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[INST] help me answer this question Prove that the expression (x + y) {mx2 + y2k (x – y)} is a factor of the expression x2k . x2 – y2k . y2, think step by step [/INST] N/A , The answer is We can prove this by expanding the expression and showing that it can be written in the form (x + y) times some other expression. First, we add and subtract y2k in the expression x2k . x2 – y2k . y2 to get x2 (x2k – y2k + y2k) – y2k. y2. From equation (1), we get x2 {m (x + y) + y2k} – y2k. y2. By multiplying the terms, we get m (x + y) x2 + y2k. x2 – y2k. y2. Taking out the common terms, we get m (x + y) x2 + y2k (x – y). Expanding using the formula, we get m (x + y) x2 + y2k (x + y) (x – y). So, we get (x + y) {mx2 + y2k (x – y)}

<s>[INST] [INST] help me answer this question Prove that the expression (x + y) {mx2 + y2k (x – y)} is a factor of the expression x2k . x2 – y2k . y2, think step by step [/INST]</s> <s>N/A , The answer is We can prove this by expanding the expression and showing that it can be written in the form (x + y) times some other expression. First, we add and subtract y2k in the expression x2k . x2 – y2k . y2 to get x2 (x2k – y2k + y2k) – y2k. y2. From equation (1), we get x2 {m (x + y) + y2k} – y2k. y2. By multiplying the terms, we get m (x + y) x2 + y2k. x2 – y2k. y2. Taking out the common terms, we get m (x + y) x2 + y2k (x – y). Expanding using the formula, we get m (x + y) x2 + y2k (x + y) (x – y). So, we get (x + y) {mx2 + y2k (x – y)}</s>

[INST] help me answer this question Find the required equation in the data., think step by step [/INST] , The answer is The required equation is 9x² + 25y² + 25z² – 225 = 0.

<s>[INST] [INST] help me answer this question Find the required equation in the data., think step by step [/INST]</s> <s>, The answer is The required equation is 9x² + 25y² + 25z² – 225 = 0.</s>

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

<s>[INST] [INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is `x ≥ 1`.</s>

[INST] help me answer this question Find the sum to n terms in the geometric progression 1, -a, a^2, -a^3 ... (if a ≠ -1), think step by step [/INST] , The answer is An expression to find the sum to n terms in the geometric progression 1, -a, a^2, -a^3 ... (if a ≠ -1)

<s>[INST] [INST] help me answer this question Find the sum to n terms in the geometric progression 1, -a, a^2, -a^3 ... (if a ≠ -1), think step by step [/INST]</s> <s>, The answer is An expression to find the sum to n terms in the geometric progression 1, -a, a^2, -a^3 ... (if a ≠ -1)</s>

[INST] help me answer this question Given x + 3y ≤ 30, what are the required points?, think step by step [/INST] , The answer is The required points are (0, 10) and (30, 0).

<s>[INST] [INST] help me answer this question Given x + 3y ≤ 30, what are the required points?, think step by step [/INST]</s> <s>, The answer is The required points are (0, 10) and (30, 0).</s>

[INST] help me answer this question If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G., think step by step [/INST] Given, G = {7, 8} and H = {5, 4, 2} We know that The Cartesian product of two non-empty sets, P and Q, is given as P × Q = {(p, q): p ∈ P, q ∈ Q} So, G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)} H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)} , The answer is [(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)], [(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)]

<s>[INST] [INST] help me answer this question If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G., think step by step [/INST]</s> <s>Given, G = {7, 8} and H = {5, 4, 2} We know that The Cartesian product of two non-empty sets, P and Q, is given as P × Q = {(p, q): p ∈ P, q ∈ Q} So, G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)} H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)} , The answer is [(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)], [(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)]</s>

[INST] help me answer this question P (k + 1) is true whenever P (k) is true., think step by step [/INST] Principle of Mathematical Induction , The answer is Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.

<s>[INST] [INST] help me answer this question P (k + 1) is true whenever P (k) is true., think step by step [/INST]</s> <s>Principle of Mathematical Induction , The answer is Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.</s>

[INST] help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step [/INST] Explaination: N/A , The answer is Answer: Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.

<s>[INST] [INST] help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step [/INST]</s> <s>Explaination: N/A , The answer is Answer: Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.</s>

[INST] help me answer this question Is the statement “(a, a) ∈ R, for all a ∈ N” true?, think step by step [/INST] For (a, a) to be an element of R, a should be equal to the square of some natural number. However, this is not true for all a belonging to N. For example, 2 ∈ N, but √2 ∉ N. , The answer is No, it is not true.

<s>[INST] [INST] help me answer this question Is the statement “(a, a) ∈ R, for all a ∈ N” true?, think step by step [/INST]</s> <s>For (a, a) to be an element of R, a should be equal to the square of some natural number. However, this is not true for all a belonging to N. For example, 2 ∈ N, but √2 ∉ N. , The answer is No, it is not true.</s>

[INST] help me answer this question Find the derivative of (iv) x5 (3 – 6x-9), think step by step [/INST] , The answer is The derivative of x5 (3 – 6x-9) is not provided.

<s>[INST] [INST] help me answer this question Find the derivative of (iv) x5 (3 – 6x-9), think step by step [/INST]</s> <s>, The answer is The derivative of x5 (3 – 6x-9) is not provided.</s>

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

<s>[INST] [INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers.</s>

[INST] help me answer this question What is the final solution area for the given inequalities?, think step by step [/INST] , The answer is The final solution area for the given inequalities is the shaded area of the graph satisfying all the given inequalities.

<s>[INST] [INST] help me answer this question What is the final solution area for the given inequalities?, think step by step [/INST]</s> <s>, The answer is The final solution area for the given inequalities is the shaded area of the graph satisfying all the given inequalities.</s>

[INST] help me answer this question , think step by step [/INST] , The answer is

<s>[INST] [INST] help me answer this question , think step by step [/INST]</s> <s>, The answer is</s>

[INST] help me answer this question A coin is tossed, and a die is thrown. Find the sample space., think step by step [/INST] , The answer is The sample space contains possible outcomes of the given experiment. Since either coin can turn up Head (H) or Tail (T), and let us assume that 1, 2, 3, 4, 5 and 6 are the possible numbers that come when the die is thrown. The total number of possible outcomes equals (2 × 6) = 12. Hence, the sample space is, S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}

<s>[INST] [INST] help me answer this question A coin is tossed, and a die is thrown. Find the sample space., think step by step [/INST]</s> <s>, The answer is The sample space contains possible outcomes of the given experiment. Since either coin can turn up Head (H) or Tail (T), and let us assume that 1, 2, 3, 4, 5 and 6 are the possible numbers that come when the die is thrown. The total number of possible outcomes equals (2 × 6) = 12. Hence, the sample space is, S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}</s>

[INST] help me answer this question If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find (i) A ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D (v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D, think step by step [/INST] , The answer is {'i': 'A ∪ B = {1, 2, 3, 4, 5, 6}', 'ii': 'A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}', 'iii': 'B ∪ C = {3, 4, 5, 6, 7, 8}', 'iv': 'B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}', 'v': 'A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}', 'vi': 'A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}', 'vii': 'B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}'}

<s>[INST] [INST] help me answer this question If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find (i) A ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D (v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D, think step by step [/INST]</s> <s>, The answer is {'i': 'A ∪ B = {1, 2, 3, 4, 5, 6}', 'ii': 'A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}', 'iii': 'B ∪ C = {3, 4, 5, 6, 7, 8}', 'iv': 'B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}', 'v': 'A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}', 'vi': 'A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}', 'vii': 'B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}'}</s>

[INST] help me answer this question Where can additional exercises related to this topic be found?, think step by step [/INST] , The answer is The other exercises of NCERT Class 11 Maths Solutions for Chapter 1 are as follows: Exercise 1.1 Solutions 6 Questions, Exercise 1.3 Solutions 9 Questions, Exercise 1.4 Solutions 12 Questions, Exercise 1.5 Solutions 7 Questions, Exercise 1.6 Solutions 8 Questions, Miscellaneous Exercise on Chapter 1 Solutions 16 Questions. Access these exercises for more practice.

<s>[INST] [INST] help me answer this question Where can additional exercises related to this topic be found?, think step by step [/INST]</s> <s>, The answer is The other exercises of NCERT Class 11 Maths Solutions for Chapter 1 are as follows: Exercise 1.1 Solutions 6 Questions, Exercise 1.3 Solutions 9 Questions, Exercise 1.4 Solutions 12 Questions, Exercise 1.5 Solutions 7 Questions, Exercise 1.6 Solutions 8 Questions, Miscellaneous Exercise on Chapter 1 Solutions 16 Questions. Access these exercises for more practice.</s>

[INST] help me answer this question tan x = √3, think step by step [/INST] , The answer is The solution to this equation is not provided in the data.

<s>[INST] [INST] help me answer this question tan x = √3, think step by step [/INST]</s> <s>, The answer is The solution to this equation is not provided in the data.</s>

[INST] help me answer this question Which numbers are elements of set A?, think step by step [/INST] Set A is defined as {1, 2, 3, 4, 5, 6}. The symbol ∈ denotes an element of, and ∉ denotes not an element of. , The answer is 5 ∈ A, 8 ∉ A, 0 ∉ A, 4 ∈ A, 2 ∈ A, 10 ∉ A

<s>[INST] [INST] help me answer this question Which numbers are elements of set A?, think step by step [/INST]</s> <s>Set A is defined as {1, 2, 3, 4, 5, 6}. The symbol ∈ denotes an element of, and ∉ denotes not an element of. , The answer is 5 ∈ A, 8 ∉ A, 0 ∉ A, 4 ∈ A, 2 ∈ A, 10 ∉ A</s>

[INST] help me answer this question A coin is tossed, and a die is thrown. Describe the sample space for this experiment., think step by step [/INST] When a coin is tossed, there are 2 possibilities (Head or Tail). When a die is thrown, there are 6 possible outcomes. So, the total number of possible outcomes is 2*6 = 12. These outcomes form the sample space S. , The answer is The sample space S contains 12 elements and is given by: S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}

<s>[INST] [INST] help me answer this question A coin is tossed, and a die is thrown. Describe the sample space for this experiment., think step by step [/INST]</s> <s>When a coin is tossed, there are 2 possibilities (Head or Tail). When a die is thrown, there are 6 possible outcomes. So, the total number of possible outcomes is 2*6 = 12. These outcomes form the sample space S. , The answer is The sample space S contains 12 elements and is given by: S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}</s>

[INST] help me answer this question 4, 7, 8, 9, 10, 12, 13, 17, think step by step [/INST] , The answer is The mean deviation about the mean for the given data is 3.

<s>[INST] [INST] help me answer this question 4, 7, 8, 9, 10, 12, 13, 17, think step by step [/INST]</s> <s>, The answer is The mean deviation about the mean for the given data is 3.</s>

[INST] help me answer this question In which quadrant will be the solution area for the given inequalities?, think step by step [/INST] , The answer is The solution area for the given inequalities will be in the 1st quadrant.

<s>[INST] [INST] help me answer this question In which quadrant will be the solution area for the given inequalities?, think step by step [/INST]</s> <s>, The answer is The solution area for the given inequalities will be in the 1st quadrant.</s>

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

<s>[INST] [INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is `x ≥ 1`.</s>

[INST] help me answer this question What are the required points on the graph for the given inequalities of 5x + 4y ≤ 20?, think step by step [/INST] These points satisfy the given inequalities. , The answer is The required points would be (0, 2), (1, 2) and so on.

<s>[INST] [INST] help me answer this question What are the required points on the graph for the given inequalities of 5x + 4y ≤ 20?, think step by step [/INST]</s> <s>These points satisfy the given inequalities. , The answer is The required points would be (0, 2), (1, 2) and so on.</s>

[INST] help me answer this question What is the domain of the relation R = {(x, x+5): x is a prime number less than 10}?, think step by step [/INST] , The answer is The relation R = {(x, x+5): x is a prime number less than 10} Prime numbers less than 10 are 2, 3, 5, and 7. So, R = {(2, 7), (3, 8), (5, 10), (7, 12)} The domain of R is the set of all first elements of the ordered pairs in the relation. Hence, Domain of R = {2, 3, 5, 7}

<s>[INST] [INST] help me answer this question What is the domain of the relation R = {(x, x+5): x is a prime number less than 10}?, think step by step [/INST]</s> <s>, The answer is The relation R = {(x, x+5): x is a prime number less than 10} Prime numbers less than 10 are 2, 3, 5, and 7. So, R = {(2, 7), (3, 8), (5, 10), (7, 12)} The domain of R is the set of all first elements of the ordered pairs in the relation. Hence, Domain of R = {2, 3, 5, 7}</s>

[INST] help me answer this question Find the mean deviation about median for the following data., think step by step [/INST] Here, the user is expected to calculate the mean deviation about the median for the given data. , The answer is The user is being asked to find the mean deviation about the median for the given data.

<s>[INST] [INST] help me answer this question Find the mean deviation about median for the following data., think step by step [/INST]</s> <s>Here, the user is expected to calculate the mean deviation about the median for the given data. , The answer is The user is being asked to find the mean deviation about the median for the given data.</s>

[INST] help me answer this question Let f be a function from R into R. Determine the range of f., think step by step [/INST] , The answer is The range of the function f(x) = x^2 / (x^2 + 1) is [0, 1).

<s>[INST] [INST] help me answer this question Let f be a function from R into R. Determine the range of f., think step by step [/INST]</s> <s>, The answer is The range of the function f(x) = x^2 / (x^2 + 1) is [0, 1).</s>

[INST] help me answer this question If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P^2 = (ab)^n, think step by step [/INST] Since the first term of the G.P. is a, and the last term is b, its product (P) can be expressed as the product of the first term, the common ratio, and the sum of n-1 terms of an arithmetic progression. We then show that P^2 = (ab)^n by using the properties of G.P. and the formula for the sum of an A.P. , The answer is P^2 = (ab)^n

<s>[INST] [INST] help me answer this question If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P^2 = (ab)^n, think step by step [/INST]</s> <s>Since the first term of the G.P. is a, and the last term is b, its product (P) can be expressed as the product of the first term, the common ratio, and the sum of n-1 terms of an arithmetic progression. We then show that P^2 = (ab)^n by using the properties of G.P. and the formula for the sum of an A.P. , The answer is P^2 = (ab)^n</s>

[INST] help me answer this question Simplify -81/4b² + 4/b² = 1 – 9/4., think step by step [/INST] , The answer is -65/4b² = -5/4

<s>[INST] [INST] help me answer this question Simplify -81/4b² + 4/b² = 1 – 9/4., think step by step [/INST]</s> <s>, The answer is -65/4b² = -5/4</s>

[INST] help me answer this question (vii) {x: x is an even natural number} … {x: x is an integer}, think step by step [/INST] Every even natural number is an integer. , The answer is {x: x is an even natural number} ⊂ {x: x is an integer}

<s>[INST] [INST] help me answer this question (vii) {x: x is an even natural number} … {x: x is an integer}, think step by step [/INST]</s> <s>Every even natural number is an integer. , The answer is {x: x is an even natural number} ⊂ {x: x is an integer}</s>

[INST] help me answer this question Solve the quadratic equation -x2 + x - 2 = 0, think step by step [/INST] Comparing the given equation with ax2 + bx + c = 0, we have a = -1, b = 1, and c = -2. The discriminant (D) is calculated as D = b2 - 4ac = 12 - 4 × (-1) × (-2) = 1 - 8 = -7. Since D < 0, there is no real solution for this equation. , The answer is There is no real solution for this equation

<s>[INST] [INST] help me answer this question Solve the quadratic equation -x2 + x - 2 = 0, think step by step [/INST]</s> <s>Comparing the given equation with ax2 + bx + c = 0, we have a = -1, b = 1, and c = -2. The discriminant (D) is calculated as D = b2 - 4ac = 12 - 4 × (-1) × (-2) = 1 - 8 = -7. Since D < 0, there is no real solution for this equation. , The answer is There is no real solution for this equation</s>

[INST] help me answer this question What is collinearity of points C (0, 1/3, 2), A, and B?, think step by step [/INST] , The answer is A, B, C are collinear.

<s>[INST] [INST] help me answer this question What is collinearity of points C (0, 1/3, 2), A, and B?, think step by step [/INST]</s> <s>, The answer is A, B, C are collinear.</s>

[INST] help me answer this question b^2 = 6a, think step by step [/INST] , The answer is Substituting the value of a from the equation a = 2 into the previously derived equation b^2 = 6a, we get b^2 = 6a.

<s>[INST] [INST] help me answer this question b^2 = 6a, think step by step [/INST]</s> <s>, The answer is Substituting the value of a from the equation a = 2 into the previously derived equation b^2 = 6a, we get b^2 = 6a.</s>

[INST] help me answer this question Expand (3x – 2/x)4, think step by step [/INST] , The answer is From the binomial theorem, the given equation can be expanded as Solution: 4C0 (3x)4 - 4C1 (3x)3 (2/x) + 6C2 (3x)2 (2/x)2 - 4C3 (3x) (2/x)3 + 4C4 (2/x)4

<s>[INST] [INST] help me answer this question Expand (3x – 2/x)4, think step by step [/INST]</s> <s>, The answer is From the binomial theorem, the given equation can be expanded as Solution: 4C0 (3x)4 - 4C1 (3x)3 (2/x) + 6C2 (3x)2 (2/x)2 - 4C3 (3x) (2/x)3 + 4C4 (2/x)4</s>

[INST] help me answer this question a^2 + 6a = 16, think step by step [/INST] , The answer is Substituting the value of b^2 from the previously derived equation b^2 = 6a, we get a^2 + 6a = 16.

<s>[INST] [INST] help me answer this question a^2 + 6a = 16, think step by step [/INST]</s> <s>, The answer is Substituting the value of b^2 from the previously derived equation b^2 = 6a, we get a^2 + 6a = 16.</s>

[INST] help me answer this question Calculate the mean deviation about median age for the age distribution of 100 persons given below., think step by step [/INST] Here, the user is expected to calculate the mean deviation about the median age for the given age distribution of 100 persons. , The answer is The user is being asked to calculate the mean deviation about the median age for the given age distribution of 100 persons.

<s>[INST] [INST] help me answer this question Calculate the mean deviation about median age for the age distribution of 100 persons given below., think step by step [/INST]</s> <s>Here, the user is expected to calculate the mean deviation about the median age for the given age distribution of 100 persons. , The answer is The user is being asked to calculate the mean deviation about the median age for the given age distribution of 100 persons.</s>

[INST] help me answer this question (i) {2, 3, 4} … {1, 2, 3, 4, 5}, think step by step [/INST] Every element of {2, 3, 4} is also an element of {1, 2, 3, 4, 5}. , The answer is {2, 3, 4} ⊂ {1, 2, 3, 4, 5}

<s>[INST] [INST] help me answer this question (i) {2, 3, 4} … {1, 2, 3, 4, 5}, think step by step [/INST]</s> <s>Every element of {2, 3, 4} is also an element of {1, 2, 3, 4, 5}. , The answer is {2, 3, 4} ⊂ {1, 2, 3, 4, 5}</s>

[INST] help me answer this question Here, the vertices are on the x-axis, think step by step [/INST] , The answer is The vertices lie on the x-axis in the given problem.

<s>[INST] [INST] help me answer this question Here, the vertices are on the x-axis, think step by step [/INST]</s> <s>, The answer is The vertices lie on the x-axis in the given problem.</s>

[INST] help me answer this question Simplify the expression: (x^2 - 1) / (x - 1), think step by step [/INST] To simplify the expression, we recognize that it is a difference of squares. Therefore, we can rewrite the expression as (x - 1)(x + 1) / (x - 1). Since the denominator and the first term in the numerator are the same, we can cancel them out to get x + 1. However, since x cannot equal 1 (because we are dividing by zero), we must exclude this value from the domain of the expression. , The answer is The simplified expression is x.

<s>[INST] [INST] help me answer this question Simplify the expression: (x^2 - 1) / (x - 1), think step by step [/INST]</s> <s>To simplify the expression, we recognize that it is a difference of squares. Therefore, we can rewrite the expression as (x - 1)(x + 1) / (x - 1). Since the denominator and the first term in the numerator are the same, we can cancel them out to get x + 1. However, since x cannot equal 1 (because we are dividing by zero), we must exclude this value from the domain of the expression. , The answer is The simplified expression is x.</s>

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

<s>[INST] [INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is `x ≥ 1`.</s>

[INST] help me answer this question (v) {a} ∈ (a, b, c), think step by step [/INST] Here {a} is a set and (a, b, c) is an ordered triple, so a set cannot be an element of an ordered triple. , The answer is False

<s>[INST] [INST] help me answer this question (v) {a} ∈ (a, b, c), think step by step [/INST]</s> <s>Here {a} is a set and (a, b, c) is an ordered triple, so a set cannot be an element of an ordered triple. , The answer is False</s>

[INST] help me answer this question Find the discriminant of the quadratic equation √2x^2 + x + √2 = 0., think step by step [/INST] , The answer is The discriminant (D) of the quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac. For the given equation, a = √2, b = 1, and c = √2. Substituting these values into the formula, we get D = (1)^2 - 4*√2*√2 = -7.

<s>[INST] [INST] help me answer this question Find the discriminant of the quadratic equation √2x^2 + x + √2 = 0., think step by step [/INST]</s> <s>, The answer is The discriminant (D) of the quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac. For the given equation, a = √2, b = 1, and c = √2. Substituting these values into the formula, we get D = (1)^2 - 4*√2*√2 = -7.</s>

[INST] help me answer this question What is the value of 3C3?, think step by step [/INST] , The answer is 3C3 = 1

<s>[INST] [INST] help me answer this question What is the value of 3C3?, think step by step [/INST]</s> <s>, The answer is 3C3 = 1</s>

[INST] help me answer this question A coin is tossed. If the outcome is a head, a die is thrown. If the die shows up an even number, the die is thrown again. Write the sample space for the experiment., think step by step [/INST] N/A , The answer is S = [S1 ∪ S2] where S1 = {(H,1), (H,3), (H,5)} and S2 = {(H,2,2), (H,2,4), (H,2,6), (H,4,2), (H,4,4), (H,4,6), (H,6,2), (H,6,4), (H,6,6)}

<s>[INST] [INST] help me answer this question A coin is tossed. If the outcome is a head, a die is thrown. If the die shows up an even number, the die is thrown again. Write the sample space for the experiment., think step by step [/INST]</s> <s>N/A , The answer is S = [S1 ∪ S2] where S1 = {(H,1), (H,3), (H,5)} and S2 = {(H,2,2), (H,2,4), (H,2,6), (H,4,2), (H,4,4), (H,4,6), (H,6,2), (H,6,4), (H,6,6)}</s>

[INST] help me answer this question (ii) (A ∩ B)’ = A’ U B’, think step by step [/INST] By De Morgan's Law , The answer is True

<s>[INST] [INST] help me answer this question (ii) (A ∩ B)’ = A’ U B’, think step by step [/INST]</s> <s>By De Morgan's Law , The answer is True</s>

[INST] help me answer this question Find the value of cosec (-1410°)., think step by step [/INST] We know that the values of cosec x repeat after an interval of 2π or 360°. Therefore, cosec (-1410°) = cosec (-1080° - 330°) = cosec 330° = cosec (30° + 360°) = cosec 30°. Hence, cosec (-1410°) = 2. , The answer is The value of cosec (-1410°) is 2.

<s>[INST] [INST] help me answer this question Find the value of cosec (-1410°)., think step by step [/INST]</s> <s>We know that the values of cosec x repeat after an interval of 2π or 360°. Therefore, cosec (-1410°) = cosec (-1080° - 330°) = cosec 330° = cosec (30° + 360°) = cosec 30°. Hence, cosec (-1410°) = 2. , The answer is The value of cosec (-1410°) is 2.</s>

[INST] help me answer this question Is f a function from A to B?, think step by step [/INST] If every first element corresponds to only one second element, then it is a function. However, here the same first element, i.e., 2, corresponds to two different images (9 and 11). , The answer is No, it is not.

<s>[INST] [INST] help me answer this question Is f a function from A to B?, think step by step [/INST]</s> <s>If every first element corresponds to only one second element, then it is a function. However, here the same first element, i.e., 2, corresponds to two different images (9 and 11). , The answer is No, it is not.</s>

[INST] help me answer this question It is given that e = 4/3, think step by step [/INST] , The answer is The eccentricity e is given as e = 4/3.

<s>[INST] [INST] help me answer this question It is given that e = 4/3, think step by step [/INST]</s> <s>, The answer is The eccentricity e is given as e = 4/3.</s>

[INST] help me answer this question 2 and 3 are prime numbers which are divisors of 6. Hence, (ii) matches with (a)., think step by step [/INST] , The answer is Set (ii) contains prime numbers that are divisors of 6, which makes it match with set (a).

<s>[INST] [INST] help me answer this question 2 and 3 are prime numbers which are divisors of 6. Hence, (ii) matches with (a)., think step by step [/INST]</s> <s>, The answer is Set (ii) contains prime numbers that are divisors of 6, which makes it match with set (a).</s>

[INST] help me answer this question Show that the products of the corresponding terms of the sequences a, ar, ar^2, …, a r^(n-1) and A, AR, AR^2, …, AR^(n-1) form a G.P., think step by step [/INST] None , The answer is To be proved: The sequence, aA, arAR, ar^2AR^2, …, a r^(n-1)AR^(n-1), forms a G.P.

<s>[INST] [INST] help me answer this question Show that the products of the corresponding terms of the sequences a, ar, ar^2, …, a r^(n-1) and A, AR, AR^2, …, AR^(n-1) form a G.P., think step by step [/INST]</s> <s>None , The answer is To be proved: The sequence, aA, arAR, ar^2AR^2, …, a r^(n-1)AR^(n-1), forms a G.P.</s>

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

<s>[INST] [INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers.</s>

[INST] help me answer this question E ∩ F, think step by step [/INST] , The answer is E ∩ F = (6) ∩ (3, 4, 5, 6) = (6)

<s>[INST] [INST] help me answer this question E ∩ F, think step by step [/INST]</s> <s>, The answer is E ∩ F = (6) ∩ (3, 4, 5, 6) = (6)</s>

[INST] help me answer this question Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m - 1)th numbers is 5:9. Find the value of m., think step by step [/INST] Explanation provided in the markdown text outlines the steps to derive the answer. It includes finding the common difference, deriving an equation on the basis of given ratio, and then solving for m. , The answer is The value of m is 11.

<s>[INST] [INST] help me answer this question Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m - 1)th numbers is 5:9. Find the value of m., think step by step [/INST]</s> <s>Explanation provided in the markdown text outlines the steps to derive the answer. It includes finding the common difference, deriving an equation on the basis of given ratio, and then solving for m. , The answer is The value of m is 11.</s>

[INST] help me answer this question A coin is tossed three times. Describe the sample space for this experiment., think step by step [/INST] When a coin is tossed three times, there are 2 possibilities (Head or Tail) for each toss. So, the total number of possible outcomes is 2*2*2 = 8. These outcomes form the sample space S. , The answer is The sample space S = {HHH, THH, HTH, HHT, TTT, HTT, THT, TTH}

<s>[INST] [INST] help me answer this question A coin is tossed three times. Describe the sample space for this experiment., think step by step [/INST]</s> <s>When a coin is tossed three times, there are 2 possibilities (Head or Tail) for each toss. So, the total number of possible outcomes is 2*2*2 = 8. These outcomes form the sample space S. , The answer is The sample space S = {HHH, THH, HTH, HHT, TTT, HTT, THT, TTH}</s>

[INST] help me answer this question Let us make the table of the given data and append other columns after calculations., think step by step [/INST] Here, the user is required to make a table of given data and add other columns after performing some calculations. , The answer is In this question, the user is being asked to create a table of the given data and append other columns after calculations.

<s>[INST] [INST] help me answer this question Let us make the table of the given data and append other columns after calculations., think step by step [/INST]</s> <s>Here, the user is required to make a table of given data and add other columns after performing some calculations. , The answer is In this question, the user is being asked to create a table of the given data and append other columns after calculations.</s>

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

<s>[INST] [INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers except `x = 4`.</s>

[INST] help me answer this question A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?, think step by step [/INST] E and F are not mutually exclusive because there is a common element in E and F. , The answer is No, E and F are not mutually exclusive events.

<s>[INST] [INST] help me answer this question A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?, think step by step [/INST]</s> <s>E and F are not mutually exclusive because there is a common element in E and F. , The answer is No, E and F are not mutually exclusive events.</s>

[INST] help me answer this question (5.2) How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?, think step by step [/INST] , The answer is The number of litres of water that has to be added will have to be more than 562.5 litres but less than 900 litres.

<s>[INST] [INST] help me answer this question (5.2) How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?, think step by step [/INST]</s> <s>, The answer is The number of litres of water that has to be added will have to be more than 562.5 litres but less than 900 litres.</s>

[INST] help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step [/INST] Explaination: N/A , The answer is Answer: Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.

<s>[INST] [INST] help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step [/INST]</s> <s>Explaination: N/A , The answer is Answer: Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.</s>

[INST] help me answer this question Evaluate the given limit:, think step by step [/INST] Apply L'Hopital's rule or factorization , The answer is a / c

<s>[INST] [INST] help me answer this question Evaluate the given limit:, think step by step [/INST]</s> <s>Apply L'Hopital's rule or factorization , The answer is a / c</s>

[INST] help me answer this question Write the sample space for the experiment in which a coin is tossed and then, if a head occurs, it is tossed a second time and if a tail occurs on the first toss, a die is thrown once., think step by step [/INST] N/A , The answer is S = {(HT), (HH), (T1), (T2), (T3), (T4), (T5), (T6)}

<s>[INST] [INST] help me answer this question Write the sample space for the experiment in which a coin is tossed and then, if a head occurs, it is tossed a second time and if a tail occurs on the first toss, a die is thrown once., think step by step [/INST]</s> <s>N/A , The answer is S = {(HT), (HH), (T1), (T2), (T3), (T4), (T5), (T6)}</s>

[INST] help me answer this question A ∪ B, think step by step [/INST] , The answer is A ∪ B = (1, 2, 3, 4, 5, 6) ∪ (φ) = (1, 2, 3, 4, 5, 6)

<s>[INST] [INST] help me answer this question A ∪ B, think step by step [/INST]</s> <s>, The answer is A ∪ B = (1, 2, 3, 4, 5, 6) ∪ (φ) = (1, 2, 3, 4, 5, 6)</s>

[INST] help me answer this question Given x2/100 + y2/400 = 1, what is the length of the minor axis?, think step by step [/INST] , The answer is Length of minor axis = 2b = 2 (10) = 20

<s>[INST] [INST] help me answer this question Given x2/100 + y2/400 = 1, what is the length of the minor axis?, think step by step [/INST]</s> <s>, The answer is Length of minor axis = 2b = 2 (10) = 20</s>

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

<s>[INST] [INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is `x ≥ 1`.</s>

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

<s>[INST] [INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is `x ≥ 1`.</s>

[INST] help me answer this question sin x + sin 3x + sin 5x = 0, think step by step [/INST] , The answer is The general solution of this equation is given by sin 3x (2 cos 2x + 1) = 0, which implies sin 3x = 0 or 2 cos 2x + 1 = 0.

<s>[INST] [INST] help me answer this question sin x + sin 3x + sin 5x = 0, think step by step [/INST]</s> <s>, The answer is The general solution of this equation is given by sin 3x (2 cos 2x + 1) = 0, which implies sin 3x = 0 or 2 cos 2x + 1 = 0.</s>

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

<s>[INST] [INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers.</s>

[INST] help me answer this question There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?, think step by step [/INST] , The answer is Let 'A' be the event of selecting a woman. n(A) = 6 P(A) = n(A)/n(S) = 6/10 = 3/5

<s>[INST] [INST] help me answer this question There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?, think step by step [/INST]</s> <s>, The answer is Let 'A' be the event of selecting a woman. n(A) = 6 P(A) = n(A)/n(S) = 6/10 = 3/5</s>

[INST] help me answer this question Let us make the table of the given data and append other columns after calculations., think step by step [/INST] Here, the user is required to make a table of given data and add other columns after performing some calculations. , The answer is In this question, the user is being asked to create a table of the given data and append other columns after calculations.

<s>[INST] [INST] help me answer this question Let us make the table of the given data and append other columns after calculations., think step by step [/INST]</s> <s>Here, the user is required to make a table of given data and add other columns after performing some calculations. , The answer is In this question, the user is being asked to create a table of the given data and append other columns after calculations.</s>

[INST] help me answer this question Are the events A and BI mutually exclusive?, think step by step [/INST] A ∩ BI = A ∩ A = A , The answer is No, they are not mutually exclusive.

<s>[INST] [INST] help me answer this question Are the events A and BI mutually exclusive?, think step by step [/INST]</s> <s>A ∩ BI = A ∩ A = A , The answer is No, they are not mutually exclusive.</s>

[INST] help me answer this question Find the sum to n terms in the geometric progression x^3, x^5, x^7, … (if x ≠ ±1 ), think step by step [/INST] , The answer is An expression to find the sum to n terms in the geometric progression x^3, x^5, x^7, … (if x ≠ ±1 )

<s>[INST] [INST] help me answer this question Find the sum to n terms in the geometric progression x^3, x^5, x^7, … (if x ≠ ±1 ), think step by step [/INST]</s> <s>, The answer is An expression to find the sum to n terms in the geometric progression x^3, x^5, x^7, … (if x ≠ ±1 )</s>

[INST] help me answer this question c/a = 4/3, think step by step [/INST] , The answer is The eccentricity e is given as the ratio of c/a, where a = 7, so c = e × a = 4/3 × 7 = 28/3.

<s>[INST] [INST] help me answer this question c/a = 4/3, think step by step [/INST]</s> <s>, The answer is The eccentricity e is given as the ratio of c/a, where a = 7, so c = e × a = 4/3 × 7 = 28/3.</s>

[INST] help me answer this question Evaluate the given limit:, think step by step [/INST] Substitute x + 1 = y, then x = y – 1 , The answer is Undefined

<s>[INST] [INST] help me answer this question Evaluate the given limit:, think step by step [/INST]</s> <s>Substitute x + 1 = y, then x = y – 1 , The answer is Undefined</s>

[INST] help me answer this question The 4th term of a G.P. is the square of its second term, and the first term is -3. Determine its 7th term., think step by step [/INST] We are given that the 4th term of the sequence is the square of its second term and the first term is -3. We first find the second term using the given relationship and then find the 7th term using the formula a_n = a * r^(n-1). , The answer is a7 = a * r^(7-1) = -3*1^6 = -3

<s>[INST] [INST] help me answer this question The 4th term of a G.P. is the square of its second term, and the first term is -3. Determine its 7th term., think step by step [/INST]</s> <s>We are given that the 4th term of the sequence is the square of its second term and the first term is -3. We first find the second term using the given relationship and then find the 7th term using the formula a_n = a * r^(n-1). , The answer is a7 = a * r^(7-1) = -3*1^6 = -3</s>

[INST] help me answer this question Find the equation of the parabola with equation x^2 = -9y., think step by step [/INST] On comparing the given equation with x^2 = -4ay, we get a = 9/4. The coordinates of the focus are (0,-a) = (0, -9/4). The given equation involves x^2, so the axis of the parabola is the y-axis. Therefore, the equation of directrix is y = a, which is y = 9/4. The length of latus rectum is 4a = 4(9/4) = 9. , The answer is x^2 = -9y.

<s>[INST] [INST] help me answer this question Find the equation of the parabola with equation x^2 = -9y., think step by step [/INST]</s> <s>On comparing the given equation with x^2 = -4ay, we get a = 9/4. The coordinates of the focus are (0,-a) = (0, -9/4). The given equation involves x^2, so the axis of the parabola is the y-axis. Therefore, the equation of directrix is y = a, which is y = 9/4. The length of latus rectum is 4a = 4(9/4) = 9. , The answer is x^2 = -9y.</s>

[INST] help me answer this question Is the `Or` used in the following statements exclusive `or` inclusive. Give reasons for your answer. (i) Sun rises or Moon sets. (ii) To apply for a driving licence, you should have a ration card or a passport. (iii) All integers are positive or negative., think step by step [/INST] , The answer is (i) It is not possible for the Sun to rise and the Moon to set together. Hence, the `or` in the given statement is exclusive. (ii) A person can have both a ration card and a passport to apply for a driving license. Hence, the `or` in the given statement is inclusive. (iii) All integers cannot be both positive and negative. Hence, the `or` in the given statement is exclusive.

<s>[INST] [INST] help me answer this question Is the `Or` used in the following statements exclusive `or` inclusive. Give reasons for your answer. (i) Sun rises or Moon sets. (ii) To apply for a driving licence, you should have a ration card or a passport. (iii) All integers are positive or negative., think step by step [/INST]</s> <s>, The answer is (i) It is not possible for the Sun to rise and the Moon to set together. Hence, the `or` in the given statement is exclusive. (ii) A person can have both a ration card and a passport to apply for a driving license. Hence, the `or` in the given statement is inclusive. (iii) All integers cannot be both positive and negative. Hence, the `or` in the given statement is exclusive.</s>

[INST] help me answer this question 1. Question: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2, think step by step [/INST] Explaination: N/A , The answer is Answer: We can write the given statement as

<s>[INST] [INST] help me answer this question 1. Question: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2, think step by step [/INST]</s> <s>Explaination: N/A , The answer is Answer: We can write the given statement as</s>

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

<s>[INST] [INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers except `x = 4`.</s>

[INST] help me answer this question An experiment consists of recording the boy-girl composition of families with 2 children. (i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births?, think step by step [/INST] , The answer is The sample space if we are interested in knowing whether it is a boy or girl in the order of their births, S = {GG, BB, GB, BG}.

<s>[INST] [INST] help me answer this question An experiment consists of recording the boy-girl composition of families with 2 children. (i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births?, think step by step [/INST]</s> <s>, The answer is The sample space if we are interested in knowing whether it is a boy or girl in the order of their births, S = {GG, BB, GB, BG}.</s>

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

<s>[INST] [INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers.</s>

[INST] help me answer this question Given, y – 5x < 30, think step by step [/INST] , The answer is The solution region of the given inequality is below the line 3y – 5x < 30. (That is, the origin is included in the region.)

<s>[INST] [INST] help me answer this question Given, y – 5x < 30, think step by step [/INST]</s> <s>, The answer is The solution region of the given inequality is below the line 3y – 5x < 30. (That is, the origin is included in the region.)</s>

[INST] help me answer this question What will Rs 500 amount to in 10 years after its deposit in a bank which pays an annual interest rate of 10% compounded annually?, think step by step [/INST] To calculate the future value of the deposit, we need to apply the formula for compound interest for 10 years, using the rate of 10% per annum. , The answer is The amount at the end of 10 years will be Rs 500 * (1.1)¹⁰.

<s>[INST] [INST] help me answer this question What will Rs 500 amount to in 10 years after its deposit in a bank which pays an annual interest rate of 10% compounded annually?, think step by step [/INST]</s> <s>To calculate the future value of the deposit, we need to apply the formula for compound interest for 10 years, using the rate of 10% per annum. , The answer is The amount at the end of 10 years will be Rs 500 * (1.1)¹⁰.</s>

[INST] help me answer this question A point is in the XZ plane. What can you say about its y-coordinate?, think step by step [/INST] , The answer is If a point is in the XZ plane, then its y-coordinate is 0.

<s>[INST] [INST] help me answer this question A point is in the XZ plane. What can you say about its y-coordinate?, think step by step [/INST]</s> <s>, The answer is If a point is in the XZ plane, then its y-coordinate is 0.</s>

[INST] help me answer this question Given P(A) = 0.75, P(A∩B) = 0.5, and P(AI∩BI) = 0.1, find the probability of passing the Hindi examination., think step by step [/INST] We have P(B) = P(A∩B) + P(AI∩BI) = 0.5 + 0.1 = 0.6, therefore, P(B) = 0.65 , The answer is 0.65

<s>[INST] [INST] help me answer this question Given P(A) = 0.75, P(A∩B) = 0.5, and P(AI∩BI) = 0.1, find the probability of passing the Hindi examination., think step by step [/INST]</s> <s>We have P(B) = P(A∩B) + P(AI∩BI) = 0.5 + 0.1 = 0.6, therefore, P(B) = 0.65 , The answer is 0.65</s>

[INST] help me answer this question In this question, we get 2 options that are (i) Either all 3 will join or (ii) None of them will join., think step by step [/INST] For option (i), we have to choose 7 students from the remaining 22 in the class. The number of ways to choose 7 students from 22 is 22C7. For option (ii), we have to choose 10 students from the remaining 22 without including the 3 students. The number of ways to choose 10 students from 22 without the 3 is 19C10. , The answer is The excursion party can be chosen in 646800 or 177100 ways, respectively

<s>[INST] [INST] help me answer this question In this question, we get 2 options that are (i) Either all 3 will join or (ii) None of them will join., think step by step [/INST]</s> <s>For option (i), we have to choose 7 students from the remaining 22 in the class. The number of ways to choose 7 students from 22 is 22C7. For option (ii), we have to choose 10 students from the remaining 22 without including the 3 students. The number of ways to choose 10 students from 22 without the 3 is 19C10. , The answer is The excursion party can be chosen in 646800 or 177100 ways, respectively</s>

[INST] help me answer this question Find the centre and radius of the circle with the given condition: x2 + y2 – 4x – 8y – 45 = 0, think step by step [/INST] , The answer is The centre of the given circle is (2, 4), and its radius is √65.

<s>[INST] [INST] help me answer this question Find the centre and radius of the circle with the given condition: x2 + y2 – 4x – 8y – 45 = 0, think step by step [/INST]</s> <s>, The answer is The centre of the given circle is (2, 4), and its radius is √65.</s>

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

<s>[INST] [INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers except `x = 4`.</s>

[INST] help me answer this question Write the additive inverse of the given number: -15, think step by step [/INST] , The answer is The additive inverse of -15 is 15.

<s>[INST] [INST] help me answer this question Write the additive inverse of the given number: -15, think step by step [/INST]</s> <s>, The answer is The additive inverse of -15 is 15.</s>

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

<s>[INST] [INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers.</s>

[INST] help me answer this question If a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d} then {a, e, i, o, u} ∩ {a, b, c, d} = {a}, think step by step [/INST] , The answer is True

<s>[INST] [INST] help me answer this question If a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d} then {a, e, i, o, u} ∩ {a, b, c, d} = {a}, think step by step [/INST]</s> <s>, The answer is True</s>

[INST] help me answer this question Given x2/25 + y2/100 = 1, what is the length of the latus rectum?, think step by step [/INST] , The answer is Length of latus rectum = 2b2/a = (2×52)/10 = (2×25)/10 = 5

<s>[INST] [INST] help me answer this question Given x2/25 + y2/100 = 1, what is the length of the latus rectum?, think step by step [/INST]</s> <s>, The answer is Length of latus rectum = 2b2/a = (2×52)/10 = (2×25)/10 = 5</s>

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

<s>[INST] [INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers except `x = 4`.</s>

[INST] help me answer this question Describe three events which are mutually exclusive and exhaustive., think step by step [/INST] , The answer is Let P be the event of getting exactly two tails. So, P = (HTT, TTH, THT). Let Q be the event of getting at least two heads. So, Q = (HHT, HTH, THH, HHH). Let R be the event of getting only one tail. So, C= (TTT). Therefore, P, Q, and R are mutually exclusive and exhaustive.

<s>[INST] [INST] help me answer this question Describe three events which are mutually exclusive and exhaustive., think step by step [/INST]</s> <s>, The answer is Let P be the event of getting exactly two tails. So, P = (HTT, TTH, THT). Let Q be the event of getting at least two heads. So, Q = (HHT, HTH, THH, HHH). Let R be the event of getting only one tail. So, C= (TTT). Therefore, P, Q, and R are mutually exclusive and exhaustive.</s>

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

<s>[INST] [INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is `x ≥ 1`.</s>

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

<s>[INST] [INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers except `x = 4`.</s>

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

<s>[INST] [INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is `x ≥ 1`.</s>

[INST] help me answer this question Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16., think step by step [/INST] The solution involves solving a system of equations using the given conditions and the general form of the equation of a circle (x – h)2 + (y – k)2 = r2. , The answer is x2 + y2 – 6x – 8y + 15 = 0

<s>[INST] [INST] help me answer this question Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16., think step by step [/INST]</s> <s>The solution involves solving a system of equations using the given conditions and the general form of the equation of a circle (x – h)2 + (y – k)2 = r2. , The answer is x2 + y2 – 6x – 8y + 15 = 0</s>

[INST] help me answer this question Solve the inequality 3x – 7 > 5x – 1. , think step by step [/INST] N/A , The answer is x < -3.

<s>[INST] [INST] help me answer this question Solve the inequality 3x – 7 > 5x – 1. , think step by step [/INST]</s> <s>N/A , The answer is x < -3.</s>

[INST] help me answer this question Evaluate: log₀ 0.1, think step by step [/INST] , The answer is log₀ 0.1 cannot be evaluated because the base (0.1) is less than 1.

<s>[INST] [INST] help me answer this question Evaluate: log₀ 0.1, think step by step [/INST]</s> <s>, The answer is log₀ 0.1 cannot be evaluated because the base (0.1) is less than 1.</s>

[INST] help me answer this question sec x = 2, think step by step [/INST] , The answer is The solution to this equation is not provided in the data.

<s>[INST] [INST] help me answer this question sec x = 2, think step by step [/INST]</s> <s>, The answer is The solution to this equation is not provided in the data.</s>

[INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

<s>[INST] [INST] help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step [/INST]</s> <s>, The answer is The domain of the function is all real numbers except `x = 4`.</s>