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All questions of Second Order Derivatives for MAT Exam

If f(x) = x + cot x,  ​
  • a)
    -4
  • b)
    2
  • c)
    4
  • d)
    -2
Correct answer is option 'C'. Can you explain this answer?

Aryan Khanna answered
 f(x) = x + cot x
f’(x) = 1 + (-cosec2 x)
f”(x) = 0 - 2cosec x(-cosec x cot x)
= 2 cosec2 x cot x
f”(π/4) = 2 cosec2 (π/4) cot(π/4)
= 2 [(2)^½]2 (1)
= 4
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  • a)
  • b)
  • c)
  • d)
Correct answer is option 'D'. Can you explain this answer?

Rocky Gupta answered
X^a y^b = (x + y)^(a + b)
taking ln on both sides :-
alnx + b lny = (a + b) ln(x + y)
diff both sides w.r.t x :-
a/x + by'/y = ( (a + b)/(x + y) ) + (((a + b) y'))/(x + y)
or,
a/x - (a +b)/(x + y) = y'[((a + b) / (x + y)) - b/y]
or,
(ax + ay - ax - bx)/x = y' [ (ay + by - bx - by)/y] (cancel (x + y))
or,
y' = dy/dx = ( y (ay - bx) )/(x( ay - bx)) = y/x
therefore we can easily say that the option (D) is the correct answer

  • a)
    2t
  • b)
    1/2a
  • c)
    -t/2a
  • d)
    t/4a
Correct answer is option 'B'. Can you explain this answer?

Knowledge Hub answered
y = at4,     x = 2at2
dy/dt = 4at3        dx/dt = 4at    => dt/dx = 1/4at
Divide dy/dt by dx/dt, we get
dy/dx = t2
d2 y/dx2 = 2t dt/dx……………….(1)
Put the value of dt/dx in eq(1)
d2 y /dx2 = 2t(1/4at)
= 1/2a

Find the second derivative of excosx​
  • a)
    -2exsinx
  • b)
    -exsinx
  • c)
    ex(sinx + cosx)
  • d)
    -2excosx
Correct answer is option 'A'. Can you explain this answer?

Rounak Nair answered
**Solution:**

To find the second derivative of the given function, we need to differentiate it twice with respect to x.

First, let's find the first derivative of the function:

f(x) = ex * cosx

Using the product rule, the derivative of f(x) is:

f'(x) = (ex * (-sinx)) + (cosx * ex)
= -ex * sinx + ex * cosx
= ex * (cosx - sinx)

Now, let's find the second derivative of the function. Taking the derivative of f'(x):

f''(x) = (ex * (-sinx)) + (ex * (-cosx))
= -ex * sinx - ex * cosx
= -ex * (sinx + cosx)

Therefore, the second derivative of excosx is -2exsinx, which is option A.

The general solution of
is
  • a)
    (c1 + c2x)e3x 
  • b)
    (c1 + c2 In x)x3 
  • c)
    (c1 + c2 x)x3 
  • d)
    (c1 + c2 In x)ex3
Correct answer is option 'B'. Can you explain this answer?

Varun Kapoor answered
The operator form of given equation is (x2D2 – 5xD + 9)y = 0 ...(*)
Let x = et ⇒ t = log x D′ ≡ d / dt
D ≡ d / dt
We have x2D2 = D′(D′ – 1) 
xD = D′ 
[D′(D′ – 1) – 5D′ + 9]y = 0 
The A.E. is m2 – 6m + 9 = 0 
(m – 3)2 = 0, m = 3, 3 
The C.F. is y = (c
1
 + c2t)e3t 
The solution of (*) is 
y = (c1 + c2 log x) x3.

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