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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 value of `cot theta-tantheta-2tan2theta-4tan4theta-8cot8theta,` is |
| Answer» Correct Answer - A | |
| 52. |
If `tan alpha=(1+2^(-x))^(-1), tan beta=(1+2^(x+1))^(-1) then alpha+beta` equalsA. `pi//6`B. `pi//4`C. `pi//3`D. `pi//2` |
| Answer» Correct Answer - B | |
| 53. |
If `cos(x-y), cosx and cos(x+y)` are in H.P., then `cos x sec (y/2)` equalsA. 1B. 2C. `sqrt2`D. none of these |
| Answer» Correct Answer - C | |
| 54. |
The value of `cosycos(pi/2-x)-cos(pi/2-y)cosx+sinycos(pi/2-x)+cosxsin(pi/2-y)`is zero if`x=0`(b) `y=0``x=y`(d) `npi+y-pi/4(n in Z)`A. `x=0`B. `y=0`C. `x=y+(pi)/(4)`D. `x=(3pi)/(4)+y` |
| Answer» Correct Answer - D | |
| 55. |
If `alpha,beta,gamma,delta`are the smallest positive angles in ascending order of magnitude whichhave their sines equal to the positive quantity `k ,`then the value of `4sinalpha/2+3sinbeta/2+2singamma/2+sindelta/2`is equal to`2sqrt(1-k)`(b) `2sqrt(1+k)``(sqrt(1-k))/2`(d) none of theseA. `2sqrt(1-k)`B. `2sqrt(1+k)`C. `2sqrtk`D. `2sqrt(k+2)` |
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Answer» Correct Answer - B It is given that `alpha, beta, tamma, delta` are the smallest positive angles in ascending order of magnitude such that `sin alpha=sinbeta=singamma=sindelta=k`(a positive quantity) `impliesbeta=pi-alpha,gamma=2pi+alphaand delta=3pi-alpha` `therefore4sin""(alpha)/(2)+3sin""(beta)/(2)+2sin""(gamma)/(2)+sin""(delta)/(2)` `=4sin""(alpha)/(2)+3cos""(alpha)/(2)-2sin""(alpha)/(2)-cos""(alpha)/(2)` `=2(cos""(alpha)/(2)+sin""(alpha)/(2))=2sqrt(1+sinalpha)=2sqrt(1+k)` |
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| 56. |
If `alpha,beta,gamma,delta`are the smallest positive angles in ascending order of magnitude whichhave their sines equal to the positive quantity `k ,`then the value of `4sinalpha/2+3sinbeta/2+2singamma/2+sindelta/2`is equal to`2sqrt(1-k)`(b) `2sqrt(1+k)``(sqrt(1-k))/2`(d) none of theseA. `2sqrt(1-k)`B. `2sqrt(1+k)`C. `(sqrt(1+k))/(2)`D. `(sqrt(1-k))/(2)` |
| Answer» Correct Answer - B | |
| 57. |
`alpha & beta ` are solutions of `a cos theta+b sin theta=c(cosalpha != cos beta)&(sin alpha != sin beta)` Then `tan((alpha+beta)/2)=?`A. `b//a`B. `c//a`C. `a//b`D. `c//b` |
| Answer» Correct Answer - A | |
| 58. |
If `cos(alpha+beta)sin(gamma+delta)=cos(alpha-beta)sin(gamma-delta)` then the value of `cot alpha cot beta cot gamma,` isA. `cot delta`B. `-cot delta`C. `tan delta`D. `-tandelta` |
| Answer» Correct Answer - A | |
| 59. |
If `A+B=C,` then `cos^(2)A+cos^(2)B+cos^(2)C-2cosAcosBcosC=`A. 1B. 2C. 0D. 3 |
| Answer» Correct Answer - A | |
| 60. |
If `alpha+beta+gamma=2 theta`,then `cos theta + cos(theta - alpha) + cos(theta - beta) + cos(theta - gamma)` =A. `4sin""(alpha)/(2)sin""(beta)/(2)sin""(gamma)/(2)B. `4cos""(alpha)/(2)cos""(beta)/(2)cos""(gamma)/(2)C. `4sin""(alpha)/(2)sin""(beta)/(2)sin""(gamma)/(2)D. `4sinalphasinbetasingamma` |
| Answer» Correct Answer - B | |
| 61. |
Write the maximum value of `12 s intheta-9sin^2 thetadot`A. 3B. 4C. 5D. 2 |
| Answer» Correct Answer - B | |
| 62. |
The value of `cosx cosy sin(x-y)+cosycosz sin(y-z)` `+cosz cosx sin(z-x)+sin(x-y)sin(y-z)sin(z-x),` is |
| Answer» Correct Answer - A | |
| 63. |
If `A = sin^2 theta+ cos^4 theta`, then for all real values of `theta`A. `alt1/2`B. `age1/2`C. `1/2lea le1`D. `age0` |
| Answer» Correct Answer - C | |
| 64. |
Which one is true ?A. `sin1 gtsin2 gt sin3`B. `sin1 ltsin2ltsin3`C. `sin1lt sin3ltsin2`D. `sin3lt sin1 ltsin2` |
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Answer» Correct Answer - D We have, `sin2-sin1=2sin""1/2cos""2/3gt0[because((1)/(2))^(c),((3)/(2))^(c)in(0,(pi)/(2))]` `impliessin1ltsin2" "...(i)` and,`sin 2-sin1=2sin1 cos 2 lt0[because2^(c)in((pi)/(2),pi)thereforecos2lt0]` `impliessin3ltsin1" "...(ii)` From (i) and (ii), we obtain `sin 3 lt sin 1 lt sin2` |
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| 65. |
`sec theta=(a^(2)+b^(2))/(a^(2)-b^(2)), where a, binR,` gives real balues of `theta` if and only ifA. `a=bne0`B. `|a|ne|b|ne0`C. `a+b=0,ane0`D. none of these |
| Answer» Correct Answer - B | |
| 66. |
`sec^(2)theta==(4ab)/((a+b)^(2)),` where a, b `inR` is true if and olny ifA. `a+b ne0`B. `a=b, a ne0`C. `a=b`D. `a ne 0, bne0` |
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Answer» Correct Answer - B We known that `sec ^(2)thetage1` `therefore sec^(2)theta=(4ab)/((a+b)^(2))` `impliesa,b ne0and-(4ab)/((a+b)^(2))ge1` `impliesa,b ne0and (4ab)/((a+b)^(2))-1ge0` `impliesa,b ne0and-((a-b)^(2))/((a+b)^(2))ge0` `impliesa,b ne0and -(a-b)^(2)ge0` `impliesa, b ne0and (a-b)^(2)le0impliesa =b and a ne0` |
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| 67. |
Statement-1: if `thetane2npi+(pi)/(2),n inZ,` then `(sec^(2)theta+tantheta)/(sec^(2)theta-tantheta)"lies between"1/3 and 3.` Statement-2: If `x inR, then 1/3le(x^(2)-x+1)/(x^(2)+x+1)le3.`A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True. |
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Answer» Correct Answer - A Let `y=(x^(2)-x+1)/(x^(2)+x+1).Then,` `x^(2)(y-1)+x(y+1)+(y-1)=0` `implies(y+1)^(2)-4(y-1)^(2)ge0" "[becausex inRthereforeDiscge0]` `implies(3y-1)(y-3)le0` `1/3leyle3` `implies1/2le(x^(2)-x+1)/(x^(2)+x+1)le3"for all"x inR.` We have, `(sec^(2)theta+tantheta)/(sec^(2)theta-tantheta)=(tan^(2)theta+tantheta-1)/(tan^(2)theta-tantheta+1)` Using statement-2, we have `1/3le (tan^(2)theta+tantheta+1)/(tan^(2)theta-tantheta+1)le3"for all"theta(ne(2n+1)(pi)/(2)),n inZ` `implies1/3le(sec^(2)theta+tantheta)/(sec^(2)theta-tantheta)le3"for all"theta ne2npi+-(pi)/(2)` Hence, both the statements are true and statement-2 is a correct explanation for statement-1. |
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| 68. |
In an actue-angled triangle ABC Statement-1: `tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2)ge1` Statement-2: `tanAtanB tanCge3sqrt3`A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True. |
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Answer» Correct Answer - B We have, `A+B+C=pi` `thereforetan(A+B+C)=tanpiand tan((A)/(2)+(B)/(2)+(C)/(2))=tan""(pi)/(2)` `implies(S_(1)-S_(3))/(1-S_(3))=0and(1-S_(2))/(S_(1)-S_(3))=0` `impliesS_(1)=S_(3)and S_(2)=1` `impliestanA+tanB+tanC=tanAtanBtanC" "...(i)` `and, tan""(A)/(2)tan""(B)/(2)+tan""(B)/(2)tan""(C)/(2)+tan""(C)/(2)tan""(A)/(2)=1" "...(ii)` Using `A.M. geG.M.,` we have `(tanAtanB+tanC)/(3)ge(tanAtanBtanC)^(1//3)` `implies(tanAtanBtanC)/(3)ge(tanA tanBtanC)^(1//3)" "["Using"(i)]` `impliestanA tanBtanCge3sqrt3` So, statement-2 is true. From (ii), we have `xy+yz+zx=1,where x=tan""(A)/(2),y=tan""(B)/(2)and z=tan""(C)/(2)` `thereforex^(2)+y^(2)+z^(2)-1` `=x^(2)+y^(2)+z^(2)-(xy+yz+zx)` `=1/2{(x-y)^(2)+(y-z)^(2)+(z-y)^(2)}ge0` `impliesx^(2)+y^(2)+z^(2)gt1impliestan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2)ge1` Hence, both the statements are true. |
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| 69. |
If `k=sin^(6)c+cos^(6)x,` then k belongs to the intervalA. `[7//8,//54]`B. `[1//2,//5//8]`C. `[1//4,//1]`D. none of these |
| Answer» Correct Answer - C | |
| 70. |
If `sin alpha -cos alpha = m`, then the value of `sin^6alpha+cos^6alpha` in terms of m isA. `(4-3(m^(2)-1)^(2))/(4)`B. `(4+3(m^(2)-1)^(2))/(4)`C. `(3+4(m^(2)-1)^(2))/(4)`D. none of these |
| Answer» Correct Answer - A | |
| 71. |
If `2 sin""A/2=sqrt(1+sinA)+sqrt(1-sinA,)then A/2` lies between,A. `2npi+(pi)/(4)and 2npi+(3pi)/(4),n inZ`B. `2npi-(pi)/(4)and 2npi+(pi)/(4),n inZ`C. `2npi-(3pi)/(4)and 2npi-(pi)/(4),n inZ`D. `-ooand +oo` |
| Answer» Correct Answer - A | |
| 72. |
If `2cos""(A)/(2)=sqrt(1+sinA)+sqrt(1-sinA), thenA/2` iles between,A. `2npi+(pi)/(4)and 2npi+(3pi)/(4)`B. `2npi+(pi)/(4)and 2npi+(pi)/(4)`C. `2npi+(3pi)/(4)and 2npi+(pi)/(4)`D. `-ooand +oo` |
| Answer» Correct Answer - B | |
| 73. |
Find the angle `theta`whose cosine is equal to its tangent.A. `cos theta=2cos18^(@)`B. `cos theta=2sin18^(@)`C. `sintheta=2 sin18^(@)`D. `sintheta=2cos18^(@)` |
| Answer» Correct Answer - C | |
| 74. |
If `tantheta=(1)/(2) and tanphi=(1)/(3)`, then the value of `theta+phi` isA. `pi//6`B. `pi`C. zeroD. `pi//4` |
| Answer» Correct Answer - D | |
| 75. |
If `tan((theta)/(2))=5/2and tan((phi)/(2))=3/4,` the value of `cos(theta+phi),` isA. `-(364)/(725)`B. `-(627)/(725)`C. `-(240)/(339)`D. `-(339)/(725)` |
| Answer» Correct Answer - B | |
| 76. |
The value of `cos""(pi)/(9)cos""(2pi)/(9)cos""(3pi)/(9),` isA. `1//8`B. `-1//8`C. 1D. 0 |
| Answer» Correct Answer - A | |
| 77. |
`cos1+cos2+cos3+...+cos180`A. 1B. 0C. 2D. `-1` |
| Answer» Correct Answer - D | |
| 78. |
The maximum and minimum values of `6 sin x cos x +4cos` are respectivelyA. `5 and -5`B. `2sqrt13 and -2 sqrt13`C. `10 and -10`D. none of these |
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Answer» Correct Answer - A We have, `6 sinx cos x+4cos2x=3sin2x+4cos2x` Now, `-sqrt(3^(2)-4^(2))le3 sin 2x+4 cos 2x lesqrt(3^(2)+4^(2))"for all" x implies-5le6sinx x cosx+4cos2x le5"for all x"` Hence, the maximum and minimum values of `3 sin2x+4cos2xare5and -5` respectively. |
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| 79. |
If `sin x+sin^2=1`, then `cos^8 x+2 cos^6 x+cos^4 x=` |
| Answer» Correct Answer - D | |
| 80. |
The value of `(3+cot76^@cot16^@)/(cot76^@+cot16^@)` isA. `cot44^(@)`B. `tan44^(@)`C. `tan2^(@)`D. `cot46^(@)` |
| Answer» Correct Answer - A | |
| 81. |
if ` sin x + sin^2 x = 1`, then the value of `cos^2 x + cos^4x` isA. 1B. 2C. `1.5`D. none of these |
| Answer» Correct Answer - A | |
| 82. |
The equation `sin^2theta=(x^2+y^2)/(2x y),x , y!=0`is possible ifA. `x=y`B. `x=-y`C. `2x=-y`D. none of these |
| Answer» Correct Answer - A | |
| 83. |
If `(sin(x+y))/(sin(x-y))=(a+b)/(a-b)` , then show that `(tanx)/(tany)=(a)/(b)`.A. `b//a`B. `a//b`C. abD. none of these |
| Answer» Correct Answer - B | |
| 84. |
The maximum value of `sin(theta+pi/6)+cos(theta+pi/6)` is attained at `theta in (0,pi/2)`A. `pi//12`B. `pi//6`C. `pi//3`D. `pi//2` |
| Answer» Correct Answer - A | |
| 85. |
If `A +B+C= pi and m/_C` is obtuse then `tan A. tan B` isA. `tanA tanB gt1`B. `tanA tanBgt1`C. `tanA tanB=1`D. none of these |
| Answer» Correct Answer - B | |
| 86. |
The value of `sin(pi+theta)sin(pi-theta)cosec^(2)theta` is equal toA. `-1`B. 0C. `sin theta`D. none of these |
| Answer» Correct Answer - A | |
| 87. |
If `cosectheta=(p+q)/(p-q),then cot(pi,//4+theta//2)=`A. `sqrt((p)/(q))`B. `sqrt((q)/(p))`C. `sqrt(pq)`D. pq |
| Answer» Correct Answer - B | |
| 88. |
If `x=cos^2theta+sin^4theta` then for all real values of `theta`A. `y in[1,2]`B. `yin[13//16,1]`C. `y in[3//4,13//16]`D. `y in[3//4,1]` |
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Answer» Correct Answer - D We have, `y-sin^(2)theta+cos^(4)theta` `impliesy=cos^(4)theta-cos^(2)theta+1` `impliesy=(cos^(2)theta-(1)/(2))^(2)+3/4` Now, `0le cos^(2)theta-1/2le 1/2` `implies0le(cos^(2)theta-1/2)^(2)+3/4le1implies3/4leyle1` |
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| 89. |
The ratio of the greatest value of `2-cos x+s in^2x`to its least value is`7/4`(2) `9/4`(3) `(13)/4`(4) `5/4`A. `7/4`B. `11/4`C. `13/4`D. none of these |
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Answer» Correct Answer - C Let `y=2-cosx+sin^(2)x.Then,` ` y=2-cosx+1-cos^(2)x` `impliesy=3-(cos^(2)x+cos)impliesy=13/4-(cosx+1/2)^(2)` Now, `-1lecosx le1` for all x `implies-1/2le(cosx+1/2)^(2)lt0"for all x"` `implies13/4-9/4le13/4(cosx+1/2)^(2)le13/4"for all x"` `implies1ley13/4` `thereforey_(max)=13/4and y_(min)=1` Hence, required ratio is `13/4` |
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| 90. |
The least value of `[cos^2 theta-6 sin theta. cos theta+3 sin^2 theta + 2]` isA. `4+sqrt10`B. `4-sqrt10`C. 0D. none of these |
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Answer» Correct Answer - B We have, `y=cos^(2)theta-6sinthetacostheta+3sin^(2)theta+2` `impliesy=((1+cos2 theta)/(2))-3sin2 theta+3((1-cos2 theta)/(2))+2` `impliesy=-cos2 theta-3sin2theta+4` Now, `-sqrt10le-cos2 theta-3sin2 thetale sqrt10"for all"theta` `implies-4sqrt10leyle4+sqrt10` Hence, the least value of y is `4-sqrt10.` |
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| 91. |
If `sinx+sin^2x+sin^3x=1` then find the value of `cos^6x-4cos^4x+8cos^2x`A. 2B. 1C. 3D. 4 |
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Answer» Correct Answer - D We have, `sinx+sin^(2)x+sin^(3)x=1` `impliessinx+sin^(3)x=1-sin^(2)x` `impliessinx+sin^(3)x=cos^(2)x` `impliessinx(1+sin^(2))=cos^(2)x` `impliessinx(2-cos^(2)x)=cos^(2)x` `impliessin^(2)x(2-cos^(2)x)^(2)=cos^(4)x` `implies(-1cos^(2)x)(4-4cos^(2)x+cos^(4))=cos^(4)x` `implies4-4cos^(2)x+cos^(4)x-4cos^(2)x+4cos^(4)x+cos^(6)x=cos^(4)x` `impliescos^(6)x-4cos^(4)x+8cos^(2)x=4.` |
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| 92. |
If `y=tan^(2)theta+sectheta, thetane(2n+1)pi//2,then`A. `y in(-oo,1]`B. `yin(-oo,-1]`C. `yin[-1,oo)`D. none of these |
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Answer» Correct Answer - C We have, `y=tan^(2)theta+sectheta` `impliessec^(2)thetasectheta-(y+1)=0` `=sectheta=(-1pmsqrt(4y+5))/(2)` Since sec `theta` is real and `sec theta ge1 or, sec thetale-1.` `therefore 4y+5ge0and((-1+sqrt(4y+5))/(2)geor,(-1-sqrt(4u+5))/(2)le-1)` `impliesy ge-5/4and (4y+5ge9or,4y+5ge1)` `implies(yge-5/4and yge1)or,(yge-5/4and yge-1)` `implies(y ge1)or, (y ge-1)impliesyge-1impliesyin[-1,oo)` |
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| 93. |
If `4nalpha =pi` then `cot alpha cot 2 alpha cot 3alpha ...cot (2n-1)alpha` `n in Z` is equal toA. 1B. `-1`C. `oo`D. none of these |
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Answer» Correct Answer - A We have, `4nalpha=pi` `implies2nalpha=(pi)/(2)` `implies2nalpha-alpha=(pi)/(2)-alpha,2n alpha-2alpha=(pi)/(2)-2alpha, 2nalpha-3alpha=(pi)/(2)-3alpha` and so on. `impliescot(2n-1)alpha=tan alpha, cot(2n-2)alpha=tan2 alphaand` so on. `thereforecot alphacot 2 alphacot 3 alpha... cot(2n-1)alpha` `=cot alpha cot2 alpha cot 3 alpha... tan2 alpha. tan alpha` `=(cotalpha.tan alpha)(cot2alpha. tan 2 alpha)(cot 3 alpha. tan3 alpha)...=1` |
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| 94. |
If `sin2 theta=3/4, then sin^(3)theta+cos^(3)theta=`A. `(sqrt5)/(8)`B. `(sqrt7)/(8)`C. `(sqrt11)/(8)`D. none of these |
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Answer» Correct Answer - D We have, `sin^(3)theta+cos^(3)theta=(sintheta+cos theta)(1-(sin2 theta)/(2))` `=sqrt(1+sin2 theta)(1-(sin2 theta)/(2))` `=sqrt(1+(3)/(4))(1-(3)/(8))=(sqrt7)/(2)xx5/8=(5sqrt7)/(16)` |
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| 95. |
If `alpha+beta+gamma=2pi` thenA. `tan""(alpha)/(2)tan""(beta)/(2)+tan""(gamma)/(2)+tan""(alpha)/(2)tan""(beta)/(2)tan""(gamma)/(2)`B. `tan""(alpha)/(2)tan""(beta)/(2)+tan""(beta)/(2)+tan""(gamma)/(2)tan""(gamma)/(2)tan""(alpha)/(2)=1`C. `tan""(alpha)/(2)tan""(beta)/(2)+tan""(gamma)/(2)+=-tan""(alpha)/(2)tan""(beta)/(2)tan""(gamma)/(2)`D. `tan""(alpha)/(2)tan""(beta)/(2)+tan""(beta)/(2)tan""(gamma)/(2)+tan""(gamma)/(2)tan""(alpha)/(2)=0.` |
| Answer» Correct Answer - A | |
| 96. |
Prove that `sin^3alpha + sin^3(120^@ +alpha)+ sin^3(240^@+ alpha)=-3/4sin3alpha`A. `3/4sin3alpha`B. `-3/4sin3 alpha`C. `4/3sin3 alpha`D. `-4/3sin3 alpha` |
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Answer» Correct Answer - B We have, `sinalpha+sin((2pi)/(3)+alpha)+sin((4pi)/(3)+alpha)` `=sin alpha+2 sin(pi+alpha)cos""(pi)/(3)=0` `thereforesin^(3)alpha+sin^(3)((2pi)/(3)+alpha)+sin^(3)((4pi)/(3)+alpha)` `=3 sinalpha sin((2pi)/(3)+alpha)+sin^(3)((4pi)/(3)+alpha)` `=3sinalphasin(120^(@)+alpha)sin(240^(@)+alpha)` `=-3sinalphasin(60^(@)-alpha)sin(60^(@)-alpha)sin(60^(@)+alpha)=-(3)/(4)sin3 alpha` |
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| 97. |
The value of `sin^(8)theta+cos^(8)theta+sin^(6)theta cos^(2)theta+3sin^(4)theta cos^(2)theta+cos^(6)theta sin^(2)theta+3sin^(2)thetacos^(4)theta` is equal toA. `cos^(2)2 theta`B. `sin^(2)2theta`C. `cos^(3)2 theta+sin^(3)2theta`D. none of these |
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Answer» Correct Answer - D We have, `sin^(8)theta+cos^(8)theta+sin^(6)thetacos^(2)theta+3 sin^(4)thetacos^(2)theta cos^(6)thetasin^(2)theta+3sin^(2)thetacos^(4)theta` `=sin^(8)theta+cos^(8)theta+sin^(2)thetacos^(2)theta(sin^(4)theta+cos^(4)theta+3sin^(2)theta+3cos^(2)theta)` `=(sin^(4)theta+cos^(4)theta)^(2)-2sin^(4)thetacos^(4)theta` `+1/4sin^(2)2 theta{(sin^(2)theta+cos^(2)theta)^(2)-1/2sin^(2)2 theta+3}` `=(1-(sin^(2)2theta)/(4))^(2)=1/8sin^(4)2theta+sin^(2)2theta-1/8sin^(4)2theta` `=1+1/2sin^(2)2theta-3/16sin^(4)2theta` `=1/16(16+8sin^(2)2theta-3sin^(4)2theta)` |
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| 98. |
The value of `sin^(6)((pi)/(49))+cos^(6)((pi)/(49))-1+3sin^(2)((pi)/(49))cos^(2)((pi)/(49))` is equal toA. `tan^(6)((pi)/(49))`B. `cot^(6)((pi)/(49))`C. 1D. 0 |
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Answer» Correct Answer - D We have, `sin^(6)((pi)/(49))+cos^(6)((pi)/(49))-1+3sin^(2)((pi)/(49))cos^(2)((pi)/(49))-sin^(6)((pi)/(49))+cos^(6)((pi)/(49))` `+3sin^(2)((pi)/(49))cos^(2)((pi)/(49))(sin^(2)""(pi)/(49)+cos^(2)""(pi)/(49))-1` `=(sin^(2)""(pi)/(49)+cos^(2)""(pi)/(29))^(3)-1=1-1=0` |
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| 99. |
In a right angled triangle, the hypotenuse is four times as long as the perpendicular drawn to it from the opposite vertex. One of the acute angle isA. `15^(@)`B. `30^(@)`C. `45^(@)`D. none of these |
| Answer» Correct Answer - A | |
| 100. |
The values of `sum_(k=1)^(13) (1)/(sin((pi)/(4)+(k-1)(pi)/(6))sin((pi)/(4)+(kn)/(6)))` is equalA. `3-sqrt3`B. `2(3-sqrt3)`C. `2(3-sqrt3)`D. `2(2+sqrt3)` |
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Answer» Correct Answer - C `underset(k=1)overset(13)(sum)(1)/(sin((pi)/(4)+(k-1)(pi)/(6))sin((pi)/(4)+(kpi)/(6)))` `=2 underset(k=1)overset(13)(sum)(sin {((pi)/(4)++(kpi)/(6))-((pi)/(4)+(k-1)(pi)/(6))})/(sin((pi)/(4)+(k-1)(pi)/(6)(pi)/(6))-sin((pi)/(4)+(kpi)/(6)))` `=2 underset(k=1)overset(13)(sum){cot((pi)/(4)+(k-1)(pi)/(6))-cot((pi)/(4)+(kpi)/(6))}` `=2{cot""(pi)/(4)cot((pi)/(4)+(13pi)/(6))}=2(1-cot""(29pi)/(12))=2(1-cot""(5pi)/(12))` `=2{1-(2-sqrt3)}=2(sqrt3-1)` |
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