InterviewSolution
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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.
| 151. |
The value of `sin12^(@)sin24^(@)sin48^(@),` isA. `cos20^(@)cos40^(@)cos60^(@)cos80^(@)`B.C. `sin20^(@)sin40^(@)sin60^(@)sin80^(@)`D. `3//15` |
| Answer» Correct Answer - A | |
| 152. |
Statement-1: For any value of `thetane0, lim_(ntooo)cos""(theta)/(2)cos""(theta)/(2^(2))cos""(theta)/(2^(3))...cos""(theta)/(2^(n))=(sintheta)/(theta)` Statement-2: `cosAcos2Acos2^(2)A...cos2^(n-1)A=(sin2^(n)A)/(2^(n)sinA) and lim_(Ato0)(sinA)/(A)=1.`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 Clearly, two given statements are true and Statement-2 is a correct explanation for statement-1. |
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| 153. |
The equaltion a `sin x+b cos x=c,` where `|c|gtsqrt(a^(2)+b^(2))` hasA. a unique solutionB. Infinite no, of solutionC. no solutionD. none of these |
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Answer» Correct Answer - C We know that `|a sinx+bcos x|lesqrt(a^(2)+b^(2))"for all" x inR` `implies|c|le sqrt(a^(2)+b^(2))` But, `|c|gtsqrt(a^(2)+b^(2))" "["Given"]` Hence, `a sin x+bcosx=c` does not hold for any `x in R.` |
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| 154. |
The least vlaue of `(1)/(5cosx+12sinx+15),` isA. `-18`B. `1//28`C. 28D. `1//18` |
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Answer» Correct Answer - B We have, `-13le 5cosx+12sin x le13 "for all"x inR` `implies2le5 cosx+12 sinx+15le 28"for all"x inR` `implies1/28le(1)/(5cosx+12sinx+15)is 1/28` |
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| 155. |
`(1+cos56^(@)+cos58^(@) -cos66^(@))/(cos28^(@)cos29^@sin33^(@)) =`A. 2B. 3C. 4D. none of these |
| Answer» Correct Answer - C | |
| 156. |
If `(cos(theta_(1)-theta_(2)))/(cos(theta_(1)+theta_(2)))+(cos(theta_(3)+theta_(4)))/(cos(theta_(3)-theta_(4)))=0,` then `tantheta_(1)tan theta_(2)tan theta_(3)tan theta_(4)=`A. 1B. 2C. `-1`D. none of these |
| Answer» Correct Answer - C | |
| 157. |
`(sin7theta+6sin5theta+17sin3theta+12sintheta)/(sin6theta+5sin4theta+12sin2theta)` is equal toA. `2cos theta`B. `cos theta`C. `2 sin theta`D. `sin theta` |
| Answer» Correct Answer - A | |
| 158. |
If `sinB=1/5sin(2A+B), then (tan(A+B))/(tanA)` is equal toA. `5//3`B. `2//3`C. `3//2`D. `3//5` |
| Answer» Correct Answer - C | |
| 159. |
The maximum value of the expression `1/(sin^2 theta + 3 sin theta cos theta + 5cos^2 theta)` isA. 2B. 3C. 4D. 6 |
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Answer» Correct Answer - A `sin^(2)theta+3sintheta costheta+5cos^(2)theta` `=1/2{1-cos2theta+3sin2theta+5(1+cos2theta}` `=3+2cos2theta+3/2sin2theta` Clearly, `-sqrt(4+(9)/(4))le2cos2theta+3/2sin2thetalesqrt(4+(9)/(4))` `implies-5/2le2cos2theta+3/2sin2thetale5/2` `implies2/11le(1)/(sin^(2)theta+3sinthetacostheta+5cos^(2)theta)le2` Hence, the maximum value of the given expression is 2. |
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| 160. |
Statement -1: If `(pi)/(12)lethetale(pi)/(3),` then `sin(theta-(pi)/(4))sin(theta-(7pi)/(12))sin(theta+(pi)/(12))"lies between"-(1)/(4sqrt2)and 1/4.` Statement-2: The value of `sin thetasin((pi)/(3)-theta)sin((pi)/(3)+theta)is 1/4sin3theta.`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 We have, `sinthetasin((pi)/(3)-theta)sin((pi)/(3)+theta)=sintheta(sin^(2)""(pi)/(3)-sin^(2)theta)` `=sin^(2)theta((3)/(4)-sin^(2)theta)=1/4sin3theta` So, statement-2 is true. On replacing `thetaby(theta-(pi)/(4))` in statement-2, we get `sin(theta-(pi)/(4))sin((7pi)/(12)-theta)sin(theta+(pi)/(12))=1/4sin(3theta-(3pi)/(4))` `impliessin(theta-(pi)/(4))sin(theta-(7pi)/(12))sin(theta+(pi)/(12))=-1/4sin(3theta-(3pi)/(4))` If `(pi)/(12)lethetale(pi)/(3),then (pi)/(4)le3thetalepi` `implies-(pi)/(2)le3theta-(3pi)/(4)le(pi)/(4)` `implies-1lesin(3theta-(3pi)/(4))le(1)/(sqrt2)` `implies-(1)/(4sqrt2)lesin(theta-(pi)/(4))sin(theta-(7pi)/(12))sin(theta+(pi)/(12))le1/4` So, statement-1 is also true and statement-2 is a correct explanation for statement-1. |
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| 161. |
Statement -1: `sin52^(@)+sin78^(@)+sin50^(@)=4cos26^(@)cos39^(@)cos25^(@)` Statement-2: If `A+B+C=pi, then sinA+sin B+sinC=4cos""(A)/(2)cos""(B)/(2)cos""©/(2)`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 If `A+B+C=pi, then sinA+sinB+sinC` `=2sin""(A+B)/(2)cos""(A-B)/(2)+2sin""(C)/(2)cos""(C)/(2)` `=2cos""(C)/(2){cos""(A-B)/(2)+cos""(A+B)/(2)}=4cos""(A)/(2)cos""(B)/(2)cos""(C)/(2)` So, statement-2 is true. On replacing A by `52^(@),Bby78^(@)and C by 50^(@),` we obtain statement-1. Hence, both the statements are true and statement-2 is a correct explanation for statement-1. |
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| 162. |
If `sinA sinB sinC+cosAcosB=1,` then the value of sinC isA. 1B. `1//2`C. 0D. `-1` |
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Answer» Correct Answer - A `As cos C le1` `therefore sinA sinB sinC+cosAcosB lesin A sinB+cosA cosB` `impliessin A sinBsinC+cosAcosBlecos(A-B)` `implies1lecos(A-B)` `impliescos(A-B) =1impliesA=B` `thereforesinA sinBsinC+cosAcosB=1` `impliessin^(2)A sinC+cos^(2)A=1` This is satisfied by `sin C =1` only. |
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| 163. |
If `alpha` is an acute angle and `sin(alpha/2)=sqrt((x-1)/(2x))` then `tan alpha` isA. `sqrt((x-1)/(x+1))`B. `(sqrt(x-1))/(x+1)`C. `sqrt(x^(2)-1)`D. `sqrt(x^(2)+1)` |
| Answer» Correct Answer - C | |
| 164. |
If A, B, C be an acute angled triangle, then the minimum value of `tan^(4)A+tan^(4)B+tan^(4)C` will beA. 729B. 27C. `81sqrt3`D. none of these |
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Answer» Correct Answer - B In a triangle ABC, we know that `tanA+tanB+tanCge3sqrt3" "...(i)` We also know that `(tan^(4)A+tan^(4)B+tan^(4)C)/(3)ge((tanA+tanB+tanC)/(3))^(4)` `implies(tan^(4)A+tan^(4)B+tan^(4)C)/(3)ge(sqrt3)^(4)" "["Using(i)"]` `impliestan^(4)A+tan^(4)B+tan^(4)C ge27` |
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| 165. |
If `tantheta=-4//3,` then `sintheta` isA. `-4//5"but not"4//5`B. `-4//5or 4//5`C. `4//5"but not"-4//5`D. none of these |
| Answer» Correct Answer - B | |
| 166. |
If `sintheta+costheta=sqrt2costhetathen costheta-sintheta` is equal toA. `sqrt2cos theta`B. `sqrt2 sintheta`C. `sqrt2(cos theta+sintheta)`D. none of these |
| Answer» Correct Answer - B | |
| 167. |
If `x=(2sintheta)/(1+costheta+sintheta),t h e n(1-costheta+sintheta)/(1+sintheta)`is equal to`1+x`(b) `1-x`(c) `x`(d) `1/x`A. `1/a`B. aC. `1-a`D. `1+a` |
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Answer» Correct Answer - B We have, `a=(2sintheta)/(1+costheta+sintheta)` `impliesa=(2sintheta(1+sintheta-costheta))/((1+costheta+sintheta)(1+sintheta-costheta))` `impliesa=(2sintheta(1+sintheta-costheta))/((1+sintheta)^(2)-cos^(2)theta)` `impliesa=(2sintheta(1+sintheta-costheta))/(2sintheta(1+sintheta))=(1+sintheta-costheta)/(1+sintheta)` |
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| 168. |
If A and B are two angles satisfying `0ltA,Blt(pi)/(2)and A+B=(pi)/(3),` then the minimum value of `sec A+sec B` isA. `(2)/(sqrt3)`B. `(4)/(sqrt3)`C. `(1)/(sqrt3)`D. none of these |
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Answer» Correct Answer - B `z=sec A+secB` will be minimum when `A=B=pi//6.` `therefore` Minimum value of `z=cos (pi)/(6)+sec""(pi)/(6)=(4)/(sqrt3)` |
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| 169. |
If `cos(theta - alpha) , costheta , cos(theta + alpha)` are in H.P. then `costheta.sec(alpha)/2 = `A. `sintheta=sqrt2cos((alpha)/(2))`B. `costheta=sqrt2cos((alpha)/(2))`C. `cos theta=sqrt2 sin((alpha)/(2))`D. `sintheta=sqrt2sin((alpha)/(2))` |
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Answer» Correct Answer - B It is given that `cos (theta-alpha), cos theta and cos (theta+alpha)` are in H.P. Therefore, `(2)/(cos theta)=(1)/(cos(theta-alpha))+(1)/(cos(theta+alpha))` `implies(2)/(cos theta)=(2costhetacosalpha)/(cos^(2)theta-sin^(2)alpha)` `cos^(2)theta(1-cosalpha)=sin^(2)alpha` `cos theta=sqrt2cos""(alpha)/(2)` |
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| 170. |
For what and only what values of `alpha` lying between `0 and pi/2` is the inequality is `sin alphacos^(3)alpha ltsin^(3) alpha cos alpha` valid?A. `alpha in(0,pi//4)`B. `alpha in(0,pi//2)`C. `alpha in(pi//4,pi//2)`D. none of these |
| Answer» Correct Answer - A | |
| 171. |
If `x sintheta=ysin(theta+(2pi)/(3))=z sin(theta+(4pi)/(3)),` thenA. `x+y+z=0`B. `xy+yz+zx=0`C. `xyz+x+y+z=1`D. none of these |
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Answer» Correct Answer - B We have, `x sintheta=y sin(theta+(2pi)/(3))=zsin(theta+(4pi)/(3))` `implies(sin theta)/(1/x)=(sin (theta+(2pi)/(3)))/(1/y)=(sin(theta+(4pi)/(3)))/(1/z)` `=(sintheta)/(1/x)=(sin(theta+(2pi)/(3)))/(1/y)=(sin(theta+(4pi)/(3)))/(1/z)` `=(sin theta+sin(theta+(2pi)/(3))+sin(theta+(4pi)/(3)))/(1/x+1/y+1/z)` `implies(sintheta)/(1/x)=(sin(theta+(2pi)/(3)))/(1/y)=(sin(theta+(4pi)/(3)))/(1/z)` `implies(sintheta)/(1/x)xx((1)/(x)+(1)/(y)+(1)/(z))=0` `implies1/x+1/y+1/z=0impliesxy+yz+zx=0` |
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| 172. |
If `2 sin alphacos beta sin gamma=sinbeta sin(alpha+gamma),then tan alpha,tan beta and gamma` are inA. A.P.B. G.P.C. H.P.D. none of these |
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Answer» Correct Answer - C We have, `2 sin alpha cos beta sin gamma=sinbeta sin(alpha+gamma)` `implies2sinalpha cos betasingamma=sinalphasinbetacosgamma+cosalpha sinbetasingamma` `implies2 cot alpha, cot beta, cot gamma` are in A.P. `impliestan alpha, beta,tangamma` are in H.P. |
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| 173. |
Given tan A and tan B are the roots of `x^2-ax + b = 0`, The value of `sin^2(A + B)` isA. `(a^(2))/(a^(2)+(1-b)^(2))`B. `(a^(2))/(a^(2)+b^(2))`C. `(a^(2))/((a+b)^(2))`D. `(a^(2))/(b^(2)+(1-a)^(2))` |
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Answer» Correct Answer - A Since tan A and tan B are the roots of the equation `x^(2)-ax+b=0.` Therefore, `tanA+tanB=aand tan A tanB=b` `thereforetan(A+B)=(a)/(1-b)` Now, `sin^(2)(A+B)=1/2[1-cos2(A+B)]` `impliessin^(2)(A+B)=1/2{1-(1-tan^(2)(A+B))/(1+tan^(2)(A+B))}` `impliessin^(2)(A+B)=1/2{1-(1-(a^(2))/((1-b)^(2)))/(1+(a^(2))/((1-b)^(2)))}` `impliessin^(2)(A+B)=1/2{(2a^(2))/(a^(2)+(1-b)^(2))}=(a^(2))/(a^(2)+(1-b)^(2))` |
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| 174. |
lf `cos 5theta-acos^5 theta + b cos^3 theta + c cos theta` then c is equal to-A. `-5`B. 1C. 5D. none of these |
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Answer» Correct Answer - C We have, `cos 5 theta+i sin5 theta=(cos theta+i sintheta)^(5)` `=cos5 theta+isin5 theta` `={""^(5)C_(0)0-""^(5)C_(2)cos^(3)thetasin^(2)theta+""^(5)C_(4)cos thetasin^(4)theta}` `" "+i{""^(5)C_(1)cos^(4)thetasintheta-""^(5)C_(3)cos^(2)thetasin^(3)theta....}` `impliescos 5 theta=cos^(5)theta-10cos^(3)thetasin^(2)theta+5 costhetasin^(4)theta` `impliescos5 theta=cos^(5)theta-10cos^(3)theta(1-cos^(2)theta)+5costheta(1-cos^(2)theta)^(2)` `impliescos5theta=16cos^(5)theta-20cos^(3)theta+5 cos theta` `thereforec=5` Differentiating with respect to `theta,` we get `-5 sin5 theta=-5a cos^(4)thetasintheta-3bcos^(2)thetasintheta-csintheta` Putting `theta=(pi)/(2),` we get `c=5.` |
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| 175. |
If `tan x =(2b)/ (a- c)`, `y = acos^2 x + 2bsin x cos x + csin^2 x`, `z=asin^2 x-2b sin x cos x + c cos^2 x`, prove that `y-z=a-c`.A. `y=z`B. `y+z=a-c`C. `y-z=a-c`D. `(y-z)=(a-c)^(2)+4b^(2)` |
| Answer» Correct Answer - C | |
| 176. |
If `sin alpha=sinbeta and cos alpha=cosbeta,` thenA. `sin""(alpha+beta)/(2)=0`B. `cos""(alpha+beta)/(2)=0`C. `sin""(alpha-beta)/(2)=0`D. `cos""((alpha-beta)/(2))=0` |
| Answer» Correct Answer - C | |
| 177. |
If `xcosalpha+ysinalpha=xcosbeta+ysinbeta=2a` then `cosalpha cosbeta=`A. `(4ax)/(x^(2)+y^(2))`B. `(4a^(2)-y^(2))/(x^(2)+y^(2))`C. `(4ay)/(x^(2)+y^(2))`D. `(4a^(2)-x^(2))/(x^(2)+y^(2))` |
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Answer» Correct Answer - B We have, `x cos alpha+ysin alpha=xcos beta+y sinbeta=2a` `impliesalpha,beta"are the roots of"x cos theta+y sintheta=2a.` Now, `x costheta+ysin theta=2a` `implies(xcostheta-2a)^(2)=y^(2)sin^(2)theta` `impliesx^(2)cos^(2)theta-4axcos theta+4a^(2)-y^(2)(1-cos^(2)theta)` `implies(x^(2)-y^(2))cos^(2)theta-4ax cos theta+4a^(2)-y^(2)=0` Clearly, `cos theta, cos beta` are the roos of this equation. `thereforecos alphacos beta=(4a^(2)-y^(2))/(x^(2)+y^(2))` |
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| 178. |
If `tanalpha`is equal to the integral solution of the inequality `4x^2-16 x+15 |
| Answer» Correct Answer - C | |
| 179. |
If `sectheta=x+1/(4x),`then `s e ctheta+t a ntheta=``x ,1/x`(b) `2x ,1/(2x)`(c) `-2x ,1/(2x)`(d) `-1/x ,x`A. `x,(1)/(x)`B. `2x,(1)/(x)`C. `-2x,(1)/(x)`D. `-(1)/(x),x` |
| Answer» Correct Answer - B | |
| 180. |
If `cos(A-B)=3/5and tan A tanB=2,` thenA. `cosAcosB=1/5`B. `sinAsinB=-2/5`C. `cos(A+B)=-1/5`D. none of these |
| Answer» Correct Answer - A | |
| 181. |
If `sinx+cosx=1/5,0lexlepi,` then tan x is equal toA. `-4/3or, -3/4`B. `3/4`C. `4/5`D. none of these |
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Answer» Correct Answer - A We have, `sinx+2x=1/5` `implies1+sin2x=1/25" "["On squaring both sides"]` `implies1+(2tanx)/(1+tan^(2)x)=1/25` `implies12tan^(2)x+25tanx+12=0` `implies(4tanx+3)(3tanx+4)=0impliestanx=-3/4or, tanx=-4/3` |
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| 182. |
if `0 |
| Answer» Correct Answer - A | |
| 183. |
The Minimum value of `27^cosx +81^sinx` is equal toA. `(2)/(3sqrt3)`B. `(2)/(9sqrt3)`C. `(4)/(3sqrt3)`D. none of these |
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Answer» Correct Answer - B We known that `A.M. ge G.M.` `therefore(27^(cosx)x+81^(sinx))/(2)gesqrt(27^(cosx)xx81^(sinx))` `implies27^(cosx)+81^(sinx)ge2sqrt(3^(3cosx+4sinx))` `implies27^(cosx) +81^(sinx)ge2xxsqrt(3^(-5))` `" "[because-5le3cosx+4sinxle5]` `implies27^(cosx)+81^(sinx)ge(2)/(9sqrt3)` Hence, the minimum value of `27^(cosx)+81^(sinx)is(2)/(9sqrt3).` |
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