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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.
| 201. |
Two forces, each equal to F, act, as shown in figure. Their resultant is A. `P//2`B. `P//4`C. `P`D. `2P` |
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Answer» Correct Answer - C `R=sqrt(A^(2)+B^(2)+2AB cos theta)` If `A=B=P` and `theta= 120^(@)` then `R=P` |
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| 202. |
A truck travelling due to north at `20m s^(-1)` turns west and travels at the same speed. Find the change in its velocity.A. `40m//s N-W`B. `20sqrt(2)m//sN-W`C. `40m//s S-W`D. `20sqrt(2) m//s S-W` |
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Answer» Correct Answer - D |
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| 203. |
If the sum of two unit vectors is a unit vector,then find the magnitude of their differences.A. `sqrt(2)`B. `sqer(3)`C. `1//sqer(2)`D. `sqrt(5)` |
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Answer» Correct Answer - B Let `hat(n)_(1)` and `hat(n)_(2)` be the two unit vector, then the sum is `vec(n)_(s)= hat(n)_(1)+hat(n)_(2)` or `n_(s)^(2)= n_(1)^(2)+n_(2)^(2)+2n_(1)n_(2) cos theta` `1+1+2 cos theta` Since it is given that `n_(s)` is also a unit vector, therefore `1= 1+1+2 cos theta implies cos theta= -1/2 :. theta= 120^(@)` Now the difference Vector is `hat(n)_(d)=hat(n)_(1)-hat(n)_(2)` or `n_(d)^(2)= n_(1)^(2)+n_(2)^(2)-2n_(1)n_(2) cos theta = 1+1-2 cos(120^(@))` `:. n_(d)^(2)= 2-2(-1//2)= 2+1= 3 impliesn_(d)= sqrt(3)` |
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| 204. |
If the sum of two unit vectors is a unit vector,then find the magnitude of their differences. |
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Answer» Let `hat(n)_(1)` and `hat(n)_(2)` are the two unit vector. Then the sum is `vec(n)_(s)=hat(n)_(1)+hat(n)_(2)` or `n_(s)^(2)=n_(1)^(2)+n_(2)^(2)+2n_(1)n_(2) cos theta` `1+1+2 cos theta` Since it is given that `n_(s)` is also a unit vector,therefore `1=1+1+2 cos theta` or `cos theta =-1/2 or theta=120^(@)` now the diffrence vector is `n_(d)=n_(1)-n_(2)` or `n_(d)^(2)=n_(1)^(2)+n_(2)^(2)-2n_(1)n_(2) cos theta=1+1-2 cos (120^(@))` `:. n_(d)^(2)=2-2(-1//2)=2+1=3rArrn_(d)=sqrt(3)` |
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| 205. |
Two forces `F_(1)=1N` and `F_(2)=2N` act along the lines x=0 and y=0, respectively. Then find the resultant of forces. |
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Answer» x=0 means y-axis `rArrvec(F)_(1)=hat(j)` y=0 means x-axis `rArr vec(F)_(2)=2hat(i)` So resultant, `vec(F)=vec(F)_(1)+vec(F)_(2)=2hat(i)+hat(j)` |
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| 206. |
Forces `F_(1) and F_(2)` act on a point mass in two mutually perpendicular directions. The resultant force on the point mass will beA. `F_(1)+F_(2)`B. `F_(1)-F_(2)`C. `sqrt(F_(1)^(2)+F_(2)^(2))`D. `F_(1)^(2)+F_(2)^(2)` |
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Answer» Correct Answer - C `F=sqrt(F_(1)^(2)+F_(2)^(2)+2F_(1)F_(2)cos 90^(@))= sqrt(F_(1)^(2)+F_(2)^(2))` |
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| 207. |
Forces `F_(1) and F_(2)` act on a point mass in two mutually perpendicular directions. The resultant force on the point mass will beA. `F_(1)+F_(2)`B. `F_(1)-F_(2)`C. `sqrt(F_(1)^(2)+F_(2)^(2)`D. `F_(1)^(2)+F_(2)^(2)` |
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Answer» Correct Answer - C |
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| 208. |
The value of `[a-b b-c c-a]` is equal to |
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Answer» Correct Answer - A `[(a-b,b-c,c-a)]` `={(a-b)xx(b-c)}.(c-a)` `=(axxb-axxc-bxxb+bxxc).(c-a)` `=(axxb+cxxa+bxxc).(c-a)` `=(axxb).c-(bxxc).a` `=[abc]-[abc]=0` |
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| 209. |
If `a+b+c=0 and |a|=sqrt(37),|b|=3, |c|=4` then the angle between b and c isA. `30^(@)`B. `45^(@)`C. `60^(@)`D. `90^(@)` |
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Answer» Correct Answer - C Given, `a+b+c=` and `|a|=sqrt(37), |b|=3` and `|c|=4` Now, `a+b+c=0` `implies a=-(b+c)` `implies |a|^(2)=|-(b+c)|^(2)` `implies |a|^(2)=|b|^(2)+|c|^(2)+2|b||c| cos theta` `=9+16+24 cos theta` `implies 37=25+24 cos theta` `implies 24 cos theta =12 implies theta = 60^(@)` |
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| 210. |
If `vecA=3hati+4hatj` and `vecB=7hati+24hatj`, find a vector having the same magnitude as `vecB` and parallel and same direction as `vecA`.A. `5hati+20hatj`B. `15hati+10hatj`C. `20hati+15hatj`D. `15hati+20hatj` |
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Answer» Correct Answer - D |
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| 211. |
Consider east as positive x-axis, north as positive y-axis and vertically upward direction as z-axis. A helicopter first rises up to an altitide of `100 m` than flies straight in north `500 m` and then suddenly takes a turn towards east and travels `1000 m` east. What is position vector of helicopter ? (Take starting point as origin)A. `1000 hat(i)-500 hat(j)+100 hat(k)`B. `1000 hat(i)+500 hat(j)-100hat(k)`C. `1000 hat(i)+ 500hat(j)+100 hat(k)`D. `-1000 hat(i)+500 hat(j)+100 hat(k)` |
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Answer» Correct Answer - C `vec(r )= 1000 hat(i)+ 500 hat(j)+ 100 hat(k)` |
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| 212. |
If `vecA=3hati+4hatj` and `vecB=7hati+24hatj`, find a vector having the same magnitude as `vecB` and parallel and same direction as `vecA`. |
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Answer» The vector parallel to `vecA` and having magnitude of `vecB` is `vecC=|vecB|hatA` `B=sqrt(7^(2)+24^(2))=25` `hatA=(vecA)/(A)=(3hati+4hatj)/sqrt(3^(2)+4^(2))=(1)/(5)(3hatj+4hatj)` `vecC=25xx(1)/(5)(3hati+4hatj)=15hati+20hatj` |
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| 213. |
If the angle between `veca` and `vecb` is `(pi)/(3)`, then angle between `2veca` and `-3vecb` is :A. `(.)/(3)`B. `(2.)/(3)`C. `(.)/(6)`D. `(5.)/(3)` |
| Answer» Correct Answer - B | |
| 214. |
Let ` vec a a n d vec b`be two unit vectors and `alpha`be the angle between them, then ` vec a+ vec b`is a unit vectors, ifA. `theta = (pi)/(4)`B. `theta = (pi)/(3)`C. `theta = (pi)/(2)`D. `theta = (2pi)/(3)` |
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Answer» Correct Answer - D Let `vec(a)` and `vec(b)` are two unit vectors and angle between them is `theta`. then, `|vec(a)|=|vec(b)| = 1` Now, `(vec(a) + vec(b))` is a unit vector if `|vec(a) + vec(b)|=1` `implies (vec(a) + vec(b))^(2)=1` `implies (vec(a)+ vec(b)). (vec(a) + vec(b)) = 1` `implies vec(a).vec(a) + vec(a).vec(b) + vec(b) . vec(b).vec(a) + vec(b) .vec(b) = 1` `implies |vec(a)|^(2) + |vec(b)|^(2) + 2vec(a).vec(b) =1 (because a.b = b.a)` `implies 1^(2) + 1^(2) + 2vec(a).vec(b) = 1` `implies vec(a).vec(b) =- (1)/(2)` `implies |vec(a)||vec(b)|cos theta =- (1)/(2)` `implies 1xx1xxcos theta = - (1)/(2)` `implies cos theta =- (1)/(2) implies theta = (2pi)/(3)` |
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| 215. |
Ifis the angle between any two vectors ` -> a`and ` -> b`,then `| -> adot -> b|=| -> axx -> b|`when `theta`isequal to(a) 0 (B)`pi/4`(C) `pi/2`(d) `pi` |
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Answer» Correct Answer - b Given that, `|vec(a).vec(b)|=|vec(a)xx vec(b)|` `implies |vec(a)||vec(b)| cos theta = |vec(a)||vec(b)|sintheta` `implies cos theta = sin theta` `implies tan theta = 1 implies theta = (pi)/(4)` |
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| 216. |
If `theta`is the angle between two vectors ` vec a a n d vec b , t h e n vec adot vec bgeq0`only when`0A. `0lt theta lt (pi)/(2)`B. `0 le theta le (pi)/(2)`C. `0 lt theta lt pi`D. `0 le theta le pi` |
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Answer» Correct Answer - B It is given that, `vec(a) . vec(b) ge 0` We know that, `vec(a). vec(b) = |vec(a)||vec(b)|cos theta ge 0` `implies cos theta ge 0 implies 0letheta le (pi)/(2)` |
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| 217. |
From figure the correct relation is A. `vec(A)+vec(B)+vec(E )=vec(0)`B. `vec(C )-vec(D)=-vec(A)`C. `vec(B)+vec(E )-vec(C )= -vec(D)`D. All of the above |
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Answer» Correct Answer - D In `DeltaMNO,vec(A)+vec(C )-vec(D)=orArrvec(C )-vec(D)=-A` Hence, (b) is correct. In `DeltaMNP,vec(A)+vec(B)+vec(E )=0` Hence ,(a) is correct. In `square MPNO, -vec(E )-vec(B)+vec(C )-vec(D)=0` `rArrvec(B)+vec(E )-vec(C )= -vec(D)` Hence, (c) is correct. |
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| 218. |
A cyclist is moving due east with a velocity of `10 km h^(-1)`. There is no wind and rain appoars to fall at an angle of `10^(@)` to the vertical. Calculate the actual speed of the rain. |
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Answer» Correct Answer - `56.72 kmh^(-1)` |
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| 219. |
A car travelling a `20 ms^(-1)` due north along the highway makes a right turn on to a side road that heads due east. It takes 50s for the car to complete the turn. At the end of 50 second, the car has a speed of `15 ms^(-1)` along the side road. Determine the magnitude of average acceleration over the 50 second interval. |
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Answer» Correct Answer - `0.5 ms^(-2)` |
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| 220. |
The angle made by the vector `vecA=2hati+3hatj` with Y-axis isA. `tan^(-1)((3)/(2))`B. `tan^(-1)((2)/(3))`C. `sin^(-1)((2)/(3))`D. `cos^(-1)((3)/(2))` |
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Answer» Correct Answer - B `cos theta=((2hati+3hatj).(hatj))/(sqrt(4+9))` |
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| 221. |
The angle made by the vector `vecA=2hati+3hatj` with Y-axis isA. `tan^(-1) (3//2)`B. `tan^(-1) (2//3)`C. `sin^(-1) (2//3)`D. `cos^(-1) (3//2)` |
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Answer» Correct Answer - B `tan theta =(x)/(y)` `theta^(@)= tan ^(-1)((2)/(3))` |
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| 222. |
What happens, when we multiply a vector by `(-2)`A. direction reverses and unit chargesB. direction reverses and megnitude is doubledC. direction remains unchanged and unit chanesD. none of the above |
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Answer» Correct Answer - B (b) When a vector is multiplied by a negative scalar nummer, then magnitude gets chnges and direction gets reversed. |
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| 223. |
`vecA + vecB` can also be written asA. A-BB. B-AC. B+AD. B-A |
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Answer» Correct Answer - C (C ) As vector addition is commutative .So A+B=B+A |
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| 224. |
A lion is at some instant a position `A(2m, 6m ,-1m)` and a goat is at position `B(1 m, 2 m, 8 m)`. The lion is free to move but the goat is unable to move due to some injury . The lion runs towards the goat and reches it in time 2 sec. A verage velocity of the lion can be expreseed as:A. `(-(hati)/(2) - 2hatj +(9)/(2)hatk)m//s`B. `((hati)/(2)-2hatj +(5)/(2) hatk)m//s`C. `(hati - (hatj)/(2)+(hatk)/(2))m//s`D. `(3hati -(5)/(2)hatj+(7)/(2)hatk)` |
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Answer» Correct Answer - A `(2hati +6hatj- hatk)+(V_(1)hati + V_(2) hatj +V_(3)hatk)xx 2 = hati + 12 hatj + 8hatk` |
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| 225. |
An insect crawls from `A` to `B` where B is the centre of the rectangular slant face. Find the (a) initial and final position vector of the insect and (b) displacement vector of the insect. . |
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Answer» Initial position vector of insect `vec(OA)=2hat(j)+hat(k)(m)` `vec(OB)=3/2hat(i)+hat(j)+1/2hat(k)(m)` Displacement of insect `vec(AB)`= position vector of `vec(B)`-position vector of `vec(A)`. `vec(AB)=(3/2-0)hat(i)+(1-2)hat(j)+(1/2-1)hat(k)=3/2hat(i)-hat(j)-1/2hat(k)(m)` Hence, magnitude of displacement `|vec(AB)|=sqrt((3/2)^(2)+(1)^(2)+(1/2)^(2))=sqrt(9/4+1+1/4)=sqrt(7/2m)`. |
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| 226. |
Assertion - small displacement is a vector quantity. Reason Pressure and surface tension are also vector quanities.A. If both Assetion and Reason are correct but Reason is the correct explanation of Assertion.B. If both Assetion and Reason are correct but Reason is not the correct explanation of Assertion.C. If Asserion is true but Reason is falseD. If Asserion is false but Reason is true |
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Answer» Correct Answer - C ( c) Pressure and surface tension are scalar quantiites. |
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| 227. |
Dot product of two mutual perpendicular vector is |
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Answer» Correct Answer - A |
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| 228. |
Two vectors are given by `veca=2vecb-vecc=0` then thrid vector `vecc` isA. `4hati-9hatj-13hatk`B. `-4hati-9hatj+13hatk`C. `4hati-9hatj-13hatj`D. `2hti-3hatj+13hatk` |
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Answer» Correct Answer - A `vecc=3veca+2vecb` |
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| 229. |
If for two vectors `vecA` and `vecB, vecA xxvecB=0`, the vectorsA. Are perpendicular to each otherB. Are parallel to each otherC. Act at an angle of `60^(@)`D. Act at an angle of `30^(@)` |
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Answer» Correct Answer - B |
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| 230. |
A spy plane is being tacked by a radar. At `t`=0, its position is reported as (100`m` ,200`m`, 1000`m`). 130`s` later, its position is reported to be (2500`m`,1200`m` ,1000 `m`). Find a unit vector in the direction of plane velocity and the magnitude of its average velocity. |
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Answer» Correct Answer - `20 ms^(-1);(12hat(i)+5hat(j))/13`; In 13 `s`, the displacement is `(2500-100)hat(i)+(1200-200)hat(j) +(100-100)hat(k)=2400hat(i)+1000hat(j)` Manitude of displacement `sqrt((2400)^(2)+(1000)^(2))=2600` `:. Velocity =2600/130=20m s^(-1)` And unit vector along velocity `=1/2600(2400hat(i)+1000hat(j))=(12hat(i)+5hat(j))/13` |
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| 231. |
A particle of `m=5kg` is momentarily at rest at `x= at `t=. It is acted upon by two forces `vec(F)_(1)` and `vec(F)_(2)`. `Vec(F)_(1)=70hat(j)` N. The direction and manitude of `vec(F)_(2)` are unknown. The particle experiences a constant acceleration, `vec(a)`,in the direction as shown in (figure) Neglect gravity. a.Find the missing force `vec(F)_(2)`. b. What is the velocity vector of the particle at `t=10 s`? c. What third force, `vec(F)_(3)` is required to make the acceleration of the particle zero? Either give magnitude and direction of `vec(F)_(3)` or its components. |
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Answer» Correct Answer - a.`30sqrt(2)`; b.`60hat(i)+80hat(j)`; c.`30hat(i)-40hat(j)` `vec(F)_(1)+vec(F)_(2)=m.vec(a)` `(70hat(j)+F_(2)hat(i)+F_(2)hat(j))=5(10 xcos53hat(i)+10sin53hat(j))` `=50(10xx3/5hat(i)+10xx4/2hat(j))` By comparing two sides. We get `F_(2)hat(i)=30hat(i)rArrF_(2)hat(j)=-30hat(j)` `:. vec(F)_(2)=30hat(i)-30hat(j)` `|F_(2)| =sqrt((30)^(2)+(30)^(2))=30sqrt(2)N` At `t`=`10s` `v=vec(u)+at=10(6hat(i)+8hat(j))=60hat(i)+80hat(j)` Acceleration will be zero if a third force makes resultant force =0 `:. vec(F)+vec(F)_(2)+vec(F)+(3)=0` or `vec(F)_(3)= -(vec(F)_(1)+vec(F)_(2))=-(70hat(j)+30hat(i)-30hat(j))=30hat(i)-40hat(j)` |
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| 232. |
A motor boat is going in a river with a velocity `vec(V) = (4 hat i-2 hat j + hat k) ms^(-1)`. If the resisting force due to stream is `vec (F)=(5 hat i-10 hat j+6 hat k)N`, then the power of the motor boat isA. 100wB. 50wC. 46wD. 23w |
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Answer» Correct Answer - C `P=vecF.vecV` |
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| 233. |
If `veca and vecb` are non-collinear vectors and `vecA= (p+4q)veca+ (2p+q+1)vecb and vecB= (-2p+q+2)veca+ (2p-3q-1)vecb`, and if `3vecA= 2vecB`, then determine `p and q`. |
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Answer» Putting the values of `vecA and vecB`, and then equating the coefficients of `veca and vecb` on both sides, we get `" "3(p+4q) = 2(-2p+q+2)` `" "3(2p+q+1)= 2(2p-3q-1)` `" "7p+10q=4 and 2p+9q=-5` Solving, we get `p=2 and q=-1` |
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| 234. |
The points with position vectors `veca + vecb, veca-vecb and veca +k vecb` are collinear for all real values of k. |
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Answer» Correct Answer - True Let position vectors of points A, B and C be `veca + vecb, veca - vecb and veca +kvecb`, respectively. Then `vec(AB) = (veca - vecb ) - (veca + vecb) = -2 vecb` Similarly, `vec(BC) = (veca + k vecb) - (veca - vecb) = (k+1)vecb` Clearly `vec(AB) "||" vec(BC) AA k in R` Hence, A, B and C are collinear `AA k inR` Therefore, the statement is true. |
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| 235. |
If the points A(3, 0, p), B (-1, q, 3) and C(-3, 3, 0) are collinear, then findThe values of p and q. |
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Answer» Putting λ = -3 in equation (2), we get 3(-3 + 1) = -3q ∴ -6 = -3q ∴ q = 2 Also, putting λ = -3 in equation (3), we get 0 = -9 + p ∴ p = 9 Hence p = 9 and q = 2. |
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| 236. |
if ` vec Ao`+ ` vec O B`= ` vec B O`+ ` vec O C`,than prove that B is themidpoint of AC. |
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Answer» `vec(AO)+vec(OB)=vec(BO)+vec(OC)` `rArr" "vec(AB)=vec(BC)` Thus, vectors `vec(AB)and vec(BC)` are collinear `rArr" "` Points A, B, C are collinear Also `" "|vec(AB)|=|vec(BC)|` `rArr" "B` is the midpoint of `AC` |
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| 237. |
If vectors a b c are non coplanar unit vectors such that a × (b × c) = (b + c)/√2 then the angle between a and b is(A) 3π/4(B) π/4(C) π/2(D) π |
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Answer» Correct answer is (A) 3π/4 |
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| 238. |
If A=5 units,B= 6 units and `|vec(A)xxvec(B)|= 15 units`, then what is the angle between `vec(A)` and `vec(B)`?A. `30^(@)`B. `60^(@)`C. `90^(@)`D. `120^(@)` |
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Answer» Correct Answer - A `sin theta =(|vec(A)xxvec(B)|)/(AB)=(15)/(5xx6)implies theta= 30^(@)` |
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| 239. |
If |A|=2 and |B|=4 and angle between then is `60^(@)` then |A-B|A. `sqrt(13)`B. `3sqrt(3)`C. `sqrt(3)`D. `2sqrt(3)` |
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Answer» Correct Answer - D (d ) `|A-B|=sqrt(A^(2)+B^(2)-2AB costheta)` `=sqrt(4+16-2xx2xx4xx(1)/(2))` `=sqrt(12)=2sqrt(3)` |
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| 240. |
What is the angle between 3a and -5a? What is the ratio of magnitude of two vectors? |
| Answer» Correct Answer - A | |
| 241. |
Two constant forces `F_(1)=2hati-3hatj+3hatk(N)` and `F_(2)=hati+hatj-2hatk(N)` act on a body and displace it from the position `r_(1)=hati+2hatj-2hatk(m)` to the position `r_(2)=7hati+10hatj+5hatk(m)`. What is the work doneA. `9J`B. `41J`C. `-3J`D. None of these |
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Answer» Correct Answer - A |
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| 242. |
If `vec(A)=4hat(i)-3hat(j)` and `vec(B)=6hat(i)+8hat(j)`,then find the magnitude and direction of `vec(A)+vec(B)`.A. `5, tan^(-1)(3//4)`B. `5 sqrt(5), tan^(-1)(1//2)`C. `10, tan^(-1)(5)`D. `25, tan^(-1)(3//4)` |
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Answer» Correct Answer - B `vec(A)+vec(B)= 4hat(i)-3hat(j)+6hat(i)+8hat(j)= 10hat(i)+5hat(j)` `|vec(A)+vec(B)|= sqrt((10)^(2)+(5)^(2))=5sqrt(5)` `tan theta= 5/(10)= 1/2implies theta= tan^(-1)(1/2)` |
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| 243. |
`vec(A)` and `vec(B)` are two Vectors and `theta` is the angle between them, if `|vec(A)xxvec(B)|= sqrt(3)(vec(A).vec(B))` the value of `theta` isA. `60^(@)`B. `45^(@)`C. `30^(@)`D. `90^(@)` |
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Answer» Correct Answer - A `vec(A)xxvec(B)= AB sin theta` and `vec(A).vec(B)= AB cos theta` Given, `|vec(A)xxvec(B)|=sqrt(3)(vec(A).vec(B))` `implies AB sin theta= sqrt(3) AB cos theta` `implies tan theta= sqrt(3)` `theta= 60^(@)` |
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| 244. |
If `|vec(A)xxvec(B)|=sqrt(3)vec(A).vec(B)`, then the value of `|vec(A)+vec(B)|` isA. `(A^(2)+B^(2)+AB)^(1//2)`B. `(A^(2)+B^(2)+(AB)/(sqrt(3)))^(1//2)`C. `A+B`D. `(A^(2)+B^(2)+sqrt(3)AB)^(1//2)` |
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Answer» Correct Answer - A Given `|vec(A)xxvec(B)|=sqrt(3)vec(A).vec(B)`….(i) but `|vec(A)xxvec(B)|=|vec(A)||vec(B)| sin theta= AB sin theta` and `vec(A).vec(B)=|vec(A)||vec(B)|cos theta= AB cos theta` Make these substitution in Eq.(i), we get `AB sin theta= sqrt(3) AB cos theta` or `tan theta =sqrt(3)implies theta= 60^(@)` The resultant of vector `vec(A)` and `vec(B)` can be given by the law of parallelogram. `:. |vec(A)+vec(B)|= sqrt(A^(2)+B^(2)+2AB cos 60^(@))` `=sqrt(A^(2)+B^(2)+2ABxx1/2)` `=(A^(2)+B^(2)+AB)^(1//2)` |
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| 245. |
Two forces, each of magnitude F have a resultant of the same magnitude F. The angle between the two forces isA. `45^(@)`B. `120^(@)`C. `150^(@)`D. `60^(@)` |
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Answer» Correct Answer - B `F^(2) = F^(2) + F^(2) + 2 F . F cos theta rArr cos theta = -(1)/(2)` `theta = 120^(@)` |
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| 246. |
Let `veca =veci -veck, vecb = xveci+ vecj + (1-x)veck and vecc =y veci +xvecj + (1+x -y)veck`. Then `veca, vecb and vecc` are non-coplanar forA. some values of xB. some values of yC. no values of x and yD. for all values of x and y |
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Answer» Correct Answer - D `veca =hati -hatk ` `" " vecb = xhati +hatj + (1-x)hatk` `" " vecc =yhati + xhatj+ (1+x-y)hatk` `" "=|{:(1,,0,,-1),(x,,1,,1-x),(y,,x,,1+x-y):}|` `" " = 1+x-y-x^(2)+y-x+x^(2)` `= 1` |
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| 247. |
Let `a, b and c` be distinct non-negative numbers. If vectos `a hati +a hatj +chatk, hati + hatk and chati +chatj+bhatk` are coplanar, then c isA. the arithmetic mean of a and bB. the geometric mean of a and bC. the harmonic mean of a and bD. equal to zero |
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Answer» Correct Answer - B a, b and c are distinct negative negative numbers and vectors `a hati + a hatj + chatk, hati +hatk and chati +c hatj +bhatk ` are coplanare `" "|{:(a,,a,,c), (1,,0,,1),(c,,c,,b):}|=0` or `ac+ c^(2) -ab -ac =0` or `c^(2) = ab` Hence, `a, c, b` are in G.P. So c is the G.M. of a and b. |
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| 248. |
If the angle between `2hati+ 2hatj- hatk` and vector `hati + chatk` is acute, then the maximum value of c is : |
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Answer» Correct Answer - 2 `cos theta = (vecA.vecB)/(|vecA||vecB|) =- ((2hati + 2hatj + hatk ) xx (hati + chatk))/(sqrt((2)^(2) + (2)^(2) + (-1)^(2) .sqrt(1+ c^(2))))` `= (2-c)/(3sqrt(1 + c^(2)))` If `theta` is acute , ` cos theta` is positive. `" "cos theta ge 0` `" "2cge 0` `" "2 le c` `" "c le 2` Thus, the maximum value of c is 2. |
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| 249. |
`|vecA + vecB|^(2) - |vecA - vecB|^(2) = n vecA. vecB` The value of n is : |
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Answer» Correct Answer - 4 `|vecA + vecB |^(2) - |vecA- vecB|^(2)` `" "A^(2) + B^(2) + 2vecA. vecB - A^(2)- B^(2) + 2vecA.vecB` `" "4vecA. vecB` |
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| 250. |
If the resultant of the vectors `3 hati +4 hatj +5 hatk` and `5 hati +3 hatj + 4hatk` makes an angle `theta` with x-axis, then find `cos theta`. |
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Answer» Correct Answer - `0.5744` |
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