(1+(1)/(b))=((a+1)(b+1))/(ab)=(1+(a+b)+ab)/(ab)=(2+ab)/(ab)` `implies" "(1+(1)/(a))(1+(1)/(b))=1+(2)/(ab)` Now, `A.M. geG.M.` `implies" "(a+b)/(2)gesqrt(ab)` `implies" "(1)/(2)gesqrt(ab)implies(1)/(ab)ge4implies1+(2)/(ab)ge9implies(1+(1)/(a))(1+(1)/(b))ge9` Hence, the least value of `(1+(1)/(a))(1+(1)/(b))` is 9. |
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| 23. |
If `agt0,bgt0,cgt0,` then the minimum value of `sqrt((4a)/(b))+root(3)((27b)/(c))+root(4)((c)/(108a)),` isA. 4B. 5C. 3D. 9 |
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Answer» Correct Answer - D Using `(la+mb+nc)/(l+m+n)ge(a^(l)b^(m)c^(n))^((1)/(l+m+n))`, we obtain `(sqrt((4a)/(b))+root(3)((27b)/(c))+root(4)((c)/(108a)))/(2+3+4)` `ge{(sqrt((4a)/(b)))^(2)xx(root(3)((27b)/(c)))^(2)xx(root(4)((c)/(180a)))^(4)}^((1)/(2+3+4))` `implies" "sqrt((4a)/(b))+root(3)((27b)/(c))+root(4)((c)/(108a))ge9` |
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| 24. |
`If-(pi)/(2)lt0lt(pi)/(2),` then the minimum value of `cos^(3)theta+sec^(3)thets` isA. 1B. 2C. 0D. none of these. |
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Answer» Correct Answer - B We know that `A.M.geG.M.` `:." "(cos^(3)theta+sec^(3)theta)/(2)gesqrt(cos^(3)thetaxxsec^(3)theta)` `implies" "cos^(3)theta+sec^(3)thetage2`. Hence, the minimum value of `cos^(3)theta+sec^(3)theta" is "2.` |
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| 25. |
The minimum value of `9^x+9^(1-x),x inR,` isA. 2B. 3C. 4D. none of these. |
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Answer» Correct Answer - D Using `A.M.geG.M.,` we have `(9x+9^(1-x))/(2)gesqrt(9^(x)xx9^(1-x))implies9^(x)+9^(1-x)ge6` Hence, the least value of `9^(x)+9^(1-x)` is 6. |
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| 26. |
If a, b and c are distinct positive numbers, then the expression `(a + b - c)(b+ c- a)(c+ a -b)- abc` is:A. positiveB. negativeC. non-positiveD. non-negative |
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Answer» Correct Answer - B Using `A.M.geG.M.,` we have `((b+c-a)+(c+a-b))/(2)gt[(b+c-a)(c+a-b)]^(1//2)` `((c+a-b)+(a+b-c))/(2)gt{(c+a-b)(a+b-c)}^(1//2)` and, `((a+b-c)+(b+c-a))/(2)gt{(a+b-c)(b+c-a)}^(1//2)` `implies" "cgt[(b+c-a)(c+a-b)]^(1//2),age{(c+a-b)(a+b-c)}^(1//2)` and, `bgt{(a+b-c)(b+c-a)}^(1//2)` `implies" "abcgt(a+b-c)(b+c-a)(c+a-b)[{:("On multiplying"),("three inequalities"):}]` `implies" "(a+b-c)(b+c-a)(c+a-b)-abclt0` |
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| 27. |
If `a > 1, b > 1,` then the minimum value of `log_b a + log_a b` is |
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Answer» Correct Answer - C Using `A.M.geG.M.,` wc have `(log_(b)a+log_(a)b)/(2)gesqrt(log_(b)alog_(a)b)` `impliescos^(3)theta+sec^(3)thetage2.` Hence, the minimum value of `log_(b)a+log_(a)b` is 2 and it is attained when a=b. |
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| 28. |
`log_100 a-log_a 0.0001 , a >1`A. 2B. 3C. 4D. none of these |
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Answer» Correct Answer - C We have, `2log_(100)a-log_(a)0.0001` `=2log_(100)a-log_(a)(100)^(-2)` `=2log_(100)a+2log_(a)100` `=2(log_(100)a+log_(a)100)` `=(log_(100)a+(1)/(log_(100)a))` `ge2xx2=4" "[becausex+(1)/(x)ge2" for all "xgt0]` |
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| 29. |
If a, b, c are positive real numbers, then the minimum value of `a^(logb-logc)+b^(logc-loga)+c^(loga-logb)`isA. 3B. 1C. 9D. 16 |
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Answer» Correct Answer - A Using `A.M.geG.M.,` we have `(a^(logb-logc)+b^(logc-loga)+c^(loga-logb))/(3)` `ge{a^(logb-logc)xxb^(logc-loga)xxc^(loga-logb)}^(1//3)` Let `a^(logb-logc)xxb^(logc-loga)xxc^(loga-logb)=lambda`.Then, `loglambda=(logb-logc)loga+(logc-loga)logb+(loga-logb)logc` `implies" "loglambda=0implieslambda=1` `a^(logb-logc)+b^(logc-loga)+c^(loga-logb)ge3` Hence, the minimum value of `a^(logb-logc)+b^(lobc-loga)+c^(loga-logb)`is 3. |
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| 30. |
Let `a_(1),a_(2)…,a_(n)` be a non-negative real numbers such that `a_(1)+a_(2)+…+a_(n)=m` and let `S=sum_(iltj) a_(i)a_(j)`, thenA. `Sle(m^(2))/(2)`B. `Sgt(m^(2))/(2)`C. `Slt(m)/(2)`D. `Sgt(m^(2))/(2)` |
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Answer» Correct Answer - A We have, `m^(2)=(a_(1)+a_(2)+...+a_(n))^(2)` `implies" "m^(2)=underset(i=1)overset(n)suma_(i)""^(2)+2underset(iltj)overset(n)suma_(i)a_(j)` `implies" "m^(2)=underset(i=1)overset(n)suma_(i)""^(2)+2" S"` `implies" "m^(2)-2S=underset(i=1)overset(n)suma_(i)""^(2)impliesm^(2)-2Sge0impliesSle(m^(2))/(2)` |
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| 31. |
If a and b are two different positive real numbers then which of the following statement is true?A. `2sqrt(ab)gta+b`B. `2sqrt(ab)lta+b`C. `2sqrt(ab)=a+b`D. none of these |
| Answer» Correct Answer - B | |
| 32. |
Statement-1 : For any real number `x,(2x^(2))/(1+x^(4)_le1` Statement-2: `A.M.gtG.M.`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - A Clearly, `(2x^(2))/(1+x^(4))le1` is true of x=0. If `xne0" then "A.MgtG.M.` `implies" "(x^(2)+(1)/(x^(2)))/(2)gesqrt(x^(2)xx(1)/(x^(2)))implies" "(x^(4)+1)/(2x^(2))ge1implies(2x^(2))/(x^(4)+1)le1` Hence, both the statements are true and statement-2 is a correct explanation for statement-1. |
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| 33. |
If a, b, c are positive real numbers such that `a+b+c=p` then, which of the following is true?A. `(p-a)(p-b)(b-c)ge(8)/(27)p^(3)`B. `(p-a)(p-b)(b-c)ge8abc`C. `(bc)/(a)+(ca)/(b)+(ab)/(c)gep`D. none of these |
| Answer» Correct Answer - B | |
| 34. |
If a, b, c are positive real numbers such that `a+b+c=2` then, which one of the following is true?A. `(2-a)(2-b)(2-c)ge8abc`B. `(1)/(a)+(1)/(b)+(1)/(c)ge2`C. `(2-a)(2-b)(2-c)lt8abc`D. `(1)/(a)+(1)/(b)+(1)/( c )=2` |
| Answer» Correct Answer - A | |
| 35. |
Statement-1 : If angles A, B, C of `DeltaABC` are acute, then `cotA cotB cotCle(1)/(3sqrt(3)).` Statement-2: If a, b, c are positive real numbers and `-ltmlt1`, then `(a^(m)+b^(m)+c^(m))/(3)lt((a+b+c)/(3))^(m)`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - B Using `A.M.geG.M.,` we have `(tanA+tanB+tanC)/(3)ge(tanAtanBtanC)^(1//3)` `implies" "(tanAtanBtanC)/(3)ge(tanAtanBtanC)^(1//3)` `" "[becausetanA+tanB+tanC=tanAtanBtanC]` `implies" "(tanAtanBtanC )^(2//3)ge3` `implies" "tanAtanBtanCge3sqrt(3)impliescotAcotBcotCle(1)/(3sqrt(3))` Hence, statement-1 is true. Statement-2 is also true but it is not a correct explanation for statement-1. |
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| 36. |
Statement-1: If a, b are positive real numbers such that `a^(3)+b^(3)=16`, then `a+ble4.` Statement-2: If a, b are positive real numbers and `ngt1`, then `(a^(n)+b^(n))/(2)ge((a+b)/(2))^(n)`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - B If `0ltnlt1,` then `(a^(n)+b^(n))/(2)le((a+b)/(2))^(n)` `:." "((a^(3))^(1//3)+(b^(3))^(1//3))/(2)le((a^(3)+b^(3))/(2))^(1//3)implies(a+b)/(2)le((16)/(2))^(1//3)` `implies" "a+ble4` Hence, statement-1 is true. Also, statement-2 is true, but it is not a correct explanation for statement-1. |
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| 37. |
If `x+y+z=1`, then the least value of `(1)/(x)+(1)/(y)+(1)/(z)`, isA. 3B. 9C. 27D. 1 |
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Answer» Correct Answer - B Using `A.M.geH.M.`, we have `((1)/(x)+(1)/(y)+(1)/(z))/(3)ge(3)/(x+y+z)` `implies" "(1)/(x)+(1)/(y)+(1)/(z)ge9` Hence, the least value of `(1)/(x)+(1)/(y)+(1)/(z)` is 9. |
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| 38. |
Statement-1: If x, y are positive real number satisfying `x+y=1`, then `x^(1//3)+y^(1//3)gt2^(2//3)` Statement-2: `(x^(n)+y^(n))/(2)lt((x+y)/(2))^(n)`, if `0ltnlt1` and `x,ygt0.`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - D Clearly, statement-2, being a standard result, is true. Using this, we obtain `(x^(1//3)+y^(1//3))/(2)lt((x+y)/(2))^(1//3)implies" "x^(1//3)+y^(1//3)lt2^(2//3)` So, statement-1 is false. |
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| 39. |
Let `a,b,c,d in R` such that `a^2+b^2+c^2+d^2=25`A. `ab+bc+c d+dale(25)/(2)`B. `ab+bc+c d+dale25`C. `ab+bc+c d+dale5`D. `ab+bc+c d+dale(5)/(2)` |
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Answer» Correct Answer - B Using `A.M.geG.M.,` we obtain `(a^(2)+b^(2))/(2)geab,(b^(2)+c^(2))/(2)gebc,(c^(2)+d^(2))/(2)gecd" and "(d^(2)+a^(2))/(2)geda` `implies" "a^(2)+b^(2)+c^(2)+d^(2)geab+bc+cd+da` `implies" "ab+bc+cd+dale25` |
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| 40. |
If x, y, z are non-negative real numbers satisfying `x+y+z=1`, then the minimum value of `((1)/(x)+1)((1)/(y)+1)((1)/(z)+1)`, isA. 8B. 16C. 32D. 64 |
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Answer» Correct Answer - D From the above example, we have `(1)/(x)+(1)/(y)+(1)/(z)ge9` Also, by using `A.M.geG.M.,` we have `implies" "(x+y+z)/(3)ge(xyz)^(1//3)` `implies" "(1)/(xyz)ge27" "[becausex+y+z=1]` Now, `((1)/(x)+1)((1)/(y)+1)((1)/(z)+1)` `=1+(1)/(x)+(1)/(y)+(1)/(z)+(x+y+z)/(xyz)+(1)/(xyz)ge1+9+27+27=64` |
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| 41. |
The minimum value of the sum of real number `a^(-5),a^(-4),3a^(-3),1,a^8,a n da^(10)w i t ha >0`isA. 6B. 7C. 8D. 9 |
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Answer» Correct Answer - C Using `AMgeGM,` we have `(a^(-5)+a^(-4)+a^(-3)+a^(-3)+a^(-3)+1+a^(8)+a^(10))/(8)` `ge(a^(-5)xxa^(-4)xxa^(-3)xxa^(-3)xxa^(-3)xx1xxa^(8)xxa^(10))^(1//8)` `implies" "(a^(-5)+a^(-4)+3a^(-3)+1+a^(8)+a^(10))/(8)ge1` `implies" "a^(-5)+a^(-4)+3a^(-3)+1+a^(8)+a^(10)ge8` Hence, the minimum value of `a^(-5)+a^(-4)+3a^(-3)+1+a^(8)+a^(10)` is 8. |
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| 42. |
If a, b, c `in` R, then which one of the following is true:A. `max(a,b)ltmax(a,b,c)`B. `min(a,b)=(1)/(2){a+b-|a-b|}`C. `min(a,b)ltmin(a,b,c)`D. none of these |
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Answer» Correct Answer - B If we assume that `clta` and `cltb` then we observe that options (a) and ( c ) do not hold. In order to check option (b), let us consider the following cases: CASE I When `altb` In this case, we have `min(a,b)=b" and "|a-b|=a-b` `:." "(1)/(2){a+b-|a-b|}=(1)/(2){a+b-(a-b)}=b` Hence, `min(a,b)=(1)/(2){a+b-|a-b|}` Thus, in both the cases, we have `min(a,b)=(1)/(2){a+b-|a-b|}` Hence, option (b) is true. |
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| 43. |
If p,q,r be three distinct real numbers, then the value of `(p+q)(q+r)(r+p),` isA. `gt8pqr`B. `lt8pqr`C. `8pqr`D. none of these |
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Answer» Correct Answer - A Using `A.M.gtG.M.,` we have `(p+q)/(2)gtsqrt(pq),(q+r)/(2)gtsqrt(qr)" and "(r+p)/(2)gtsqrt(rp)` `implies" "p+qgt2sqrt(pq),q+rgt2sqrt(qr)" and "r+pgt2sqrt(rp)` `implies" "(p+q)(q+r)(r+p)gt2sqrt(pq)xx2sqrt(qr)xx2sqrt(rp)` `implies" "(p+q)(q+r)(r+p)gt8pqr` |
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| 44. |
If a, b are postitive real numbers such that ab=1, then the least value of the expression `(1+a)(1+b)` isA. 2B. 3C. 4D. none of these. |
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Answer» Correct Answer - C We have, `ab=1impliesb=(1)/(a)` `:." "(1+a)(1+b)=(1+a)(1+(1)/(a))` `implies" "(1+a)(1+(1)/(a))=2+a+(1)/(a)ge2+2=4" "[because a+(1)/(a)ge2]` Hence, the least value of `(1+a)(1+b)` is 4. |
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| 45. |
If x and y are positive real numbers such that `x^(2)y^(3)=32` then the least value of `2x+3y` isA. 5B. 10C. 20D. 15 |
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Answer» Correct Answer - B Let `S=2x+3y.` Then, `S=x+x+y+y+y` Clearly, S is the sum of five terms and the product of these five terms `i.e. x^(2)y^(3)` is given as a constant equal to 32. Therefore, S is least when all these factors are equal `i.e.x=y.` `:." "x^(2)y^(3)=32impliesx^(5)=2^(5)impliesx=2` Hence, the least value of S is given by `S=2xx2+3xx2=10.` |
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| 46. |
If `mgt1` and `ninN`, such that `1^(m)+2^(m)+3^(m)+...+n^(m)gtn((n+1)/(k))^(m)` Then, k=A. 2B. nC. mD. 1 |
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Answer» Correct Answer - A For `mgt1,` we have `A.M. "of mth powers"gt"mth power of "A.M.` `:." "(1^(m)+2^(m)+3^(m)+...+n^(m))/(n)gt((1+2+3...+n)/(n))^(m)` `implies" "(1^(m)+2^(m)+3^(m)+n^(m))/(n)gt((n+1)/(2))^(m)` `implies" "1^(m)+2^(m)+...+n^(m)gtn((n+1)/(2))^(m)` Hence, k=2. |
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| 47. |
If `a^2+b^2+c^2=1` then `ab+bc+ca` lies in the intervalA. [0,1]B. `[-1//2,1]`C. `[0,1//2]`D. [1,2] |
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Answer» Correct Answer - B We have, `a^(2)+b^(2)+c^(2)-ab-bc-ca=(1)/(2)[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]ge0` `implies" "a^(2)+b^(2)+c^(2)geab+bc+ca` `implies" "1geab+bc+ca" "[becaise a^(2)+b^(2)+c^(2)=1]` `implies" "ab+bc+cale1" "...(i)` Also, `(a+b+c)^(2)ge0` `implies" "a^(2)+b^(2)+c^(2)+2(ab+bc+ca)ge0` `implies" "1+2(ab+bc+ca)ge0" "[because a^(2)+b^(2)+c^(2)=1]` `implies" "ab+bc+ca ge-(1)/(2)` From (i) and (ii), we obtain `-(1)/(2)geab+bc+cale1impliesab+bc+ca in[-1//2,1].` |
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| 48. |
If `a ,b ,c ,d`are positive real umbers such that `a=b+c+d=2,t h e nM=(a+b)(c+d)`satisfies the relation`0lt=Mlt=1``1lt=Mlt=2``2lt=Mlt=3``3lt=Mlt=4`A. `0leMle1`B. `1leMle2`C. `2leMle3`D. `3leMle4` |
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Answer» Correct Answer - A Using `A.M.geG.M.,` we have `((a+b)+(c+d))/(2)ge{(a+b)(c+d)}^(1//2)` `implies" "(2)/(2)geM^((1)/(2))` `implies" "Mle1` As a, b, c, d`gt`0. Therefore, `M=(a+b)(c+d)gt0.` Hence, `0leMle1.` |
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| 49. |
If a, b, c are positive real number such that `lamba` abc is the minimum value of `a(b^(2)+c^(2))+b(c^(2)+a^(2))+c(a^(2)+b^(2))`, then `lambda`=A. 1B. 2C. 3D. 6 |
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Answer» Correct Answer - D Using `A.M.geG.M.`, we have `(a(b^(2)+c^(2))+b(c^(2)+a^(2))+c(a^(2)+b^(2)))/(6)ge(ab^(2)ac^(2)bc^(2)ba^(2)ca^(2)cb^(2))^(1//6)` `implies" "(a(b^(2)+c^(2))+b(c^(2)+a^(2))+c(a^(2)+b^(2)))/(6)geabc` `implies" "a(b^(2)+c^(2))+b(c^(2)+a^(2))+c(a^(2)+b^(2))ge6abc` `implies" "lambda=6` |
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