InterviewSolution
Saved Bookmarks
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. |
The integrating factor of the differentialequation `(dy)/(dx)(x(log)_e x)+y=2(log)_e x`is given by(a)`( b ) x (c)`(d)(b) `( e ) (f) (g) e^(( h ) x (i))( j ) (k)`(l)(c) `( m ) (n) (o)(( p )log)_( q ) e (r) (s) x (t)`(u)(d) `( v ) (w) (x)(( y )log)_( z ) e (aa) (bb)(( c c )(( d d )log)_( e e ) e (ff) (gg) x)( h h )`(ii)A. `e^(x)`B. log xC. log(log x)D. x |
|
Answer» Correct Answer - b ` (dy)/(dx) (xlog x) +y = 2 log x` ` rArr (dy)/(dx) +y/(x log x ) = 2/x` Here , ` P = 1/(x log x ),Q = 2/x` ` :. IF = e^(intPdx) = e^(int(dx)/(x logx) )= log x ` |
|
| 202. |
The integrating factor of the differential equation ` (y log y) dx = (log y - x) dy ` isA. `1/(logy)`B. log(logy)C. `1+logy`D. logy |
|
Answer» Correct Answer - d Given differential eqution can be rewritten as ` (dx)/(dy) = ((log y -x))/((y log y)) rArr (dx)/(dy) + x/(y log y) = 1/y` ` :. IF = e^(int 1/(y log y)dy)=e^(log log y) = log y` |
|
| 203. |
solution of differential equation `xcosx(dy)/(dx)+y(xsinx+cosx)=1` isA. `xy=sin x+C cos x`B. `xy+cosx+C sin x = 0`C. ` xy +sec x+C sin x=0`D. None of the above (where , C is arbitrary constant) |
|
Answer» Correct Answer - a Given , `x cos x ((dy)/(dx)) + y ( x sin x + cos x ) = 1` We can write , ` (dy)/(dx) +y (tanx.+1/x)=(1/(xcosx))` which is linear differential equation ltbRgt Here, ` P = tan x + 1/x , Q = 1/(x cos x)` ` :. IF = e^((tan x + 1/x)dx) = e ^(log sec x + log x ) = e^(log ( x sec x))= x sec x ` Now, required solution is `y (IF) = int IF * Q dx +C` ` rArr y ( x sec x) = int x sec x * 1/(x cos x) dx +C` ` rArr xy sec x = int sec^(2) x dx+C` ` rArr xy sec x = tan x +C` On multiplying both sides by cos x , we get `xy = sin x + C cos x` |
|
| 204. |
Solution of the differential equation `x(dy)/(dx)-y+xsin((y)/(x))=0` isA. `x(1-cos((y)/(x)))=csin((y)/(x))`B. `x(1+cos((y)/(x)))=csin((y)/(x))`C. `x(1-sin((y)/(x)))=c cos((y)/(x))`D. `x(1+sin((y)/(x)))=c cos((y)/(x))` |
| Answer» Correct Answer - A | |
| 205. |
Solution of the differential equation `(x(dy)/(dx)-y)sin((y)/(x))=x^(2)cosx` isA. `sinx-cos((y)/(x))=c`B. `sinx+cos((y)/(x))=c`C. `cosx+sin((y)/(x))=c`D. `cosx-sin((y)/(x))=c` |
| Answer» Correct Answer - B | |
| 206. |
Bacteria increases at the rate proportional to the number of bacteria present. If the original number N doubles in 3 hours, find in how many hours the number of bacteria will by 4N ?A. 6 hoursB. 4 hoursC. 5 hoursD. 5.5 hours |
| Answer» Correct Answer - A | |
| 207. |
An integrating factor of the differential equation ` x(dy)/(dx) +y log x = xe^(x)x^((-1)/2log x), (x gt 0 ), ` isA. `x^(logx)`B. ` (sqrt(x))^(logx)`C. ` (sqrt(e))^((logx)^(2))`D. ` e^(x^(2))` |
|
Answer» Correct Answer - c Given differential equation is ` x(dy)/(dx) + y log x = xe^(x) x ^((-1//2)logx)` ` (dy)/(dx) + y. 1/x log x = e^(x) x ^((-1//2))logx` Here , ` P = 1/x log x and Q = e^(x)x^((-1//2)logx)` ` :. IF = e ^(int 1/xlogxdx) = e^((logx)^(2)/2)=(sqrt(e))^((logx)^(2))` |
|
| 208. |
Solution of the differential equation `(y+x(dy)/(dx))sin(xy)=cosx` isA. sinx+cos(xy)=cB. sinx-cos(xy)=cC. cosx+sin(xy)=cD. cosx-sin(xy)=c |
| Answer» Correct Answer - A | |
| 209. |
Solution of the differential equation `tan y.sec^(2) x dx + tan x. sec^(2)y dy = 0` isA. tan x tan y = cB. tan x =c tan yC. `sec^(2)xsec^(2)y=c`D. tan y = c tan x |
| Answer» Correct Answer - A | |
| 210. |
Solution of the differential equation `x+y(dy)/(dx)=sec(x^(2)+y^(2))` isA. `sin(x^(2)+y^(2))=x+c`B. `cos(x^(2)+y^(2))=x+c`C. `cos(x^(2)+y^(2))=2x+c`D. `sin(x^(2)+y^(2))=2x+c` |
| Answer» Correct Answer - D | |
| 211. |
The integrating factor of linear differential equation` (dy)/(dx) + y sec x = tan x ` isA. sec x-tan xB. sec x tan xC. sec x +tan xD. sec x cot x |
| Answer» Correct Answer - C | |
| 212. |
The solution of the differential equation `(dy)/(dx)-y tan x=e^(x)sec x` isA. `y= e^(x) cos x +C`B. `y cos x =e^(x) +C`C. ` y = e^(x) sin x +C`D. `y sin x = e^(x) +C` |
|
Answer» Correct Answer - b Given , ` (dy)/(dx) - y tan x = e^(x) sec x ` ` :. IF = e^(- int tan x dx ) = e^(-logsec x)=1/(secx)` ` :. ` General solution is ` rArr y * 1/(sec x) = int e^(x) sec x * 1/(sec x) dx` ` rArr y cos x = e^(x) + C` |
|
| 213. |
Solution of the differential equation `sec^(2)(x-2y)(1-2(dy)/(dx))=1` isA. tan(x-2y)+x=cB. tan(x-2y)-x=cC. 2tan(x-2y)+x=cD. 2tan(x-2y)-x=c |
| Answer» Correct Answer - B | |
| 214. |
Show that the differential equation: `(xcosy/x)(ydx+xdy)=(ysiny/x)(xdy-ydx)` is homogenous and solve it.A. `cos((y)/(x))=cxy`B. `xycos((y)/(x))=c`C. `xcos((y)/(x))=cy`D. `ycos((y)/(x))=cx` |
| Answer» Correct Answer - B | |
| 215. |
Solution of the differential equation `(xdy)/(x^(2)+y^(2))=((y)/(x^(2)+y^(2))-1)dx`, isA. `tan^(-1)((y)/(x))-x=c`B. `tan^(-1)((y)/(x))+x=c`C. `tan^(-1)((y)/(x))-log|x|=c`D. `tan^(-1)((y)/(x))+log|x|=c` |
| Answer» Correct Answer - B | |
| 216. |
The differential equation of the family of lines where length of the normal form origin is p and the inclination of the normal is `alpha`, isA. `(d^(2)y)/(dx^(2))=cotalpha`B. `(d^(2)y)/(dx^(2))=0`C. `(dy)/(dx)=cotalpha`D. `(dy)/(dx)=-cotalpha` |
| Answer» Correct Answer - B | |
| 217. |
The solution of differential equation `(xy^(5)+2y)dx-xdy =0,` isA. `9 x^(8) + 4x^(9) y^(4) = 9y^(4)C`B. ` 9x^(8) - 4x^(9) y^(4) - 9y^(4) C = 0 `C. ` x^(8) (9+4y^(4)) = 10y^(4)C`D. None of these |
|
Answer» Correct Answer - a `(xy^(5)+2y)dx = xdy` ` rArr x (dy)/(dx) - 2y = xy^(5)` ` rArr (dy)/(dx) -(2y)/x = y^(5)` ` rArr y^(-5) (dy)/(dx) - (2y^(-4))/x = 1" "` …(i) Put , `y^(-4) =t` ` rArr -4y^(-5) (dy)/(dx) = (-1)/4 (dt)/(dx)" "` …(ii) From Eqs. (i) and (ii) , ` -1/4 (dt)/(dx) - (2t)/x = 1 rArr (dt)/(dx) + (8t)/x =-4` ` IF = e^(int 8/xdx) = e^(8logx) = x^(8)` ` :. t* x^(8) int (-4)x^(8) dx +C` ` rArr (x^(8))/(y^(4))= -(4*x^(9))/9 +C` ` rArr 9x^(8) + 4x^(9)* y^(4) = 9y^(4)C` |
|
| 218. |
The solution of ` (y - (xdy)/dx)= 3 (1-x^(2)(dy)/(dx))` isA. `(y+3)(1-3x)=Cx`B. `(y-3)=3xy+Cx`C. `(y+3)(1+3x)=Cx`D. `y+3=3xy-Cx` |
|
Answer» Correct Answer - b ` (ydx - xdy)/(dx) = 3(dx - x^(2)dy)/(dx)` ` rArr (ydx - xdy )/(x^(2)) = 3 ((dx)/(x^(2)) - dy)` ` rArr int - d (y/x) = int 3 ((dx)/(x^(2))-dy)` ` rArr -y/3 = -3/x -3y -C` ` y = 3 +3xy + Cx` ` rArr y - 3 = 3xy Cx` |
|
| 219. |
solution of the differential equation `xdy-ydx=sqrt(x^2+y^2 )dx` isA. ` y + sqrt(x^(2)+y^(2))=Cx`B. ` y + sqrt(x^(2)+y^(2))=Cx^(2)`C. ` y + sqrt(x^(2)+y^(2))=C`D. None of these |
|
Answer» Correct Answer - b Given equation can be written as ` x dy = ( sqrt(x^(2)+y^(2))+y)dx,` I.e ` (dy)/(dx) = (sqrt(x^(2)+y^(2))+y)/x + y" "` …(i) This is a homogenous differential equation . To simplify it , Put ` y - vx rArr (dy)/(dx) = v+x (dv)/(dx)` ` v+ x (dv)/(dx) = (sqrt(x^(2)+v^(2)x^(2))+vx)/x` i.e `v + x (dv)/(dx) = sqrt(1+v^(2))+v` ` x (dv)/(dx) = sqrt(1+v^(2))` ` rArr (dv)/(sqrt(1+v^(2)))=(dx)/x" " ` ...(ii) On integrating both sides of Eq. (ii) ,we get ` log ( v + sqrt(1+v^(2))) = log x + log C` ` rArr v + sqrt(1+v^(2))= Cx` ` rArr y/x + sqrt(1+(y^(2))/(x^(2)))= Cx` `rArr y + sqrt(x^(2) + y^(2)) = Cx^(2)` |
|
| 220. |
Solution of the differential equation `tan y.sec^(2) x dx + tan x. sec^(2)y dy = 0` isA. `tan y tan x = C`B. `(tan y)/(tan x) =C`C. `(tan^(2)x)/(tan y) = C`D. None of these |
|
Answer» Correct Answer - a Given , ` (sec^(2)x)/(tanx) dx = -(sec^(2)y)/(tany) dy` On integrating , we get ` rArr int (sec^(2)x)/(tanx) dx =- int (sec^(2)y)/(tany) dy " "` …(i) Put tan x = u ` rArr sec^(2) x dx = du and tan y = v ` `rArr sec^(2) y dy = dv` From Eq. (i) ` int (du)/u = - int (du)/v` ` rArr log u = - log v + log C rArr uv = C` ` :. tan x * tany = C` |
|
| 221. |
The solution of `(dy)/(dx)+1 = e^(x+y) ` isA. ` e^(-(x+y))+x+C=0`B. ` e^((-x+y)) - x + C = 0 `C. ` e^(x+y) + x+C = 0`D. ` e^(x+y)- x + C = 0 ` |
|
Answer» Correct Answer - a Given, ` (dy)/(dx) +1 = e^(x+y)` `rArr 1 + (dy)/(dx) = (dz)/(dx)` ` :. (dz)/(dx) = e^(z)` On integrating , we get ` rArr int e^(-z) dz = int dx` `rArr - e^(-z) = x+C` ` rArr x + e^(-(x+y)) + C = 0` |
|
| 222. |
If `xdy = y(dx + ydy); y(1) = 1 and y(x) > 0`, then what is `y(-3)` equal to?A. 3B. 2C. 1D. 0 |
|
Answer» Correct Answer - a Given , `xdy = y (dx + ydy), y gt 0` ` rArr xdy - ydx = y^(2)dy ` ` rArr (xdy-ydx)/(y^(2)) = dy rArr d(x/y) = -dy` On integrating both sides , we get ` x/y = -y +C " "` …(i) Since , ` y (1) =1 rArr x = 1 , y =1` ` :. C = 2` Now , Eq, (i) becomes Again for x=-3 rArr ` -3 +y^(2) = 2Y` ` rArr y^(2) - 2y - 3 = 0` ` rArr (y+1)(y-3)=0` As `y gt 0` take y = 3 , neglecting y = -1 |
|
| 223. |
Let F be the family of ellipse whose centre is the origin and major axis is the y-axis. Then the differential equation of family F isA. ` (d^(2)y)/(dx^(2))+(dy)/(dx)(x(dy)/(dx)-y)=0`B. ` xy(d^(2)y)/(dx^(2)) -(dy)/(dx) (x(dy)/(dx)-y)=0`C. ` xy(d^(2)y)/(dx^(2))+(dy)/(dx)(x(dy)/(dx)-y)=0`D. ` (d^(2)y)/(dx^(2))-(dy)/(dx)(x (dy)/(dx)-y)=0` |
|
Answer» Correct Answer - c Equation of family of ellipse is ` (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 ` On differentiating , we get ` rArr (2x)/(a^(2))+(2y)/(b^(2)) * (dy)/(dx) = 0` ` rArr x/(a^(2)) + y/(b^(2))* (dy)/(dx) = 0 ` Again , on differentiating , we get ` 1/(a^(2))+y/(b^(2))*(d^(2)y)/(dx^(2))+ ((dy)/(dx))^(2)1/(b^(2))=0` ` rArr (b^(2))/(a^(2)) + y ((d^(2)y)/dx^(2)) + ((dy)/dx)^(2) = 0 ` ` rArr -y/x * (dy)/(dx) + y (d^(2)y)/(dx^(2)) + ((dy)/dx)^(2) = 0 ` ` [ "from Eq. "(i) (b^(2))/(a^(2))= - y/x *(dy)/(dx)]` ` rArr xy (d^(2)y)/(dx^(2)) + (dy)/(dx) (x (dy)/(dx)-y )=0` |
|
| 224. |
The equation of the ellips is `(x^(2))/(36)+(y^(2))/(16)=k.` The differential equation of the ellips whose length of major and minor axes half the lenghts of the given ellipse respectively, isA. `2x+3y(dy)/(dx)=0`B. `2x-3y(dy)/(dx)=0`C. `4x+9y(dy)/(dx)=0`D. `4x-9y(dy)/(dx)=0` |
| Answer» Correct Answer - C | |
| 225. |
The solution of the differential equation `(dy)/(dx)=secx-ytanx` is :A. `e^(tanx)`B. `e^(cosx)`C. sec x +tan xD. cosec x |
| Answer» Correct Answer - C | |
| 226. |
The differential equation `(dy)/(dx) = (x(1+y^(2)))/(y(1+x^(2))` represents a family ofA. parabolaB. hyperbolaC. circleD. ellipse |
|
Answer» Correct Answer - b Given , `(dy)/(dx) = ((1+y^(2))x)/(y(1+x^(2)))` ` rArr int (2y)/(1+y^(2)) dy = int (2x)/(1+x^(2)) dx` ` rArr log (1+y^(2))= log (1+x^(2)) +log k` ` rArr (1+y^(2)) = (1+x^(2)k` This equation represents family of hyperbola . |
|
| 227. |
The solution of the differential equation `(dy)/(dx)=(a x+g)/(b y+f)`represents a circle when`a=b`b. `a=-b`c. `a=-2b`d. `a=2b`A. ` e^(x) - e^(-siny) + (x^(3))/3 = C`B. ` e^(-x) e^(-siny) + (x^(3))/3 = C`C. ` e^(x) +e^(-siny) + (x^(3))/3 = C`D. ` e^(x) - e^(siny) - (x^(3))/3 = C` |
|
Answer» Correct Answer - c Given , ` cos y (dy)/(dx) = e^(x+siny) +x^(2)e^(siny)` ` rArr cos y (dy)/(dx) = e^(siny) (e^(x)+x^(2))` ` rArr int (cos y)/(e^(siny)) dy = int (e^(x)+x^(2))dx` On integrating both sides , we get ` int (dt)/(e^(t))dx = int (e^(x)+x^(2))dx` ` rArr -e^(-t) = e^(x) +(x^(3))/3 -C` ` rArr e^(x) +e^(-siny) + (x^(3))/3 =C` |
|
| 228. |
The solution of the differential equation `(dy)/(dx)=(a x+g)/(b y+f)`represents a circle when`a=b`b. `a=-b`c. `a=-2b`d. `a=2b`A. a=bB. `a=-b`C. `a=-2b`D. `a=2b` |
|
Answer» Correct Answer - b We have , ` (dy)/(dx) = (ax +g)/(by+f)` ` rArr (by + f) dy = (ax + g ) dx` Onintegrating ,we get ` (by^(2))/2 +fy = (ax^(2))/2 + gx + C` `rArr ax^(2) - by^(2) + 2gx - 2fy +C = 0` This represents a circle , if a = -b |
|
| 229. |
The differential equation for all ellips whose major axis is twice its minor axis isA. `2y(dy)/(dx)-x=0`B. `2y(dy)/(dx)+x=0`C. `4y(dy)/(dx)-x=0`D. `4y(dy)/(dx)+x=0` |
| Answer» Correct Answer - D | |
| 230. |
The differential equation `y(dy)/(dx) + x = c` representsA. hyperbolasB. parabolasC. ellipsesD. circles |
|
Answer» Correct Answer - d Given that, `y(dy)/(dx) +x=C` ` rArr y dy = (C -x)dx` On intgrating ,we get ` rArr int" y " dy = int (C-x) dx` ` (y^(2))/2 = Cx - (x^(2))/2 + d` ` rArr x^(2) +y^(2) - 2Cx - 2d =0` which represents a family of circle . |
|
| 231. |
The differential equation for `y=c_(1)sinx+c_(2)cosx` isA. `(d^(2)y)/(dx^(2))-y=0`B. `(d^(2)y)/(dx^(2))+y=0`C. `(d^(2)y)/(dx^(2))=0`D. `(d^(2)x)/(dy^(2))=0` |
| Answer» Correct Answer - B | |
| 232. |
If `dy/dx+2ytanx=sinx` and `y=0`, when `x=pi/3`, show that the maximum value of `y` is `1/3`A. `(1)/(4)`B. `(1)/(8)`C. `(1)/(16)`D. `(3)/(8)` |
| Answer» Correct Answer - B | |
| 233. |
The differential equation `(d^(2)y)/(dx^(2))=2` represents which of the following curve?A. A straight lineB. A cirleC. A parabola whose axis is parallel to Y-axisD. A parallel to X-axis |
| Answer» Correct Answer - C | |
| 234. |
The differential equation of all parabolas whoseaxis are parallel to the y-axis is(a)`( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )`(s)(b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)`(kk)(c)`( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )`(ii)(d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)`(ggg)A. `y_(2)=2y_(1)+x`B. ` y_(3) = 2y_(1)`C. ` y_(2)^(3)=y_(1)`D. None of these |
|
Answer» Correct Answer - d The equation of the family of parabolas with axis parallel to axis of Y is `(x-1)^(2) = A ( y -b)` On differenting w.r.t x, we get ` 2(x-a) = Ay_(1)` Again , on differenting w.r.t. x, we get `2 = Ay_(2)` ltbRgt And again , on differentiating w.r.t x, we get ` y _(3) = 0` |
|
| 235. |
The differential equation of all parabolas whoseaxis are parallel to the y-axis is(a)`( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )`(s)(b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)`(kk)(c)`( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )`(ii)(d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)`(ggg)A. ` (d^(3)y)/(dx^(3))= 0`B. `(d^(2)x)/(dy^(2))=C`C. ` (d^(3)y)/(dx^(3))+(d^(2)x)/(dy^(2))=0`D. `(d^(2)y)/(dx^(2))+2(dy)/(dx)=C` |
|
Answer» Correct Answer - a The equation of a member of the family of parabolas having axis to Y - axis is ` y = Ax^(2) + Bx +C` ltbRgt ` rArr (dy)/(dx) = 2Ax + B` ` (d^(2)y)/(dx^(2)) = 2A rArr (d^(3)y)/(dx^(3))= 0` |
|
| 236. |
If `(2+sinx)(dy)/(dx)+(y+1)cosx=0andy(0)=1,` then `y((pi)/(2))` is equal toA. `(4)/(3)`B. `(1)/(3)`C. `(-2)/(3)`D. `(-1)/(3)` |
| Answer» Correct Answer - B | |
| 237. |
The particular solution of the differntial equation ` x dy` + `2y dx = 0 `, when x=2 , y=1 isA. `xy=4`B. `x^(2)y=4`C. `xy^(2)=4`D. `x^(2)y^(2)=4` |
| Answer» Correct Answer - B | |
| 238. |
The doctor took the temperature of a dead body at11.30 Pm which was `94. 6^0Fdot`He took the temperature of the body again afterone hour, which was `93. 4^0Fdot`If the temperature of the room was`70^0F`, estimate the time of death. Taking normaltemperature of human body as `98. 6^0Fdot`[Given: `log(143)/(123)=0. 15066 ,log(123)/(117)=0. 05`]A. 8.30 a.m.B. 8.30 p.m.C. 2.30 a.m.D. 9.30 p.m. |
| Answer» Correct Answer - B | |
| 239. |
The differential equation of all parabolas whoseaxis are parallel to the y-axis is(a)`( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )`(s)(b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)`(kk)(c)`( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )`(ii)(d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)`(ggg)A. `x(d^(2)y)/(dx^(2))-(dy)/(dx)=0`B. `x(d^(2)y)/(dx^(2))+(dy)/(dx)=0`C. `x(d^(2)y)/(dx^(2))-y=0`D. `x(d^(2)y)/(dx^(2))-(dy)/(dx)=0` |
| Answer» Correct Answer - A | |
| 240. |
Show that the differential equation of all parabolas `y^2 = 4a(x - b)` is given byA. `y(d^(2)y)/(dx^(2))+(dy)/(dx)=1`B. `y(d^(2)y)/(dx^(2))+(dy)/(dx)=0`C. `y(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)=0`D. `y(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)=1` |
| Answer» Correct Answer - C | |
| 241. |
The differential equation of all parabolas whoseaxis are parallel to the y-axis is(a)`( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )`(s)(b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)`(kk)(c)`( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )`(ii)(d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)`(ggg)A. `(d^(3)y)/(dx^(3))=0`B. `(d^(2)y)/(dx^(2))=0`C. `(d^(3)y)/(dx^(3))+(d^(2)x)/(dy^(2))=0`D. `(d^(2)y)/(dx^(2))+2(dy)/(dx)=0` |
| Answer» Correct Answer - A | |
| 242. |
Water at ` 100^(@) C` cools in 10 minutes to ` 88^(@)C` in a room temperature of `25^(@)C` .Find the temperature of water after 20 minutes.A. `67^(@)C`B. `68^(@)C`C. `77^(@)C`D. `78^(@)C` |
| Answer» Correct Answer - D | |
| 243. |
A right circular cone has height 9 cm and base radius is 5 cm. It is inverted and water is poured into it. If at any instant, the level of the water riese at the surface at that instant, then vessel will be full inA. 25 secondsB. 50 secondsC. 75 secondsD. 100 seconds |
| Answer» Correct Answer - C | |
| 244. |
The rate of decay of certain substance is directly proportional to the amount present at that instant . Initially there are 27 gm of certain substance and three hours later it is found that 8 gm are left. Find the amount left after one more hour.A. `(2)/(3)` gramsB. `(4)/(3)` gramsC. `(8)/(3)` gramsD. `(16)/(3)` grams |
| Answer» Correct Answer - D | |
| 245. |
The uranium disintegrates at a rate proportional to the amount present at any instant. If `m_(1)andm_(2)` gms of uranium are present at time `t_(1)andt_(2)` respectively, then half life of uranium isA. `((t_(1)-t_(2))log2)/log((m_(1))/(m_(2)))`B. `((t_(2)-t_(1))log2)/log((m_(1))/(m_(2)))`C. `((t_(1)-t_(2))log(m_(1)/m_(2)))/log2`D. `((t_(2)-t_(1))log(m_(1)/m_(2)))/log2` |
| Answer» Correct Answer - B | |
| 246. |
The order of the differential equation of the family of parabolas whose axis is the X-axis isA. 2 and 1B. 1 and 2C. 3 and 2D. 3 and 1 |
| Answer» Correct Answer - B | |
| 247. |
The rate of decay of the mass of a radioactive substance at any time is k times its mass at that time. The differential equation satisfied by the mass of the substance isA. `(dm)/(dt)+K=0,Kgt0`B. `(dm)/(dt)-K=0,Kgt0`C. `(dm)/(dt)+Km=0,Kgt0`D. `(dm)/(dt)-Km=0,Kgt0` |
| Answer» Correct Answer - C | |
| 248. |
Findthe equation of a curve passing through the origin given that the slope ofthe tangent to the curve at any point (x, y) is equal to the sum of thecoordinates of the point.A. `x+y+1=e^(x)`B. `x+y-1=e^(x)`C. `x-y+1=e^(x)`D. `x-y-1=e^(x)` |
| Answer» Correct Answer - A | |
| 249. |
The equation of the curve passing through the point `(1,pi/4)` and having a slope of tangent at any point (x,y) as `y/x - cos^2(y/x)` isA. ` x = e^(1+tan(y/x))`B. ` x = e^(1-tan(y/x))`C. ` x = e^(1+tan(x/y))`D. ` x = e^(1-tan(x/y))` |
|
Answer» Correct Answer - c Given, `(dy)/(dx) = y/x - (cos^(2)) (y/x)` ` rArr (x-dy-y dx)/x = - (-cos^(2). y/x)dx` ` sec^(2) (y/x) ((x-dy-ydx)/(x^(2))) = -(dx)/2` `rArr sec^(2). y/2 .d(y/x) = -(dx)/x` On integrating both sides we get ltbRgt ` rArr tan. y/x = - log x+C` when x = 1, `y = pi/4 rArr C = 1` ` :. tan.(y/x) = 1- log x rArr x = e^(1-tan.(y/x))` |
|
| 250. |
If radium decoposes at a rate proportional to tha amount Q present, then the differential equation isA. `(dQ)/(dt)=kQ,kgt0`B. `(dQ)/(dt)=-kQ,kgt0`C. `(dQ)/(dt)=Q`D. `(dQ)/(dt)=-Q` |
| Answer» Correct Answer - B | |