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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. |
The maximum values of `3 costheta+5sin(theta-(pi)/(6))` for any real value of `theta` is:A. `sqrt(19)`B. `sqrt(79)/2`C. `sqrt(31)`D. `sqrt(34)` |
| Answer» Correct Answer - 1 | |
| 52. |
Maximum and minimum value of `2sin^(2)theta-3sintheta+2` is-A. `1/4, -7/4`B. `1/4, 21/4`C. `21/4, -3/4`D. `7, 7/8` |
| Answer» Correct Answer - D | |
| 53. |
The number of solutions of the pair of equations`2s in^2theta-cos2theta=0``2cos^2theta-3sintheta=0`in the interval `[0,2pi]`is0 (b)1 (c) 2(d) 4A. zeroB. oneC. twoD. four |
| Answer» Correct Answer - C | |
| 54. |
Find maximum and minium value of `5costheta+3sintheta(theta+pi/6)` for all real values of `theta`. |
| Answer» Correct Answer - 7 and `-7` | |
| 55. |
The value of `cos(pi/2^(2)).cos(pi/2^(3))……….cos(pi/2^(10)).sin(pi/2^(10))` isA. `1/256`B. `1/2`C. `1/512`D. `1/1024` |
| Answer» Correct Answer - 3 | |
| 56. |
If x and y are real number such that `x^2 +2xy-y^2=6`, find the minimum value of `(x^2+y^2)^2` |
| Answer» Correct Answer - 18 | |
| 57. |
The minimum value of `f(x)=|x-1|+|x-2|+|x-3|` is equal toA. 1B. 2C. 3D. 0 |
| Answer» Correct Answer - B | |
| 58. |
Solve the inequation `2^(1/(cos^2 x))sqrt(y^2-y+1/2) |
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Answer» `2^(1/cos^(2)x) sqrt(y^(2)-y+1//2) le1`…………(i) `2^(1/(cosp^(2)x)) sqrt((y-1/2)^(2)+(1/2)^(2)) le1` Miniumum value of `2^((1)/(cos^(2)x)=2` Minimum value of `sqrt((y-1/2)^(2)+(1/2)^(2))=1/2` `rArr` Minimum value of `2^(1/(cos^(2)x)) sqrt(y^(2)-y+1/2)` is 1 `rArr` (i) is possible when `2^(1/(cos^(2)x)) sqrt((y-1/2)^(2)+(1/2)^(2))=1` `rArr cos^(2)x=1` and `y=1/2 rArr cosx = +1 rArr x=npi`, where `n in I`, Hence, `x=npi, n in I` and `y=1//2`. |
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| 59. |
Find the solution set oif inequation `cosxge 1/2` |
| Answer» `overset(n in I)cup [2npi-(2pi)/3, 2npi+(2pi)/3]` | |
| 60. |
The complete solution set of the inequation `sqrt(x+18)le 2-x` is equal to-A. `[-18,-2]`B. `(-infty,-2) cup (7,infty)`C. `(-18,2) cup(7, infty)`D. `[-18,-2]` |
| Answer» Correct Answer - D | |
| 61. |
In any triangle ABC, `sinA -cosB=cosC`, then angle B isA. `pi/2`B. `pi/3`C. `pi/4`D. `pi/6` |
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Answer» Correct Answer - a We have, `sinA-cosB=cosC` `sinA=cosB+cosC` `rArr 2sinA/2cosA/2=2cos(B+C)/2cos(B-C)/2` `rArr 2sinA/c cosA/2=2cos(pi-A)/2 cos(B-C)/(2)` `therefore A+B+C=pi` `rArr 2sinA/2 cosA/2=2sinA/2cos(B-C)/(2)` `rArr cosA/2=cos(B-C)/(2)` or `A=B-C`, But A+B+C=`pi` Therefore `2B=pi rArr B=pi/2` |
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| 62. |
If `sinA=3/5` and `cosB=9/41` then find values of the following compound anglesA. `sin(A+B)`B. `sin(A-B)`C. `cos(A+B)`D. `cos(A-B)` |
| Answer» a) `187/205`, b) `133/205`, c) `-84/205`, d) `156/205` | |
| 63. |
Find the value of (a) `sin((pi)/(8))` (b) `cos((pi)/(8))` (c) `tan ((pi)/(8))` |
| Answer» Correct Answer - `-9/8` | |
| 64. |
Simplify `(sin75^(@)-sin15^(@))/(cos75^(@)+cos15^(@))` |
| Answer» Correct Answer - `1/sqrt(3)` | |
| 65. |
If `2tan^(2)theta=sec^(2)theta`, then the general solution of `theta`-A. `npi+pi/4(n in I)`B. `npi-pi/4(n in I)`C. `npi +- pi/4(n in I)`D. `2npi+-pi/4(n in I)` |
| Answer» Correct Answer - C | |
| 66. |
Number of principle solution(s) of the equation `4.16^(sin^(2)x) = 2^(6sinx)` isA. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - C | |
| 67. |
Find the values of `alpha` lying between 0 and `pi` for which of the inequality: `tanalpha gt tan^(3) alpha` is valid. |
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Answer» We have : `tanalpha-tan^(3)alpha gt 0 rArr tanalpha(1-tan^(2)alpha) lt 0` So, `tanalpha lt -1,0 gt tan alpha gt 1` `therefore` (Given inequality holds for) `alpha in (0,pi/4) cup (pi/2, (3pi)/4)` |
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| 68. |
The value of `sin 78^@ - sin 66^@ - sin 42^@ + sin 6^@` is |
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Answer» The expression `=(sin78^(@)-sin42^(@))-(sin66^(@)-sin6^(@))=2cos(60^(@))sin(18^(@))-2cos36^(@).sin30^(@)` `=sin18^(@)-cos36^(@)=(sqrt(5)-1)/(4) -(sqrt(5)+1)/(4)=-1/2` |
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| 69. |
sin(67`1/2)^(@) + cos(`67`1/2)^(@)` is equal to A) `1/2sqrt(4+2sqrt(2))`, B) `1/2sqrt(4-2sqrt(2))`, C) `1/4(sqrt(4+2sqrt(2)))`, D) `1/4(sqrt(4-2sqrt(2)))` |
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Answer» `sin67^(@)1/2^(@)+cos67^(@)1/2^(@) = sqrt(1+sin135^(@)) = sqrt(1+1/sqrt(2))` (using `cosA+sinA=sqrt(1+sin2A))` `=1/2sqrt(4+2sqrt(2))` Ans |
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| 70. |
Prove that `(2cos2A+1)/(2cos2A-1)=tan(60^0+A)tan(60^0-A)dot` |
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Answer» RHS `=tan(60^(@)+A)tan(60^(@)-A)` `=((tan60^(@)+tanA)/(1-tan60^(@)tana))((tan60^(@)-tanA)/(1+tan60^(@)tanA)) = ((sqrt(3)+tanA)/(1-sqrt(3)tanA))((sqrt(3)-tanA)/(1+sqrt(3)tanA))` `=(3-tan^(2)A)/(1-3tan^(2)A) = (3-(sin^(2)A)(cos^(2)A))/(1-3(sin^(2)A)/(cos^(2)A)) = (3cos^(2)A-sin^(2)A)/(cos^(2)A-3sin^(2)A) = (2cos^(2)A+cos^(2)A-2sin^(2)A+sin^(2)A)/(2cos^(2)A-2sin^(2)A-sin^(2)A-cos^(2)A)` `=(2(cos^(2)A-sin^(2)A)+cos^(2)A+sin^(2)A)/(2(cos^(2)A-sin^(2)A)-cos^(2)A)` `=(2(cos^(2)A-sin^(2)A)+cos^(2)A+sin^(2)A)/(2(cos^(2)A-sin^(2)A))-(sin^(2)A+cos^(2)A) = (2cos2A+1)/(2cos2A-1)`= LHS |
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| 71. |
The sum of all the solutions to the equations `2log_(10)x-log_(10)(2x-75)=2`A. 30B. 350C. 75D. 200 |
| Answer» Correct Answer - D | |
| 72. |
Let `x= 2^(log 3)` and `y=3^(log 2)` where base of the logarithm is 10,then which one of the following holds good. (A) `2x lt y` (B) `2y lt x` (C) `3x = 2y` (D) `y = x`.A. `2x lt y`B. `2y lt x`C. `3x=2y`D. `y=x` |
| Answer» Correct Answer - D | |
| 73. |
Find the square of the sum of the roots of the equation `log_(3)x.log_(4)x.log_(5)x=log_(3)x.log_(4)x+log_(4)x. log_(5)x+log_(5)x.log_(3)x` |
| Answer» Correct Answer - 3721 | |
| 74. |
If a,b,c are distinct real number different from 1 such that `(log_(b)a. log_(c)a-log_(a)a) + (log_(a)b.log_(c)b.log_(c)b-log_(b)b) +(log_(a)c.log_(b)c-log_(c)C)=0`, then abc is equal to |
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Answer» `(log_(b)alog_(c)a-1) + (log_(a)b.log_(c)b-1)+(log_(a)clog_(b)c-1)=0` `rArr (loga)/(logb).(loga)/(logc) +(logb)/(loga). (log b)/(log c) + (log c)/(log a).(log c)/(log b) =3` `rArr (loga+logb+logc)=0 [therefore "if" a^(3)+b^(3)+c^(3)-3abc=0`, then a+b+c=0 if `a ne b ne c` `rArr logabc = log1 rArr abc=1` |
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| 75. |
Consider the function f(x) `=|x-1|-2|x+2|+|x+3|` |
| Answer» `A to Q, B to r, C to p, D to s` | |
| 76. |
Prot that theequation `k cos x-3s in x=k+1`possess a solution if`k in (-oo,4]dot` |
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Answer» Here, `kcosx-3sinx=k+1`, could be re-written as: `k/sqrt(k^(2)+9) cosx -3/sqrt(k^(2)+9) sinx = (k+1)/sqrt(k^(2)+9)` or `cos(x+phi) = (k+1)/sqrt(k^(2)+9)`, where `tanphi=3/k` which possess a solution only is `-1 le(k+1)/sqrt(k^(2)+9) le1` i.e., `|(k+1)/sqrt(k^(2)+9)| le1` i.e., `(k+1)^(2)lek^(2)+9` i.e., `k^(2)+2k+1 lek^(2)+9` or `k le4` The interval of k for which the equation `(kcosx-3sinx=k+1)` has a solution is `(-infty,4)`. |
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| 77. |
If `3^x=4^(x-1)`then x=A. `(2log_(3)2)/(2log_(3)2-1)`B. `2/(2log_(2)3)`C. `1/(1-log_(4)3)`D. `(2log_(2)3)/(2log_(2)3-1)` |
| Answer» Correct Answer - A,B,C | |
| 78. |
If `log_(a)x = p` and `log_(b)x^(2) =q,` then `log_(x)sqrt(ab)` is equal to (where, a,b, `x in R^(+)-{1})-` A) `1/p+1/q`, B) `1/(2p)+1/q`, C) `1/p+1/(2q)`, D) `1/(2p)+1/(2q)` |
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Answer» `log_(a)x = p rArr a^(p)= x rArr a=x^(1//p)` similarly `b^(q) = x^(2) rArr b=x^(2/q)` Now, `log_(x)sqrt(ab) = log_(x) sqrt(x^(1//)x^(2//q)) = log_(x)^(1/p+2/q.1/2) = 1/(2p) +1/q` |
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| 79. |
Solve the following equations where `x in R` a) `(x-1)|x^(2)-4x+3|+2x^(2)+3x-5=0` b) `|x^(2)+4x+3|+2x+=0` c) `|x+3|(x+1)+|2x+5|=0` |
| Answer» a) 1, b) `-4, sqrt(3)-1, c) -4,-2 , -sqrt(3)-1` | |
| 80. |
The value of `((log_(2)9)^(2))^(1/(log_(2)(log_(2)9))) xx (sqrt(7))1/(log_(4)7) ` is .................... |
| Answer» Correct Answer - 8 | |
| 81. |
if ` sin x + sin^2 x = 1`, then the value of `cos^2 x + cos^4x` is |
| Answer» Correct Answer - C | |
| 82. |
Prove that: `tanA + tan(60^(@)+A) + tan(120^(@)+A)=3tan3A` |
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Answer» LHS `=tanA+tan(60^(@)+A)+tan(120^(@)+A)` `=tanA+tan(60^(@)+A)-tan(60^(@)-A)` `(therefore tan(180^(@)-theta) = -tantheta)` `=tanA+(tan60^(@)+tanA)/(1+tan60^(@)tanA) = tanA+(sqrt(3)+tanA)/(1-sqrt(3)tanA)-(sqrt(3)-tanA)/(1+sqrt(3)tanA)` `=tanA+(sqrt(3)+tanA+3tanA+sqrt(3)tan^(2)A-sqrt(3)+tanA+3tanA-sqrt(3)tan^(2)A)/((1-sqrt(3)tanA)(1+sqrt(3)tanA)` `=tan+(8 tanA)/(1-3tan^(2)A) = (tanA-3tan^(3)A+8tanA)/(1-3tan^(2)A)` `=(9tanA-3tan^(3)A)/(1-3tan^(2)A) = (tanA-3tan^(3)A+8tanA)/(1-3tan^(2)A)` `=(9tanA-3tan^(3)A)/(1-3tan^(2)A) = 3(3tanA-tan^(3)A)/(1-3tan^(2)A) = 3tan3A`=RHS |
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| 83. |
The value for `2(sin^6theta+cos^6theta)-3(sin^4theta+cos^4theta)`+1 isA. 2B. 0C. 4D. 6 |
| Answer» Correct Answer - B | |
| 84. |
If `tanA=-1/2` and `tanB=-1/3`, (where A, `B gt 0)`, then A+B can beA. `pi/4`B. `(3pi)/4`C. `(5pi)/4`D. `(7pi)/4` |
| Answer» Correct Answer - D | |
| 85. |
If `tanA+tanB+tanC=tanA.tanB.tanC`, then-A. A,B,C must be angles of a triangleB. the sum of any two of A,B,C is equal to the thirdC. A+B+C must be n integral multiple of `pi`D. None of these |
| Answer» Correct Answer - C | |
| 86. |
The number of real solutions of the equation `sin(e^x)=2^x+2^(-x)` isA. 1B. 0C. 2D. Infinite |
| Answer» Correct Answer - B | |
| 87. |
The value of `sin10^(@)+2sin20^(@)+sin30^(@)+….+sin360^(@)` is equal to |
| Answer» Correct Answer - A | |
| 88. |
`cos^6(pi/16)+cos^6(3pi/16)+cos^6(5pi/16)+cos^6(7pi/16)` |
| Answer» Correct Answer - `5/4` | |
| 89. |
`(2cos4 0^(@)-cos2 0^(@))/(sin2 0^(@)` |
| Answer» Correct Answer - `sqrt(3)` | |
| 90. |
Let `S={xepsilon(-pi,pi):x!=0,+pi/2}`The sum of all distinct solutions of the equation `sqrt3secx+cosecx+2(tan x-cot x)=0` in the set S is equal toA. `(-7pi)/9`B. `-(2pi)/9`C. 0D. `(5pi)/9` |
| Answer» Correct Answer - C | |
| 91. |
Find the number of solutions of `tanx + secx = 2cosx` in `[0,2pi]` |
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Answer» Here, `tanx + secx = 2cosx rArr sinx +1=2cos^(2)x` `rArr 2sin^(2)x+sinx-1=0 rArr sinx=1/2, -1` But `sinx=-1 rArr x=(3pi)/(2)` for which `tanx+secx =2 cosx` is not defined. Thus, `sinx=1/2 rArr x=pi/6, (5pi)/(6)` `rArr` number of solutions of `tanx+secx = 2cos x` is 2. Ans. |
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