This is a question on application of derivatives. We have to take log on both the sides, and then differentiate with respect to x on all the sides. The value of y will be substituted in terms of x.

Equating the first differential to 0, and knowing the fact that x^-x will be always non negative, we can find the value of x for which the given function will be minimum.

Note. If we got 2 values of x from first differential, then we have to go for double differential and check individually which one of them will give minimum value.

In the given question, we can take Common and check what are the factors left as summation. Now, in  immaginary cases, we know square of iota is -1 and cube of iota is negative of iota.

Thus, the remaining terms left after taking common results in 0, and the answer becomes 0.

Note. iota is nothing but the immaginary term 'i'

The coefficient of variation(CV) is nothing but the ratio of the standard deviation and mean of the particular set, when expressed as percentage.

Thus, we can find out the value of both the mean of the given sets by using the formula we have described.

We have find rate of change of x and y with respect to t, that is dx/dt and dy/dt respectively.

thus we can get the value of dy/dx by dividing dy/dt with dx/dt.

In order to get the value of slope, we need to find the value of t. We are given values of x and y. Since we can see t is a quadratic function for both x and y, we have solve both x and y to get that particular value of t.

Thus, the putting the common value of t,we will get the value of the slope.

Since we have to find out the angles between a and b, we will rearrange the equation such that magnitude of (a+b) is represented in terms of magnitude of c.

Squaring both the sides and representing square of (a+b) with the given formula, we will get the value of the cosine of the angle as we know the magnitude of a,b,c.

Thus,we can get the angle.

Note. Here a,b,c are 3 vectors and not scalars.

A binary operation is said to be commutative if a*b yields same answer as b*a.

An operation is said to be associative if a*(b*c) yields the same result as (a*b)*c.

Now, all we need to do is to check whether the properties for the given binary operation holds valid or not.

In order to solve these questions, we have to take log on both sides to avoid differentiation of terms in the form of x^a or y^b

Now, differentiating both sides with respect to x and rearranging the results such that the coefficient of dy/dx and dy/dx is on one side and rest terms on other side.

Thus, we will be able to find out the value of dy/dx

 The basic formulae we need to know from inverse trigonometry is that cos^-1x= sec^-1(1/x) and vice versa

Multiplying x and y, and rearranging terms,

Now, e^a*e^b= e^a+b

and, sin^-1x+cos^-1x=90 degrees or pi/2

Thus, now we differentiate both sides with respect to x, and we get the required differential equation.

This is question of sequence and series, where we need to have optimum knowledge about the summation of an infinite series(having infinite terms) of a geometric progression.

Thus, after we found out the summation,we equate both the sides and find the required angle of the trigonometric identity

First, we need to find the value of the cosine part. In order to get that, we will use the cyclic property, that is, after every 360 degrees, the value of cosine repeats itself.

Now, after we simplified the angle of cosine part, we will convert that into sine part as sin x= cos(90-x) where angles are in degrees.

Now, sin^-1[sin(x)]= x if and only if x lies between -180 degrees to +180 degrees

As per the question states, the angle between any 2 vectors is 90 degrees, that is, cosine of any angle will be 0

Which means that, the dot product of any 2 vectors will be 0

Thus, the squaring the sum of the vectors is nothing but the sum of square of the magnitude of each vectors

Firstly, by the properties of indices, we find that the numerator is 1 and denominator is in the form of a+ib where a is the real part and b is the immaginary part.

Now, the conjugate of a+ib is a-ib.

Hence, we will multiply with (a-ib) on both numerator and denominator.Denominator becomes a²+b² and we get the real and immaginary parts by seperating them

Area between f(x) and g(x) curves are given by  the integration of {f(x)-g(x)}dx with the limits found out the from the points of intersection,specifically,the x-coordinate.

From the given 2 curves, we will find out the x-coordinates of points of intersection. These points will be the limits of integration.

The result of integration, when taken with positive sign, is the area enclosed by 2 curves.

The first thing we have to do is to rearrange the lines in the form of y=mx+c, where m is the slope. Thus, we get 2 slopes for the 2 lines.

For 2 lines to be perpendicular, the product of the slopes of the lines will always be equal to -1

Thus, on solving, we will get the value of the unknown term "k"

In order to solve these type of questions, we need to have a very clear knowledge on matrix operations and inverse trigonometric properties.

sin^-1x+ cos^-1x= 90 degrees or half of pi radians.

Now, taking in common form the solution matrix, we will find that the solution is a multiple of an identity matrix I.

In order to find integrating factor, first, we have to rearrange the differential equation in the form of:

(dy/dx)+[P(x)]x= Q(x), where P(x) and Q(x) are functions of x.

From there, integrating factor can be found out as the e to the power, integration of P(x) with respect to dx.

Now, from logarithmic properties, we know, e^{ln a}= a

Hence, we can find out the result

This question is a simple question on matrix multiplication.

The square of matrix A is the multiplication of 2 same matrices, which is, A. The matrix multiplication is done as shown in the solution.

Whenever a matrix is multiplied with a constant value, then each of the elements in that matrix is multiplied with that specific value.

Then, the operations are done as given in the question and we get the result.

This is a question based on choosing of cards and probability.In order to find out the required probability, we need to find out the sample space first, which is the number of ways 2 cards can be selected from the deck of 52 cards.Then we have to find the number of ways 2 cards can be selected from the 4 aces in the deck. The ratio of latter to former will give us the probability of the required event.

This is a question on definite integration. Though initially it looks to be a difficult one, it is basically solved within few time if we know what is an odd or an even function.

f(-x)=-f(x) means a function is an odd function.

integration of definite function f(x) from a to b is same as integration of f(a+b-x)

From the above equation, we find that f(x) is an odd function, and integration of an odd function results in 0

This is a basic question on limits.

In order to solve these questions, first we need to see that whether the function is a 0/0 limit or not.

If it is a 0/0 form, we can use L'Hospital Rule where we can differentiate the numerator and denominator individually  till the 0/0 discontinuity is removed.

Thus, we get the value of the required limit.

We know, for any angle, sin²x+cos²x=1

To get the equation of the circle, all we need to do from both equations, we need to rearrange them in the form of sin x and cos x

After we have rearranged, squaring and adding both the equations will give us the equation.From the equation of the circle, we can easily find out the centre coordinates and radius

We know, one of the basic principles of Inverse trigonometry is that the sum of inverse of sin and cos results in 90 degrees.

Moreover, cos^-1x=sin^-1{(1-x²)^0.5}

From both the equations, we can substitute and rearrange to get the value of X in terms of Y in the given question.

We know, sin A+ sin B= 2sin(A+B/2) cos (A-B/2)

Clubbing sin 1 with sin 359, sin 2 with sin 358 and so on, we get one common term in each case, which is sin 180. The value of sin 180 is 0

That is, basically, the continued sum of  zeros will yield 0 only.

This is a very basic question on the expansion of terms in Binomial expansion.

All we have to do is find out the magnitude of the term using the given formula for expansion of Binomial theorem

This is a very basic question on the multiplication of matrices and their rules involved.

For matrix multiplication AB' to be defined, column of A should be equal to row of B' 

Similarly for B'A to be defined, column of B' should be equal to the row of A 

Thus, using these rules, we will get the required answer.

In this question, firstly we have to convert log_ab as log b/log a

Then, in each case, we have to differentiate with respect to x.

Dividing the equations will imply differentiation of first function with respect to second function.

In order to find out the angle cut by 2 curves, firstly we need to take the common point between the 2 curves as (h,k)

Then, we need to find out the slopes at (h,k) using applications of differentiation.

2 curves are said to cut at right angle if their product of the slopes at (h,k) results in -1.

Since, the product is -1, we can say the curves cut at right angle

We will firstly change the inverse trigonometric identity to trigonometric identity, thus tan alpha is a constant.

Now, we will differentiate both sides with respect to x, and right hand side will fetch 0 as differentiation of a constant is 0.

Thus, after differentiation, we will rearrange equations to get the values of dy/dx.

This is a basic question on the application of derivatives.

We need to know the formula of area of circle in terms of its radius, and then we have to differentiate Area with respect to radius.

After differentiation, we will substitute the value of radius, if any, with the given magnitude in question and get the answer.

Note. Rate of change of y with respect to x at a particular value of x is nothing but dy/dx at that specific value of x

Integration of f(x) over limits a to b is same as integration of f(a+b-x) over limits a to b.

Thus any power of sin x over limits 0 to 90 degrees become that specific power of cos x. This holds true for any power of cos x too.

From the property, we find that the value of integral remains unchanged while the numerator contains terms of cos x and denominator remains same.

Adding the integrals, we will get twice the value we have to find out, the numerator and denominator gets cancelled. Thus, we get the value of the required integral.

We know 22.5 degrees is the half of 45 degrees, whose value is known to us.

Thus, we will use the formula of tan (2*theta) in terms of tan(theta) where theta is 22.5 degrees.

Thus, from that, we will get a quadratic equation in terms of tan (2*theta) where tan 45=1

Hence, we get the required value of tan 22.5 

This is a basic question on differential equation as both the components- x and y are already arranged.

We will differentiate both the left and right sides, right hand side being 0, while we separate the integrations for dx/x and dy/y

Thus we get the function of x and y, and yes, do not forget to add the constant with the function as "+c"

This is a question on inverse trigonometry properties.

We will focus on tan^-1(x-y/x+y)

Dividing both the numerator and denominator with y, we will the term with "y" as 1 and "x" as x/y.

Now, the term can be re-simplified with the formulae associated with tan inverse, and we get: tan^-1(x/y) - tan^-11

Thus, from first term, if we subtract this, we will be left with only tan^-11, which is 45 degrees. 

Since, we have the matrix A with us, we can easily find out the transpose of Matrix A, which is represented as A^T .

After that, we will do the matrix addition of both the matrix and its transpose.

In the final matrix, we will compare the term in first row and first column and equate it to 1 as we know the resultant matrix is an identity matrix.

On solving, we will get the value of the angle required.

The basic concept associated with this question is that e^alnx= x^a

Thus, we can change all the terms in the containing numerator and denominator.

Then, we will take common, and after cancelling out the common factors from numerator and denominator, we will be left with a very basic integration.

We know that sum of tan inverse and cot inverse is 90 degrees or pi/2 radians.

From the 3 tan inverse, we take out one of them and club with the cot inverse to get 90 degrees,thus the equation becomes very simple, and we are easily able to get the value of x

In the numerator, we will club terms containing the tan inverse, and then we will break the total term into 2 parts- one containing tan inverse terms and other containing rest terms.

In the term containing tan inverse, we will use integration by parts rule, and we simplify the integration like this, and we see that some terms get cancelled.

Since x is common, the vector is actually (x)i+(x)j+(x)k

The Magnitude of a vector is shown as the square root of the sum of squares of the coefficients of i,j and k.

Since, it is said that the given vector is a unit vector, its magnitude is 1.

Using the 2 given constraints, we will get the value of "x"

cos 2x= 1- 2cos²x

From this formula, we can get the value of cos 2x.

The sum of squares of direction cosines result to 1.

Squaring the sum,doubling it, and subtracting it from 3, we will get the required value.

In this given determinant, all we need to do simply the calculation for magnitude using the row and column-operations.

Now, if any 2 rows or columns are equal, the value of the determinant becomes 0.

After we have simplified the determinant, we find that value of the determinant to be 0.

In order to check continuity and discontinuity of a specific function, we can draw its graph to check at which specific points they are discontinuous.

The greatest integer function is a special type of function which is discontinuous at integral points only and continuous elsewhere, thus as per the options, for any value between n and n+1, where n is an integer,the given function is continuous

This is a very basic question on vectors.

It involves on how to open the dot product of two vectors. The dot product of 2 vectors is nothing but the multiplication of 3 quantities, the magnitude of both the vectors and the cosine of the angle between the 2 vectors.

From this observation above, on equating with the given result, we will be able to find the cosine of the angle, and hence, the angle can be found out.

In order to get the latus rectum, we need to arrange the equation in the form of (y-b)²=4a(x-a).

Once we have rearranged the equation in the form of this, we will get the latus rectum as "4a"

Thus, the major step while solving these questions is how smartly we are arranging the variables of x and y, as the arrangement will simplify the equation in terms of basic form of the parabola 

On simplifying the discriminant using the row and column operations, we will get the solution in terms of xy,yz,xz and xyz. 

Dividing each of the terms with "xyz", we will be left with 1/z,1/x,1/y and 1 respectively.

Thus, rearranging them, we will get the value of x^-1+y^-1+z^-1, as asked in the question

This is a question on integral whose trickery lies on the basic concept of substitution. The better substitution anyone does, the simpler the problem it becomes.

In the question, if we take t=xe^x to be substituted, we see that on differentiating,dt is nothing but the numerator.

Similarly, the denominator,which looks a bit complex, becomes simpler with the substitution.

In order to solve this, we need to know the sample space,that is, total outcomes in rolling 2 dice, which is 6*6=36 cases

Now, in order to get a score of 5, which is the event space here, we need to check from the sample space, which rolls sum up to 5.

The ratio of the event to the sample space will give us the required probability.

We need to know how to find out the magnitude of sum or subtraction of 2 vectors. The magnitude of the vector depends on the magnitudes of both the vectors individually, and the angle included by them.

In the question we are given the information that vectors a,b and a(3)^0.5-b are unit vectors, hence all we need to do is to substitute the values in the equation and get the value of the angle included between vectors a and b. 

In order to solve this, we have 2 steps, one of them is converting tan and cot into sin/cos and cos/sin respectively, and solve as sin²x+cos²x=1

Another step to solve this is turning cot x into 1/tan x, and then multiplying each sides by tan x and rearranging them on one side, we get a quadratic equation on tan x.

Solving them, we get the value of tan x, and hence, we get the value of x.

Integration of f(x) over limits a to b is same as integration of f(a+b-x) over limits a to b.

From the property, we find that the value of integral remains unchanged while the numerator becomes {8+2-(10-x)}^0.5 ,that is, x^0.5 and denominator remains same.

Adding the integrals, we will get twice the value we have to find out, the numerator and denominator gets factorised,and becomes 1. Thus, we get the value of the required integral.

By keen observation, we can find that first term is the ratio of square of 1 to 1, second term is ratio of sum of square upto 2 to the sum upto 2.

Thus, we can say, nth term of the sequence is the ratio of the sum of squares from 1 to n to the sum from 1 to n.

Now, by using the expansion formula of the sum of natural numbers and sum of square of natural numbers, we get the value of nth term.

Now, sum of the sequence is the sum of nth term from n=1 to n, and hence, we will use the formula for expansion of the sum of terms again, and we will get the result.

Apparently, this question looks difficult as it involves a complex term x^y  

In order to avoid the complexity, we will take ln on both sides, left side being y lnx, while right side becomes x-y.

Now, differentiating both sides with respect to x, and rearranging the equation, we will get the required differential equation.

f^-1(x) means that the function f(x), which is defined in terms of x, now, has to be rearranged in such a way that we will define x as f^-1(x), in terms of other variable(here, y) which we represented as f(x)

Remember, after writing x in terms of y, when we define f^-1(x), we will convert each and every term containing y into x, as shown in the options.