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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.
| 1. |
An insect fly start from one corner of a cubical room and reaches at diagonally opposite corner. The magnitude or displacement of the insect is `40sqrt(3)` ft. Find the volume of cube.A. `64sqrt(3)ft^(3)`B. `1600ft^(3)`C. `64000ft^(3)`D. None of these |
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Answer» (c) Here, `sqrt(3)a=40sqrt(3)ft` `:. a=40ft` `:.` Volume `=a^(3)=(40ft)^(3)=64000ft^(3)` |
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| 2. |
If `A.B=AxxB`, then angle between `A` and `B` isA. `45^(@)`B. `30^(@)`C. `60^(@)`D. `90^(@)` |
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Answer» (a) `A.B=AxxB=ABcos theta=AB sin theta` `tan theta=1=tan theta =tan 45^(@) impliestheta=45^(@)` |
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| 3. |
If `|AxxB| = sqrt3 A.B,` then the value of |A+B| isA. `(A^(2)+B^(2)+(AB)/(sqrt(3)))^(1//2)`B. `A+B`C. `(A^(2)+B^(2)+sqrt(3)AB)^(1//2)`D. `(A^(2)+B^(2)+AB)^(1//2)` |
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Answer» Correct Answer - D (d) `ABsintheta=sqrt(3)ABcosthetaortantheta=sqrt(3)` `:.theta=60^(@)` Now `|A+B|=sqrt(A^(2)+B^(2)+2ABcos60^(@))` `=sqrt(A^(2)+B^(2)AB)` |
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| 4. |
Find the vector that must be added to the vector `hat(i)-3hat(j)+2hat(k)` and `3hat(i)+6hat(j)-7hat(k)` so that the resultant vector is a unit vector along the y-axis.A. `-4hat(i)-2hat(j)+5hat(k)`B. `-4hat(i)+2hat(j)+5hat(k)`C. `4hat(i)-2hat(j)+5hat(k)`D. `4hat(i)-2hat(j)-5hat(k)` |
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Answer» Correct Answer - A (a) `(hat(i)-3hat(j)+2hat(k))+(3hat(i)+6hat(j)-7hat(k))+r=hat(j)` `rArr r=-4hat (i)-2hat(j)+5hat(k)` |
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| 5. |
The angle between `A` the resultant of `(A+B)` and `(A-B)` will beA. `0^(@)`B. `tan^(-1)(A/B)`C. `tan^(-1)(B/A)`D. `tan^(-1)((A-B)/(A+B))` |
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Answer» (a) `R=(A+B)+(A-B)` ltbrLgt `R=2A` The angle between `A` and `2A` is zero, because they are parallel vectors. |
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| 6. |
If `P+Q=0`. Then which of the following is necessarily ture?A. `P=0`B. `P= -Q`C. `Q=0`D. `P=Q` |
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Answer» Correct Answer - B The sum of two vectors is zero only when one vector is equal in jmagnitude but oppositein direction to that of other vector. |
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| 7. |
`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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| 8. |
What happens, when we multiply a vector by `(-2)`A. Direction reverses and unit chargesB. Direction reverses and magnitude is doubledC. Direction remains unchanged and unit changesD. None of the above |
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Answer» Correct Answer - B (b)When a vector is multiplied by a negative scalar number, then its magnitude gets change and direction gets reversed. |
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| 9. |
A vector multiplied by the number 0, results into |
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Answer» Correct Answer - C (c) When a vector is multiplied by zero, it results into zero vector, i.e.,a vector having zero magnitude. It is written as 0. |
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| 10. |
what is the dot product of two vectors of magnitudes 3 and 5,if angle between them is `60^@`?A. 5.2B. 7.5C. 8.4D. 8.6 |
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Answer» Correct Answer - B (b) We have,`A.B=ABcostheta` `=(5)(3)cos60^(@)=75` |
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| 11. |
If `(|a+b|)/(|a-b|)=1`, then the angle between `a` and `b` isA. `0^(@)`B. `45^(@)`C. `90^(@)`D. `60^(@)` |
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Answer» (c) `(|a+b|)/(|a-b|)=1` or `|a+|=|a-b|` or `sqrt(a^(2)+b^(2)+2ab cos alpha)=sqrt(a^(2)+b^(2)-2ab cos alpha)` or `a^(2)+b^(2)+2ab cos alpha=a^(2)+b^(2)-2ab cos alpha` or` 4ab cos alpha=0` or `cos alpha=0` or `alpha=90^(@)` |
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| 12. |
If two forces of equal magnitude 4 units acting at a point and the angle between them is `120^(@)`, then find the magnitude of direction of the sum of the two vectorsA. `4, theta=tan^(-10(1.73)`B. `4, theta=tan^(-1)(0.73)`C. `2, theta=tan^(-1)(1.73)`D. `6, theta^(-1)(0.73)` |
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Answer» (a) `|R|=|a+b|=sqrt(4^(2)+4^(2)+2(4)cos 120^(@))` `|R|=4` Let `theta=` angle between `a+b` with `x`-axis `theta=tan^(-1)((4sin (120^(@)))/(4+4cos(120^(@))))="tan"^(-1)(3.646)/2=tan^(-1)(1.73)` |
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| 13. |
If `A` and `B` denote the sides of a parallelogram and its area is `1/2AB` (`A` and `B` are magnitude of `A` and `B` respectively), the angle between `A` and `B` isA. `30^(@)`B. `45^(@)`C. `60^(@)`D. `90^(@)` |
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Answer» (a) Area of parallelogram `=|AxxB|` `AB sin theta=1/2 ABimpliessin theta =1/2 implies theta =30^(@)` |
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| 14. |
A cat is situated at a point `A(0,3,4)` and rat is situated at point `B(5,0,-8)`. The car is free to move but the rat is always at rest. Find the minimum distance travelled by cat to catch the rat.A. 5 unitB. 12 unitC. 13 unitD. 17 unit |
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Answer» (c) The minimum distance `=` The magnitude of displacement of cat `=|r_(B)-r_(A)|` Here `r_(B)=5hati-8hatk, r_(A)=3hati+4hatk` `:.s=r_(B)-r_(A)=5hati-3hatj-12hatk` `:.|s|=sqrt((5)^(2)+(-3)^(2)+(-12)^(2))` `=sqrt(25+9+144)` `=13.34~~13` unit |
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| 15. |
Calculate the distance travelled by the car, if a car travels `4km` towards north at an angle of `45^(@)` to the east and then travels a distance of `2km` towards north at an angle of `135^(@)` to the est.A. `6km`B. `8km`C. `5km`D. `2km` |
| Answer» (a) As the distance is a scalar quantity. So, total distance travelled `=4+2=6km` | |
| 16. |
Find the area oif the parallelogram determined `A=2hati+hatj-3hatk` and `B=12hatj-2hatk`A. 42B. 56C. 38D. 74 |
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Answer» (a) `AxxB=|(hati, hatj, hatk),(2, 1, -3),(0, 12, -2)|` `=hati(-2+36)-hatj(-4-0)+hatk(24-0)` `=34hati+4hatj+24hatk` `|AxxB|=sqrt(1156+16+576)=4181~~42` |
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| 17. |
A force `F=(2hati+3hatj-hatk)N` is acting on a body at a position `r=(6hati-3hatj-2hatk)`. Calculate the torque about the originA. `(3hati+2hatj+12hatk)Nm`B. `(9hati+2hatj+7hatk)Nm`C. `(hati+2hatj+12hatk)Nm`D. `(3hati+12hatj+hatk)Nm` |
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Answer» (a) `tau=rxxF=|(hati, hatj, hatk),(6,3,-2),(2,3,-1)|` `=hati(-3+6)-hatj(-6+4)_hatk(18-6)` `=tau=(3hati+2hatj+12hatk)Nm` |
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| 18. |
If the three vectors are coplanar, then find `x`. `A=hati-2hatj+3hatk, B=xhatj+3hatk, C=7hati+3hatj-11hatk`A. `36//21`B. `-51//32`C. `51//32`D. `-36//21` |
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Answer» (b) The three vectors are coplanar, if their scalar tripler product is zero. i.e. `A.(BxxC)=0=|(1, -2, 3),(0, x, 3),(7, 3, -11)|=0` `(1)[-x(11)-9=+2(0-21)+3(0-7x)=0` `-11x-9-42-21x=0implies-32x-51=0impliesx=- 51/32` |
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| 19. |
The three conterminous edges of parallelopiped are `a=2hati-6hatj+3hatk, b=5hatj,c=-2hati+hatk` Calculate the volumeof parallelopipedA. 36 cubic unitsB. 45 cubic unitsC. 40 cubic unitsD. 54 cubic units |
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Answer» (c) Volume `V=a.(bxxc)=|(2, -6, 3),(0, 5, 0),(-2, 0, 1)|` `=2(5-0)+6(0)+3(+10)=10+30=40` cubic units. |
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| 20. |
Find the values of `x` and `y` for which vectors `A=(6hati+xhatj-2hatk)` and `B(5hati-6hatj-yhatk)` are be parallelA. `x-=0, y=2/3`B. `x=-36/5,y=5/3`C. `x=-15/3, y=23/5`D. `x=36/5, y=15/4` |
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Answer» (b) Condition for parallel vectors `AxxB=0` `=|(hati, hatj, hatk),(6, x, -2), (5, -6, y)|=0` `-hati(-xy-12)-hatj(-6y+10)+hatk(-36-5x)=0` or `xy=-12, +6y=10-36=5ximpliesx=-36/5,y=5/3` |
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| 21. |
Find the component of vector A+B along i. X-axis, ii. C. Given, `A=hat(i)-2hat(j)+3hat(k)andC+hat(i)+hat(j).`A. `3,(1)/(sqrt(2))`B. `2,(3)/(sqrt(2))`C. `5,(2)/(3)`D. `4.1, (2)/(sqrt(2))` |
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Answer» `(a) A+B=(hat(i)-2hat(j))+(2hat(i)+3hat(j))=3hat(i)-2hat(j)+3hat(k)` i. Component of A+B along X -axis is 3. ii. Component of A+B=R (say) along C is `:. R*C=RC cos theta` `Rcostheta+(R*C)/C=((31hati-2hat(j)+3hat(k))*(hati+hat(j)))/sqrt((1)^(2)+(1)^(2))` `=(3-2)/(sqrt(2))=(1)/(sqrt(2))` |
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| 22. |
Given that `P+Q+R=0`. Which of the following statement is true?A. `|P|+|Q|=|R|`B. `|P+Q|=|R|`C. `|P|-|Q|+|R|`D. `|P-Q|=|R|` |
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Answer» Correct Answer - B if` P+Q+R=0,` then `|P+Q|=|R|` |
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| 23. |
At what angle should the two forces 2P and `sqrt(2P)` and `Psqrt(2)P` act so that resultant force is `Psqrt(10)`?A. `45^(@)`B. `60^(@)`C. `90^(@)`D. `120^(@)` |
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Answer» Correct Answer - A (a) `Psqrt(10)=sqrt(4P^(2)+2P^(2)+4sqrt(2)P^(2)costheta)` :. `theta=45^(@)` |
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| 24. |
What are minmum number or unequal fores whose vector sum is zero ?A. twoB. threeC. fourD. Any |
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Answer» Correct Answer - B (b) Minimum three forces of unequal magnitude are required to make vector sum equal to zero. |
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| 25. |
`vec(P)+vec(Q)` is a unit vector along x-axis. If `vec(P)= hat(i)-hat(j)+hat(k)`, then what is `vec(Q)`?A. `hat(i)+hat(j)-hat(k)`B. `hat(j)-hat(k)`C. `hat(i)+hat(j)+hat(k)`D. `hat(j)+hat(k)` |
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Answer» Correct Answer - B (b) Given `P+1=hat(i)-hat(j)+hat(k)` and `P+Q=hat(i)-hat(i)+hat(j)+hat(k)=hat(j)-hat(k)` |
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| 26. |
If `a_(1) and a_(2)` aare two non- collineaar unit vectors and if `|a_(1)+a_(2)|=sqrt(3),` ,then value of `(a_(1)-a_(2)).(2a_(1)-a_(2))` isA. 2B. `(3)/(2)`C. `(1)/(2)`D. 1 |
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Answer» Correct Answer - B (b) Since, `a_(1)anda_(2)` are non-collinear `:.a_(1)=a_(2)=1` ` "and "|a_(1)+a_(2)|=sqrt(3)` `rArra_(1)^(2)+a_(2)^(2)+2a_(1)a_(2)costheta=(sqrt(3))^(2)` `rArr1+1+2costheta=3` `rArrcostheta=(1)/(2)` Now, `(a_(1)-a_(2)).(2a_(1)-a_2)=2a_(1)^(2)-a_(1).a_(2)-2a_(1).a_(2)+a_(2)^(2)` `=2a_(1)^(2)+a_(2)^(2)-3a_(1)a_(2)costheta` `=2+1-(3)/(2)=3//2` |
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| 27. |
The expression `(1/(sqrt(2))hat(i)+1/(sqrt(2))hat(j))` is aA. unit vectorB. null vectorC. vector of magnitude `sqrt(2)`D. scalar |
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Answer» Correct Answer - A (a) Let `A=(1)/(sqrt(2))hat(i)+(1)/(sqrt(2))hat(j)` `|A|=sqrt(((1)/(sqrt(2)))^(2)+((1)/(sqrt(2)))^(2))=sqrt((2)/(2))=1` As, the magnitude of given vector is 1 `:.` It is a unit vector. |
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| 28. |
The value of `hat(i)xx(hat(i)xxa)+hat(j)xx(hat(j)xxa)+hat(k)xx(hat(k)xxa)` isA. aB. `axxhat(k)`C. `-2a`D. `-a` |
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Answer» Correct Answer - C (c) Suppose, `a=a_(1)hat(i)+a_(2)hat(j)+a_(3)hat(k)` Now, `(hat(i)xxa)=a_(2)hat(k)-a_(3)hat(j)` Now `hat(i)xx(hat(i)xxa)=-a_(2)hat(j)-a_(3)hat(k)` Similarly, `hat(i)xx(hat(j)xxhat(a))=-a_(1)hat(i)-a_(3)hat(k)` and `hat(k)xx(hat(j)xxa)=-a_(1)hat(i)-a_(2)hat(j)` `:.hat(i)(hat(i)xxa)+hat(j)(hat(j)xxa)+hat(k)(hat(k)xxa)=-2a` |
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| 29. |
The area of the parallenlogram determined by two adjacentt sides as `A=2hat(i)+hat(j)-3hat(k)andB=12hat(j)-2hat(k)` is approximatelyA. 43B. 56C. 38D. 74 |
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Answer» Correct Answer - A (a) Area of parallelogram `=|AxxB|` `AxxB=|{:(hat(i),hat(j),hat(k)),(2,1,-3),(0,12,-2):}|` ` =hat(i)(-2+36)-hat(j)(-4)+hat(k)(24)` `=34hat(i)+4hat(j)+24hat(k)` `|AxxB|=sqrt((34)^(2)+(4)^(2)+(24)^(2))` `=418~~43` |
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| 30. |
What is the angle between `vec(P)` and the resultant of `(vec(P)+vec(Q))` and `(vec(P)-vec(Q))` ?A. zeroB. `tan^(-1)(P//Q)`C. `tan^(-1)(Q//P)`D. `tan^(-1)(P-Q)//(P+Q)` |
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Answer» Correct Answer - A (a) Resultant of `(P+Q) and (P-Q) is P+Q+P-Q or 2P` which is parallel to P. So, angle between P and 2P will be zero. |
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| 31. |
For the resultant of two vectors to be maximum , what must be the angle between them ?A. `0^(@)`B. `60^(@)`C. `90^(@)`D. `90^(@)` |
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Answer» Correct Answer - A (a) Resultant of two vectors will be maximum when they are parallel,i.e., angle between them is zero. |
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| 32. |
Find the torque of a force `F=-3hat(i)+2hat(j)+hat(k)` acting at the point `r=8hat(i)+2hat(j)+3hat(k),(iftau=rxxF)`A. `14hat(i)-38hat(j)+16hat(k)`B. `4hat(i)+4hat(j)+6hat(k)`C. `-14hat(i)+38hat(j)-16hat(k)`D. `-4hat(i)-17hat(j)+22hat(k)` |
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Answer» Correct Answer - D (d) Torque of the force, `tau=rxxF` `tau=|{:(hat(i),hat(j),hat(k)),(8,2,3),(-3,2,1):}|` `=hat(i)(2-6)-hat(j)(8+9)+hat(k)(16+6)` `=-4hat(i)-17hat(j)+22hat(k)` |
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| 33. |
The condition `(a.b)^(2)=a^(2)b^(2)` is satisfied whenA. a is paraller to bB. `aneb`C. `a.b=1`D. `abotb` |
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Answer» Correct Answer - A (a) `(a.b)^(2)=a^(2)b^(2)cos^(2)theta=a^(2)b^(2)` `:.theta=0^(@)` `rArr` a is parallel to b. |
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| 34. |
Given `A=hat(i)+hat(j)+hat(k)andB=-hat(i)-hat(j)-hat(k)` then (A-B) will make angle with AA. `0^(@)`B. `180^(@)`C. `90^(@)`D. `60^(@)` |
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Answer» Correct Answer - A (a) `A-B=(hat(i)+hat(j)+hat(k))-(-hat(i)-hat(j)-hat(k))` `=(2hat(i)+2hat(j)+2hat(k)=2A)` i.e., A - B and A are parallel. |
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| 35. |
If two vectors are equal and their resultant is also equal to one of them, then the angle between the two vectors isA. `60^(@)`B. `120^(@)`C. `90^(@)`D. `0^(@)` |
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Answer» Correct Answer - B (b) We have ,`R=A=B` `:.R^(2)=R^(2)+R^(2)+2R Rcostheta` `"or "costheta=-(1)/(2)` `:.theta=120^(@)` |
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| 36. |
If three vectors along coordinate axes represent the adjacent sides of a cubie of length `b`, then the unit vector along its diagonal passing through the origin will beA. `(hati+hatj+hatk)/(sqrt(2))`B. `(hati+hatj+hatk)/(sqrt(3)b)`C. `hati+hatj+hatk`D. `(hati+hatj+hatk)/(sqrt(3))` |
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Answer» (d) Diagonal vector `A=bhati+bhatj+bhatk` or `A=sqrt(b^(2)+b^(2)+b^(2))=sqrt(3)b` ltbrrgt `:. hatA=A/A=(hati+hatj+hatk)/(sqrt(3))` |
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| 37. |
If three vectors along coordinate axis represent the adjacent sides of a cube of length b, then the unit vector along its diagonal passing through the origin will beA. `(hat(i)+hat(j)+hat(k))/(sqrt(2))`B. `(hat(i)+hat(j)+hat(k))/(sqrt(36))`C. `hat(i)+hat(j)+hat(k)`D. `(hat(i)+hat(j)+hat(k))/(sqrt(3))` |
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Answer» Correct Answer - D (d) Diagonal vector, `A=bhat(i)+bhat(j)+bhat(k)` ` "or "A=sqrt(b^(2)+b^(2)+b^(2))=sqrt(3)b` `:.A=(A)/|A|=(hat(i)+hat(j)+hat(k))/(sqrt(3))` |
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| 38. |
The modulus of the vector product of two vectors is `(1)/(sqrt(30))` times their scalar product. The angle between vectors isA. `(pi)/(6)`B. `(pi)/(2)`C. `(pi)/(4)`D. `(pi)/(3)` |
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Answer» Correct Answer - A (a) `ABsintheta=(1)/(sqrt(3))ABcostheta` `:.tantheta=(1)/(sqrt(3))ortheta=30^(@)=(pi)/(6)` |
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| 39. |
If the sum of two unit vectors is a unit vector, then magnitude of difference is-A. `sqrt(2)`B. `sqrt(3)`C. `1//sqrt(2)`D. `sqrt(5)` |
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Answer» Correct Answer - B (b)Sum of two unit vectors is a unit vector, means angle between those two unit vectors is `120^(@)` `:.|S|=sqrt(1+1-2xx1xx1xxcos120^(@))=sqrt(3)` |
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| 40. |
A vector `vecF_(1)` is along the positive `X`-axis. If its vectors product with another vector `vecF_(2)` is zero then `vecF_(2)` could beA. `4hatj`B. `(hatk+hatj)`C. `(hatj+hatk)`D. `-4hati` |
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Answer» (d) Let `F_(1)=xhati` As, `F_(1)xxF_(2)=0` and only `hatixxhati=0` `:. F_(2)=-4hati` |
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| 41. |
The component of vector `A=a_(x)hati+a_(y)hatj+a_(z)hatk` and the directioin of `hati-hatj` isA. `a_(x)-a_(y)+a_(z)`B. `a_(x)-a_(y)`C. `(a_(x)-a_(y))//sqrt(2)`D. `(a_(x)+a_(y)+a_(z))` |
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Answer» (c) Let `B=hati-hatj` Then component of vector `A` along `B=(A.B)/(|B|)` `=((a_(x)hati+a_(y)hatj+a_(z)hatk).(hati-hatj))/(|hati-hatj|)=(a_(x)-a_(y))/(sqrt(2))` |
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| 42. |
The unit vector perpendicular to vectors `a=3hati+hatj` and `=2hati-hatj-5hatk` isA. `+-((hati-3hatj+hatk))/(sqrt(11))`B. `+-(3hati+hatj)/(sqrt(11))`C. `+-((2hati-hatj-5hatk))/(sqrt(30))`D. none of these |
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Answer» (a) `a=3hati+hatj` and `b=2hati-hatj-5hatk` `axxb=|(hati, hatj, hatk),(3,1,0),(2,-1,-5)|=(-5)hati-(-15-0)hatj+(-3-2)hatk` `=-5hati+15hatj-5hatk` `R=axxb=-5(hati-3hatj+hatk)` `hatR=(hati-3hatj+hatk)/(sqrt((1)^(2)+(-3)^(2)+(1)^(2)))=(hati-3hatj+hatk)/(sqrt(11))` |
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| 43. |
If a particle is moving on an parallel path given by `r=b cos omegat hati+asinomegat hatj`, then find its radial acceleration along `r`A. `omega r`B. `omega^(2)r`C. `-omega^(2)r`D. none of these |
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Answer» (a) `(dr)/(dt)=-bomegat sin hati+aomega cos omegat hatj` `(d^(2)r)/(dt^(2))=-bomega^(2) sin omegat hati+aomega^(2) cos omegat hatj` `(d^(2)r)/(dt^(2))=-omega^(2) [b cos omegat hati+a sin omegat hatj]=-omega^(2)r` |
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| 44. |
If three vectors `xa-2b+3c, -2a+yb-4c` and `-zb+3c` are coplanar, where `a,b` and `c` are unit (or any) vectors thenA. `xy+3zx-3z=4`B. `2xy-3zx-3z-4=0`C. `4xy-3zx-3z=4`D. `xy-2zx-3z-4=0` |
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Answer» (d) Condition of coplanarity, `|(x,-2,3),(-2,y,-4),(0,-z,3)|=0` `impliesx(2-4z)+2(-4-0)+3(2z-0)=0` `implies2xy-4zx-8+6z=0` ltbrrgt `impliesxy-2zx-4+3z=0` |
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| 45. |
In the given figure `O` is the centre of regular pentagon `ABCDE`. Five forces each of magnitude `F_(0)` are acted as shown in figure. The resultant force is A. `5F_(0)`B. `5F_(0) cos 72^(@)`C. `5F_(0) sin 72^(@)`D. zero |
| Answer» (d) According to polygon law, resultant force will be zero. | |