We know that the solution of first order linear differential equation of the standard form

where P (x) and Q(x) are real differentiable functions, C is a real constant of integration and the integrating factor is 

The given differential equation is,

We have to convert the above equation to the standard form but we can see that the differential equation is not linear as the variable y is with power 2.

Now the above result is a linear differential equation of the standard form where

In this question, they are given,

From the given question we get,

That is each set of four elements.

It is given in the question

So, in each set all the elements are different. 

So, the number of ways to partition is given below,

We can cancel the term8! in the numerator and the term8! in the denominator. 

So, we get

We know that,

As we know that if all gifts are maximum (i.e., 9) then the sum of digits is 63 which is 2 more than 61.

So, all digits cannot be 9.

But if all digits are the second-largest digit (i.e., 8 ) then the sum is 56 which is 5 less than 61.

So, out of seven digits, five digits must be equal to 9.

So, the sum of five digits will be equal to 5×9=45.

So, the sum of the remaining two digits must be equal to 61−45=16.

Now as we know that if the sum of two digits is 16 then these two digits must be 8 and 8 or 9 and 7.

Now as we know that according to the identity of permutation which states that if there is are r persons and total n identical items and all n items are distributed to r persons and let each of the i^tℎ person is given ^m1 balls then the number of ways for this distribution is given as 

So, therefore there are two possible cases.

Case 1:- (five digits are 9 and two digits are 8)

Thus, the total possible even digits number with the sum of its digit as 61 will be 21+7=28.

Hence, the correct answer is 28

L'Hospital's Rule:

Apply L'Hospital's Rule, we get

Given that:

Point R lie on the line segment PQ.

And x - Coordinate of R is 4.

Let Point R be (4, b,c).

Let R divide line segment PR in the ratio k:1.

We know that:

Coordinate of Point that divides line in the ratio m:n is 

Here,

Putting values:

y - Coordinate:

Well, first one should know that this is a concept of the algebra of complex numbers. The first basic concept is what is the algebra of complex numbers. First, know that complex numbers are an algebraic expression including the factor These numbers have two parts, one is called the real part denoted by Re(z) and the other part is called the imaginary part. Imaginary part is denoted by Im(z) for the complex number represented by z′. Also one should know another important thing: the equation of the circle which is basically the main part of the question.

Given that:

Given:

Equation of the pair of straight lines is

We know that the standard equation of the pair of straight lines is

Comparing the given equation with the standard equation, we get

We also know that the condition for the pair of straight lines is

Given:

Comparing the given equations, we obtain:

Substituting all the values in equation (1), we obtain:

Since distance is always non-negative, the distance between the given lines is  units.

We know that the distance of the point (x₁,y₁,z₁) from the plane ax+by+cz+d =0 is given by

Given, the point is (2,3,−5).

= 3 units

Hence, the correct answer is 3.

Consider the equations,

Divide each side of the equation (1) by the equation (2).

Apply the trigonometric formula, 

Combine the terms whose sum is the even number so we can use the formulas,

Apply the trigonometric formula; 

Substitute values of A and B into the formula.

Comparing the angles we get,