All questions of Mathematics for JEE Exam

No

Solution:

Given, (82)1/4.

Let's solve step by step:

Finding the prime factorization of 82,
82 = 2 × 41

Now, applying the exponent 1/4 to both sides,
(82)1/4 = (2 × 41)1/4

Using the property of exponents, (ab)c = acbc,
(2 × 41)1/4 = 21/4 × 41/4

Evaluating the values,
21/4 = 1.1892
41/4 = 2.2974

Multiplying the values,
(82)1/4 = 1.1892 × 2.2974

(82)1/4 = 2.735

Approximating the value to two decimal places,

(82)1/4 ≈ 3.01

Therefore, the correct option is (c) 3.00925.

(sec-1x) = cos-1(1/x) = tan-1(√(x^2-1)/x)

Properties of Invertible Matrices:

1. (AA-1) = (A-1 A) = I
2. AB = BA = I
3. (A-1)-1 = A
4. (kA)-1 = (1/k)A-1, where k is a non-zero scalar
5. (AB)-1 = B-1 A-1

Explanation:

Option A states that (AB)-1 = A-1 B-1, which is not a property of invertible matrices. This statement is false because in general, (AB)-1 ≠ A-1 B-1.

To see why this is true, consider the case where A and B are both 2x2 matrices:

A = [a b]
[c d]

B = [e f]
[g h]

Then AB is given by:

AB = [ae+bg af+bh]
[ce+dg cf+dh]

The inverse of AB, assuming it exists, is given by:

(AB)-1 = 1/det(AB) [dh -bh -cf af]
[-dg ae ce -af]

where det(AB) = (ae+bg)(cf+dh)-(af+bh)(ce+dg)

On the other hand, the inverse of A and B are given respectively by:

A-1 = 1/det(A) [d -b]
[-c a]

B-1 = 1/det(B) [h -f]
[-g e]

where det(A) = ad-bc and det(B) = eh-fg

Now, if we compute A-1 B-1, we get:

A-1 B-1 = 1/det(A)det(B) [(dh-bgf-eh+fg) -(bh-ae-hf+cf)]
[(-dg+ce+bg-af) (ag-ce-df+ae)]

Notice that A-1 B-1 is not equal to (AB)-1 in general, unless A and B commute (i.e. AB = BA).

Therefore, option A is not a property of invertible matrices.

Solution:

To find the derivative of xn, we can use the power rule of differentiation.

The power rule states that if we have a function of the form f(x) = xn, then the derivative of f(x) with respect to x is given by:

f'(x) = n*x^(n-1)

Now let's apply this rule to the function xn:

Derivative of xn:

Using the power rule, we have:

f'(x) = n*x^(n-1)

In this case, the function is xn, so we substitute n for the exponent:

f'(x) = n*x^(n-1)

Therefore, the derivative of xn is n*x^(n-1).

Explanation:

The power rule is a fundamental rule in calculus that allows us to find the derivative of a function raised to a power. It tells us that when differentiating a function of the form f(x) = xn, the exponent n becomes the coefficient in front of the term, and the exponent is reduced by 1.

This rule can be derived using the limit definition of the derivative and the binomial theorem. However, it is commonly taught as a basic rule in calculus because it simplifies the process of finding derivatives for polynomial functions.

In the case of xn, the derivative is n*x^(n-1). This means that the coefficient in front of the term is n, and the exponent is reduced by 1 to (n-1).

The power rule is a powerful tool in calculus as it allows us to find the derivative of a wide range of functions easily. By applying this rule, we can find the derivative of xn, which is n*x^(n-1).

Equivalence relation

An equivalence relation is a relation that satisfies three properties: reflexivity, symmetry, and transitivity.

Reflexive relation
A reflexive relation is a relation where every element is related to itself. In this case, the relation R defined as R = {(I1, I2):I1 is parallel to I2} is not reflexive because a line cannot be considered parallel to itself.

Symmetric relation
A symmetric relation is a relation where if (a, b) is in the relation, then (b, a) is also in the relation. In this case, the relation R defined as R = {(I1, I2):I1 is parallel to I2} is not symmetric because if line I1 is parallel to line I2, it does not necessarily mean that line I2 is parallel to line I1.

Transitive relation
A transitive relation is a relation where if (a, b) and (b, c) are in the relation, then (a, c) is also in the relation. In this case, the relation R defined as R = {(I1, I2):I1 is parallel to I2} is transitive because if line I1 is parallel to line I2, and line I2 is parallel to line I3, then it follows that line I1 is parallel to line I3. Therefore, it satisfies the transitive property.

Equivalence relation
To be an equivalence relation, a relation must satisfy all three properties: reflexivity, symmetry, and transitivity. Since the given relation R satisfies the transitive property, it can be considered an equivalence relation.

Therefore, the correct answer is option 'B' - Equivalence relation.

B)A^T = A
c)A = A^-1
d)A = 0

b) A^T = A holds true for a symmetric matrix.

The volume of a cube is given by the formula V = s^3, where s is the length of one side of the cube. We are given that the cube changes its volume while maintaining its shape, so the side length also changes. Let's denote the rate of change of the side length as ds/dt, and the rate of change of the volume as dV/dt.

To find the rate of change of the volume, we can differentiate the volume formula with respect to time:

dV/dt = d/dt (s^3)

To simplify the differentiation, we can use the chain rule:

dV/dt = 3s^2 * ds/dt

Now we need to relate the rate of change of the volume to the rate of change of the area of any face of the cube. The area of a face of the cube is given by the formula A = s^2. We can differentiate this formula with respect to time to find the rate of change of the area:

dA/dt = d/dt (s^2)

Using the chain rule again:

dA/dt = 2s * ds/dt

We can rewrite this equation as:

ds/dt = (1/2s) * dA/dt

Substituting this expression for ds/dt into the equation for dV/dt, we get:

dV/dt = 3s^2 * [(1/2s) * dA/dt]

Simplifying:

dV/dt = 3/2 * s * dA/dt

Since the volume of the cube is 1 (given in the question as "unit volume"), we have s = 1. Substituting this value into the equation, we get:

dV/dt = 3/2 * dA/dt

Therefore, the rate of change of the volume is equal to 3/2 times the rate of change of the area of any face of the cube. Hence, the correct answer is option 'D' - 3/2 times the rate of change of the area of any face of the cube.

To find the length of a tangent from a point to a circle, we can use the formula for the length of a tangent:

Length of tangent = √(r^2 - d^2)

where r is the radius of the circle and d is the distance from the center of the circle to the given point.

Given the equation of the circle as 2x^2 + 2y^2 + 5x + y = 15, we need to first rewrite it in the standard form of a circle equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

1. Convert the given equation into standard form:
2x^2 + 2y^2 + 5x + y = 15
Divide the equation by 2 to simplify:
x^2 + y^2 + (5/2)x + (1/2)y = 7.5
Rearrange the terms:
x^2 + (5/2)x + y^2 + (1/2)y = 7.5
Complete the square for x terms by adding (5/4)^2 = 25/16 inside the parentheses:
x^2 + (5/2)x + 25/16 + y^2 + (1/2)y = 7.5 + 25/16
Complete the square for y terms by adding (1/4)^2 = 1/16 inside the parentheses:
x^2 + (5/2)x + 25/16 + y^2 + (1/2)y + 1/16 = 7.5 + 25/16 + 1/16
Simplify:
(x + 5/4)^2 + (y + 1/4)^2 = 127/16

2. Compare the equation with the standard form:
(x - h)^2 + (y - k)^2 = r^2

We can see that the center of the circle is (-5/4, -1/4) and the radius squared is 127/16.

3. Find the distance from the center to the given point:
Using the distance formula, the distance from the center (-5/4, -1/4) to the point (7, 2) is:
d = √((7 - (-5/4))^2 + (2 - (-1/4))^2)
Simplify:
d = √((7 + 5/4)^2 + (2 + 1/4)^2)
d = √((28/4 + 5/4)^2 + (8/4 + 1/4)^2)
d = √((33/4)^2 + (9/4)^2)
d = √(1089/16 + 81/16)
d = √(1170/16)
d = √73.125

4. Calculate the length of the tangent:
Using the formula for the length of a tangent, we can find the length of the tangent from the point (7, 2) to the circle:
Length of tangent = √(r^2 - d^2)
Length of tangent = √(127/16 - 73.125)
Length of tangent = √

Differential Equation of Parabolas with Directrices Parallel to x-axis

To derive the differential equation for all parabolas whose directrices are parallel to the x-axis, we can use the standard equation of a parabola with directrix at y = k:

(y - k)^2 = 4p(x - h)

where (h, k) is the vertex and p is the distance from the vertex to the focus.

Since the directrix is parallel to the x-axis, we have k = -p. Substituting this into the equation, we get:

(y + p)^2 = 4p(x - h)

Differentiating both sides with respect to x, we get:

2(y + p) (dy/dx) = 4p

Simplifying, we get:

dy/dx = 2p/(y + p)

To eliminate p, we can use the fact that the distance from the vertex to the focus is equal to the distance from the vertex to the directrix. Since the directrix is parallel to the x-axis, this distance is simply the absolute value of p. Therefore, we have:

p = |(y - k)/2|

Substituting this into the previous equation, we get:

dy/dx = 2|(y - k)/2|/(y + |(y - k)/2|)

Simplifying, we get:

dy/dx = (y - k)/|y - k|

Taking the derivative of both sides with respect to x, we get:

d^2y/dx^2 = d/dx[(y - k)/|y - k|]

Using the quotient rule, we get:

d^2y/dx^2 = [(dy/dx)|y - k| - (y - k) d/dx|y - k|]/|y - k|^2

Since d/dx|y - k| = 0 when y = k, we have:

d^2y/dx^2 = [(dy/dx)|y - k|]/|y - k|^2

To eliminate the absolute value, we can consider two cases:

Case 1: y > k

In this case, we have |y - k| = y - k. Substituting this into the previous equation, we get:

d^2y/dx^2 = [(dy/dx)(y - k)]/(y - k)^2

Simplifying, we get:

d^2y/dx^2 = dy/dx/(y - k)

Taking the derivative of both sides with respect to x, we get:

d^3y/dx^3 = d/dx[dy/dx/(y - k)]

Using the quotient rule, we get:

d^3y/dx^3 = [(d^2y/dx^2)(y - k) - dy/dx]/(y - k)^2

Substituting d^2y/dx^2 = dy/dx/(y - k), we get:

d^3y/dx^3 = -2(dy/dx)/(y - k)^3

Case 2: y < />

In this case, we have |y - k| = -(y - k). Substituting this into the previous equation, we get:

d^2y/dx^2 = [(dy/d

There seems to be a mistake in the question. Can you please provide more information or clarify the question?

Explanation:

Given,
Distance travelled, x = x
Time taken, t = x2/2x

To find the acceleration and retardation of the particle, we need to differentiate the distance equation with respect to time.

Differentiating x with respect to time, we get the velocity equation, v.

Differentiating v with respect to time, we get the acceleration equation, a.

Differentiating a with respect to time, we get the retardation equation, r.

Finding Velocity:

We know that,
v = dx/dt

Differentiating x with respect to time, we get
v = x/t

Substituting the value of t from the given equation, we get
v = 2x/x2
v = 2/x

Finding Acceleration:

We know that,
a = dv/dt

Differentiating v with respect to time, we get
a = -2/x2

Finding Retardation:

We know that,
r = da/dt

Differentiating a with respect to time, we get
r = 4/x3

Comparing the options with the derived equations, we can see that Option A is correct.

Therefore, the retardation of the particle is the cube of its velocity.

Understanding the Relationship Between a Plane and a Line
When we have a plane defined by the equation \( ax + by + cz + d = 0 \) and a line characterized by direction ratios \( a_1, b_1, c_1 \), it is essential to understand how their orientations relate when the plane and the line are parallel.

Condition for Parallelism
For a line to be parallel to a plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. The normal vector of the plane given by \( ax + by + cz + d = 0 \) is represented by the coefficients \( (a, b, c) \).

Mathematical Representation
The mathematical condition for the perpendicularity of the line's direction ratios \( (a_1, b_1, c_1) \) to the normal vector \( (a, b, c) \) can be expressed as:
\[ a_1a + b_1b + c_1c = 0 \]
This equation states that the dot product of the line's direction ratios and the plane's normal vector equals zero, confirming that they are perpendicular. Hence, if this condition holds, the line is parallel to the plane.

Why Option 'B' is Correct
- **Option (b)**: \( a_1a + b_1b + c_1c = 0 \) indicates that the line is parallel to the plane since the dot product condition for perpendicular vectors is satisfied.
Other options do not satisfy this critical condition for parallelism.

Conclusion
Understanding this relationship is vital in geometry and vector analysis, ensuring accurate interpretations of the spatial relationships between various geometric entities. Thus, option 'B' is the correct choice, confirming the necessary condition for the parallelism of the line to the plane.

There is not enough information to determine which value is similar to sin-1sin(6.

Given data:
x = t^3 + 6t^2 - 15t + 18

Calculating velocity:
To find the velocity of the particle, we need to differentiate x with respect to t.

dx/dt = d/dt (t^3 + 6t^2 - 15t + 18)
dx/dt = 3t^2 + 12t - 15

Calculating acceleration:
Similarly, to find the acceleration, we differentiate the velocity with respect to time.

d^2x/dt^2 = d/dt (3t^2 + 12t - 15)
d^2x/dt^2 = 6t + 12

Calculating acceleration at t=2:
To find the acceleration at t=2 seconds, we substitute t=2 into the acceleration equation.

d^2x/dt^2 = 6(2) + 12
d^2x/dt^2 = 12 + 12
d^2x/dt^2 = 24 cm/sec^2

Therefore, the acceleration of the particle at the end of 2 seconds is 24 cm/sec^2, which corresponds to option C.

The equation x = t^4/12 represents the motion of the particle as it moves in a straight line. The distance x from a fixed point on the line at any time t seconds is given by this equation.

To understand the meaning of this equation, let's break it down:

- The variable x represents the distance of the particle from a fixed point on the line. It can be measured in any unit of length (e.g., meters, feet).
- The variable t represents time, measured in seconds.
- The equation x = t^4/12 indicates that the distance x is equal to t raised to the power of 4, divided by 12.

This equation suggests that the distance of the particle from the fixed point increases as time passes, and the rate of increase is proportional to the fourth power of time. The constant factor of 1/12 determines the scale of this relationship.

For example, if t = 2 seconds, then x = (2^4)/12 = 16/12 = 4/3. This means that at 2 seconds, the particle is located 4/3 units away from the fixed point.

Similarly, if t = 3 seconds, then x = (3^4)/12 = 81/12 = 6.75. This means that at 3 seconds, the particle is located approximately 6.75 units away from the fixed point.

In summary, the equation x = t^4/12 describes the distance x of a particle from a fixed point on a straight line as a function of time t.

The given equation is:
(y(dy/dx) + 2x)^2 = (y^2 - 2x^2)(1 - (dy/dx)^2)

Expanding the left-hand side of the equation:
(y^2(dy/dx)^2 + 4xy(dy/dx) + 4x^2) = (y^2 - 2x^2)(1 - (dy/dx)^2)

Expanding the right-hand side of the equation:
y^2(dy/dx)^2 + 4xy(dy/dx) + 4x^2 = y^2 - 2x^2 - y^2(dy/dx)^2 + 2x^2(dy/dx)^2

Combining like terms:
2y^2(dy/dx)^2 + 6x^2(dy/dx) - 2x^2 = 0

Simplifying the equation:
y^2(dy/dx)^2 + 3x^2(dy/dx) - x^2 = 0

This is a quadratic equation in terms of (dy/dx). We can solve this equation using the quadratic formula:

(dy/dx) = (-3x^2 ± √(9x^4 + 4x^2y^2)) / 2y^2

Therefore, the solution is:
(dy/dx) = (-3x^2 ± √(9x^4 + 4x^2y^2)) / 2y^2

Thus, the solution of the given equation is:
(dy/dx) = (-3x^2 ± √(9x^4 + 4x^2y^2)) / 2y^2

Note that "c" is not the correct answer.

The differential equation that represents the family of solutions y = ae^3x * be^x is:

d^2y/dx^2 - 4dy/dx + 3y = 0

Differentiation is a mathematical operation that measures the rate at which a function changes with respect to its independent variable. In this question, we need to differentiate the given expression, which consists of two terms: 8e^(-x) and 2ex, with respect to x.

To differentiate each term, we can use the differentiation rules for exponential functions and constant multiples.

Differentiating 8e^(-x):
- The derivative of e^(-x) with respect to x is -e^(-x), according to the chain rule.
- Since 8 is a constant, its derivative with respect to x is 0.
- Multiplying these two derivatives together gives us -8e^(-x).

Differentiating 2ex:
- The derivative of ex with respect to x is ex, according to the chain rule.
- Since 2 is a constant, its derivative with respect to x is 0.
- Multiplying these two derivatives together gives us 2ex.

Adding the derivatives of the two terms together gives us the final result:

-8e^(-x) + 2ex

Thus, the correct answer is option 'D'.

To find the coordinates of the foot of the normal to the parabola y^2 = 3x, which is perpendicular to the line y = 2x + 4, we can use the properties of perpendicular lines and the concept of the slope of a line.

Let's break down the problem into steps:

Step 1: Find the slope of the given line.
The given line is y = 2x + 4, which is in the slope-intercept form y = mx + b, where m is the slope of the line. Comparing this equation with the standard form, we can see that the slope of the line is 2.

Step 2: Find the slope of the normal to the parabola.
The slope of the normal to the parabola is the negative reciprocal of the slope of the tangent to the parabola at a given point. To find this slope, we need to find the tangent to the parabola at a point.

Step 3: Find the point of tangency.
To find the point of tangency, we can equate the slope of the tangent to the slope of the parabola at a given point. Let's consider a point (h, k) on the parabola. The slope of the parabola at this point can be found by differentiating the equation of the parabola with respect to x.

Differentiating y^2 = 3x with respect to x, we get:
2yy' = 3
y' = 3/(2y)

Since the slope of the tangent is the derivative of the y-coordinate with respect to the x-coordinate, we can substitute the coordinates of the point (h, k) into y' to find the slope of the tangent at that point.

Substituting h and k into y', we get:
m_tangent = 3/(2k)

Step 4: Find the slope of the normal.
The slope of the normal is the negative reciprocal of the slope of the tangent. Therefore, the slope of the normal is:
m_normal = -2k/3

Step 5: Find the point of intersection.
We know that the normal to the parabola is perpendicular to the given line y = 2x + 4. Therefore, the product of the slopes of the two lines should be -1.

(-2k/3) * 2 = -1
-4k/3 = -1
k = 3/4

Substituting k = 3/4 into the equation of the parabola, we can solve for h:

(3/4)^2 = 3h
9/16 = 3h
h = 3/16

Therefore, the point of intersection is (3/16, 3/4).

Step 6: Find the foot of the normal.
The foot of the normal is the point on the parabola that is closest to the given line. Since the normal is perpendicular to the line, the foot of the normal is the projection of the point of intersection onto the line.

To find the foot of the normal, we need to find the equation of the line passing through the point of intersection (3/16, 3/4) and perpendicular to the given line y = 2x + 4.

The equation of a line passing through a point (x1, y

Relation Types

- Reflexive
- Symmetric
- Transitive
- Surjective

Explanation

A relation is a set of ordered pairs that relate elements from two sets. The four types of relations are reflexive, symmetric, transitive, and surjective.

Reflexive Relation: A relation is reflexive if every element of the set is related to itself. For example, the relation "is equal to" is reflexive because every element is equal to itself.

Symmetric Relation: A relation is symmetric if whenever (a,b) is in the relation, then (b,a) is also in the relation. For example, the relation "is a sibling of" is symmetric because if A is a sibling of B, then B is also a sibling of A.

Transitive Relation: A relation is transitive if whenever (a,b) and (b,c) are in the relation, then (a,c) is also in the relation. For example, the relation "is an ancestor of" is transitive because if A is an ancestor of B and B is an ancestor of C, then A is also an ancestor of C.

Surjective Relation: A relation is surjective if every element in the second set is related to at least one element in the first set. For example, the function f(x) = x^2 is surjective because every non-negative number has a square root.

Answer

Therefore, the correct answer is option 'B' (Surjective) because it is not a type of relation. It is a property of functions.

The correct option is (a) 1/v2 = 1/2u2 + kt.

We can use the equation of motion for uniform retardation, which is given by:

v2 = u2 - 2ax

where v is the velocity at time t, u is the initial velocity, a is the retardation, and x is the displacement at time t.

In this case, the retardation is kv3, so we can substitute a = kv3 into the equation of motion:

v2 = u2 - 2kxv3

We can rearrange this equation to get:

1/v2 = 1/u2 - 2kx/u2v3

We know that x is the displacement at time t, so we can substitute x = vt into the equation above:

1/v2 = 1/u2 - 2kt

Rearranging this equation gives us:

1/v2 = 1/2u2 + kt

Therefore, option (a) is correct.

Given: The edge of a cube is increasing at a rate of 7 cm/s and x=6 cm.

To find: The rate of change of area of the cube.

Solution:

Let the edge of the cube be x.

Volume of the cube, V = x^3

Differentiating both sides w.r.t time t, we get

dV/dt = 3x^2 dx/dt

Surface area of the cube, A = 6x^2

Differentiating both sides w.r.t time t, we get

dA/dt = 12x dx/dt

Given, dx/dt = 7 cm/s and x=6 cm

Substituting the values in the above equations, we get

dV/dt = 3(6^2)×7 = 756 cm^3/s

dA/dt = 12×6×7 = 504 cm^2/s

Therefore, the rate of change of area of the cube when x=6 cm is 504 cm^2/s.

Answer: Option C (504 cm^2/s)

Possible Operations on Matrices

There are six elementary operations that can be performed on matrices:

1. Interchange two rows or columns.
2. Multiply a row or column by a non-zero scalar.
3. Add a multiple of one row or column to another row or column.
4. Transpose (switch rows and columns).
5. Invert (find the inverse of) a matrix.
6. Multiply two matrices.

Explanation

1. Interchange two rows or columns: This operation involves swapping the positions of two rows or columns in a matrix. For example, if we have a matrix A with two rows, we can interchange the first and second rows to get a new matrix A' where the first row of A is now the second row of A' and vice versa.

2. Multiply a row or column by a non-zero scalar: This operation involves multiplying all the elements in a row or column of a matrix by a non-zero scalar. For example, if we have a matrix A and we multiply the second row by 2, we get a new matrix A' where all the elements in the second row of A are multiplied by 2.

3. Add a multiple of one row or column to another row or column: This operation involves adding a multiple of one row or column of a matrix to another row or column. For example, if we have a matrix A and we add 3 times the first row to the second row, we get a new matrix A' where the second row of A' is equal to the original second row of A plus 3 times the first row of A.

4. Transpose: This operation involves switching the rows and columns of a matrix. For example, if we have a matrix A with two rows and three columns, we can transpose it to get a new matrix A' with three rows and two columns.

5. Invert: This operation involves finding the inverse of a matrix. The inverse of a matrix A is denoted by A^-1 and is a matrix such that the product of A and A^-1 is the identity matrix.

6. Multiply two matrices: This operation involves multiplying two matrices together. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix. The elements of the resulting matrix are obtained by multiplying the corresponding elements of the rows and columns of the two matrices and then summing the products.

This is a pair of values, both of which are represented by the letter "a". Without any additional context or information, it is impossible to determine the meaning or significance of this particular pair. It could represent anything from a coordinate on a graph to a value in a dataset to a pair of parameters in a mathematical equation.

Transitivity and Reflexivity

To determine if a relation is transitive, we need to check if whenever there are two pairs (a, b) and (b, c) in the relation, then the pair (a, c) is also in the relation. In other words, if (a, b) and (b, c) are in the relation, then (a, c) should also be in the relation.

On the other hand, reflexivity means that every element in the set is related to itself. In other words, for every element a in the set, (a, a) should be in the relation.

Given Set and Relations

The given set S = {3, 4, 6} consists of three elements.

Let's analyze each relation to determine if it is transitive and reflexive:

Option a) R = {(3, 4), (4, 6), (3, 6)}
- This relation is transitive because if we have (3, 4) and (4, 6), it implies that (3, 6) should also be in the relation.
- However, this relation is not reflexive because it does not contain the pairs (3, 3), (4, 4), and (6, 6).

Option b) R = {(1, 2), (1, 3), (1, 4)}
- This relation is not transitive because there are no pairs (a, c) such that (a, b) and (b, c) are both in the relation.
- This relation is also not reflexive because it does not contain the pairs (3, 3), (4, 4), and (6, 6).

Option c) R = {(3, 3), (4, 4), (6, 6)}
- This relation is reflexive because it contains the pairs (3, 3), (4, 4), and (6, 6).
- However, this relation is not transitive because there are no pairs (a, c) such that (a, b) and (b, c) are both in the relation.

Option d) R = {(3, 4), (4, 3)}
- This relation is not transitive because it does not contain the pair (3, 3) or (4, 4) to satisfy the transitive property.
- This relation is not reflexive because it does not contain the pairs (3, 3), (4, 4), and (6, 6).

Conclusion

Among the given options, only option a) R = {(3, 4), (4, 6), (3, 6)} is transitive because it satisfies the condition that if (a, b) and (b, c) are in the relation, then (a, c) should also be in the relation. However, it is not reflexive because it does not contain the pairs (3, 3), (4, 4), and (6, 6).

Explanation:

To understand the number of arbitrary constants in the general solution of a second-order differential equation, let's first review the concept of a general solution.

General solution of a differential equation:
The general solution of a differential equation is a solution that contains all possible solutions of the equation. It typically contains arbitrary constants that can take any value, which allows for a wide range of solutions.

Second-order differential equation:
A second-order differential equation involves a second derivative (d²y/dx²) of a dependent variable y with respect to an independent variable x. It can be expressed in the form:
a(d²y/dx²) + b(dy/dx) + cy = f(x)

Order of a differential equation:
The order of a differential equation is the highest order of derivative present in the equation. For a second-order differential equation, the highest order of derivative is 2.

Number of arbitrary constants:
The number of arbitrary constants in the general solution of a second-order differential equation is equal to the order of the equation. In this case, since we have a second-order differential equation, the number of arbitrary constants in the general solution will be 2.

Therefore, the correct answer is option C - 2.

Explanation:

To determine the correct answer, we need to understand the concept of a matrix and the number of elements it contains.

Matrix:
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The number of rows and columns in a matrix determines its dimensions. For example, a matrix with m rows and n columns is called an m × n matrix.

Number of elements in a matrix:
The number of elements in a matrix is equal to the product of its number of rows and columns. For example, a matrix with m rows and n columns will have m × n elements.

In this question, we are given that the matrix has 6 elements. Therefore, the product of the number of rows and columns must be equal to 6.

Solution:
Let's analyze each option to determine if it is a possible ordered pair for a matrix with 6 elements:

a) (2,3):
If the matrix has 2 rows and 3 columns, the number of elements will be 2 × 3 = 6. So, option a) is a possible ordered pair for a matrix with 6 elements.

b) (3,1):
If the matrix has 3 rows and 1 column, the number of elements will be 3 × 1 = 3. This does not match the given requirement of having 6 elements. Therefore, option b) is not a possible ordered pair for a matrix with 6 elements.

c) (1,6):
If the matrix has 1 row and 6 columns, the number of elements will be 1 × 6 = 6. So, option c) is a possible ordered pair for a matrix with 6 elements.

d) (3,2):
If the matrix has 3 rows and 2 columns, the number of elements will be 3 × 2 = 6. So, option d) is a possible ordered pair for a matrix with 6 elements.

Therefore, the correct answer is option b) because (3,1) is not a possible ordered pair for a matrix with 6 elements.