Partial Derivatives MCQ Quiz in मल्याळम - Objective Question with Answer for Partial Derivatives - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Last updated on Mar 24, 2025

നേടുക Partial Derivatives ഉത്തരങ്ങളും വിശദമായ പരിഹാരങ്ങളുമുള്ള മൾട്ടിപ്പിൾ ചോയ്സ് ചോദ്യങ്ങൾ (MCQ ക്വിസ്). ഇവ സൗജന്യമായി ഡൗൺലോഡ് ചെയ്യുക Partial Derivatives MCQ ക്വിസ് പിഡിഎഫ്, ബാങ്കിംഗ്, എസ്എസ്‌സി, റെയിൽവേ, യുപിഎസ്‌സി, സ്റ്റേറ്റ് പിഎസ്‌സി തുടങ്ങിയ നിങ്ങളുടെ വരാനിരിക്കുന്ന പരീക്ഷകൾക്കായി തയ്യാറെടുക്കുക

Latest Partial Derivatives MCQ Objective Questions

Top Partial Derivatives MCQ Objective Questions

Partial Derivatives Question 1:

If u=loge(x4+y4x+y), the value of xux+yuy is

  1. 6
  2. 5
  3. 4
  4. 3

Answer (Detailed Solution Below)

Option 4 : 3

Partial Derivatives Question 1 Detailed Solution

Concept:

A function f(x, y) is said to be homogeneous of degree n in x and y, it can be written in the form

f(λx, λy) = λn f(x, y)

Euler’s theorem:

If f(x, y) is a homogeneous function of degree n in x and y and has continuous first and second-order partial derivatives, then

xfx+yfy=nf

x22fx2+2xy2fxy+y2fy=n(n1)f

If z is homogeneous function of x & y of degree n and z = f(u), then

xux+yuy=nf(u)f(u)

Calculation:

Given, u=loge(x4+y4x+y)

z=x4+y4x+y

z is a homogenous function of x & y with a degree 3.

Now, z = eu

Thus, by Euler’s theorem:

xux+yuy=3eueu

xux+yuy=3

Partial Derivatives Question 2:

Consider a function f(x, y, z) given by f(x, y, z) = (4x3y – 3y2x2 + 4z) lny. The value of fy = (fy)at point x = 2, y = 1, z = 2 is

Answer (Detailed Solution Below) 28

Partial Derivatives Question 2 Detailed Solution

f(x, y, z) = (4x3y – 3y2x2 + 4z) lny

fy=fy=(4x36yx2)lny+(4x3y3y2x2+4z)1y

At point x = 2, y = 1, z = 2

fy = 0 + (4(8)(1) - 3(1)(4) + 4(2))

= 28

Partial Derivatives Question 3:

The time period of simple pendulum is T=2πlg. Find maximum error % in T due to possible error upto 1% in l and 2.5% in g.

Answer (Detailed Solution Below) 1.73 - 1.78

Partial Derivatives Question 3 Detailed Solution

T=2πlg

logT=log2π+12logl12logg

1TδT=0+12δll12δgg

δTT100=12(δll100δgg100)

=12(1±2.5)

= 1.75% or -0.75%

Maximum error in T = 1.75%

Partial Derivatives Question 4:

The tangent to the curve represented by y = x log x is required to have 45° inclination with the x axis. The coordinates of the tangent point would be

  1. (1, 0)
  2. (0, 1)
  3. (1, 1)
  4. (2,2)

Answer (Detailed Solution Below)

Option 1 : (1, 0)

Partial Derivatives Question 4 Detailed Solution

Target is having inclination of 45° with x-axis

dydx=tan45d(xlnx)dx=1lnx+xx=1

∴ At x = 1, y = 1 × ln1 = 0

Partial Derivatives Question 5:

If u = exyz, then 3uxyz at (1, 1, 1) is _____.

  1. 5e
  2. 3e
  3. 2e
  4. 4e

Answer (Detailed Solution Below)

Option 1 : 5e

Partial Derivatives Question 5 Detailed Solution

uz=xyexyz

y(uz)=y(xyexyz)

2uyz=xyy(exyz)+exyzy(xy)

= xy(xz)exyz + xexyz

x(2uyz)=x(x2yz+x)exyz

3uxyz=(x2yz+x)yzexyz+exyz(2xyz+1)

=exyz(x2y2z2+xyz+2xyz+1)

= (1 + 3xyz + x2y2z2) exyz

putting x, y, z = 1, 1, 1, we get

3uxyz=(1+3+1)e

= 5e

Partial Derivatives Question 6:

If H=tan1xy, x = u + v, y = u - v then HVis 

  1. uu2+v2
  2. vu2+v2
  3. ux2+y2
  4. 2vx2+y2

Answer (Detailed Solution Below)

Option 1 : uu2+v2

Partial Derivatives Question 6 Detailed Solution

Concept:

Hv=Ht×tv

Calculation:

Given: 

H=tan1xy

x = u + v

y = u - v

substituting values of x and y in H, we get
H=tan1[u+vuv]

let t=u+vuv------------(1)

H=tan1t

differentiating w.r.t 't'

Ht=11+t2------------(2)

In equation (1), differentiate 't' w.r.t v

tv=(uv)(1)(u+v)(1)(uv)2=2u(uv)2---------(3)

Multiplying (2) and (3), we get

Hv=11+t2[2u(uv)2]

By substituting 't=u+vuv' we get

Hv=uu2+v2

Partial Derivatives Question 7:

If z = xy In(xy), then

  1. xzx+yzy=0
  2. yzx=xzy
  3. xzx=yzy
  4. yzx+xzy=0

Answer (Detailed Solution Below)

Option 3 : xzx=yzy

Partial Derivatives Question 7 Detailed Solution

Taking the partial derivative on both the sides, we get:

xzx=yzy

In partial derivative, we take the other variable as a constant.

Partial Derivatives Question 8:

The partial derivative of the function

f(x,y,z)=e1xcosy+xze1/(1+y2)

With respect to x at the point (1, 0, e) is

  1. -1
  2. 0
  3. 1
  4. 1e

Answer (Detailed Solution Below)

Option 2 : 0

Partial Derivatives Question 8 Detailed Solution

Concept:

When the input of a function is made up of multiple variables, partial derivative is used to understand how the function changes as we let just one of the variable changes, keeping all the other variables as constant.

Application:

Given f(x,y,z)=e1xcosy+xze11+y2

Differentiating the above with respect to x and treating y and z as a constant, we get:

fx=e(1xcosy)ddx(1xcosy)+ze11+y2x(x)

fx=e(1xcosy)(cosy)+ze1y2+1

fx=cosye(1xcosy)+ze1y2+1

At point (1, 0, e), the partial derivative gives a value:

fx|(1,0,e)=cos(0)e(1cos0)+e.e11

= -1e0 + e.e-1

= -1 + 1

fx|(1,0,e)=0

Partial Derivatives Question 9:

If Z = eax + by F(ax - by); the value of bZx+aZy is

  1. 2Z
  2. 2a
  3. 2b
  4. 2abZ

Answer (Detailed Solution Below)

Option 4 : 2abZ

Partial Derivatives Question 9 Detailed Solution

We are given z = eax + by⋅ f(ax - by)

Now, zx=aeax+byF(axby)+aeax+byF(axby)

bzx=ab[z+eax+byF(axby) ---(1)

Now, zy=beax+byF(axby)beax+byF(axby)

azy=ab[ze(ax+by)F(axby)] ---(2)

Add equation (1) & (2), we get

bzx+azy=2abz

Partial Derivatives Question 10:

The contour on the x-y plane, where the partial derivative of x2 + y2 with respect to y is equal to the partial derivative of 6y + 4x with respect to x is

  1. y = 2
  2. x = 2
  3. x + y = 4
  4. x – y = 0

Answer (Detailed Solution Below)

Option 1 : y = 2

Partial Derivatives Question 10 Detailed Solution

CONCEPT:

In Partial Differentiation, all variables are considered as a constant except the independent derivative variable i.e If f(x,y,z) is a function, then its partial derivative with respect to x is calculated by keeping y and z as constant.

CALCULATION:

Given x2 + y2 = F (say…)

and 6y + 4x = G (say…)

Differentiating F with respect to y and G with respect to x and then equating them according to the given condition, we get;

2y = 4

So, y = 2.

Hence option 1 is correct 
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