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151.

The differential equation for `y=Ae^(3x)+Be^(2x)` isA. `(d^(2)y)/(dx^(2))-5(dy)/(dx)+6y=0`B. `(d^(2)y)/(dx^(2))+5(dy)/(dx)-6y=0`C. `(d^(2)y)/(dx^(2))-5(dy)/(dx)-6y=0`D. `(d^(2)y)/(dx^(2))+5(dy)/(dx)+6y=0`

Answer» Correct Answer - A
152.

The differential equation for `y=ae^(x)+be^(-2x)` isA. `(d^(2)y)/(dx^(2))+(dy)/(dx)-2y=0`B. `(d^(2)y)/(dx^(2))-(dy)/(dx)-2y=0`C. `(d^(2)y)/(dx^(2))+(dy)/(dx)-2y=0`D. `(d^(2)y)/(dx^(2))-(dy)/(dx)+2y=0`

Answer» Correct Answer - C
153.

The differential equation for `y=Ae^(3x)+Be^(-3x)` isA. `(d^(2)y)/(dx^(2))-9y=0`B. `(d^(2)y)/(dx^(2))-3y=0`C. `(d^(2)y)/(dx^(2))+9y=0`D. `(d^(2)y)/(dx^(2))+3y=0`

Answer» Correct Answer - A
154.

The order of the differential equation whose solution is `x^(2)+y^(2)+2gx+2fy+c=0` isA. 1B. 2C. 3D. 4

Answer» Correct Answer - C
155.

The order of the differential equation ` (d^(2)y)/(dx^(2)) = sqrt(1+((dy)/(dx))^(2))` isA. 3B. 2C. 1D. 4

Answer» Correct Answer - b
Given differential equation is
` (d^(2)y)/(dx^(2)) = sqrt(1+((dy)/dx)^(2)) = 1+ (dy/dx)^(2)`
Hence, order is 2
156.

The differentiala equation representing the family of curves `y^(2)=2k(x+sqrtk)` where k is a positive parameter, is ofA. order=1,degree=2B. order 2, degree=2C. order=1,degree=3D. order2,degree=1

Answer» Correct Answer - C
157.

The order of the differential equation whose solution is `ae^(x) + be^(2x) + ce^(3x) + d = 0` isA. 4B. 2C. 3D. 1

Answer» Correct Answer - A
158.

The order and degree of the differential equation `rho=({1+((dy)/(dx))^(2)}^(3//2))/((d^(2)y)/(dx^(2)))` are respectivelyA. 2 and 2B. 2 and 3C. 2 and 1D. 1 and 4

Answer» Correct Answer - A
159.

The order and the degree of the differential equation ` sqrt(y+ (d^(2)y)/(dx))= x + ((dy)/dx)^(3//2)` areA. 2,2B. 2,1C. 1,2D. 2,3

Answer» Correct Answer - a
The given differential equation can be rewritten as
` y + (d^(2)y)/(dx^(2)) = [ x + ((dy)/dx)^(3//2) ]^(2)`
` rArr y + (d^(2)y)/(dx^(2)) = x^(2)+ ((dy)/dx)^(3) + 2x ((dy)/dx)^(3//2)`
` rArr [ y + (d^(2)y)/(dx^(2)) - x^(2) - ((dy)/(dx))^(3)]^(2) = [ 2x ((dy)/dx)^(3//2) ]^(2) = [ 4x ((dy)/dx)^(3)]`
` :. ` Order and degree of the given differential equation and 2 respectively .
160.

Order and degree of the differential equation `((d^(3)y)/(dx^(3))+x)^(5/2)=(d^(2)y)/(dx^(2))` respectively areA. 3 and 5B. 2 and 2C. 2 and 1D. 3 and 2

Answer» Correct Answer - A
161.

The differential equation representing the family of curves ` y^(2) = 2c (x +c^(2//3))` , where c is a positive parameter , is ofA. order 3, degree 3B. order2,degree4C. order 1,degree 5D. order 5,degree 1

Answer» Correct Answer - c
` y^(2) = 2c (x+c^(2//3))`
On differentiating both sides , we get op
` rArr 2y (dy)/(dx) = 2c rArr c = y (dy)/(dx)`
` :. y^(2) = 2y (dy)/(dx) [ x + (y (dy)/dx)^(2//3)]`
` rArr (y/(2(dy)/dx)-x) = (y (dy)/(dx))^(2//3)`
` rArr (y - 2x (dy)/(dx))^(3) = (2 (dy)/dx)^(3) (y (dy)/(dx))^(2)`
` rArr (y - 2x (dy)/(dx))^(3) = 8y^(2)(dy/dx)^(5)`
Here , order = 1, degree = 5
162.

The order and degree of the differential equation `e^((dy)/(dx))+(dy)/(dx)=x` respectively areA. 1 and 1B. both order and degree can not be definedC. order =1, degree can not be definedD. degree =1, order can not be defined

Answer» Correct Answer - C
163.

Order and degree of the differential equation `(d^(4)y)/(dx^(4))=(1+((dy)/(dx))^(2))^(3)` respectively areA. 4 and 6B. 1 and 6C. 4 and 3D. 4 and 1

Answer» Correct Answer - D
164.

The order and degree of the differential equation `(d^(2)y)/(dx^(2))+cos((dy)/(dx))=0` respectively areA. both degree and other can not be definedB. order =2, degree can not be definedC. 2 and 1D. degree =2, order can not be defined

Answer» Correct Answer - B
165.

Order and degree of the differential equation `((d^(3)y)/(dx^(3)))^(1/6)((dy)/(dx))^(1/3)=5` respectively areA. `3 and (1)/(6)`B. `1and (1)/(3)`C. 1 and 3D. 3 and 1

Answer» Correct Answer - D
166.

Order and degree of the differential equation `(d^(3)y)/(dx^(3))=5sqrt(1-((dy)/(dx))^(2))` respectively areA. 3 and 1B. 3 and 3C. 3 and 2D. 3 and 5

Answer» Correct Answer - D
167.

The order and the degree of the differential equation ` y = x (dy)/(dx) +2/(dy//dx)` areA. 1,2B. 1,3C. 2,1D. 1,1

Answer» Correct Answer - a
Given differential equation is
` y = x(dy)/(dx) + 2/(dy//dx) rArr y (dy)/(dx) = x ((dy)/(dx))^(2) +2`
Here , order = 1 degree = 2
168.

The order and degree of the differential equation `[1+((dy)/(dx))^(2)]^(3//4)=((d^(2)y)/(dx^(2)))^(1//3)`A. (2,4)B. (2,3)C. (6,4)D. (6,9)

Answer» Correct Answer - a
Given differential equation can be rewritten as
` [ 1+((dy)/dx)^(2)]^(3/4 xx12) = ((d^(2)y)/(dx^(2)))^(1/3 xx12) rArr [ 1 + ((dy)/(dx))^(2)]^(9) = ((d^(2)y)/dx^(2))^(4)`
Here, we see that order of highest derivative is 2 and degree is 4 .
169.

The degree and order of the differential equation `y=px+root3(a^(2)p^(2)+b^(2)),` where `p = (dy)/(dx) ` are respectivelyA. 3,1B. 1,3C. 1,1D. 3,3

Answer» Correct Answer - b
Given differential equation is
` y = x(dy)/(dx)+ [ a^(2) ((dy)/(dx))^(2)+b^(2)]^(1/3)`
` rArr ( y - x(dy)/dx)^(3) = a^(2) ((dy)/dx)^(2) +b^(2)`
` :. ` Order and degree of the above differential equaiton are 1 and 3 respectively .
170.

Find the differential equation whose general solution is given by `y=(c_(1)+c_(2))cos(x+c_(3))-c_(4)e^(x+c)`, where `c_(1),c_(2), c_(3), c_(4), c_(5)` are arbitary constants.A. 5B. 6C. 3D. 2

Answer» Correct Answer - c
` y (c_(1) +c_(2)) cos (x+c_(3))=c_(4e^(x+c_(5))`
` y = (c_(1) +c_(2)) cos (x+c_(3))-c_(4)e^(C_(5))*e^(x)`
` rArr y = A cos (x+c_(3))- Be^(x)`
where ` A= c_(1) + c_(2) and B = c_(4)e^(5)`
It has three arbitrary constant so, the order of differential equation is 3.
171.

The order of the diferential equation whose general solution is given by ` y = c_(1)e^(2x+c_(2)) + c_(3)e^(x)+c_(4)sin (x+c_(5))` isA. 5B. 4C. 3D. 2

Answer» Correct Answer - b
Given equation is ` y = c_(1)e^(2x+c_(2)) +c_(3)e^(x) +c_(4)sin (x+c_(5))`
`= c_(1)e^(c_(2))e^(2x)c_(3)e^(x)+ c_(4)(sin x cos c_(5)+cos x sin c_(5))`
` = Ae^(2x) +c_(3)e^(x)+B sin x +D cos x`
Here, ` A = c_(1) e^(c_(2)) , B = c_(4) cos c_(5) , D c_(4) sin c_(5)` ,
Since , equation consists of four arbitrary constants So, the order of differential equation is 4.
172.

Let I be the purchase value of an equipment and V(t) be the value after ithas been used for t years. The value V(t) depreciates at a rate given bydifferential equation `(d V(t)/(dt)=-k(T-t)`, where `k"">""0`is a constant and T is thetotal life in years of the equipment. Then the scrap value V(T) of theequipment is :(1) `T^2-1/k`(2) `I-(k T^2)/2`(3) `I-(k(T-t)^2)/2`(4) `e^(-k T)`A. ` I-(kT^(2))/2`B. `(dy)/(dx) = (x(1+y^(2)))/(y(1+x^(2)))`C. `e^(-kT)`D. `T^(2) -1/k`

Answer» Correct Answer - a
Given , ` (d{V(t)})/(dt) = -k (T-t)`
` :. d{V(t)} = - k (T-t) dt" "` …(i)
when t = 0, then V (t)=I
` int_(0)^(T)d {V(t)} = int_(0)^(T)- k(T-t)dt`
` rArr V (T) - V(0) = k [ (t-T)^(2)/2]_(0)^(T)`
` rArr V(T) -I = k/2 [{(T-T)^(2)-(0-T)^(2)]`
` :. V (T) = I - k/2 T^(2)`
173.

If a curve passes through the point `(2,7/2)` and has slope `(1-1/(x^(2)))` at any point (x,y) on ity , then the ordinate of the point on the curve , whose abscissa is -2, isA. `-3/2`B. `3/2`C. `5/2`D. `-5/2`

Answer» Correct Answer - a
Given, ` (dy)/(dx) = 1- 1/(x^(2)) rArr dy (1-1/(x^(2)))dx`
`rArry = x + 1/x +C` [ on integration ]
since , the curve passing through the point `(2,7/2)`
i.e ` 7/2 = 2+1/2 + C rArr C = 1`
` :. Y = x+1/x +1 " "` …(i)
Given also `x =-2`
` :. " Ordinate " , y = -2 - 1/2 +1 = 3//2`
174.

If `(dy)/(dx)=y+3 and y(0)=2`, then y(ln 2) is equal toA. 5B. 13C. `-2`D. 7

Answer» Correct Answer - d
Here, ` (dy)/(dx) = y + 3 gt and y (0) = 2`
` rArr int (dy)/(y+3) = int dx`
` rArr log | y + 3 | = x +C`
But , given `y(0) =2`
` rArr log | y + 3| = x + log_(e)5`
when ` x = log _(e)2`
` rArr log |y +3| = log_(e)2 + log_(e) 5 = log_(e) 10`
` :. y + 3 = 10`
` rArr y = 7`
175.

The solution of `dy/dx = (x^2+y^2+1)/(2xy)` satisfying `y(1)=0` is given byA. hyperbolaB. circleC. ellipseD. parabola

Answer» Correct Answer - d
`(dy)/(dx) = (x^(2)+y^(2)+1)/(2xy)`
` rArr 2xydy = (x^(2) +1) dx +y^(2)dx`
` rArr (xd(y^(2))-y^(2)dx)/(x^(2)) = ((x^(2)+1)/(x^(2)))dx`
` rArr int d ( y^(2) //x) = int (1+1/(x^(2)))dx rArr (y^(2))/x = x - 1/x +C`
` rArr y^(2) = (x^(2) - 1+Cx)`
Given, x=1 , y = 0
` :. 0 = 1 - 1 +C rArr C = 0 `
` rArr y^(2) = x^(2) -1 rArr x^(2) - y^(2) =1`
which is the equation of a hyperbola .
176.

The solution of the differential equation `{1/x-y^(2)/(x-y)^(2)}dx+{x^(2)/(x-y)^(2)-1/y}dx=0` isA. ` In |x/y|+(xy)/((x-y))=C`B. ` In |xy|+(xy)/((x-y))=C`C. ` (xy)/((x-y))=Ce^(x//y)`D. ` (xy)/((x-y))=Ce^(xy)`

Answer» Correct Answer - a
The given equation can be written as
` ((dx)/y-(dy)/y )+(x^(2)dy-y^(2)dx)/((x-y)^(2))=0`
` rArr ((dx)/x - (dy)/y)+(((dy)/(y^(2)) -(dx)/(x^(2))))/ ((1/y-1/x)^(2))=0`
` rArr ((dx)/x - (dy)/y)+(((dy)/(y^(2)) -(dx)/(x^(2))))/ ((1/x-1/y)^(2))=0`
On integrating both sides , we get
` In |x| - In |y| - 1/((1/x-1/y))=C`
` rArr In |x/y| = (xy)/((y-x))= C rArr In |x/y| +(xy)/((x-y)) = C`
177.

Thesolution of the differential equation `(dy)/(dx)=(x+y)/x`satisfying the condition `y""(1)""=""1`is(1) `y""="ln"x""+""x`(2) `y""=""x"ln"x""+""x^2`(3) `y""=""x e(x-1)`(4) `y""=""x"ln"x""+""x`A. ` y = x log x +x`B. ` y = log x +x`C. ` y = x log x+x^(2)`D. ` y = xe^(x-1)`

Answer» Correct Answer - a
Given equation cn be writtn as
` (dy)/(dx)-1/x*y = 1`
` :. If = ^(-f 1/x dx) = e ^(-logx) =1/x `
`:. ` Required solution is
` y (1/x) = int 1/x dx+ C = log x + C`
Since , y (1) =1
` rArr C = 1`
` :. Y = x log x +x`
178.

The degree of the differential equation satisfying the relation `sqrt(1+x^2) + sqrt(1+y^2) = lambda (x sqrt(1+y^2)- ysqrt(1+x^2))` isA. 1B. 2C. 3D. None of these

Answer» Correct Answer - a
On putting x = tan A and y = tan B in the given relation ,
we get ` cos A + cos B = lambda ( sin A - sinB)`
` rArr tan.((A-B)/2) = 1/lambda `
` rArr tan^(-1) x - tan^(-1) y = 2 tan^(-1) (1/lambda )`
On differentiating w.r.t x, we get
`1/(1+x^(2)) - 1/(1+y^(2)) = 0 `
` rArr (dy)/(dx) = (1+y^(2))/(1+x^(2))`
Clearly , it is a differential equation of degree `
179.

Intergrating factor of the differential equaiton `(x^(2)+1)(dy)/(dx)+2xy=x^(2)-1` isA. `(x^(2)-1)/(x^(2)+1)`B. `(2x)/(x^(2)+1)`C. `x^(2)+1`D. `-(x^(2)+1)`

Answer» Correct Answer - C
180.

The differential equation of the family of circles with fixed radius 5 units and centre on the line y=2 isA. `(y-10)^2((dy)/(dx))^2+y^2 + 20y =0`B. `(y-10)^2((dy)/(dx))^2-y^2 + 20y =0`C. `(y-10)^2((dy)/(dx))^2+y^2 - 20y =0`D. `(y-10)^2((dy)/(dx))^2-y^2 - 20y =0`

Answer» Correct Answer - C
181.

The differential equation of the family of fixed radii r with centers on the X-axis isA. `y(dy)/(dx)+y= r`B. `y(dy)/(dx)-y= r`C. `y^2((dy)/(dx))^2+y^2= r^2`D. `y^2((dy)/(dx))^2-y^2= r^2`

Answer» Correct Answer - C
182.

The differential equation of all circles which passes through the origin and whose centers lie on Y-axis isA. `(x^(2)-y^(2))(dy)/(dx)-2xy=0`B. `(x^(2)-y^(2))(dy)/(dx)+2xy=0`C. `(x^(2)-y^(2))(dy)/(dx)-2xy=0`D. `(x^(2)-y^(2))(dy)/(dx)+2xy=0`

Answer» Correct Answer - A
183.

Assume that a spherical rain drop evaporates at a rate proportional to its surfaceradius originally is 3 mm and 1 hour later has been reduced to 2 mm, find an expression for the radius of the rain drop at any time.A. 2-t mmB. 2+t mmC. 3-t mmD. 3+t mm

Answer» Correct Answer - C
184.

The integrating factor of the differentiable equation ` (xy-1)(dy)/(dx)+y^(2) =0` isA. `1/x`B. `1/y`C. `1/(xy)`D. xy

Answer» Correct Answer - b
The given equation can be written as
` xy dy + y^(2) dx - dy =0`
` rArr x dy + y dx - 1/y dy = 0`
` rArr d (xy) = (dy)/y`
` rArr int d (xy) = int (dy)/y`
`rArr xy - log y = C`
` rArr 1/y` is integrating factor .
185.

Solution of differential equation `cosy(dy)/(dx)=e^(x+siny)+x^(2)e^(siny)`isA. `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
186.

Findthe particular solution of the differential equation `log(dy)/(dx)=3x+4y`given that `y" "=" "0`when`x" "=" "0`.A. `4e^(3x)+3e^(-4y)+7=0`B. `4e^(3x)+3e^(-4y)-7=0`C. `4e^(3x)-3e^(-4y)+7=0`D. `4e^(3x)-3e^(-4y)+7=0`

Answer» Correct Answer - B
187.

The particular solution of the differential equation `log ((dy)/(dx)) = x`, when x = 0 , y = 1 is …..A. `y=e^(x)`B. `y=-e^(x)`C. `y=e^(x)+2`D. `y=-e^(x)+2`

Answer» Correct Answer - A
188.

The population of a city increase at a rate proportional to the population at that time. In 40 years the population is increased from 30,000 to 40,000 then after time t, the population isA. `(40000)((4)/(3))^(t/40)`B. `(40000)((3)/(4))^(t/40)`C. `(30000)((4)/(3))^(t/40)`D. `(30000)((3)/(4))^(t/40)`

Answer» Correct Answer - C
189.

The population of a city increases at a rate proportional to the population at that time. If constant of proportionality is 0.04, then population of a city after 25 years, when initial polulation is 10,000 is (e=2.72)A. 27200B. 13600C. 2720D. 1360

Answer» Correct Answer - A
190.

Integrating factor of the differential equation `(x.logx)(dy)/(dx)+y=2logx` isA. `e^(x)`B. log xC. log(logx)D. x

Answer» Correct Answer - b
Given differential equation can be rewritten as
` (dy)/(dx) + 1/(x log x) *y = 2/x`
` (dy)/(dx) + 1/(x log x) *y = 2/x`
191.

The population grows in at the rate of 8% per year. Find the time taken for the population to become double . (Given : log 2 = 0.6912)A. 12.98 yearsB. 4.32 yearsC. 17.28 yearsD. 8.64 years

Answer» Correct Answer - D
192.

The population of a city increases at a rate proportional to the population at that time. If the polultion of the city increase from 20 lakhs to 40 lakhs in 30 years, then after another 15 years, the polulation isA. `10sqrt2 lakhs`B. `20sqrt2 lakhs`C. `30sqrt2`D. `40sqrqt2 lakhs`

Answer» Correct Answer - D
193.

The rate of growth of a population is proportionalto the number present if the population of a city doubled in the past 25years, and the present population is 100000, when will the city have apopulation of 500000?A. 60 yearsB. 58 yearsC. 48 yearsD. 54 years

Answer» Correct Answer - B
194.

The solution of the differential equation `(dy)/(dx) = 1/(x+y^(2))` isA. `y = -x^(2) - 2x - 2 +Ce^(x)`B. `y = x^(2) + 2x +2 - Ce^(x)`C. ` x = -y^(2) - 2y + 2 - Ce^(y)`D. ` x = -y^(2) - 2y - 2 +Ce^(y)`

Answer» Correct Answer - d
Given differential equation is
` (dy)/(dx) = 1/(x+y^(2))`
` rArr (dx)/(dy)- x = y^(2)`
Here, `P = -1 , Q = y^(2)`
` :. IF = e^(int - dy) = e^(-y)`
195.

The differential equation whose solution is `(x-h)^2+ (y-k)^2=a^2` is (a is a constant)A. ` [ 1+ ((dy)/(dx))^(2)]=a^(2)(d^(2)y)/(dx^(2))`B. ` [ 1+ ((dy)/(dx))^(2)]^(3) = a^(2) ((d^(2)y)/(dx^(2)))^(2)`C. ` [ 1+((dy)/(dx))]^(3)=a^(2)((d^(2)y)/(dx^(2)))`D. None of the above

Answer» Correct Answer - b
Given , `(x-h)^(2) +(y-k)^(2) = a^(2) `
` rArr2(x-h)+2 (y-k)(dy)/(dx) =0`
` rArr (x-h) +(y-k) (dy)/(dx) =0" " ` … (ii)
Again , on differentiating , we get
` 1+ (y-k) (d^(2)y)/(dx^(2)) +((dy)/(dx))^(2)+((dy)/(dx))^(2)=0`
` rArr (y-k) = -(1+((dy)/(dx))^(2))/((d^(2)y)/(dx^(2)))`
Putting the value of `(y-k) ` in Eq. (ii) , we get
` x-h =-(y-k)(dy)/(dx)=([1+((dy)/(dx))^(2)](dy)/(dx))/((d^(2)y)/(dx^(2)))`
Putting the values of `(x-h) and (y-k)` in Eq.(i) , we get
` ([ 1((dy)/(dx))^(2)]^(2)((dy)/(dx))^(2))/(((d^(2)y)/(dx^(2)))^(2))+ ([1+((dy)/(dx))^(2)]^(2))/(((d^(2)y)/dx^(2))^(2))=a^(2)`
` rArr [1+((dy)/(dx))^(2)]^(2)[((dy)/(dx))^(2)+1] = a^(2) ((d^(2)y)/(dx^(2)))^(2)`
` rArr [ 1+ ((dy)/(dx))^(2)]^(3)=a^(2)((d^(2)y)/(dx^(2)))^(2)`
196.

The rate of increase in the number of bacterai in a certain becteria culture is propotional to the number present at that time. If is found that the number doubles in 4 hours, then at the end of 12 hours, the number of bacteria areA. 4 times the originalB. 6 time the originalC. 8 times the originalD. 10 times the original

Answer» Correct Answer - C
197.

The polulation of a city increases at a rate propotional to the population at that time. If the population of the city is doubled in 60 years, then population will be triplet in (log2=0.6912,log 3=1.0986)A. 95.4 yearsB. 95.3 yearsC. 94.5 yearsD. 95.5 years

Answer» Correct Answer - A
198.

The rate of increase in the number of bacteria in a certain becteria culture is propotional to the number present at that time. If initiallly there are 300 bacteria and after 2 hours, the bacteria polulation is increased by 20% then after 24 houre, the number of bacteria are `(log 1.2=0.18232,e^(2.18784)=8.9166)`A. 2675B. 2674C. 3210D. 3209

Answer» Correct Answer - A
199.

The rate of increase in the number of bacteria in a certain culture is propotional to the number present at that time. After 2 hours there are 600 bacteria and after 8 hours the count is 75000, then the population will be 200000 afterA. 9.21 hoursB. 9.12 hoursC. 9.22 hoursD. 9.23 hours

Answer» Correct Answer - C
200.

The rate of increase in the number of bacteriad in a certain bacteria culture is propotional to the number present at that time. After 2 hours there are 600 becteria and after 8 hours the count is 75000, then the initial population isA. 102B. 120C. 124D. 142

Answer» Correct Answer - B