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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.
| 101. |
If `tan((alphapi)/(4))=cot((betapi)/(4)),` thenA. `alpha +beta=0`B. `alpha+beta=2n`C. `alpha+beta+2n+1`D. `alpha+beta=2(2n+1), n inZ` |
| Answer» Correct Answer - D | |
| 102. |
Let `alpha, beta` be such that `pi lt alph-betalt3piif sin alpha+sinbeta=-21/65and cos alpha+cos beta =-27/65,` then the value of `cos""(alpha-beta)/(2),` isA. `-(6)/(65)`B. `(3)/(sqrt130)`C. `6/65`D. `-(3)/(sqrt130)` |
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Answer» Correct Answer - D We have, `sin alpha+sinbeta=-21/65and cos alpha+cosbeta=-27/65` `implies2sin""(alpha+beta)/(2)cos""(alpha-beta)/(2)=-21/65` `and 2cos""(alpha+beta)/(2)cos""(alpha-beta)/(2)=-27/65` `implies4(sin^(2)""(alpha+beta)/(2)+cos^(2)""(alpha+beta)/(2))cos^(2)""(alpha-beta)/(2)=(-(21)/(65))^(2)+(-(27)/(65))^(2)` `implies4cos^(2)((alpha-beta)/(2))=(1170)/(65^(2))` `implies4cos^(2)((alpha-beta)/(2))=(1170)/(4xx65^(2))` `implies|cos""(alpha-beta)/(2)|=(sqrt(1170))/(130)` `impliescos""(alpha-beta)/(2)=-(sqrt(1170))/(130)" "[{:(,because(pi)/(2)lt(alpha-beta)/(2)lt(3pi)/(2)),(,thereforecos""(alpha-beta)/(2)lt0):}]` `impliescos""(alpha-beta)/(2)=-sqrt((1170)/(130xx130))=-(3)/(sqrt(130))` |
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| 103. |
The number of integral triplets (a, b, c) such thet `a+b cos 2x+c sin^(2)x=0` for all x, is |
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Answer» Correct Answer - D We have, `a+bcos2x+c sin^(2)x=0` for all x `impliesa+b+(c-2b)sin^(2)x=0` for all x `impliesa+b=0and c-2b=0` `impliesa=-bandc =2b` Thus, the triplets are `(-b,b,2b), where b in R.` Hence, there are infinitely many triplets. |
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| 104. |
If A > 0, B > 0 and `A+B=pi/6`, then the minimum value of `tan A + tan B` isA. `2-sqrt3`B. `(2)/(sqrt3)`C. `sqrt3-sqrt2`D. `4-2sqrt3` |
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Answer» Correct Answer - D `tan A +tan D` `=tanA+tan((pi)/(6)-A)=tanA+(tan""(pi)/(6)-tanA)/(1+tan""(pi)/(6)tanA)` `=tanA+(1-sqrt3tanA)/(sqrt3+tanA)=(1+tan^(2)A)/(sqrt3+tanA)=(1)/(sqrt3cos^(2)A+sinA cos A)` `=(2)/(sqrt3(1+cos2A)+sin2A)=(2)/(sqrt3+(sqrt3cos2A+sin2A))` `=(1)/((sqrt3)/(2)+sin(2A+(pi)/(3))` `Now, 0ltAlt(pi)/(6)implies0lt2Alt(pi)/(3)` `implies(pi)/(3)lt2A+(pi)/(3)lt(2pi)/(3)implies(sqrt3)/(2)ltsin(2A+(pi)/(3))le1` `impliessqrt3lt(sqrt3)/(2)+sin(2A+(pi)/(3))le(sqrt3)/(2)+1` `implies(2)/(2+sqrt3)le(1)/((sqrt3)/(2)+sin(2A+(pi)/(3)))lt(1)/(sqrt3)` `implies2(2-sqrt3)le tanA+tanBlt(1)/(sqrt3)` lt |
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| 105. |
If `A+B+C=0,` then the value of `sum cot (B+C-A) cot (C+A-B)` is equal to |
| Answer» Correct Answer - B | |
| 106. |
If `a sin x+b cos(x+theta)+b cos(x-theta)=d,` then the minimum value of `|cos theta|` is equal toA. `(1)/(2|b|)sqrt(d^(2)-a^(2))`B. `(1)/(2|a|)sqrt(d^(2)-a^(2))`C. `(1)/(|d|)sqrt(d^(2)-a^(2))`D. none of these |
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Answer» Correct Answer - A We have, `a sinx+bcos(x+theta)+b cos(x-theta)=d` for some real x `impliesa sinx+2bcos x costheta=d` `implies|d|lesqrt(a^(2)+4b^(2)cos^(2)theta)` `impliesd^(2)lea^(2)+4b^(2)cos^(2)theta` `implies(d^(2)-a^(2))/(4b^(2))lecos^(2)thetaimpliescosthetage""(1)/(2|b|)sqrt(d^(2)-a^(2))` Hence, the minimum value of `cos thetais (1)/(1|b|)sqrt(d^(2)-a^(2))` |
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| 107. |
If x,y,z are variables and `3tan x+4tany+5tanz=20,` then the minimum value of `tan^(2)x+tan^(2)y+tan^(2)z,` isA. 10B. 15C. 8D. 12 |
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Answer» Correct Answer - C `Let veca=3hati+4hatj+5hatk` and, `hatb=(tanx)hati+(tany)hatk+(tanz)hatk.` Then, `(veca,vecb)^(2)le|veca|^(2)|vecb|^(2)` `implies(3tanx+4tany+5tanz)^(2)` `" "le(9+16+25)(tan^(2)x+tan^(2)y+tan^(2)z)` `implies(20)^(2)le50(tan^(2)x+tan^(2)y+tan^(2)z)` `impliestan^(2)x+tan^(2)y+tan^(2)zge8` |
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| 108. |
In `(0,pi//2)tan^(m)x+cot^(m)x` attainsA. a minimum value which is independent of mB. a minimum value which is a function of mC. the minimum value of 2D. the minimum value at the some point indiependent of m. |
| Answer» Correct Answer - B | |
| 109. |
Find the Value of `tan 82 1/2^@`A. `sqrt2+sqrt3+sqrt4+sqrt6`B. `(sqrt3+sqrt2)(sqrt2-1)`C. `-(sqrt3+sqrt2)(sqrt2+1)`D. none of these |
| Answer» Correct Answer - A | |
| 110. |
if `y=cos^2theta+sec^2theta` thenA. `y=0`B. `y le2`C. `y ge-2`D. `y ne2` |
| Answer» Correct Answer - D | |
| 111. |
If `sin2theta=cos3theta` and `theta` is an acute angle , then `sintheta` is equal toA. `(sqrt5-1)/(4)`B. `-((sqrt5-1)/(4))`C. `(sqrt5+1)/(4)`D. `(-sqrt5-1)/(4)` |
| Answer» Correct Answer - A | |
| 112. |
If `A+B=pi/3 and cos A+cos B=1,` then which of the following is/are true ?A. `cos(A-B)=1/3`B. `|cosA-cosB|=sqrt(2/3)`C. `cos(A-B)=-2/3`D. `|cosA-cosB|=(1)/(2sqrt3)` |
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Answer» Correct Answer - B::C We have, `cosA+cos B=1` `implies2 cos ""(A+B)/(2)cos""(A-B)/(2)=1` `impliescos""(A-B)/(2)=(1)/(sqrt(3))" "[becauseA+B=(pi)/(3)]` `implies2 cos^(2)""((A-B)/(2))-1=2/3-1impliescos(A-B)=-1/3` Now, `|cosA-cosB|` `=|2sin((A+B)/(2))sin((B-A)/(2))|` `=|2sin""(pi)/(6)sin((A-B)/(2))|` `= |sin((A-B)/(2))|=sqrt(1-cos^(2)((A-B)/(2)))=sqrt(1-(1)/(3))=sqrt((2)/(3))` So, option (b) is also true, |
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| 113. |
Which of the following statements about `tan10^(@)` is true?A. It is a rational numberB. It is an inrrational number less than 2C. It is an irrational number greater than 2D. It is greater than 2 |
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Answer» Correct Answer - B::C Since tan `theta` is an increasing function on `(0,pi//2)` `thereforetan10^(@)lttan15^(@)=2-sqrt3lt2.` Thus option (c ) is true. Now, `tan30^(@)=(3 tan10^(@)-tan^(3)10^(@))/(1-3tan^(2)10^(@))` If `tan 10^(@)` is a trational number, then `(3tan10^(@)-tan^(3)10^(@))/(1-3tan^(2)10^(@))` must be a rational number, which is not true, Hence, `tan10^(@)` is an irrational number. |
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| 114. |
The value of `sec 40^(@)+sec 80^(@)+sec 160^(@)` will beA. 4B. `-4`C. 6D. 8 |
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Answer» Correct Answer - C We find that `cos40^(@)+cos80^(@)cos160^(@)` `2cos60^(@)cos20^(@)-cos20^(@)=0` `cos40^(@)cos80^(@)+cos80^(@)cos160^(@)+160^(@)cos40^(@)` `=1/2[2cos80^(@)cos40^(@)+2cos160^(@)cos80^(@)+2cos160^(@)cos40^(@)]` `=1/2[cos120^(@)+cos40^(@)+cos240^(@)+cos80^(@)+cos200^(@)+cos120^(@)]` `=1/2(-1/2+2cos40^(@)-1/2+cos80^(@)-cos20^(@)""1/2)` `=1/2(-(3)/(2)+cos60^(@)cos20^(@)-cos20^(@))=-3/4` and, `cos40^(@)cos80^(@)cos160^(@)=(sin(2^(3)xx40^(@)))/(2^(3)sin40^(@))=1/8(sin320^(@))/(sin40^(@))=-1/8` So, the equation having `cos40^(@),cos80^(@)and cos160^(@)` at its roots is `x^(3)-x^(2)xx0+x xx-3/4-(-(1)/(8))=0or,8x^(3)-6x +1=0` The equation having `sec 40^(@), sec80^(@) and sec 160^(@)` as its roots is `x^(3)-6x^(2)+8=0` `thereforesec40^(@)+sec80^(@)+sec160^(@)=6` |
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| 115. |
If `theta=(2pi)/(2009), then costhetacos 2thetacos3theta... cos1004theta` is |
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Answer» Correct Answer - C Let `P=costhetacos2thetacos 3theta...sin1004theta` Then, `2^(1004)PQ=sin2theta sin4thetasin6theta...sin2008 theta` `implies2^(1004)PQ=(sin2thetasin4thetasin60...sin2008theta)(sin1006thetasin1008theta...sin2008theta)` `2^(1004)PQ=(sin2thetasin4thetasin6theta...sin1004theta)` `{sin(2pi-1003theta)sin(2pi-1001theta)...sin(2pi-theta)}` `implies2^(1004)PQ=(-1)^(502)sinthetasin2thetasin3thetasin4theta.....sin1003thetasin1001theta` `implies2^(1004)PQ=QimpliesP=(1)/(2^(1004))` |
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| 116. |
Statement-1: `cos36^(@)gttan36^(@)` Statement-2: `cos36^(@)gtsin36^(@)`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 For `0lethetalt(pi)/(4),"we have" cos thetagtsintheta.` So, statement-2 is true. Now, `cos36^(@)-tan36^(@)` `=(cos^(2)36^(@)-sin36^(@))/(cos36^(@))` `=(1+cos72^(@)-2sin36^(@))/(2cos36^(@))` `=(1+sin18^(@)-2sin(30^(@)+6))/(2cos36^(@))` `=(1+2sin9^(@)cos9^(@)-2(sin30^(@)cos6^(@)+cos30^(@)sin60^(@)))/(2cos36^(@))` `=1+2sin9^(@)cos9^(@)-cos6^(@)-2cos30^(@)sin6^(@)` `=(1-cos6^(@))+2(sin9^(@)cos9^(@)-cos30^(@)sin6^(@))` `gt[because1-cos6^(@)gt0and sin9^(@)cos9^(@)gtcos30^(@)sin6^(@)]` `thereforecos36^(@)-tan36^(@)gt0impliescos36^(@)gttan36^(@)` So, statement-2 is true. But statement-2 is not a correct explanation for statement-1. |
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| 117. |
Statement-1: `(cos36^(@)-cos752^(@))/(cos36^(@)cos72^(@))=2` Statement-2: `sin15^(@)=(sqrt6-sqrt7)/(4)`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, `(cos36^(@)-cos72^(@))/(cos36^(@)cos72^(@))=(cos36^(@)-sin18^(@))/(cos36^(@)sin18^(@))=((sqrt5+1)/(4)-(sqrt5-1)/(4))/((sqrt5+1)/(4)xx(sqrt5-1)/(4))=2` So, statement-1 is true. `sin15^(@)=sin(45^(@)-30^(@))=(sqrt3-1)/(2sqrt2)=(sqrt6-sqrt2)/(4)` So, statement-2 is also true. But it is not a correct explanation for statement-1. |
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| 118. |
if `a=cos 2 and b =sin7,` thenA. `a gt0, b gt0`B. `ab lt0`C. `a gtb`D. `a ltb` |
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Answer» Correct Answer - B We have, `(pi)/(2)le2^(c)le(3pi)/(4)and 2pilt7^(c)(5pi)/(2)` `implies2^(c)"is in second quadrant and"7^(c) "is in first quadrnat"` `impliesa =cos2^(c)lt0and b=sin7^(c)gt0` `impliesab lt0` |
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| 119. |
`sqrt(3)cossec2 0^0-sec2 0^0`A. 2B. 1C. 4D. `-4` |
| Answer» Correct Answer - C | |
| 120. |
The vlaue of `sqrt3cot20^(@)-4cos20^(@),` isA. 1B. `-1`C. 4D. `-4` |
| Answer» Correct Answer - A | |
| 121. |
Biggest among `(sin1+cos1),(sqrt(sin1))+sqrt(cos1)),sqrt(1+sin2)and 1, is`A. `sin1 +cos1`B. `sqrt(sin1)+sqrt(cos1)`C. `sqrt(1+sin2)`D. 1 |
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Answer» Correct Answer - B We have, `sqrt(sin1)gtsin1gtsin^(2)1 and, sqrt(cos1)gtcos1 gtcos^(2)1` `thereforesqrt(sin)1+sqrt(cos1)gtsin1+cos+gt1` `Also, sqrt(1+sin2)=sqrt(sin^(2)1+cos^(2)1+2 sin1 cos1)=sin1+cos1` `thereforesqrt(sin1)+sqrt(cos1)gtsqrt(1+sin2)gt1.` Hence, ` sqrt(sin1)+sqrt(cos1)` is the greatest. |
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| 122. |
The maximum value of `1+sin(pi/4+theta)+2cos(pi/4-theta)` for real values of `theta` isA. 3B. 5C. 4D. none of these |
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Answer» Correct Answer - C We have, `y=1+sin ((pi)/(4)+theta)+2cos((pi)/(4)-theta)` `impliesy=1+(1)/(sqrt2)(sin theta+costheta)+sqrt2(cos theta+sin theta)` `impliesy=1+(sqrt2+(1)/(sqrt2))sintheta+(sqrt2+(1)/(sqrt2))costheta [becausecostheta+sinthetalesqrt2]` `impliesy=1+(sqrt2+(1)/(sqrt2))(cos theta+sintheta)` `impliesyle1(sqrt2+(1)/(sqrt2))sqrt2impliesyle4` Hence, the maximum value is 4, |
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| 123. |
In a `DeltaABC, if tanA+2tanB=0,` then which one is correct?A. `0lttan^(2)Cle1/8`B. `1/8lttan^(2)Cle1/2`C. `1/2lttan^(2)Clt1`D. none of these |
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Answer» Correct Answer - A We have, `tanA+2tanB=0` But in a `DeltaABC` `tanA+tanB+tanC=tanAtanBtanC` `implies-2tanB+tanB+tanC=-2tan^(2)B tanC [because tanA=-2tanB]` `implies2 tan ^(2)BtanC-tanB+tanC=0` Since the B is real. Therefore, `1-8 tan^(2)c ge0` `impliestan^(2)Cle1/8lttan^(2)Cle1/8` |
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| 124. |
`tan""(2pi)/(5)-tan""(pi)/(15)-sqrt3tan""(2pi)/(5)tan""(pi)/(15)` is equal toA. `-sqrt3`B. `1//sqrt3`C. 1D. `sqrt3` |
| Answer» Correct Answer - D | |
| 125. |
Let `f_k(x) = 1/k(sin^k x + cos^k x)` where `x in RR` and `k gt= 1.` Then `f_4(x) - f_6(x)` equalsA. `1/4`B. `1/12`C. `1/6`D. `1/3` |
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Answer» Correct Answer - B We have, `f_(k)(x)=1/k(sin^(k)x+cos^(k)x)` `thereforef_(4)(x)-f_(6)(x)` `=1/4(sin^(4)x+cos^(4)x)-1/6(sin^(6)x+cos^(6)x)` `=1/4{(sin^(2)x+cos^(2)x)^(2)-2sin^(2)xcos^(2)x}` `-1/6{(sin^(2)x+cos^(2)x)^(3)-3sin^(2)x cos^(2)(sin^(2)x+cos^(2)x)}` `=1/4(1-2sin^(2)xcos^(2)x)-1/6(1-3sin^(2)xcos^(2)x)=1/12` |
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| 126. |
If `sec alpha and cosec alpha` are the roots of the equation `x^(2)-ax+b=0,` thenA. `a^(2)=b(b-2)`B. `a^(2)=b(b+2)`C. `a^(2)+b^(2)=2b`D. none of these |
| Answer» Correct Answer - B | |
| 127. |
If `tan(x/2)=cosec x - sin x` then the value of `tan^2(x/2)` isA. `2-sqrt5`B. `2+sqrt5`C. `-2-sqrt5`D. none of these |
| Answer» Correct Answer - D | |
| 128. |
The value of `cos 10^@ - sin 10^@` isA. positiveB. negativeC. 0D. 1 |
| Answer» Correct Answer - A | |
| 129. |
If `sinx+cosesx=2,thensin^(n)x+cosec^(n)x` is equal toA. 2B. 2nC. `2n-1`D. `2n-2` |
| Answer» Correct Answer - A | |
| 130. |
The value of `cos 1^@ cos 2^@ cos 3^@... cos 179^@` isA. `(1)/(sqrt2)`B. 0C. 1D. none of these |
| Answer» Correct Answer - B | |
| 131. |
`1+sinx+sin^(2)x+..."to"oo=2sqrt3 + 4, if` x = ?A. `x=(3pi)/(3)or,(pi)/(3)`B. `x=(7pi)/(6)`C. `x=(pi)/(6)`D. `x=(pi)/(4)` |
| Answer» Correct Answer - A | |
| 132. |
Which of the following is correct `sin 1^@ > sin 1`A. `sin1^(@)gtsin1`B. `sin1^(@)ltsin1`C. `sin1^(@)=sin1`D. `sin1^(@)=(pi)/(180)sin1` |
| Answer» Correct Answer - B | |
| 133. |
The maximum value of `cos^2(pi/3-x)-cos^2(pi/3+x)`, isA. `(sqrt3)/(2)`B. `1/2`C. `-(sqrt3)/(2)`D. `3/2` |
| Answer» Correct Answer - A | |
| 134. |
If `1+sinx+sin^2x+sin^3x+oo`is equal to `4+2sqrt(3),0A. `(pi)/(6)`B. `(pi)/(4)`C. `(pi)/(3)or (pi)/(6)`D. `(pi)/(3)or (2pi)/(3)` |
| Answer» Correct Answer - D | |
| 135. |
the value of `tan9^@-tan2 7^@-tan6 3^@+tan8 1^@` is equal toA. 2B. 3C. 4D. 1 |
| Answer» Correct Answer - C | |
| 136. |
Find the value of log `tan1^0logtan2^0logtan89^0` |
| Answer» Correct Answer - A | |
| 137. |
The value of `tan1^@ tan 2^@ tan 3^@ ... tan 89^@` isA. 1B. 0C. `oo`D. `1//2` |
| Answer» Correct Answer - A | |
| 138. |
The expression `cosec^(2)A cot^(2)A-sec^(2)A tan^(2)A-(cot^(2)A-tan^(2)A)(sec^(2)A cosec^(2)A-1)` is equal toA. 1B. `-1`C. 0D. 2 |
| Answer» Correct Answer - C | |
| 139. |
The expression `3{sin^(6)""((pi)/(2)+alpha)+sin^(6)(5pi-alpha)` is equal to |
| Answer» Correct Answer - B | |
| 140. |
If `A=130^(@)and x=sinA+cosA,` thenA. `x gt0`B. `xlt0`C. `x=0`D. `x ge0` |
| Answer» Correct Answer - A | |
| 141. |
If `tan theta=a/b` then `b cos 2theta+asin 2theta=`A. aB. bC. `b//a`D. `a//b` |
| Answer» Correct Answer - B | |
| 142. |
If `abs(cos theta{sin theta+sqrt(sin^2theta+sin^2alpha)})lek`, then the value of kA. `sqrt(1+cos^(2)alpha)`B. `sqrt(1+sin^(2)alpha)`C. `sqrt(2+sin^(2)alpha)`D. `sqrt(2+cos^(2)alpha)` |
| Answer» Correct Answer - B | |
| 143. |
The value of `sin10^@+sin20^@+sin30^@...+sin360^@` is equal to -A. 1B. 0C. `-1`D. `1//2` |
| Answer» Correct Answer - B | |
| 144. |
`cos.(2pi)/(7)+cos.(4pi)/(7)+cos.(6pi)/(7)`A. 1B. `-1`C. `1//2`D. `-1//2` |
| Answer» Correct Answer - D | |
| 145. |
if `sin(alpha+beta)=1 and sin(alpha-beta)=1/2 " "0lealpha,beta,lepi/2,then` find `tan(alpha+2beta)and tan(2alpha+beta)`A. 1B. `-1`C. 0D. `1//2` |
| Answer» Correct Answer - A | |
| 146. |
If ABCD is a cyclic quadrilateral, then the value of cosA-cosB+cosC-cosD is equal toA. 1B. 0C. `-1`D. none of these |
| Answer» Correct Answer - B | |
| 147. |
If `x+y+z=pi,,tan x tanz=2and tanytanz=18,then tan^(2)z=`A. 15B. 16C. 19D. 20 |
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Answer» Correct Answer - B Whe have, `x+y+z=pi` `impliestanx+tany+tanz=tanxtanyz` `impliestanx+tany+tanz=2tany" "[becausetanxtanz=2]` `impliestanx+tanz=tany` `(tanx+tanz)tanz=tanytanz` `implies(tanx+tnz)tanz=18` `impliestanx tanz+tan^(2)z=18` `implies+tan^(2)z=18impliestan^(2)z=16.` |
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| 148. |
In a cyclic quadrilateral ABCD, the value of `2+sum cos A soc B,` isA. `sin^(2)A+sin^(2)B,` isB. `sin^(2)B+sin^(2),` DC. `sin^(2)A+sin^(2)C`D. `sin^(2)B+sin^(2)C` |
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Answer» Correct Answer - A In the cyclin quadrilateral ABCD, we have `A+C=pi=B+D` `thereforecos A+cosB+cos C+cosD)^(2)=0` `implies(cosA+cosB+cosC+cosD)^(2)=0` `impliescos^(2)A+cosB+cosC+cos^(2)D+2sumcosAcosB=0` `implies2 cos^(2)A+2cos^(2)B+2sumcosAcosB=0` `" "[becausecosC=-cosA and cosD=-cosB]` `impliessum cos A cos B=-[cos^(2)A+cos^(2)B]` `sumcosAcosB=-2+sin^(2)A+sin^(2)B` `implies2+sumcos A cos B=sin^(2)A+sin^(2)B=sin^(2)C+sin^(2)D.` |
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| 149. |
The value of `cot36^(@)cot72^(@),` isA. `1//5`B. `1//sqrt5`C. 1D. `1//3` |
| Answer» Correct Answer - B | |
| 150. |
The minimum value of `9tan^2theta+4cot^2theta`is`6``12``4`none of these``A. 13B. 9C. 6D. 12 |
| Answer» Correct Answer - D | |