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All questions of August Week 1 for JEE Exam

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Two concentric coils carry the same current in opposite directions. The diameter of the inner coil is half that of the outer coil. If the magnetic field produced by the outer coil at the common centre are 1 T, the net field at the centre is
  • a)
    4T
  • b)
    2T
  • c)
    1T
  • d)
    3T
Correct answer is option 'C'. Can you explain this answer?

Krishna Iyer answered

The magnetic field produced by a current-carrying coil at its center is given by the formula,
B = μ0 * (N*I/R),
where,
B is the magnetic field,
μ0 is the permeability of free space,
N is the number of turns in the coil,
I is the current through the coil, and
R is the radius of the coil.
In this case, both the coils carry the same current but in opposite directions. So, the fields produced by them will be in opposite directions. Also, the diameter of the inner coil is half that of the outer coil. Thus, the radius of the inner coil will be half that of the outer coil.
Therefore, the field at the center due to the inner coil will be double that due to the outer coil (because the magnetic field is inversely proportional to the radius).
Since the fields are in opposite directions, the net field at the center will be the difference between the two fields. That is, 2B (due to the inner coil) - B (due to the outer coil) = B.
So, if the field due to the outer coil is 1 T (Tesla), the net field at the center will also be 1 T.
Hence, the correct answer is 3. 1T.

Wire of length l, carries a steady current I. It is bent first to form a circular coil of one turn. The same wire of same length is now bent more sharply to give two loops of smaller radius the magnetic field at the centre caused by the same current is
  • a)
    one third of its initial value
  • b)
    nine times of its initial value
  • c)
    four times of its initial value
  • d)
    unaltered
Correct answer is option 'C'. Can you explain this answer?

Om Desai answered
Let the radii be r1​ and r2​ respectively.
Since there are two turns of radius r2​, r1​=2r2​
Magnetic field B at the centre of  the coil of radius r1​ B1​=​μo​i/2r1​=​μo​i​/4r2
Magnetic field B at the center of the coil of radius r2​ B2​=2×​μo​i​/2r2
∴ B2/B1 =(2× μo​i/2r2​)/(μo​i /4r2​)​ ​​=4
Hence the answer is option C, four times its initial value.
 

A circular coil of radius r carries current I. The magnetic field at its center is B. at what distance from the center on the axis of the coil magnetic field will be B/8
  • a)
    √3R
  • b)
    √2R
  • c)
    2R
  • d)
    3R
Correct answer is option 'A'. Can you explain this answer?

Shilpa Saha answered
As you know that magnetic field at point on the axis of current carrying ring is 


where x is the point on the axis of ring, R is the radius of ring , i is the current carrying on ring and N is the number of turns .



This is possible only when x = +-√3R 
Hence, √3R distance from the centre magnetic field is equal to magnetic field at centre .

  • 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)
  • b)
  • c)
  • d)
Correct answer is option 'D'. Can you explain this answer?

Knowledge Hub answered
ey =1/ (x+5)
Taking log both side we get
log ey = log {1/(5+x)}
then, y = log {1/(5+x)}
Differentiating both side ,
dy/dx =( x+5) . {-1/(5+x)²}
dy/dx = -1/(5+x) ……..( 1)
Again Differentiating, d²y/dx²= 1/(5+x)²
d²y/dx²= {-1/(5+x)}²
From equation (1)
d²y/dx² = {dy/dx}²

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

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.

Find the differential coefficient y = (sec 5x)5x 
  • a)
    log ( sec 5x) + 5x log (tan 5x)
  • b)
    5(sec 5x)5x [log (sec 5x) + 5x tan 5x ]
  • c)
    (sec 5x)5x [log( sec 5x) + 5x log (tan 5x)]
  • d)
    log (sec 5x) + 5x tan 5x
Correct answer is option 'B'. Can you explain this answer?

Mehul Kulkarni answered
Explanation:

Given: y = (sec 5x)^5x

Step 1: Apply the chain rule to find the differential coefficient of y with respect to x.

Step 2: Differentiate the function (sec 5x)^5x using the chain rule.

Step 3: Apply the chain rule to differentiate (sec 5x)^5x.

Step 4: The differential coefficient of y is given by 5(sec 5x)^5x[log(sec 5x) + 5x tan 5x].
Therefore, the correct answer is option B - 5(sec 5x)^5x[log(sec 5x) + 5x tan 5x].

The magnetic field B on the axis of a circular coil at distance x far away from its centre are related as:
  • a)
    Bαx-1
  • b)
    Bαx-4
  • c)
    Bαx-3
  • d)
    Bαx-2
Correct answer is option 'C'. Can you explain this answer?

Naina Bansal answered
The formula for the magnetic field B (flux density) at P at a distance d from the center of O the coil on the axis of the coil of radius r :

Put appropriate punctuation marks in the following sentences.
Sorry to disturb you could I speak to you for a moment
 
Correct answer is 'Sorry to disturb you – could I speak to you for a moment?'. Can you explain this answer?

Gowri Chakraborty answered
In the sentence "Sorry to disturb you – could I speak to you for a moment?", the appropriate punctuation marks are a hyphen (-) after "disturb you" and a question mark (?) after "moment".
The hyphen is used to join two words together, in this case "disturb" and "you", to form a compound adjective. A compound adjective is an adjective made up of two or more words that describes a noun. In this sentence, the compound adjective "sorry to disturb you" describes the speaker and indicates that they feel apologetic about interrupting the person they are speaking to.

The question mark is used to indicate that the sentence is a question. In this case, the speaker is asking if they can speak to the person for a moment.

Overall, the punctuation in the sentence helps to clarify the meaning and structure of the sentence, making it easier for the reader to understand.

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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