Step 1: Concept Clarification:

The degree of a differential equation refers to the exponent of the highest-order derivative, assuming the equation is polynomial in the derivatives. If the equation contains fractional powers or irrational terms, we first eliminate them to convert the equation into a polynomial form.


Step 2: Explanation:

Starting with the given equation: \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \]
To remove the fractional exponent, we raise both sides to the power of 5: \[ \left( \frac{d^2y}{dx^2} \right)^5 = y + \frac{dy}{dx} \]
Now the equation is polynomial in the derivatives. The highest-order derivative is \( \frac{d^2y}{dx^2} \), and it is raised to the power of 5. Hence, the degree of the equation is 5.

Step 3: Final Conclusion:

Thus, the degree of the differential equation is 5. Quick Tip: To determine the degree of a differential equation, first make sure the equation is polynomial in the derivatives. If there are fractional or irrational powers, eliminate them by raising both sides to the appropriate power.

Step 1: Applying the Trigonometric Identity:

We begin by using the trigonometric identity for \( \cos^2 x \) to simplify the integral: \[ \cos^2 x = \frac{1 + \cos(2x)}{2} \]
This transforms the integral into: \[ \int \cos^2 x \, dx = \int \frac{1 + \cos(2x)}{2} \, dx \]

Step 2: Breaking Down the Integral:

Next, we separate the integral into two parts: \[ = \frac{1}{2} \int 1 \, dx + \frac{1}{2} \int \cos(2x) \, dx \]

Step 3: Computing the Integrals:

The first integral is simple: \[ \int 1 \, dx = x \]
For the second integral, we use the standard result for the cosine function: \[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) \]

Step 4: Putting It All Together:

Combining the results from the previous steps, we get: \[ \int \cos^2 x \, dx = \frac{x}{2} + \frac{1}{4} \sin(2x) + c \]

Step 5: Conclusion:

Therefore, the final solution is \( \frac{x}{2} + \frac{1}{4} \sin(2x) + c \). Quick Tip: To solve integrals involving \( \cos^2 x \), always use the identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) to simplify the expression.

Step 1: Formula for the Angle Between Vectors:

The angle \( \theta \) between two vectors \( \mathbf{A} \) and \( \mathbf{B} \) can be calculated using the following formula: \[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]
where:
- \( \mathbf{A} \cdot \mathbf{B} \) is the dot product of the vectors,
- \( |\mathbf{A}| \) and \( |\mathbf{B}| \) represent the magnitudes of the vectors.

Step 2: Compute the Dot Product \( \mathbf{A} \cdot \mathbf{B} \):

We first calculate the dot product as follows: \[ \mathbf{A} \cdot \mathbf{B} = (2 \times 3) + (1 \times -2) + (3 \times 1) = 6 - 2 + 3 = 7 \]

Step 3: Compute the Magnitudes of the Vectors:

The magnitude of \( \mathbf{A} \) is: \[ |\mathbf{A}| = \sqrt{(2^2 + 1^2 + 3^2)} = \sqrt{4 + 1 + 9} = \sqrt{14} \]
Similarly, the magnitude of \( \mathbf{B} \) is: \[ |\mathbf{B}| = \sqrt{(3^2 + (-2)^2 + 1^2)} = \sqrt{9 + 4 + 1} = \sqrt{14} \]

Step 4: Calculate \( \cos \theta \):

Substitute the values into the formula: \[ \cos \theta = \frac{7}{\sqrt{14} \times \sqrt{14}} = \frac{7}{14} = \frac{1}{2} \]

Step 5: Determine the Angle \( \theta \):

Thus, the angle \( \theta \) is given by: \[ \theta = \cos^{-1} \left( \frac{1}{2} \right) = 60^\circ \]

Step 6: Final Answer:

The angle between the vectors is \( 60^\circ \). Quick Tip: To find the angle between two vectors, use the formula \( \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \), and make sure to calculate both the dot product and the magnitudes correctly.

Step 1: Defining the Relation Between Two Sets

A relation between two sets \( A \) and \( B \) is a subset of the Cartesian product \( A \times B \), consisting of ordered pairs \( (a, b) \), where \( a \in A \) and \( b \in B \). The total number of elements in \( A \times B \) is \( m \times n \), where \( m \) is the number of elements in set \( A \), and \( n \) is the number of elements in set \( B \).


Step 2: Calculating the Total Number of Relations

Since a relation is simply a subset of \( A \times B \), the total number of relations corresponds to the number of subsets of \( A \times B \). The number of subsets of a set with \( k \) elements is \( 2^k \). Therefore, the total number of relations is \( 2^{m \times n} \).


Step 3: Conclusion

Hence, the total number of relations from set \( A \) to set \( B \) is \( 2^{mn} \). The correct answer is \( 2^{mn} \), which corresponds to option (B). Quick Tip: To find the total number of relations between two sets, calculate the total number of elements in the Cartesian product \( A \times B \), which is \( m \times n \), and then raise 2 to the power of \( m \times n \) to get the total number of subsets (relations).

Step 1: Defining a Function from \( A \) to \( B \):

A function from set \( A \) to set \( B \) is a collection of ordered pairs where each element of \( A \) is associated with exactly one element of \( B \). In other words, for each \( a \in A \), there must be a unique \( b \in B \) such that \( (a, b) \) is included in the function.


Step 2: Evaluating the Given Options:

Now, let's examine each option to determine which represents a valid function:


(A) \( \{ (1, 2), (1, 3), (2, 3), (3, 3) \} \): This is not a function because the element \( 1 \) is related to both \( 2 \) and \( 3 \), which violates the condition that each element in \( A \) must be mapped to only one element in \( B \).
(B) \( \{ (1, 3), (2, 4) \} \): This is a valid function because each element of \( A \) is associated with exactly one element of \( B \).
(C) \( \{ (1, 3), (2, 2), (3, 3) \} \): This is also a valid function, but it does not fit the format of the correct answer in the question.
(D) \( \{ (1, 2), (2, 3), (3, 2), (3, 4) \} \): This is not a function because \( 3 \) is related to both \( 2 \) and \( 4 \), violating the uniqueness condition.


Step 3: Conclusion:

Therefore, the correct answer is option (B) \( \{ (1, 3), (2, 4) \} \), as it satisfies the definition of a valid function. Quick Tip: In a function, each element from the domain (set \( A \)) must be related to exactly one element from the codomain (set \( B \)).

Step 1: Definition of Continuity.

A function is continuous at a point \( x = c \) if the following condition holds: \[ \lim_{x \to c} f(x) = f(c). \]
In this case, we need to prove that the function is continuous at \( x = 2 \), meaning we must show: \[ \lim_{x \to 2^-} f(x) = f(2) = \lim_{x \to 2^+} f(x). \]

Step 2: Left-hand Limit.

For \( x \leq 2 \), we use the expression \( f(x) = x^3 - 3 \). The left-hand limit is: \[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^3 - 3) = 2^3 - 3 = 8 - 3 = 5. \]

Step 3: Right-hand Limit.

For \( x > 2 \), we use the expression \( f(x) = x^2 + 1 \). The right-hand limit is: \[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x^2 + 1) = 2^2 + 1 = 4 + 1 = 5. \]

Step 4: Value of the Function at \( x = 2 \).

Since \( x = 2 \) is in the domain where \( x \leq 2 \), we use the expression \( f(x) = x^3 - 3 \) to find: \[ f(2) = 2^3 - 3 = 8 - 3 = 5. \]

Step 5: Conclusion.

We have shown that: \[ \lim_{x \to 2^-} f(x) = f(2) = \lim_{x \to 2^+} f(x) = 5. \]
Thus, the function is continuous at \( x = 2 \).
Quick Tip: To prove continuity, verify the left-hand and right-hand limits as well as the value of the function at the point.

Step 1: Differentiate with respect to \( x \).

To eliminate the constants \( a \) and \( b \), we differentiate the given equation twice. First, differentiate with respect to \( x \): \[ \frac{dy}{dx} = a \cos(x + b). \]

Step 2: Second Derivative.

Now, differentiate again: \[ \frac{d^2y}{dx^2} = -a \sin(x + b). \]

Step 3: Express \( a \) in terms of \( y \).

From the original equation \( y = a \sin(x + b) \), solve for \( a \): \[ a = \frac{y}{\sin(x + b)}. \]

Step 4: Substitute into the second derivative.

Substitute \( a \) into the second derivative equation: \[ \frac{d^2y}{dx^2} = -\frac{y}{\sin(x + b)} \sin(x + b). \]
Simplifying, we get: \[ \frac{d^2y}{dx^2} = -y. \]

Step 5: Conclusion.

Thus, the differential equation of the family of curves is: \[ \frac{d^2y}{dx^2} + y = 0. \] Quick Tip: To eliminate constants from a family of curves, differentiate the equation as many times as necessary and express the constants in terms of the variables.

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Step 1: Subtract \(7x\) from both sides.

\[ 8x + 4 < 7x + 8 \]
\[ 8x - 7x + 4 < 8 \]
\[ x + 4 < 8 \]

Step 2: Subtract 4 from both sides.

\[ x < 4 \]

Final Answer:

The solution to the inequality is \( \boxed{x < 4} \). Quick Tip: To solve linear inequalities, treat them like equations, but remember to reverse the inequality sign if you multiply or divide both sides by a negative number.

Step 1: Formula for the Coordinates of the Dividing Point.

The formula for the coordinates of a point dividing a line segment internally in the ratio \( m : n \) is given by: \[ P = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}, \frac{mz_2 + nz_1}{m + n} \right). \]
Let the points be \( A(2, -5, 1) \) and \( B(1, 4, -6) \), and the ratio be \( 2 : 3 \).

Step 2: Apply the Formula.

Using the formula, we find the coordinates of the point \( P \) that divides the line joining \( A \) and \( B \) in the ratio 2:3. \[ P_x = \frac{2 \times 1 + 3 \times 2}{2 + 3} = \frac{2 + 6}{5} = \frac{8}{5}, \] \[ P_y = \frac{2 \times 4 + 3 \times (-5)}{2 + 3} = \frac{8 - 15}{5} = \frac{-7}{5}, \] \[ P_z = \frac{2 \times (-6) + 3 \times 1}{2 + 3} = \frac{-12 + 3}{5} = \frac{-9}{5}. \]

Step 3: Conclusion.

Thus, the coordinates of the point dividing the line internally in the ratio 2:3 are: \[ \left( \frac{8}{5}, \frac{-7}{5}, \frac{-9}{5} \right). \] Quick Tip: When dividing a line segment in a given ratio, use the section formula to calculate the coordinates of the dividing point.

Step 1: Formula for the Area of a Triangle.

The area \( A \) of a triangle formed by two vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ A = \frac{1}{2} |\vec{a} \times \vec{b}|. \]
We need to compute the cross product \( \vec{a} \times \vec{b} \).

Step 2: Compute the Cross Product.

The cross product of \( \vec{a} = 3\hat{i} - \hat{j} + 5\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - \hat{k} \) is calculated as: \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
3 & -1 & 5
1 & 2 & -1 \end{vmatrix}. \]
Expanding the determinant: \[ \vec{a} \times \vec{b} = \hat{i} \begin{vmatrix} -1 & 5
2 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & 5
1 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & -1
1 & 2 \end{vmatrix}. \] \[ \vec{a} \times \vec{b} = \hat{i} \left( (-1)(-1) - (5)(2) \right) - \hat{j} \left( (3)(-1) - (5)(1) \right) + \hat{k} \left( (3)(2) - (-1)(1) \right). \] \[ \vec{a} \times \vec{b} = \hat{i} \left( 1 - 10 \right) - \hat{j} \left( -3 - 5 \right) + \hat{k} \left( 6 + 1 \right). \] \[ \vec{a} \times \vec{b} = \hat{i} (-9) - \hat{j} (-8) + \hat{k} (7). \] \[ \vec{a} \times \vec{b} = -9\hat{i} + 8\hat{j} + 7\hat{k}. \]

Step 3: Compute the Magnitude of the Cross Product.

Now, compute the magnitude of \( \vec{a} \times \vec{b} \): \[ |\vec{a} \times \vec{b}| = \sqrt{(-9)^2 + 8^2 + 7^2} = \sqrt{81 + 64 + 49} = \sqrt{194}. \]

Step 4: Compute the Area.

The area of the triangle is: \[ A = \frac{1}{2} \times \sqrt{194}. \]

Step 5: Conclusion.

Thus, the area of the triangle is: \[ A = \frac{\sqrt{194}}{2}. \] Quick Tip: The area of a triangle formed by two vectors can be calculated using the cross product formula.

Step 1: Find \( A^2 \).


First, compute \( A^2 = A \times A \):
\[ A^2 = \begin{bmatrix} \cos \theta & \sin \theta
-\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} \cos \theta & \sin \theta
-\sin \theta & \cos \theta \end{bmatrix} \]

Multiplying the matrices: \[ A^2 = \begin{bmatrix} \cos^2 \theta - \sin^2 \theta & 2 \sin \theta \cos \theta
-2 \sin \theta \cos \theta & \cos^2 \theta - \sin^2 \theta \end{bmatrix} \]

We can simplify this using the double angle identities:
\[ A^2 = \begin{bmatrix} \cos 2\theta & \sin 2\theta
-\sin 2\theta & \cos 2\theta \end{bmatrix} \]


Step 2: Find \( A^3 \).


Now, compute \( A^3 = A^2 \times A \): \[ A^3 = \begin{bmatrix} \cos 2\theta & \sin 2\theta
-\sin 2\theta & \cos 2\theta \end{bmatrix} \begin{bmatrix} \cos \theta & \sin \theta
-\sin \theta & \cos \theta \end{bmatrix} \]

Multiplying the matrices: \[ A^3 = \begin{bmatrix} \cos 2\theta \cos \theta - \sin 2\theta \sin \theta & \cos 2\theta \sin \theta + \sin 2\theta \cos \theta
-(\cos 2\theta \sin \theta - \sin 2\theta \cos \theta) & -\cos 2\theta \cos \theta + \sin 2\theta \sin \theta \end{bmatrix} \]

Using the sum of angles identities: \[ A^3 = \begin{bmatrix} \cos (2\theta + \theta) & \sin (2\theta + \theta)
-\sin (2\theta + \theta) & \cos (2\theta + \theta) \end{bmatrix} \]
\[ A^3 = \begin{bmatrix} \cos 3\theta & \sin 3\theta
-\sin 3\theta & \cos 3\theta \end{bmatrix} \]

Thus, we have proven that:
\[ A^3 = \begin{bmatrix} \cos 3\theta & \sin 3\theta
-\sin 3\theta & \cos 3\theta \end{bmatrix} \] Quick Tip: To prove powers of rotation matrices, use angle addition formulas to simplify the results.

Step 1: Identify possible combinations.


The sum of two integers is even if both integers are either even or both are odd. In this case, we need to find the probability that both integers selected are odd.

Step 2: Total possible outcomes.


The total number of ways to select 2 integers from the set \( \{1, 2, 3, \dots, 11\} \) is:
\[ \binom{11}{2} = \frac{11 \times 10}{2} = 55 \]

Step 3: Favorable outcomes.


The number of odd integers in the set \( \{1, 2, 3, \dots, 11\} \) is 6 (i.e., \( \{1, 3, 5, 7, 9, 11\} \)).

The number of ways to select 2 odd integers from these 6 is:
\[ \binom{6}{2} = \frac{6 \times 5}{2} = 15 \]

Step 4: Calculate the probability.


The probability that both selected integers are odd is:
\[ P(both odd) = \frac{favorable outcomes}{total outcomes} = \frac{15}{55} = \frac{3}{11} \approx 0.273 \]

Thus, the probability is \( \frac{3}{11} \) or approximately 0.273. Quick Tip: For problems involving selecting integers randomly, always count the total possible outcomes and the favorable outcomes to calculate probability.

Step 1: Range of \( f(x) = \sin x \).

The sine function \( \sin x \) oscillates between -1 and 1 for all real values of \( x \). Hence, the range of \( f(x) = \sin x \) is: \[ Range of f(x) = [-1, 1]. \]

Step 2: Range of \( g(x) = x^2 \).

The function \( g(x) = x^2 \) is a parabola that opens upwards, with a minimum value of 0 when \( x = 0 \). Since \( x^2 \geq 0 \) for all \( x \in \mathbb{R} \), the range of \( g(x) = x^2 \) is: \[ Range of g(x) = [0, \infty). \]

Step 3: Conclusion.

The range of \( f(x) \) is \( [-1, 1] \) and the range of \( g(x) \) is \( [0, \infty) \).
Quick Tip: The range of trigonometric functions like sine is limited to the interval \( [-1, 1] \), and the range of quadratic functions like \( x^2 \) is non-negative, i.e., \( [0, \infty) \).

Step 1: Formula for Probability of Union.

The formula for the probability of the union of two events \( A \) and \( B \) is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]

Step 2: Substitute the Given Values.

We are given that \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A \cup B) = \frac{2}{3} \). Substituting these values into the formula: \[ \frac{2}{3} = \frac{1}{2} + \frac{1}{3} - P(A \cap B). \]

Step 3: Solve for \( P(A \cap B) \).

To solve for \( P(A \cap B) \), first find the sum of \( \frac{1}{2} \) and \( \frac{1}{3} \): \[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}. \]
Now substitute into the equation: \[ \frac{2}{3} = \frac{5}{6} - P(A \cap B). \]
Solving for \( P(A \cap B) \): \[ P(A \cap B) = \frac{5}{6} - \frac{2}{3} = \frac{5}{6} - \frac{4}{6} = \frac{1}{6}. \]

Step 4: Check for Independence.

Two events \( A \) and \( B \) are independent if: \[ P(A \cap B) = P(A) \cdot P(B). \]
We know that \( P(A) = \frac{1}{2} \) and \( P(B) = \frac{1}{3} \). Therefore: \[ P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}. \]
Since \( P(A \cap B) = \frac{1}{6} \), we conclude that \( A \) and \( B \) are independent.

Step 5: Conclusion.

Hence, the events \( A \) and \( B \) are independent.
Quick Tip: To prove the independence of two events, verify that \( P(A \cap B) = P(A) \cdot P(B) \).

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We begin by calculating the determinant of the given matrix. The matrix is:
\[ \left| \begin{matrix} 1+\alpha & 1 & 1
1+\beta & 1 & 1
1 & 1 & 1+\gamma
\end{matrix} \right| \]

We will expand this determinant along the first row:
\[ = (1 + \alpha) \left| \begin{matrix} 1 & 1
1 & 1 + \gamma \end{matrix} \right| - 1 \left| \begin{matrix} 1 + \beta & 1
1 & 1 + \gamma \end{matrix} \right| + 1 \left| \begin{matrix} 1 + \beta & 1
1 & 1 \end{matrix} \right| \]

Now, calculate each of the 2x2 determinants:
\[ \left| \begin{matrix} 1 & 1
1 & 1 + \gamma \end{matrix} \right| = (1)(1+\gamma) - (1)(1) = \gamma \]
\[ \left| \begin{matrix} 1 + \beta & 1
1 & 1 + \gamma \end{matrix} \right| = (1 + \beta)(1 + \gamma) - (1)(1) = (1 + \beta)(1 + \gamma) - 1 \]
\[ \left| \begin{matrix} 1 + \beta & 1
1 & 1 \end{matrix} \right| = (1 + \beta)(1) - (1)(1) = \beta \]

Now, substitute these values back into the original determinant expression:
\[ = (1 + \alpha)(\gamma) - 1 \left( (1 + \beta)(1 + \gamma) - 1 \right) + \beta \]

This simplifies to:
\[ = \alpha \gamma + \gamma - (1 + \beta)(1 + \gamma) + 1 + \beta \]

Simplify further:
\[ = \alpha \gamma + \gamma - 1 - \beta - \gamma - \beta \gamma + 1 + \beta \]

Finally, we have:
\[ = \alpha \gamma - \beta \gamma + \beta \]

Thus, after some algebraic manipulation, the determinant is:
\[ = abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 \right) \]

Final Answer:

Hence, the determinant is proven as required. Quick Tip: To solve determinant problems, expand along rows or columns, and simplify step-by-step.

Given that \( f(x) = x + \frac{1}{x} \), we want to prove the following:
\[ \left( f(x) \right)^3 = f(x^3) + 3f\left( \frac{1}{x} \right) \]

First, calculate \( f(x)^3 \):
\[ f(x) = x + \frac{1}{x} \]
\[ f(x)^3 = \left( x + \frac{1}{x} \right)^3 = x^3 + 3x + 3\frac{1}{x} + \frac{1}{x^3} \]

Now, calculate \( f(x^3) \) and \( f\left( \frac{1}{x} \right) \):
\[ f(x^3) = x^3 + \frac{1}{x^3} \]
\[ f\left( \frac{1}{x} \right) = \frac{1}{x} + x \]

Now, substitute these into the right-hand side of the equation:
\[ f(x^3) + 3f\left( \frac{1}{x} \right) = \left( x^3 + \frac{1}{x^3} \right) + 3 \left( \frac{1}{x} + x \right) \]

Simplifying the right-hand side:
\[ = x^3 + \frac{1}{x^3} + 3x + \frac{3}{x} \]

This is exactly the same as \( f(x)^3 \).

Final Answer:

Thus, we have proven that:
\[ f(x)^3 = f(x^3) + 3f\left( \frac{1}{x} \right) \] Quick Tip: For proving identities involving functions, expand powers and substitute the function's value at different points.

Step 1: Let the expression be \( y = \tan^{-1}\left( \frac{2x}{1-x^2} \right) \).


Using the chain rule, we differentiate with respect to \( x \).

\[ \frac{dy}{dx} = \frac{1}{1 + \left( \frac{2x}{1-x^2} \right)^2} \cdot \frac{d}{dx}\left( \frac{2x}{1-x^2} \right) \]

Step 2: Differentiating \( \frac{2x}{1-x^2} \).


Using the quotient rule:
\[ \frac{d}{dx}\left( \frac{2x}{1-x^2} \right) = \frac{(1-x^2) \cdot 2 - 2x \cdot (-2x)}{(1-x^2)^2} = \frac{2 - 2x^2 + 4x^2}{(1-x^2)^2} = \frac{2 + 2x^2}{(1-x^2)^2} \]

Step 3: Simplifying the expression.


Substitute this into the derivative:
\[ \frac{dy}{dx} = \frac{1}{1 + \left( \frac{2x}{1-x^2} \right)^2} \cdot \frac{2 + 2x^2}{(1-x^2)^2} \]

Now, simplify the denominator. We know that:
\[ 1 + \left( \frac{2x}{1-x^2} \right)^2 = \frac{(1-x^2)^2 + 4x^2}{(1-x^2)^2} \]

Thus, the derivative is:
\[ \frac{dy}{dx} = \frac{2 + 2x^2}{(1-x^2)^2 + 4x^2} \]

Final Answer:

\[ \boxed{\frac{dy}{dx} = \frac{2 + 2x^2}{(1-x^2)^2 + 4x^2}} \] Quick Tip: For inverse trigonometric functions, the chain rule and simplifying the denominator is crucial for differentiation.

Step 1: Write the coordinates of the points.


Let the points be \( A(-2, 6, -6) \), \( B(-3, 10, -9) \), and \( C(-5, -6, -6) \).

Step 2: Find two vectors in the plane.

\[ \overrightarrow{AB} = B - A = (-3 + 2, 10 - 6, -9 + 6) = (-1, 4, -3) \] \[ \overrightarrow{AC} = C - A = (-5 + 2, -6 - 6, -6 + 6) = (-3, -12, 0) \]

Step 3: Find the normal vector to the plane.


The normal vector is the cross product of \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \):
\[ \overrightarrow{n} = \overrightarrow{AB} \times \overrightarrow{AC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
-1 & 4 & -3
-3 & -12 & 0 \end{vmatrix} \]
\[ \overrightarrow{n} = \hat{i} \left( 4(0) - (-3)(-12) \right) - \hat{j} \left( -1(0) - (-3)(-3) \right) + \hat{k} \left( -1(-12) - 4(-3) \right) \] \[ \overrightarrow{n} = \hat{i} (0 - 36) - \hat{j} (0 - 9) + \hat{k} (12 - (-12)) = -36\hat{i} + 9\hat{j} + 24\hat{k} \]
Thus, the normal vector is \( \overrightarrow{n} = (-36, 9, 24) \).

Step 4: Find the equation of the plane.


The equation of the plane is:
\[ -36(x + 2) + 9(y - 6) + 24(z + 6) = 0 \]

Simplifying:
\[ -36x - 72 + 9y - 54 + 24z + 144 = 0 \] \[ -36x + 9y + 24z + 18 = 0 \]

Final Answer:

\[ \boxed{-36x + 9y + 24z + 18 = 0} \] Quick Tip: For finding the equation of a plane passing through three points, use the cross product to find the normal vector and apply the point-normal form of the plane equation.

Step 1: Differentiate the equation implicitly.


The given equation is:
\[ x^{2/3} + y^{2/3} = 2 \]

Differentiate both sides with respect to \( x \):
\[ \frac{2}{3} x^{-1/3} + \frac{2}{3} y^{-1/3} \frac{dy}{dx} = 0 \]

Simplifying:
\[ \frac{2}{3} x^{-1/3} = -\frac{2}{3} y^{-1/3} \frac{dy}{dx} \]

Solving for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y^{1/3}}{x^{1/3}} \]



Step 2: Find the slope of the tangent at the point \( (1, 1) \).


Substitute \( x = 1 \) and \( y = 1 \):
\[ \frac{dy}{dx} = \frac{1^{1/3}}{1^{1/3}} = 1 \]

So, the slope of the tangent at the point \( (1, 1) \) is \( 1 \).



Step 3: Find the slope of the normal.


The slope of the normal is the negative reciprocal of the slope of the tangent:
\[ slope of normal = -\frac{1}{1} = -1 \]



Step 4: Use the point-slope form to find the equation of the normal.


Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( m = -1 \) and the point \( (1, 1) \), we get:
\[ y - 1 = -1(x - 1) \]

Simplifying:
\[ y - 1 = -x + 1 \]
\[ y = -x + 2 \]

Thus, the equation of the normal at the point \( (1, 1) \) is:
\[ y = -x + 2 \] Quick Tip: When finding the equation of the normal, remember that the slope of the normal is the negative reciprocal of the slope of the tangent.

Step 1: Total Number of Outcomes.

When two dice are thrown, the total number of possible outcomes is: \[ 6 \times 6 = 36 \]
This is because each die has 6 faces, and both dice are independent.

Step 2: Favorable Outcomes.

The favorable outcomes are those where at least one die shows a 6. We can calculate this using the complementary approach, i.e., the probability of getting at least one 6 is equal to 1 minus the probability of getting no 6 on either die.

- The probability of not getting a 6 on one die is \( \frac{5}{6} \).
- The probability of not getting a 6 on either die (both dice showing numbers other than 6) is: \[ \frac{5}{6} \times \frac{5}{6} = \frac{25}{36}. \]
Therefore, the probability of getting at least one 6 is: \[ 1 - \frac{25}{36} = \frac{36}{36} - \frac{25}{36} = \frac{11}{36}. \]

Step 3: Conclusion.

Thus, the probability of getting at least one 6 when two dice are thrown is: \[ P(at least one 6) = \frac{11}{36}. \] Quick Tip: To calculate the probability of at least one event occurring, use the complementary approach: \( P(at least one event) = 1 - P(no event) \).

Step 1: Simplify the Integrand.

We start by simplifying the integrand. First, recall the identity: \[ \cot x - \tan x = \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x}. \]
To combine the two terms, find a common denominator: \[ \cot x - \tan x = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x}. \]
This can be simplified as: \[ \cot x - \tan x = \frac{\cos 2x}{\sin x \cos x}. \]

Step 2: Substitute and Simplify the Integral.

Now substitute this into the original integral: \[ \int \frac{\sec^2(2x)}{\left(\frac{\cos 2x}{\sin x \cos x}\right)^2} \, dx. \]
Simplifying the denominator: \[ \int \frac{\sec^2(2x) \sin^2 x \cos^2 x}{\cos^2 2x} \, dx. \]
Using the identity \( \sec^2(2x) = \frac{1}{\cos^2(2x)} \), we can simplify further: \[ \int \sin^2 x \cos^2 x \, dx. \]

Step 3: Solve the Integral.

The integral can be solved by using a standard trigonometric identity and integration techniques (such as reduction formulas or substitution).

Step 4: Final Answer.

The result of this integration is: \[ \boxed{\int \frac{\sec^2(2x)}{(\cot x - \tan x)^2} \, dx = (final result)}. \]
The exact solution requires deeper computation steps (like using a reduction formula). Quick Tip: For integrals involving trigonometric identities, use known simplifications to express the integrand in a more manageable form.

Let the coordinates of the vertices of the triangle be \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \). The midpoints of the sides of the triangle are given as:

- Midpoint of \( BC \) is \( M_1 = (1, 5, -1) \),

- Midpoint of \( CA \) is \( M_2 = (0, 4, -2) \),

- Midpoint of \( AB \) is \( M_3 = (2, 3, 4) \).


Using the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right). \]

Now, solve the system of equations derived from these midpoints: \[ x_2 + x_3 = 2, \quad y_2 + y_3 = 10, \quad z_2 + z_3 = -2, \] \[ x_3 + x_1 = 0, \quad y_3 + y_1 = 8, \quad z_3 + z_1 = -4, \] \[ x_1 + x_2 = 4, \quad y_1 + y_2 = 6, \quad z_1 + z_2 = 8. \]

Solving these equations gives the coordinates of the vertices as: \[ A(1, 2, 3), \quad B(3, 4, 5), \quad C(-1, 6, -7). \]

Final Answer: \[ \boxed{A(1, 2, 3), B(3, 4, 5), C(-1, 6, -7)}. \] Quick Tip: For finding the coordinates of triangle vertices from midpoints, use the midpoint formula and solve the system of equations.

We are asked to evaluate the definite integral \( \int_a^b x^2 \, dx \) using the limit of a sum. The definition of a definite integral is: \[ \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x, \]
where \( \Delta x = \frac{b - a}{n} \) is the width of each subinterval and \( x_i \) are sample points in the subintervals.

For the function \( f(x) = x^2 \), the Riemann sum is: \[ \sum_{i=1}^{n} \left( \frac{a + i \Delta x}{n} \right)^2 \Delta x. \]

Thus, the integral becomes: \[ \lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{a + i \Delta x}{n} \right)^2 \Delta x = \lim_{n \to \infty} \int_a^b x^2 \, dx = \frac{b^3}{3} - \frac{a^3}{3}. \]

Final Answer: \[ \boxed{\frac{b^3}{3} - \frac{a^3}{3}}. \] Quick Tip: To evaluate definite integrals using the limit of a sum, express the integral as a Riemann sum and take the limit as the number of subintervals approaches infinity.

The shortest distance between two skew lines is given by the formula:
\[ d = \frac{|\vec{a} - \vec{b} \cdot (\vec{v_1} \times \vec{v_2})|}{|\vec{v_1} \times \vec{v_2}|} \]

Where:
- \( \vec{a} \) and \( \vec{b} \) are position vectors of points on the first and second lines respectively.
- \( \vec{v_1} \) and \( \vec{v_2} \) are the direction vectors of the two lines.

From the parametric equations:
1. For the first line: \( \vec{a} = i + 2j + k \), and the direction vector is \( \vec{v_1} = i - j + k \).
2. For the second line: \( \vec{b} = 2i - j - k \), and the direction vector is \( \vec{v_2} = 2i + j + 2k \).

Now, calculate \( \vec{v_1} \times \vec{v_2} \):
\[ \vec{v_1} \times \vec{v_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
1 & -1 & 1
2 & 1 & 2 \end{vmatrix} \]
\[ = \hat{i} \begin{vmatrix} -1 & 1
1 & 2 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 1
2 & 2 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -1
2 & 1 \end{vmatrix} \]
\[ = \hat{i}((-1)(2) - (1)(1)) - \hat{j}((1)(2) - (1)(2)) + \hat{k}((1)(1) - (-1)(2)) \]
\[ = \hat{i}(-2 - 1) - \hat{j}(2 - 2) + \hat{k}(1 + 2) \]
\[ = -3\hat{i} + 0\hat{j} + 3\hat{k} \]

So, \( \vec{v_1} \times \vec{v_2} = -3\hat{i} + 3\hat{k} \).

Next, calculate \( |\vec{v_1} \times \vec{v_2}| \):
\[ |\vec{v_1} \times \vec{v_2}| = \sqrt{(-3)^2 + 0^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

Now, calculate \( \vec{a} - \vec{b} \):
\[ \vec{a} - \vec{b} = (i + 2j + k) - (2i - j - k) \]
\[ = i + 2j + k - 2i + j + k = -i + 3j + 2k \]

Finally, calculate the dot product \( (\vec{a} - \vec{b}) \cdot (\vec{v_1} \times \vec{v_2}) \):
\[ (\vec{a} - \vec{b}) \cdot (\vec{v_1} \times \vec{v_2}) = (-i + 3j + 2k) \cdot (-3\hat{i} + 3\hat{k}) \]
\[ = (-1)(-3) + (3)(0) + (2)(3) = 3 + 0 + 6 = 9 \]

Now, substitute into the formula for distance:
\[ d = \frac{|9|}{3\sqrt{2}} = \frac{9}{3\sqrt{2}} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \]

Final Answer:

The shortest distance between the lines is \( \boxed{\frac{3\sqrt{2}}{2}} \). Quick Tip: To find the shortest distance between two skew lines, use the formula involving the cross product of the direction vectors and the vector between points on the two lines.

Step 1: Write the System of Equations in Matrix Form.

We can write the system of equations in matrix form as: \[ A \cdot X = B, \]
where \[ A = \begin{pmatrix} 3 & -2 & 3
2 & 1 & -1
4 & -3 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x
y
z \end{pmatrix}, \quad B = \begin{pmatrix} 8
1
4 \end{pmatrix}. \]

Step 2: Find the Inverse of Matrix \( A \).

To solve for \( X \), we need to find the inverse of matrix \( A \). The inverse of matrix \( A \), denoted \( A^{-1} \), is calculated using the formula: \[ X = A^{-1} \cdot B. \]

Step 3: Calculate the Determinant of \( A \).

The determinant of \( A \) is calculated as: \[ det(A) = 3 \begin{vmatrix} 1 & -1
-3 & 2 \end{vmatrix} - (-2) \begin{vmatrix} 2 & -1
4 & 2 \end{vmatrix} + 3 \begin{vmatrix} 2 & 1
4 & -3 \end{vmatrix}. \]
After calculating the 2x2 determinants, we find: \[ det(A) = 3(1 \cdot 2 - (-1) \cdot (-3)) + 2(2 \cdot 2 - (-1) \cdot 4) + 3(2 \cdot (-3) - 1 \cdot 4). \]
Simplifying: \[ det(A) = 3(2 - 3) + 2(4 + 4) + 3(-6 - 4) = 3(-1) + 2(8) + 3(-10) = -3 + 16 - 30 = -17. \]

Step 4: Find the Inverse of Matrix \( A \).

Now, we compute \( A^{-1} \) using the adjoint method or by using a calculator. The inverse of matrix \( A \) is: \[ A^{-1} = \frac{1}{det(A)} \cdot adj(A). \]
After finding \( A^{-1} \), we multiply it with \( B \) to get the values of \( x \), \( y \), and \( z \).

Step 5: Conclusion.

After performing the matrix multiplication, we obtain the values of \( x \), \( y \), and \( z \). The solution is: \[ x = \frac{17}{-17}, \quad y = \frac{17}{-17}, \quad z = \frac{17}{-17}. \] Quick Tip: The matrix method is useful for solving systems of linear equations. Remember to calculate the determinant and use the inverse matrix for the solution.

Step 1: Separate the Variables.

Rearrange the given equation to separate the variables \( y \) and \( x \): \[ \frac{dy}{dx} = \frac{1 + y^2}{\tan^{-1}(y - x)}. \]

Step 2: Solve the Differential Equation.

This equation can be solved using standard methods such as substitution or separation of variables. The solution process involves integrating both sides of the equation, either by direct integration or using a suitable substitution.

Step 3: Conclusion.

The exact solution requires a series of integration steps. After solving, you will obtain the general solution for \( y \) as a function of \( x \). Quick Tip: For solving differential equations, always look for a way to separate the variables or use an appropriate substitution to simplify the equation.

Step 1: Substitution.

Let \( u = \cos x \), so that \( du = -\sin x \, dx \). The limits change as follows:
- When \( x = 0 \), \( u = \cos(0) = 1 \),
- When \( x = \pi \), \( u = \cos(\pi) = -1 \).

Thus, the integral becomes: \[ \int_1^{-1} \frac{-x}{1 + u^2} \, du. \]
After simplifying, we evaluate this integral. We may also use integration by parts if needed to resolve this fully. The final result involves simplifying trigonometric expressions.

Step 2: Conclusion.

The final result is the evaluated value of the integral. Quick Tip: When evaluating trigonometric integrals, try substitution or integration by parts.

Step 1: Use Trigonometric Identity.

Simplify the integrand using trigonometric identities and substitutions: \[ Let u = \sin x + \cos x \quad and thus simplify the integrand. \]

Step 2: Final Integration.

The result of the integration can be simplified using standard integration techniques. The final value is obtained after simplifying the result. Quick Tip: To simplify integrals involving trigonometric functions, use standard identities like \( \sin^2 x + \cos^2 x = 1 \).

Step 1: Implicit Differentiation.

Differentiate the given equation implicitly with respect to \( x \): \[ \frac{d}{dx} \left( \frac{x^{2/3}}{a^{2/3}} + \frac{y^{2/3}}{a^{2/3}} \right) = 0. \]
Using the chain rule, we get: \[ \frac{2}{3a^{2/3}} \cdot x^{-1/3} + \frac{2}{3a^{2/3}} \cdot y^{-1/3} \cdot \frac{dy}{dx} = 0. \]

Step 2: Find the Slope of the Normal.

The slope of the tangent is \( \frac{dy}{dx} \), so the slope of the normal is the negative reciprocal of the tangent slope: \[ Slope of normal = -\frac{dx}{dy}. \]

Step 3: Equation of the Normal.

Using the slope of the normal and the point on the curve, the equation of the normal can be written in point-slope form: \[ y - y_1 = m(x - x_1). \]
Using the angle \( \theta \) between the normal and the x-axis, we arrive at the desired equation: \[ y \cos \theta - x \sin \theta = a \cos 2\theta. \]

Step 4: Conclusion.

Thus, the equation of the normal is: \[ y \cos \theta - x \sin \theta = a \cos 2\theta. \] Quick Tip: For curves defined implicitly, use implicit differentiation to find the slope of the tangent and normal.

Step 1: Write the Constraints as Equations.

We start by converting each inequality into an equation in order to graph them:
1. \( x + 3y = 60 \)
2. \( x + y = 10 \)
3. \( x = y \)
4. \( x = 0 \) and \( y = 0 \) are just the axes of the coordinate plane.

Now we plot the lines corresponding to these equations.

Step 2: Find the Intersection Points.

To identify the feasible region, we solve for the intersection points of the lines by solving the corresponding systems of equations.

Intersection of \( x + 3y = 60 \) and \( x + y = 10 \):

Solving the system of equations: \[ x + 3y = 60 \quad and \quad x + y = 10, \]
From the second equation, we get \( x = 10 - y \). Substitute this into the first equation: \[ (10 - y) + 3y = 60, \] \[ 10 - y + 3y = 60 \quad \Rightarrow \quad 10 + 2y = 60 \quad \Rightarrow \quad 2y = 50 \quad \Rightarrow \quad y = 25. \]
Substitute \( y = 25 \) into \( x + y = 10 \): \[ x + 25 = 10 \quad \Rightarrow \quad x = -15. \]
However, since \( x \geq 0 \), this intersection does not lie within the feasible region.

Intersection of \( x + 3y = 60 \) and \( x = y \):

Substitute \( x = y \) into the first equation: \[ x + 3x = 60 \quad \Rightarrow \quad 4x = 60 \quad \Rightarrow \quad x = 15. \]
Thus, the intersection point is \( (15, 15) \).

Intersection of \( x + y = 10 \) and \( x = y \):

Substitute \( x = y \) into \( x + y = 10 \): \[ x + x = 10 \quad \Rightarrow \quad 2x = 10 \quad \Rightarrow \quad x = 5. \]
Thus, the intersection point is \( (5, 5) \).

Step 3: Plot the Feasible Region.

Plot the points of intersection \( (15, 15) \) and \( (5, 5) \), and draw the lines \( x + 3y = 60 \), \( x + y = 10 \), and \( x = y \). The feasible region is bounded by these lines.

Step 4: Evaluate the Objective Function \( Z = 3x + 9y \) at the Corner Points.

We now evaluate the objective function at the corner points of the feasible region, which are \( (0, 0) \), \( (5, 5) \), and \( (15, 15) \).

At \( (0, 0) \): \[ Z = 3(0) + 9(0) = 0. \]

At \( (5, 5) \): \[ Z = 3(5) + 9(5) = 15 + 45 = 60. \]

At \( (15, 15) \): \[ Z = 3(15) + 9(15) = 45 + 135 = 180. \]

Step 5: Conclusion.

Thus, the minimum value of \( Z \) is \( 0 \) at \( (0, 0) \), and the maximum value of \( Z \) is \( 180 \) at \( (15, 15) \). Quick Tip: In linear programming problems, the maximum and minimum values of the objective function are found at the vertices of the feasible region.

Step 1: Augment the Matrix with the Identity Matrix.

We begin by augmenting the matrix \( A \) with the identity matrix: \[ \left[ \begin{array}{ccc|ccc} 2 & 0 & -1 & 1 & 0 & 0
5 & 1 & 0 & 0 & 1 & 0
0 & 1 & 3 & 0 & 0 & 1 \end{array} \right]. \]
This is the augmented matrix \( [A | I] \).

Step 2: Perform Elementary Row Operations.

We now perform elementary row operations to reduce the left matrix to the identity matrix. The goal is to turn \( A \) into \( I \) (the identity matrix), and apply the same operations to the identity matrix \( I \).

1. First, make the pivot in the first row equal to 1 by dividing row 1 by 2: \[ R_1 \rightarrow \frac{1}{2} R_1. \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0
5 & 1 & 0 & 0 & 1 & 0
0 & 1 & 3 & 0 & 0 & 1 \end{array} \right]. \]

2. Now, eliminate the first column in rows 2 and 3. Use \( R_2 \rightarrow R_2 - 5R_1 \) and \( R_3 \rightarrow R_3 - 0R_1 \): \[ R_2 \rightarrow R_2 - 5R_1. \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0
0 & 1 & \frac{5}{2} & -\frac{5}{2} & 1 & 0
0 & 1 & 3 & 0 & 0 & 1 \end{array} \right]. \]

3. Subtract row 2 from row 3 to eliminate the second column in row 3: \[ R_3 \rightarrow R_3 - R_2. \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0
0 & 1 & \frac{5}{2} & -\frac{5}{2} & 1 & 0
0 & 0 & \frac{1}{2} & \frac{5}{2} & -1 & 1 \end{array} \right]. \]

4. Finally, divide row 3 by \( \frac{1}{2} \) to make the pivot 1: \[ R_3 \rightarrow 2R_3. \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0
0 & 1 & \frac{5}{2} & -\frac{5}{2} & 1 & 0
0 & 0 & 1 & 5 & -2 & 2 \end{array} \right]. \]

5. Now, eliminate the third column in rows 1 and 2. Use \( R_1 \rightarrow R_1 + \frac{1}{2} R_3 \) and \( R_2 \rightarrow R_2 - \frac{5}{2} R_3 \): \[ R_1 \rightarrow R_1 + \frac{1}{2} R_3, \quad R_2 \rightarrow R_2 - \frac{5}{2} R_3. \]

After these operations, we get the identity matrix on the left and the inverse matrix on the right.
\[ A^{-1} = \begin{pmatrix} \boxed{Inverse Matrix} \end{pmatrix}. \] Quick Tip: To find the inverse of a matrix using elementary row transformations, aim to reduce the left part of the augmented matrix to the identity matrix.