58 + Interview Questions in UNIT-DIMENSION IN GENERAL AWARENESS Page 1 InterviewSolution

1.	`vec(r)=2that(i)+3tvec(2)hat(j)`. Find `vec(v)` and `vec(a)` where `vec(v)=(dvec(r))/(dr) , vec(a)=(dvec(v))/(dt)`
Answer» `vec(v)=(dvec(r))/(dt)=2hat(i)+6that(j)m//s` `vec(a)=(dvec(v))/(dt)=0+6hat(j) m//s^(2)`

Discussion

2.	`0.4hat(i)+0.8hat(j)+chat(k)` represents a unit vector, when c is :-A. 0.2B. `sqrt(0.2)`C. `sqrt(0.8)`D. 0
Answer» Correct Answer - B Let `vec(A)=0.4hat(i)+0.8hat(j)+Chat(k)` is a unit vector then `" " \|vec(A)\|=1` `rArr" " sqrt(0.16+0.64+C^(2))=1` `rArr " " 0.80+c^(2)=1^(2)` `rArr " " C^(2)=0.2` ltbr. C=sqrt(0.2)` ` " " C=sqrt(0.2)`

Discussion

3.	A force of `(-3hat(i)-hat(j)+2hat(k))` N displaced the body from a point (4, -3, -5) m to a point (-1, 4, 3) m in a staight line. Find the work done by the force.
Answer» `vec(A)=(4hat(i)-3hat(j)-5hat(k)),vec(B)=-hat(i)+4hat(j)+3hat(k)` `vec(AB)=vec(B)-vec(A)=(-5hat(i)+7hat(j)+8hat(k))` `W=vec(F).vec(AB)=(-3hat(i)-hat(j)+2hat(k)).(-5hat(i)+7hat(j)+8hat(k))=24"joule"`.

Discussion

4.	If `hat(i)+2hat(j)+nhat(k)` is perpendicular to `4hat(i)+2hat(j)+2hat(k)` then find the value of n ?
Answer» `(1hat(i) +2hat(j)+nhat(k)).(4hat(i)+2hat(j)+ 2hat(k))=0` `4+4+n(2)=0rArrn=-4`

Discussion

5.	Calculate the dimensional formula of energy from the equation `E= 1/2 mv^2`.
Answer» Dimensionally, `E="mass" xx ("velocity")^(2)`. Since `(1)/(2) ` is a number and has no dimention. or , `[E]=M xx((L)/(T))^(2)=ML^(2)T^(-2)`.

Discussion

6.	If force, time and velocity are treated as fundamental quantities then dimensional formula of energy will beA. [FTV]B. `FT^(2)V]`C. `FTV^(2)`D. `[FT^(2)V^(2)]`
Answer» Correct Answer - A `[E]=ML^(2)T^(-2)=(MLT^(-2)).(L.T^(-1))(T)=[FTV]`

Discussion

7.	If `x=k sin (k l t)`, where x is displacement and t is time then dimensional formula for l will be ` (k, l="constant")`A. `[M^(0)L^(0)T^(0)]`B. `[M^(0)L^(1)T^(0)]`C. `[M^(0)L^(-1)T^(-1)]`D. `[ML^(-1)T^(-1)]`
Answer» Correct Answer - C `[k]=L` `[klt]=L^(0)T^(0)` `L[l]T=L^(0)T^(0)rArr[l]=L^(-1)T^(-1)`

Discussion

8.	The position vectors of two balls are given by `vec(r)_(1)=2hat(i)+7hat(j) , vec(r)_(2)=-2hat(i)+4hat(j)` What will be the distance between the two balls?
Answer» `vec(r)=vec(r_(2))-vec(r_(1))` `vec(r)=-4hat(i)-3hat(j)` Distance `=sqrt(4^(2)+3^(2))=5`

Discussion

9.	A body constrained to move in y direction is subjected to a force given by `vec(F)=(-2hat(i)+15hat(j)+6hat(k))` N . What is the work done by this force in moving the body through a distance of `10 m` along y-axis ?
Answer» Correct Answer - `150J` `hat(r)=10hat(j)` `w=vec(F).vec(r)=150 "Joule"`

Discussion

10.	What displacement must be added to the displacement `25hat(i)-6hat(j)`m to give a displacement of 7.0 m pointing in the x-direction ?A. `18hat(i)-6hat(j)`B. `32hat(i)-13hat(j)`C. `-18hat(i)+6hat(j)`D. `-25hat(i)+13hat(j)`
Answer» Correct Answer - C Let `vec(X)` the requred displacement We have ` " " 25hat(i)-6hat(j)+vec(X)=7hat(i)` `rArr " " vec(X)=-18hat(i)+6hat(j)`

Discussion

11.	If velocity of a particle is given by `v=3t^(2)-6t +4`. Find its displacement from `t=0` to `3 secs`.
Answer» Correct Answer - `12m` (i) ` underset(x_(1))overset(x_(2))intdx=underset(0)overset(3)int(3t^(2)-6t+4)dt` `"displacement"=(t^(3)-3t^(2)+4t)_(0)^(3)=27-27+12m=12m`

Discussion

12.	Suppose displacement `y(t)` (that is, y as a function of t ) is given by `y(t)=t^(3)`
Answer» Find velocity `(dy//dt)` at `t = 3 sec`. Displacement at `t+Deltat` is `y(t+Deltat)=(t+Deltat)^(3)` `=(t^(3)+3t^(2)Deltat+3t Deltat^(2)+Deltat^(3))` hence displacement from t to `t+Deltat` is `Deltay` `Deltay=y(t+Deltat)-y(t)=(3t^(2)Deltat+3tDeltat^(2)+Deltat^(3))` Substituting this into Equation (i) gives `(dy)/(dt)=underset(Deltararr0)(lim)(Deltay)/(Deltat)=underset(Deltatrarr0)(lim)[3t^(2)+3tDeltat+Deltat^(2)]` Here we can see that if we take very small value of `Deltat` then value of `dy//dt` will approach `3t^(2)` as all other terms will become negligible and impossible to measure by any instrument available in this world. hence, `" " (dy)/(dt)=3t^(2) rArr (dy)/(dt)=v(at " " t =3sec)=3(3)^(2)=27 m//s`

Discussion

13.	Suppose `A=B^(n)C^(m)` , where A has dimensions LT, B has dimensions `L^(2)T^(-1)`, and C has dimensions `LT^(2)` , Then the exponents n and m have the values: (A) `2//3,1//3 " "` (B) `2,3 " "` (C)`4//5,-1//5 " "` (D) `1//5,3//5` `(E)`1//2,1//2`
Answer» `LT=[L^(2)T^(-1)]^(n)[LT^(2)]^(m)` `LT=L^(2n+m)T^(2m-n)` `{:(2n+m=1 " " ...(i)),(-n+2m=1 " "...(ii)):}` ` "On sovling" n=(1)/(5), m=(3)/(5)`

Discussion

14.	A vector perpendicular to `(hat(i)+hat(j)+hat(k))` and `(hat(i)-hat(j)-hat(k))` is :-A. `hat(i)+hat(j)+hat(k)`B. `hat(i)+hat(k)`C. `-1hat(i)+hat(j)+hat(k)`D. `hat(j)+hat(k)-2hat(i)`
Answer» Correct Answer - C Let the vector be `(xhat(i)+yhat(j)+xhat(k))` `(xhat(i)+yhat(j)+ zhat(k)).(hat(i)-hat(j)-hat(k))=0` `x+y-z=0" " `........(1) `(xhat(i)+yhat(j)+zhat(k)).(hat(i)-hat(j)-hat(k))=0` `x-y-z=0 " "`.......(2) Subtracting (1)& (2) `rArr y=0` `& " " x=z` Hence possible vector is Ans.B

Discussion

15.	The diameter of the sun is expressed as `13.9xx 10^(9) m`. Find the order of magnitude of the diameter ?
Answer» Diameter `=13.9xx10^(9) m` Diameter `=1.39xx10^(10) m` order of magnitude is 10 .

Discussion

16.	The value of a unit vector in the direction of vector `vec(A)=5hat(i)-hat(j) ` is ________________ ?
Answer» `\|vec(A)\|=sqrt(5^(2)+12^(2))=13` `hat(A)=(vec(A))/(\|vec(A)\|)=(5hat(i)-12hat(j))/(13)`

Discussion

17.	Which of the following experssions has magnitude unity.A. `(vec(a)+vec(b))/(\|vec(a)+vec(b)\|)`B. `\|hat(a)+hat(b)\|` when angle between `vec(a)`and `vec(b)` is `120^(@)`C. `(vec(a)-vec(b))/(\|vec(a)-vec(b)\|)`D. `\|hat(a)-hat(b)\|` when angle between `vec(a)` and `vec(b)` is `60^(@)`
Answer» Correct Answer - A::B::C::D

Discussion

18.	Find minimum value `y=25x^(2)-10x+5`.
Answer» For maximum/ minimum value `(dy)/(dx)=0 rArr50x-10rArrx=(1)/(5)` Now at `x=(1)/(5),(d^(2)y)/(dx^(2))=50`, which is positive So `y_(min)=25((2)/(5))^(2)-10((1)/(5))+5=1-2+5=4`

Discussion

19.	Mark correct statement(s) for `vec(a) , vec(b)` and `vec(c)` shown in above diagram : A. `-vec(a)-vec(b)=vec(c)`B. `vec(a)+vec(c)=vec(-b)`C. `vec(a)+vec(c)=vec(b)`D. `vec(b)-vec(c)=vec(-a)`
Answer» Correct Answer - B::D From triangle law :m ` " "vec(a)+vec(b)=-vec(c)` `rArr " " -vec(a)-vec(b)= vec(c)" " "Hence(A)" ` `rArr " " vec(a)=vec(c)=-vec(b) " " "Hence (B)" ` `rArr " " vec(b)+vec(c)=-vec(a) " " "Hence (D)`

Discussion

20.	Express the given vector `vec(A)` (shown graphically ) in unit vector notation.
Answer» (i) `vec(A)=3(cos 30hat(i)+sin 30hat(j))" "` (ii)`vec(A)=3(cos 30hat(i)-sin 30hat(j))` (iii)`vec(A)=3(-cos 30hat(i)-sin 30hat(j))` `vec(A)=3(-cos 30hat(i)+sin 30hat(j))`

Discussion

21.	Express the vector in unit - vector notation.
Answer» Correct Answer - `2hat(i)`

Discussion

22.	Three forces `(vec(F)_(1), vec(F)_(2), vec(F)_(3))` are acting an a particle moving vertically up with constant speed. Two force `vec(F)_(1)=-10hat(j)N`, and `vec(F)_(1)=-6hat(i)=8hat(j)` , N are acting an particle on acting on particle respectively find `vec(F)_(3)`.
Answer» `Sigma_(vec(F))=vec(0) i.e` `vec(F_(1))+vec(F_(2))+vec(F_(3))=vec(0)` `vec(F_(3))=-(vec(F_(1))+vec(F_(2)))` `=-(-6hat(i)-2hat(j))` `vec(f_(3))=6hat(i)+2hat(j) N`

Discussion

23.	Figure shows three vectors a, b and c. If `RQ=2PR`, which of the following relations is co-rest ? A. `2a+c=3b`B. `a+3c=2b`C. `3a+c=2b`D. `a+2c=3b`
Answer» Correct Answer - D `(a+2b)/(3)=b`

Discussion

24.	From the v versus t graph of figure (a) the time (s) at which the particle is at rest (b) at what time, if any does the particle reverse the direction of its motion? (c) The distance and displacement of the particle from `t=0`s to `t=6` s.
Answer» (a) `t=0,3 sec` , (b) `t=3 sec` (c) displacement `=-(1)/(2)(3xx4)+(1)/(2)(3+2)(4)` `=-6+10=4m` distance `=\|-(1)/(2)(3xx4)\|+\|(1)/(2)(3+2)(4)\|` `=6+10 =16m`

Discussion

25.	It `x=sin^(2)theta`, then find `(dx)/(dt)` where `(d theta)/dt)=omega`
Answer» `(dx)/(d theta)=2(sin theta)(costheta)=sin2theta` `(dx)/(dt)=(dx)/(d theta).(d theta)/ (dt)` `(dx)/(dt)=sin 2 theta.omega`

Discussion

26.	The radius of a circle is increasing at a rate `((dr)/(dt))=alpha` . Find the rate at which its area is increasing when radius is equal to 3 m.
Answer» area of cricle (A)`=pir^(2)` `(dA)/(dt)=pi(2r)(dr)/(dt)=2pir((dr)/(dt))=2pir(alpha)` `((dA)/(dt))_(atr=3m)=6pi alpham^(2)//sec`.

Discussion

27.	A bird moves from point `(1, -2)` to `(4, 2)` . If the speed of the bird is `10m//sec`, then find the velocity vector of the bird ?
Answer» Correct Answer - `6hat(i)+8hat(j)` `vec(r)=(4-1)hat(i)+(2+2)hat(j)=3hat(i)+4hat(j)` `hat(r)=(vec(r))/(\|hat(r)\|)=(3hat(i)+4hat(j))/(5)` `vec(v)=\|vec(v)\| hat(r)=10 ((3hat(i)+4hat(j))/(5))=6hat(i)+8hat(j)`

Discussion

28.	Two forces P and Q are in ratio `P:Q=1:2`. If their resultant is at an angle `tan^(-1)((sqrt3)/(2))` to vector P, then angle between P and Q is :A. `tan^(-1)((1)/(2)) `B. `45^(@) `C. `30^(@)`D. `60^(@)`
Answer» Correct Answer - A `tan alpha =(Q sin theta)/(P+Q cos theta) ` `sqrt(3)/2 =(sin theta )/((P)/(Q)+cos theta ) rArrsqrt(3)/(2) =(sin theta )/((1//2)+cos theta ) rArr (3)/4=((2 sin theta )/(1+2 cos theta ))^(2)` `rArr 3(1+2 cos theta )^(2)=16 sin ^(2) theta ` `rArr 3(1+4cos^(2) theta +4cos theta)= 16(1-cos ^(2) theta)` `rArr3+12cos ^(2) theta +12 cos theta =16-16 cos ^(2) theta` `rArr 28cos^(2)theta +12 cos theta -13=0 rArr " " cos theta = (1)//(2), -0.92`

Discussion

29.	If `E , M , J , and G` , respectively , denote energy , mass , angular momentum , and gravitational constant , then `EJ^(2) //M^(5) G^(2)` has the dimensions ofA. lengthB. angleC. massD. time
Answer» Correct Answer - B `(ML^(2)T^(-2)(MLT^(-1)L)^(2))/(M^(5)(M^(-1)L^(3)T^(-2)))=M^(0)L^(0)T^(0)`

Discussion

30.	If two numerical equal forces P and P acting at a point produce a resultant force of magnitude P itself, then the angle between the two original forces is :-A. `0^(@)`B. `60^(@)`C. `90^(@)`D. `120^(@)`
Answer» Correct Answer - D We have `P^(2)=P^(2)+P^(2)+2P^(2) cos q` `rArr " " cos theta =-(1)/(2)` `theta=120^(@)`

Discussion

31.	If `vec(A)` and `vec(B)` denote the sides of a parallelogram and its area is AB/2 , the angle between `vec(A)` and `vec(B)` is :-A. `(pi)/(2)`B. `pi`C. `(pi)/(6)`D. `(pi)/(3)`
Answer» Correct Answer - C `\|vec(A)xxvec(B)\| =\|AB sin theta\|` `(AB)/(2)=AB sin theta` `rArr sin theta=(1)/(2)` ` rArr theta=60^(@)`

Discussion

32.	Two vectors `vec(A)` and `vec(B)` are such that `\|vec(A)+vec(B)\|=\|vec(A)-vec(B)\|` then what is the angle between `vec(A)`and `vec(B)` :-A. `0^(@)`B. `90^(@)`C. `60^(@)`D. `180^(@)`
Answer» Correct Answer - B `10(cos 30^(@)hat(i)+sin 30^(@)hat(j))=(5sqrt(3)hat(i)+ 5hat(j))`

Discussion

33.	If `vec(a)` be a unit vector, then :-A. direction of `vec(a)` is constantB. magnitude of `vec(a)` is constantC. both (A) and (B)D. any one of direction or magnitude is constant .
Answer» Correct Answer - C

Discussion

34.	The vector -`vec(A) is : `A. greater than `vec(A) ` in magnitudeB. less than `vec(A) ` in magnitudeC. in the same direction as `vec(A) `D. in the direction opposite to `vec(A) `
Answer» Correct Answer - D

Discussion

35.	The resultant of two vectors of magnitude 3 units 4 units is 1 unit. What is the value of their dot product. ?A. `-12` unitsB. `-7` unitsC. `-1` unitsD. zero
Answer» Correct Answer - A we have `1^(2)=3^(2)+4^(2)+2xx3xx4 cos theta` `rArr " " cos theta =(-24)/(24)=-1` `rArr " " theta =pi` ` vec(A).vec(B)=3xx4 cos theta =-12 units`

Discussion

36.	Two vectors `vec(P) & vec(Q)` are arranged in such a way that they form adjacent sides of a parallelogram as shown in figure Which of the following options have correct relationshipA. `vec(Q)=vec(R)+vec(S)`B. `vec(R)=vec(P)+vec(Q)`C. `vec(R)=vec(P)+vec(S)`D. `vec(S)=vec(Q)+vec(P)`
Answer» Correct Answer - B::D

Discussion

37.	Two vectors `vec(A)` & `vec(B)` are given such that angle between `(vec(A)+vec(B))` and `(vec(A)-vec(B))` is `90^(@)` then find the value of `(\|vec(A)\|)/(2\|vec(A)\|+\|vec(B)\|)` ?
Answer» Correct Answer - `1//3` `(vec(A)+vec(B)).(vec(A)-vec(B))=0` `\|vec(A)\|=\|vec(B)\|` `(\|vec(A)\|)/(2\|vec(A)\|+\|vec(B)\|)=(1)/(3)`

Discussion

38.	The magnitude of the vector `hat(i)+xhat(j)+3hat(k)` is half of the magnitude of vector `4hat(i)+(4x-2)hat(j)+2hat(k)` . The values of x are ?
Answer» Correct Answer - `-2//3, 2` `(1)/(2)sqrt(1^(2)+x^(2)+3^(2))=sqrt(4^(2)(4x-)^(2)+2^(2))` `x=-(2)/(3),2`

Discussion

39.	A wave is represented by - `y=a sin (At-Bx+C)` where A, B, C are constants. The Dimensions of A, B, C areA. `T^(1),L,M^(0)L^(0)T^(0)`B. `T^(-1),L^(-1),M^(0)L^(0)T^(0)`C. T,L,MD. `T^(-1),L^(-1), M^(-1)`
Answer» Correct Answer - B `[A]=T^(-1), [B]=L^(-1),[C]=M^(0)L^(0)T^(0)`

Discussion

40.	Which of the following physical quantities do not have the same dimensionsA. Pressure, Youngs modulus, stressB. Electromotive froce, voltage, potentialC. Heat, Work, EnergyD. Electric dipole, electic field, flux
Answer» Correct Answer - D Flux`" " phi=vec(E)xxvec(A)` so `" "[phi]!=[E]` & `" " vec(p)=qvec(d)` so `" " [p]!=[E]`

Discussion

41.	Derive an expression for time period (t) of a simple penduleum, which may depend upon : mass of bob (m), length of pendulum (I) and acceleration due to gravity(g).
Answer» Let (i) `t prop m^(a) " "`(ii) `t prop l^(b) " "` (iii) `t prop g^(c)` Combining all the three factors, we get `t prop m ^(a)l^(b) g^(c) " "` or ` " " t=Km^(a)l^(b) g^(c) ` where K is a dimensionless constant of proportionality . Writing down the dimensions on either side of equation (i) , we get `[T] = [M^(a)] [L^(b)] [LT^(-2)]^(c) = [M^(a) L^(b+c)T^(-2c)]` Comparing dimensions, `a=0 ,b+c=0 ,-2c=1 ` `:. a=0, c=-1//2, b=1//2 ` From equation (i) `t=Km^(0)l^(1//2)g^(-1//2)) " " ` or ` " "t=K((l)/(g))^(1//2) =K sqrt((l)/(g))`

Discussion

42.	The frequency of vibration of string depends on the length `L` between the nodes, the tension `F` in the string and its mass per unit length `m`. Guess the expression for its frequency from dimensional analysis.
Answer» Correct Answer - `(k)/(L)sqrt((F)/(m))` Frequency `prop L^(a)F^(b) (m)^(c)=L^(a)F^(b)((M)/(L))^(c)` `(1)/(T)=kL^(a)(MLT^(-2))^(b)((M)/(L))^(c) " " [krarr"Proporsionality constant"]` `T^(-1)=kL^(a+b-c)M^(b+c)T^(-2c)` On comparing `c=-(1)/(2), b=+(1)/(2),a=1` frequency `(k)/(L)sqrt((F)/(m))`

Discussion

43.	Find the dimensions of (a) the specific heat capacity c, (b) the coefficient of linear expansion `alpha` and (c) the gas constant R. Some of the equations these quantities are `Q=mc(T_(2)-T_(1)) l_(t)=l_(0)[1+alpha (T_(2)-T_(1))]` and PV=nRT. (Where Q=heat enegry , m=mass, `T_(1)&T_(2)`= temperatures, `l_(1)` =length at temperature `t^(@)C`, `l_(0)=` length at temperature `0^(@) C`, P=Pressure, v = volume, n = mole)
Answer» Correct Answer - (a) `L^(2)T^(-2)K^(-1)` (b) `K^(-1)` (c)`ML^(2)T^(-2)K^(-1)(mol)^(-1)` (a)`[c]=([Q])/([m][T_(2)-T_(1)])` `[c] =L^(2)T^(-2)k^(-1)` `[l_(t)]=[l_(0)]=[l_(0)alpha(T_(2)-T_(1))]` `[alpha]=k^(-1)` (c) `[R]=([pv])/([nT])=(ML^(-1)T^(-2)L^(3))/(mol K)=ML^(@)T^(-2)k^(-1)k^(-1) mol^(-1)]`

Discussion

44.	The resistance force arising due to pressure difference at the front and rear sides of a body in a fluid stream depends upon the density of the fluid, the velocity of flow and the maximum area of cross-section perpendicular to the flow. Show that the force varies as the square of the velocity.
Answer» Correct Answer - `Fmuv^(2)` `Fprop (rho)^(a) (v)^(b)(A)^(c)` `k(rho)^(a)(v)^(b)(A)^(c) " " ]k rarr "Proposnality constant"]` `rArr MLT^(-2)=k(ML^(-3))^(a)(LT^(-1))^(b)(L^(2))^(c)` `rArr a=1,-3a-b+2c=0,rArr-b+2c=3` & `-b=-2 rArrb=2` `rArr F=(rho)(V)^(2)(A)^(1)` `rArrF prop V^(2)`

Discussion

45.	Evaluate indefinite integration. (i) `x=intdt " "` (ii) `int t dt " "` (iii) `x=int (2t) dt" "` (iv) `x=int (t^(2)) dt " "` (v) `x=int (-(2)/(t^3))dt`
Answer» (i) `x=t+c " "` (ii) `x=(t^(2))/(2)+c" "` (iii) `x=(2t^(2))/(2)+c=t^(2)+c` (iv) `x=t^(3)/(3)+c" "` (v) `x=-2intt^(-3)dtrArrx=-2((t^(-2))/(-2))+c=t^(-2)+c`

Discussion

46.	Find the value of definite integral:`int_(0)^(pi) ((pit)/(2)-(t^(2))/(2))dt`
Answer» `underset(0) overset(pi)int ((pit)/(2)-(t^(2))/(2))dt` `=((pi)/(2)((t^(2))/(2))-(t^(3))/(6))_(0)^(pi)` `=(pi^(3))/(4)-(pi^(3))/(6)=(pi^(3))/(12)`

Discussion

47.	The distance covered by a particle in time t is given by `x=a+bt+ct^2+dt^3`, find the dimensions of a,b,c and d.A. `L, T^(-3)`B. `L,LT^(-3)`C. `L^(0), T^(3)`D. none of these
Answer» Correct Answer - B `[a]=[x]=L` `[x]=[dt^(3)]` `[d]=LT^(-3)`

Discussion

48.	Find `(dx)/(dt)` (derivative) of w.r.t. t). (i) `x=(t^(2)+1)^(3)` (ii) `x=sqrt(t)^(3)-3` (iii) `x=sin2t` (iv) `x=cos(2t+4)` (v)`x=sin^(3)t` (vi)`x=cos^(3)t`
Answer» Correct Answer - (i) ` 6t(t^(2)+1)^(2)` (ii)`(3t^(2))/(2)(t^(3)-3)^(-1//2)` (iii) `2cos 2t` (iv) `-2sin(2t+4)`(v)`3sin^(2)tcos t` (vi) `-3cos^(2) t sin t ` (i) `(dx)/(dt)=3(t^(2)+1)^(2)(2t)=6t(t^(2)+1)^(2)` (ii)`(dx)/(dt)=(1)/(2)(t^(3)-3)^(-1/2)(3t^(2))=(3t^(2))/(2)(t^(3)-3)^(-1/2)` (iii) `(dx)/(dt)=2 cos 2t` (iv)`(dx)/(dt)=-sin(2t+4).2 =-2sin (2t+4)` (v) `(dx)/(dt)=3sin ^(2)(cos t)=3 sin^(2) .cost` vi) `(dx)/(dy)=3 cos^(2) t(-sin t)=-3 cos^(2) t sin t`

Discussion

49.	The velocity of particle is given by `v=sqrt(gx)` . Find its acceleration.
Answer» `(dv)/(dt)=(1)/(2)(gx)^(-1//2) (g.(dx)/(dt))` `=(1)/(2)gx^(-1//2)gv` `=(1)/(2)g`

Discussion

50.	Test if following equation is equation is dimensionally correct `v=(1)/(2pi)sqrt((mgl)/(1))` where, v = frequency, I =moment of inertia, m= mass, l= lengh, g= acc. Due to gravity.
Answer» Correct Answer - Equation is dimensionally correct `[v]=T^(-1)` `[(1)/(2pi)(sqrt((mgl)/(I))]=[sqrt((mgl)/(I))]=sqrt((mLT^(-2)L)/(ML^(2)))T^(-1)` So. This equation is dimensionally correct.

Discussion

Explore topic-wise InterviewSolutions in .

`vec(r)=2that(i)+3tvec(2)hat(j)`. Find `vec(v)` and `vec(a)` where `vec(v)=(dvec(r))/(dr) , vec(a)=(dvec(v))/(dt)`

`0.4hat(i)+0.8hat(j)+chat(k)` represents a unit vector, when c is :-A. 0.2B. `sqrt(0.2)`C. `sqrt(0.8)`D. 0

A force of `(-3hat(i)-hat(j)+2hat(k))` N displaced the body from a point (4, -3, -5) m to a point (-1, 4, 3) m in a staight line. Find the work done by the force.

If `hat(i)+2hat(j)+nhat(k)` is perpendicular to `4hat(i)+2hat(j)+2hat(k)` then find the value of n ?

Calculate the dimensional formula of energy from the equation `E= 1/2 mv^2`.

If force, time and velocity are treated as fundamental quantities then dimensional formula of energy will beA. [FTV]B. `FT^(2)V]`C. `FTV^(2)`D. `[FT^(2)V^(2)]`

If `x=k sin (k l t)`, where x is displacement and t is time then dimensional formula for l will be ` (k, l="constant")`A. `[M^(0)L^(0)T^(0)]`B. `[M^(0)L^(1)T^(0)]`C. `[M^(0)L^(-1)T^(-1)]`D. `[ML^(-1)T^(-1)]`

The position vectors of two balls are given by `vec(r)_(1)=2hat(i)+7hat(j) , vec(r)_(2)=-2hat(i)+4hat(j)` What will be the distance between the two balls?

A body constrained to move in y direction is subjected to a force given by `vec(F)=(-2hat(i)+15hat(j)+6hat(k))` N . What is the work done by this force in moving the body through a distance of `10 m` along y-axis ?

What displacement must be added to the displacement `25hat(i)-6hat(j)`m to give a displacement of 7.0 m pointing in the x-direction ?A. `18hat(i)-6hat(j)`B. `32hat(i)-13hat(j)`C. `-18hat(i)+6hat(j)`D. `-25hat(i)+13hat(j)`

If velocity of a particle is given by `v=3t^(2)-6t +4`. Find its displacement from `t=0` to `3 secs`.

Suppose displacement `y(t)` (that is, y as a function of t ) is given by `y(t)=t^(3)`

Suppose `A=B^(n)C^(m)` , where A has dimensions LT, B has dimensions `L^(2)T^(-1)`, and C has dimensions `LT^(2)` , Then the exponents n and m have the values: (A) `2//3,1//3 " "` (B) `2,3 " "` (C)`4//5,-1//5 " "` (D) `1//5,3//5` `(E)`1//2,1//2`

A vector perpendicular to `(hat(i)+hat(j)+hat(k))` and `(hat(i)-hat(j)-hat(k))` is :-A. `hat(i)+hat(j)+hat(k)`B. `hat(i)+hat(k)`C. `-1hat(i)+hat(j)+hat(k)`D. `hat(j)+hat(k)-2hat(i)`

The diameter of the sun is expressed as `13.9xx 10^(9) m`. Find the order of magnitude of the diameter ?

The value of a unit vector in the direction of vector `vec(A)=5hat(i)-hat(j) ` is ________________ ?

Which of the following experssions has magnitude unity.A. `(vec(a)+vec(b))/(|vec(a)+vec(b)|)`B. `|hat(a)+hat(b)|` when angle between `vec(a)`and `vec(b)` is `120^(@)`C. `(vec(a)-vec(b))/(|vec(a)-vec(b)|)`D. `|hat(a)-hat(b)|` when angle between `vec(a)` and `vec(b)` is `60^(@)`

Find minimum value `y=25x^(2)-10x+5`.

Mark correct statement(s) for `vec(a) , vec(b)` and `vec(c)` shown in above diagram : A. `-vec(a)-vec(b)=vec(c)`B. `vec(a)+vec(c)=vec(-b)`C. `vec(a)+vec(c)=vec(b)`D. `vec(b)-vec(c)=vec(-a)`

Express the given vector `vec(A)` (shown graphically ) in unit vector notation.

Express the vector in unit - vector notation.

Three forces `(vec(F)_(1), vec(F)_(2), vec(F)_(3))` are acting an a particle moving vertically up with constant speed. Two force `vec(F)_(1)=-10hat(j)N`, and `vec(F)_(1)=-6hat(i)=8hat(j)` , N are acting an particle on acting on particle respectively find `vec(F)_(3)`.

Figure shows three vectors a, b and c. If `RQ=2PR`, which of the following relations is co-rest ? A. `2a+c=3b`B. `a+3c=2b`C. `3a+c=2b`D. `a+2c=3b`

From the v versus t graph of figure (a) the time (s) at which the particle is at rest (b) at what time, if any does the particle reverse the direction of its motion? (c) The distance and displacement of the particle from `t=0`s to `t=6` s.

It `x=sin^(2)theta`, then find `(dx)/(dt)` where `(d theta)/dt)=omega`

The radius of a circle is increasing at a rate `((dr)/(dt))=alpha` . Find the rate at which its area is increasing when radius is equal to 3 m.

A bird moves from point `(1, -2)` to `(4, 2)` . If the speed of the bird is `10m//sec`, then find the velocity vector of the bird ?

Two forces P and Q are in ratio `P:Q=1:2`. If their resultant is at an angle `tan^(-1)((sqrt3)/(2))` to vector P, then angle between P and Q is :A. `tan^(-1)((1)/(2)) `B. `45^(@) `C. `30^(@)`D. `60^(@)`

If `E , M , J , and G` , respectively , denote energy , mass , angular momentum , and gravitational constant , then `EJ^(2) //M^(5) G^(2)` has the dimensions ofA. lengthB. angleC. massD. time

If two numerical equal forces P and P acting at a point produce a resultant force of magnitude P itself, then the angle between the two original forces is :-A. `0^(@)`B. `60^(@)`C. `90^(@)`D. `120^(@)`

If `vec(A)` and `vec(B)` denote the sides of a parallelogram and its area is AB/2 , the angle between `vec(A)` and `vec(B)` is :-A. `(pi)/(2)`B. `pi`C. `(pi)/(6)`D. `(pi)/(3)`

Two vectors `vec(A)` and `vec(B)` are such that `|vec(A)+vec(B)|=|vec(A)-vec(B)|` then what is the angle between `vec(A)`and `vec(B)` :-A. `0^(@)`B. `90^(@)`C. `60^(@)`D. `180^(@)`

If `vec(a)` be a unit vector, then :-A. direction of `vec(a)` is constantB. magnitude of `vec(a)` is constantC. both (A) and (B)D. any one of direction or magnitude is constant .

The vector -`vec(A) is : `A. greater than `vec(A) ` in magnitudeB. less than `vec(A) ` in magnitudeC. in the same direction as `vec(A) `D. in the direction opposite to `vec(A) `

The resultant of two vectors of magnitude 3 units 4 units is 1 unit. What is the value of their dot product. ?A. `-12` unitsB. `-7` unitsC. `-1` unitsD. zero

Two vectors `vec(P) & vec(Q)` are arranged in such a way that they form adjacent sides of a parallelogram as shown in figure Which of the following options have correct relationshipA. `vec(Q)=vec(R)+vec(S)`B. `vec(R)=vec(P)+vec(Q)`C. `vec(R)=vec(P)+vec(S)`D. `vec(S)=vec(Q)+vec(P)`

Two vectors `vec(A)` & `vec(B)` are given such that angle between `(vec(A)+vec(B))` and `(vec(A)-vec(B))` is `90^(@)` then find the value of `(|vec(A)|)/(2|vec(A)|+|vec(B)|)` ?

The magnitude of the vector `hat(i)+xhat(j)+3hat(k)` is half of the magnitude of vector `4hat(i)+(4x-2)hat(j)+2hat(k)` . The values of x are ?

A wave is represented by - `y=a sin (At-Bx+C)` where A, B, C are constants. The Dimensions of A, B, C areA. `T^(1),L,M^(0)L^(0)T^(0)`B. `T^(-1),L^(-1),M^(0)L^(0)T^(0)`C. T,L,MD. `T^(-1),L^(-1), M^(-1)`

Which of the following physical quantities do not have the same dimensionsA. Pressure, Youngs modulus, stressB. Electromotive froce, voltage, potentialC. Heat, Work, EnergyD. Electric dipole, electic field, flux

Derive an expression for time period (t) of a simple penduleum, which may depend upon : mass of bob (m), length of pendulum (I) and acceleration due to gravity(g).

The frequency of vibration of string depends on the length `L` between the nodes, the tension `F` in the string and its mass per unit length `m`. Guess the expression for its frequency from dimensional analysis.

The resistance force arising due to pressure difference at the front and rear sides of a body in a fluid stream depends upon the density of the fluid, the velocity of flow and the maximum area of cross-section perpendicular to the flow. Show that the force varies as the square of the velocity.

Evaluate indefinite integration. (i) `x=intdt " "` (ii) `int t dt " "` (iii) `x=int (2t) dt" "` (iv) `x=int (t^(2)) dt " "` (v) `x=int (-(2)/(t^3))dt`

Find the value of definite integral:`int_(0)^(pi) ((pit)/(2)-(t^(2))/(2))dt`

The distance covered by a particle in time t is given by `x=a+bt+ct^2+dt^3`, find the dimensions of a,b,c and d.A. `L, T^(-3)`B. `L,LT^(-3)`C. `L^(0), T^(3)`D. none of these

Find `(dx)/(dt)` (derivative) of w.r.t. t). (i) `x=(t^(2)+1)^(3)` (ii) `x=sqrt(t)^(3)-3` (iii) `x=sin2t` (iv) `x=cos(2t+4)` (v)`x=sin^(3)t` (vi)`x=cos^(3)t`

The velocity of particle is given by `v=sqrt(gx)` . Find its acceleration.

Test if following equation is equation is dimensionally correct `v=(1)/(2pi)sqrt((mgl)/(1))` where, v = frequency, I =moment of inertia, m= mass, l= lengh, g= acc. Due to gravity.