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[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 β₯ 1`.
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[INST] help me answer this question From the sample space, calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts., think step by step
[/INST] These amounts are calculated by taking into account the different combinations of heads and tails that can occur when tossing a coin four times. The calculation for each amount is provided in the solution. , The answer is The different amounts of money that can be won are βΉ 4, βΉ 1.50, βΉ 1.50, βΉ 1.50, βΉ 1.50, -βΉ 1, -βΉ 1, -βΉ 1, -βΉ 1, -βΉ 3.50, -βΉ 3.50, -βΉ 3.50, -βΉ 3.50, -βΉ 6.
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[INST] help me answer this question Write the decimal 0.6 as a fraction., think step by step
[/INST] To convert a decimal to a fraction, we can rewrite the decimal as a ratio of its whole number and fractional parts. In this case, 0.6 can be rewritten as 6/10, which simplifies to 3/5. , The answer is The decimal 0.6 as a fraction is 3/5.
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[INST] help me answer this question What is the value of x in the following equation: 2x + 3 = 11?, think step by step
[/INST] To find the value of x, you would first subtract 3 from both sides of the equation to get 2x = 8, and then divide both sides by 2 to get x = 4. , The answer is The value of x is 4.
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[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`.
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[INST] help me answer this question We know that there are 5 vowels and 21 consonants in the English alphabet. Choosing two vowels out of 5 would be done in 5C2 ways. Choosing 2 consonants out of 21 can be done in 21C2 ways., think step by step
[/INST] This is a combination problem because the order of choosing the vowels and consonants does not matter. We need to find the number of ways to select 2 vowels from 5 and 2 consonants from 21. This can be calculated using the combination formula: nCr = n! / (r! Γ (n-r)!), where n is the total number of objects, r is the number of objects to choose, and '!' denotes factorial. , The answer is The total number of ways to select 2 vowels and 2 consonants = 5C2 Γ 21C2
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[INST] help me answer this question b^2 = 6a, think step by step
[/INST] , The answer is Further simplifying b^2 = 12a/2, we get b^2 = 6a.
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[INST] help me answer this question A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of exactly 3 girls?, think step by step
[/INST] , The answer is The number of ways of forming the committee with exactly three girls is 4C3 Γ 9C4 = 126.
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[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 β₯ 1`.
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[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.
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[INST] help me answer this question x + x/2 + x/3 < 11, think step by step
[/INST] The solutions of the given inequality are defined by all the real numbers less than 6. Hence, the solution set is (-β, 6). , The answer is x < 6
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[INST] help me answer this question How does the order of elements in a set affect its equality?, think step by step
[/INST] , The answer is The order in which the elements of a set are listed is not significant.
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[INST] help me answer this question 15x >= 2 (4x β 1), think step by step
[/INST] (Blank) , The answer is x >= -2/7
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[INST] help me answer this question For the following frequency distributions, find the mean., think step by step
[/INST] , The answer is The mean is 26.8.
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[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.
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[INST] help me answer this question 2b^2 = 12a, think step by step
[/INST] , The answer is By comparing with the formula for the length of the latus rectum, we can equate 2b^2 to 12a.
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[INST] help me answer this question Find the mean for the data 6, 10., think step by step
[/INST] The mean is calculated as the sum of all observations divided by the number of observations, which comes out to be 8. , The answer is The mean of the data is 8.
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[INST] help me answer this question Exercise 1.2 Solutions, think step by step
[/INST] None , The answer is Not applicable, as this is a section header, not a question.
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[INST] help me answer this question Question: Insert two numbers between 3 and 81 so that the resulting sequence is a G.P., think step by step
[/INST] Explanation:
Given the first term (3) and the fourth term (81), and the fact that the sequence is a G.P., we can find the common ratio (r) by solving for the cube root of the ratio between the first and fourth terms (r^3 = 81/3, hence r = 3). Using this ratio, we can find the two numbers, which are indeed the second and third terms of the sequence: 3 * 3 = 9 (second term) and 3 * 3^2 = 27 (third term). , The answer is Answer: 9 and 27
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[INST] help me answer this question As x β₯ 1 β (x β 1) β₯ 0 β β(x β 1) β₯ 0 Thus, the range of f is the set of all real numbers greater than or equal to 0. Range of f = [0, β)., think step by step
[/INST] , The answer is The range of the function f(x) = β(x β 1) is the set of all real numbers greater than or equal to 0.
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[INST] help me answer this question Evaluate the given limit:, think step by step
[/INST] Given, substitute x = 3, we get (3 + 3) = 6 , The answer is 6
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[INST] help me answer this question Prove that (cos x + cos y)^2 + (sin x + sin y)^2 = 2 + 2cos(x β y), think step by step
[/INST] This involves the use of trigonometric formulas and simplification. , The answer is LHS = (cos^2 x + 2cos x cos y + cos^2 y) + (sin^2 x + 2sin x sin y + sin^2 y) = 2 + 2cos(x β y) = RHS
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[INST] help me answer this question (1 β i) β (β1 + i6), think step by step
[/INST] , The answer is 2 β i7
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[INST] help me answer this question What is A U B?, think step by step
[/INST] , The answer is The value of A U B is the empty set (Ο).
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[INST] help me answer this question Show that the statement \"For any real numbers a and b, a2 = b2 implies that a = b\" is not true by giving a counter-example., think step by step
[/INST] The given statement is a conditional statement in the form of 'if and only if' (iff). To show that the statement is not true, a counter-example is provided, where a and b are real numbers which satisfy a2 = b2 but a eq b. , The answer is Let a = 1 and b = -1<br>a2 = (1)2 = 1 and<br>b2 = (-1)2 = 1<br>Hence, a2 = b2<br>However, a eq b<br>Therefore, it can be concluded that the given statement is false.
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[INST] help me answer this question The function βtβ, which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = 9/5C + 32. Find, think step by step
[/INST] , The answer is (i) t(0) = 32
(ii) t(28) = 82
(iii) t(β10) = 14
(iv) C when t(C) = 212 = 100
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[INST] help me answer this question b^2 = 12a/2, think step by step
[/INST] , The answer is Simplifying 2b^2 = 12a, we get b^2 = 12a/2.
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[INST] help me answer this question A man starts repaying a loan as the first instalment of Rs. 100. If he increases the instalment by Rs. 5 every month, what amount will he pay in the 30th instalment?, think step by step
[/INST] Given, The first instalment of the loan is Rs. 100. The second instalment of the loan is Rs. 105, and so on, as the instalment increases by Rs. 5 every month. Thus, the amount that the man repays every month forms an Arithmetic Progression (A.P.) with the first term a = 100 and the common difference d = 5. So, the 30th term in this A.P. will be A30 = a + (30 β 1)d = 100 + (29) (5) = 100 + 145 = 245 Therefore, the amount to be paid in the 30th instalment will be Rs. 245. , The answer is The amount to be paid in the 30th instalment will be Rs. 245.
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[INST] help me answer this question Find the domain of the function f(x) = (x - 1)/(x - 2)., think step by step
[/INST] , The answer is The domain of the function f(x) is R - {2}, where R is the set of all real numbers.
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[INST] help me answer this question A card is selected from a pack of 52 cards.
(a) How many points are there in the sample space?, think step by step
[/INST] , The answer is The number of points in the sample space is 52.
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[INST] help me answer this question What are the coordinates of the foci?, think step by step
[/INST] , The answer is The coordinates of the foci are (0, β21) and (0, -β21).
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[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 β₯ 1`.
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[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 β₯ 1`.
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[INST] help me answer this question (sin 2x + cos x)(sin 2x - cos x), think step by step
[/INST] Simplify the given expression using the formula (a - b)(a + b) = a^2 - b^2 and (sin^2 x + cos^2 x) = 1. , The answer is sin4 x + cos2 x - 1
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[INST] help me answer this question For all the values of y, the value of x would be the same in the given inequality, which would be the region above the x-axis on the graph.
, think step by step
[/INST] , The answer is N/A
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[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 β₯ 1`.
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[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.
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[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`.
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[INST] help me answer this question Write the balanced chemical equation for the reaction between Sodium Hydroxide and Hydrochloric Acid., think step by step
[/INST] , The answer is The balanced chemical equation is as follows:
NaOH + HCl -> NaCl + H2O
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[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.
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[INST] help me answer this question If <img src='https://latex.codecogs.com/svg.image?log_{10}x=2' title='log_{10}x=2' />, find x., think step by step
[/INST] , The answer is x = 3
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[INST] help me answer this question (3) How is the solution region of the inequality described?, think step by step
[/INST] N/A , The answer is The solution region of the given inequality is right to the line x > β 3, and the origin is included in the region
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[INST] help me answer this question In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?, think step by step
[/INST] This is derived from n(C βͺ T) = n(C) + n(T) - n(C β© T) by substituting the given values and solving for n(C β© T). , The answer is 19 people like both coffee and tea.
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[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 β₯ 1`.
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[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.
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[INST] help me answer this question If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find (i) A β© B (ii) B β© C (iii) A β© C β© D (iv) A β© C (v) B β© D (vi) A β© (B βͺ C) (vii) A β© D (viii) A β© (B βͺ D) (ix) (A β© B) β© (B βͺ C) (x) (A βͺ D) β© (B βͺ C), think step by step
[/INST] , The answer is {'i': 'A β© B = {7, 9, 11}', 'ii': 'B β© C = {11, 13}', 'iii': 'A β© C β© D = Ξ¦', 'iv': 'A β© C = {11}', 'v': 'B β© D = Ξ¦', 'vi': 'A β© (B βͺ C) = {7, 9, 11}', 'vii': 'A β© D = Ξ¦', 'viii': 'A β© (B βͺ D) = {7, 9, 11}', 'ix': '(A β© B) β© (B βͺ C) = {7, 9, 11}', 'x': '(A βͺ D) β© (B βͺ C) = {7, 9, 11, 15}'}
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[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`.
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[INST] help me answer this question Is 3 a member of set C?, think step by step
[/INST] , The answer is Yes, 3 is a member of set C.
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[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 β₯ 1`.
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[INST] help me answer this question Describe two events which are not mutually exclusive., think step by step
[/INST] , The answer is Let A be the event of getting at least two heads. So, A = (HHH, HHT, THH, HTH). Let B be the event of getting only head. So, B= (HHH). Therefore, A β© B = (HHH), and A and B are not mutually exclusive.
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[INST] help me answer this question Explain the roster form of a set., think step by step
[/INST] , The answer is The roster form of a set lists all its elements between braces {}.
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[INST] help me answer this question Is 3! + 4! = 7?, think step by step
[/INST] LHS = 3! + 4! = 30<br>RHS = 7! = 5040<br>Therefore, LHS =RS , The answer is No
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[INST] help me answer this question Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola 9y2 β 4x2 = 36, think step by step
[/INST] Given the equation 9y2 β 4x2 = 36 or y2/4 β x2/9 = 1, we can compare it with the standard equation of a hyperbola y2/a2 β x2/b2 = 1. By doing so, we find that a = 2 and b = 3. We then calculate c using a2 + b2 = c2, which gives us c = β13. The coordinates , The answer is The coordinates of the foci are (0, β13) and (0, -β13), the coordinates of the vertices are (0, 2) and (0, -2), the eccentricity is e = c/a = β13/2, and the length of the latus rectum is (9/2).
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[INST] help me answer this question Is the origin included in the solution area for the second inequality?, think step by step
[/INST] , The answer is No, the origin is not included in the solution area for the second inequality.
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[INST] help me answer this question Find the centre and radius of the circle with the given condition: (x + 5)2 + (y β 3)2 = 36, think step by step
[/INST] , The answer is The centre of the given circle is (-5, 3), and its radius is 6.
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[INST] help me answer this question Is the origin included in the solution area for the third inequality?, think step by step
[/INST] , The answer is Yes, the origin is included in the solution area for the third inequality.
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[INST] help me answer this question What are the required points for the second inequality?, think step by step
[/INST] , The answer is The required points are (0, 1) and (1, 0).
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[INST] help me answer this question (iii) {x: x is a student of Class XI of your school} β¦ {x: x student of your school}, think step by step
[/INST] Every element of {x: x is a student of Class XI of your school} is an element of {x: x student of your school}. , The answer is {x: x is a student of Class XI of your school} β {x: x student of your school}
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[INST] help me answer this question Prove that 9n+1 - 8n+1 - 9 is divisible by 64 whenever n is a positive integer., think step by step
[/INST] , The answer is Not applicable.
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[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 β₯ 1`.
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[INST] help me answer this question Simplify the given expression: (x^2)^3, think step by step
[/INST] , The answer is The simplified form of the given expression is x^6.
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[INST] help me answer this question (12.3 Exercises Solutions), think step by step
[/INST] The data contains a link to access the solutions for Exercise 12.3, but does not include the solutions directly. , The answer is Not provided in the data.
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[INST] help me answer this question Write the contrapositive of the given statement: "If x is not divisible by 4, then x is not an even number.", think step by step
[/INST] The contrapositive statement is created by negating both the hypothesis and conclusion of the original statement and then interchanging them. , The answer is If x is an even number, then x is divisible by 4.
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[INST] help me answer this question Find the mean and variance for the first n natural numbers., think step by step
[/INST] The mean is calculated using the formula for the sum of an arithmetic series. The variance is calculated using the formula for the variance of an arithmetic series. , The answer is The mean is (n + 1)/2 and the variance is (n^2 β 1)/12.
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[INST] help me answer this question Write the chemical formula for Compound X and Y, which are composed of Sodium, Chlorine, Calcium, and Oxygen?, think step by step
[/INST] Sodium (Na) has an atomic mass of 23 u, and Chlorine (Cl) has an atomic mass of 35.5 u. Thus, the molar mass of Sodium is 23 g/mol, and the molar mass of Chlorine is 35.5 g/mol. To find the number of moles of each element, we divide the mass of each element by its molar mass. So, we get 0.2 moles of Sodium and 0.16 moles of Chlorine. Since both Sodium and Chlorine have a valence of 1, the chemical formula for Compound X is NaCl. Calcium (Ca) has an atomic mass of 40 u, and Oxygen (O) has an atomic mass of 16 u. Thus, the molar mass of Calcium is 40 g/mol, and the molar mass of Oxygen is 16 g/mol. To find the number of moles of each element, we divide the mass of each element by its molar mass. So, we get 0.056 moles of Calcium and 0.11375 moles of Oxygen. Since Calcium has a valence of 2, and Oxygen has a valence of 2, we need to divide the number of moles of Oxygen by 2 to balance the valence of Calcium. Therefore, the chemical formula for Compound Y is CaO. , The answer is The chemical formula for Compound X is NaCl and the chemical formula for Compound Y is CaO.
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[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 β₯ 1`.
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[INST] help me answer this question Find the equation of the circle with the given condition: 36x2 + 36x + 36y2 β 18y + 11 = 0, think step by step
[/INST] , The answer is The equation of the circle is 36x2 + 36y2 + 36x β 18y + 11 = 0
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[INST] help me answer this question cos 6x = 32 cos6 x β 48 cos4 x + 18 cos2 x β 1, think step by step
[/INST] cos 6x = cos 3(2x), and using the formula cos 3A = 4 cos3 A β 3 cos A, it can be written as 4 cos3 2x β 3 cos 2x. Further simplification yields the given expression. , The answer is True
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[INST] help me answer this question Define the terms 'domain' and 'codomain' of a relation. , think step by step
[/INST] The domain of a relation is the set of all inputs for the relation, while the codomain is the set of all possible outputs. , The answer is The domain of a relation is the set of all first elements of the ordered pairs in the relation. The codomain of a relation is the set of all possible values that the second elements of the relation can take.
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[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.
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[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`.
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[INST] help me answer this question Write the first five terms of the sequence whose nth term is an = n/n+1, think step by step
[/INST] , The answer is 1/2, 2/3, 3/4, 4/5, 5/6
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[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`.
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[INST] help me answer this question A β C, think step by step
[/INST] , The answer is A β C = (1, 2, 3, 4, 5, 6) - (3, 6) = (1, 2, 4, 5)
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[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`.
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[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.
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[INST] help me answer this question Is set A equal to set B?, think step by step
[/INST] , The answer is No, set A is not equal to set B.
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[INST] help me answer this question If in two circles, arcs of the same length subtend angles 60Β° and 75Β° at the centre, find the ratio of their radii., think step by step
[/INST] , The answer is To find the ratio of the radii, we first need to convert the angles to radians and then use the ratio of the arc-lengths formula. The final answer would be a simplified ratio of the radii based on the given angles.
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[INST] help me answer this question Find the coefficient of x5 in the product (1 + 2x)6 (1 β x)7 using binomial theorem., think step by step
[/INST] After expanding both expressions and multiplying them, the coefficient of x5 in the resulting expression is 171 , The answer is 171
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[INST] help me answer this question Substitute the value of 1/aΒ² in equation (3), 36/aΒ² + 4/bΒ² = 1., think step by step
[/INST] , The answer is 9/4 β 81/4bΒ² + 4/bΒ² = 1 β 9/4
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[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`.
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[INST] help me answer this question Is set B equal to set C?, think step by step
[/INST] , The answer is No, set B is not equal to set C.
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[INST] help me answer this question The number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination is , think step by step
[/INST] N/A , The answer is 778320
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[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 β₯ 1`.
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[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.
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[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`.
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[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 β₯ 1`.
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[INST] help me answer this question a11 if an = (-1)^n-1 n, think step by step
[/INST] , The answer is a11 = 11
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[INST] help me answer this question Given x2/25 + y2/100 = 1, what is the eccentricity?, think step by step
[/INST] , The answer is Eccentricity, e = c/a = β3/2
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[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.
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[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`.
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[INST] help me answer this question Find the variance of the following data: 2.25, 21, 24, 27, 30, 16.5., think step by step
[/INST] The variance can be calculated by first finding the mean (average) of the data, then finding the deviation of each number from the mean, squaring those deviations, adding them together, and dividing by the number of data points less one. The mean is 16.5 and the variance is 74.25. , The answer is The variance is 74.25.
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[INST] help me answer this question What is the coefficient of variation (C.V) for group A?, think step by step
[/INST] , The answer is The coefficient of variation (C.V) for group A is 33.83.
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[INST] help me answer this question Question : y + 8 >= 2x, think step by step
[/INST] The question is incomplete and missing the solution region information. , The answer is N/A
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[INST] help me answer this question (a) What is the role of a catalyst in a chemical reaction?, think step by step
[/INST] A catalyst works by providing an alternative reaction pathway with a lower energy barrier, allowing more reactant molecules to overcome the activation energy threshold and undergo a reaction. , The answer is A catalyst is a substance that increases the rate of a chemical reaction by lowering the activation energy required for the reaction to occur.
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[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 β₯ 1`.
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[INST] help me answer this question If A = {β1, 1}, find A Γ A Γ A., think step by step
[/INST] , The answer is The A Γ A Γ A for a non-empty set A is given by A Γ A Γ A = {(a, b, c): a, b, c β A} Here, it is given A = {β1, 1} So, A Γ A Γ A = {(β1, β1, β1), (β1, β1, 1), (β1, 1, β1), (β1, 1, 1), (1, β1, β1), (1, β1, 1), (1, 1, β1), (1, 1, 1)}
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[INST] help me answer this question Find the sum to n terms of the series whose nth term is given by n (n + 1) (n + 4)., think step by step
[/INST] , The answer is The sum to n terms of the series can be expressed as (1/3)n(n+1)(n+4)(n+5).
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[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 β₯ 1`.
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[INST] help me answer this question What is set A?, think step by step
[/INST] , The answer is Set A = {2, 4, 8, 12}
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