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nwadgeri

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cs541_dataset

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[INST] help me answer this question What is the mean for Y?, think step by step [/INST] , The answer is The mean (ȳ) for Y is 105.

[INST] help me answer this question Divide the given numbers and express the result in standard form: 5 × 10⁵ ÷ 2 × 10³, think step by step [/INST] , The answer is The result of the given division in standard form is 2.5 × 10².

[INST] help me answer this question Solve the given equation: 2x - 3 = 0, think step by step [/INST] , The answer is The solution of the given equation is x = 3/2.

[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.

[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`.

[INST] help me answer this question Find the number of litres that the owner of a milk store could sell weekly at Rs. 17/litre., think step by step [/INST] , The answer is The owner can sell 1340 litres weekly at Rs. 17/litre.

[INST] help me answer this question What is the mean for X?, think step by step [/INST] , The answer is The mean (x̄) for X is 51.

[INST] help me answer this question (ii) Subsets of {a, b} are, think step by step [/INST] , The answer is Φ, {a}, {b}, and {a, b}.

[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 ∩ B = (HHH), and A and B are not mutually exclusive.

[INST] help me answer this question In the figure given, PQ || RS. Find the measure of ∠R., think step by step [/INST] Since PQ is parallel to RS, then ∠PRQ = ∠QRS. Also, ∠PRQ and ∠R are vertical angles, so they are equal in measure. Therefore, ∠R = ∠PRQ = 60 degrees. , The answer is ∠R measures 60 degrees.

[INST] help me answer this question Given, n(A) = 3 and n(B) = 2, and (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements., think step by step [/INST] , The answer is Set A = {x, y, z}, Set B = {1, 2}

[INST] help me answer this question Is it a statement?, think step by step [/INST] It is not evident to whom 'he' is referred. , The answer is No, it is not a statement.

[INST] help me answer this question Write the fraction 5/6 as a decimal., think step by step [/INST] To convert a fraction to a decimal, we divide the numerator by the denominator. In this case, 5 divided by 6 equals 0.833333333333 when expressed as a repeating decimal. , The answer is The fraction 5/6 as a decimal is 0.833333333333.

[INST] help me answer this question Which of the conditions for a valid probability distribution do not hold for the assignment of throwing a die?, think step by step [/INST] , The answer is The second condition, which states that the sum of probabilities should be greater than or equal to 1, does not hold for the assignment of throwing a die.

[INST] help me answer this question If X and Y are two sets such that X has 40 elements, X ∪Y has 60 elements, and X ∩Y has 10 elements, how many elements does Y have?, think step by step [/INST] This is derived from n(X ∪ Y) = n(X) + n(Y) - n(X ∩ Y) by substituting the given values and solving for n(Y). , The answer is Y has 30 elements.

[INST] help me answer this question (ii) (a, b) ∈ R, implies (b, a) ∈ R, think step by step [/INST] , The answer is No, it is not true that (a, b) ∈ R, implies (b, a) ∈ R.

[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`.

[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`.

[INST] help me answer this question For the inequality x + y ≥ 4, 2x – y < 0, think step by step [/INST] , The answer is The shaded region is the required solution to the inequalities.

[INST] help me answer this question Is it a statement?, think step by step [/INST] It is not evident to whom 'he' is referred. , The answer is No, it is not a statement.

[INST] help me answer this question Find the indicated term in the sequence whose nth term is an = n2/2n ; a7, think step by step [/INST] On substituting n = 7, we get a7 = 72/27 = 49/128. , The answer is a7 = 49/128

[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`.

[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.

[INST] help me answer this question Given x2/49 + y2/36 = 1, what is the length of the major axis?, think step by step [/INST] , The answer is Length of major axis = 2a = 2 (7) = 14

[INST] help me answer this question **Question**: If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets: (i) A = {a, b, c}, think step by step [/INST] , The answer is A' = {d, e, f, g, h}

[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.

[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`.

[INST] help me answer this question (-2 – 1/3i)3, think step by step [/INST] Not available in the text , The answer is -22/3 – 107/27i

[INST] help me answer this question Which of the following cannot be a valid assignment of probabilities for outcomes of sample Space S = {ω1, ω2, ω3, ω4, ω5, ω6, ω7}?, think step by step [/INST] The axiomatic approach to probability requires that each probability assigned to the outcomes be non-negative and the sum of all the probabilities of all outcomes should be equal to 1. , The answer is

[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.

[INST] help me answer this question (a + 8) (a – 2) = 0, think step by step [/INST] , The answer is Factoring the quadratic equation a^2 + 8a – 2a – 16 = 0, we get (a + 8) (a – 2) = 0.

[INST] help me answer this question Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. , think step by step [/INST] The process involves comparing the given equation with the standard form of the ellipse, i.e., x2/a2 + y2/b2 = 1, to find the values of a, b, and c. Then, using these values, the coordinates of the foci, vertices, and other parameters are calculated. , The answer is {'1.1': {'coordinates_foci': '(2√5, 0) and (-2√5, 0)', 'coordinates_vertices': '(6, 0) and (-6, 0)', 'length_major_axis': 12, 'length_minor_axis': 8, 'eccentricity': '√5/3', 'length_latus_rectum': '16/3'}, '1.2': {'coordinates_foci': '(0, √21) and (0, -√21)', 'coordinates_vertices': '(0, 5) and (0, -5)', 'length_major_axis': 10, 'length_minor_axis': 4, 'eccentricity': '√21/5'}}

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

[INST] help me answer this question If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation., think step by step [/INST] Let the roots of the quadratic equation be a and b. Given the arithmetic mean (A.M.) and geometric mean (G.M.) of the roots, we can use the properties of A.M. and G.M. to determine the values of (a + b) and ab. Subsequently, we can form a quadratic equation using the formula x² - x(a + b) + (ab) = 0. , The answer is The quadratic equation is x² - 16x + 25 = 0.

[INST] help me answer this question Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis. (i) x – √3y + 8 = 0 (ii) y – 2 = 0 (iii) x – y = 4, think step by step [/INST] The equation of a line in normal form is x cos θ + y sin θ = p, where θ is the angle between the perpendicular and the positive x-axis and p is the perpendicular distance from the origin. To reduce the given equations into normal form, we need to divide both sides of the equation by the constant term so that the coefficient of x and y are 1 and -1, respectively. We then find the value of p by taking the absolute value of the constant term on the right-hand side of the equation. To find the angle θ, we use the fact that the tangent of the angle is the negative reciprocal of the coefficient of y. , The answer is Perpendicular distance from origin = 4, Angle with positive x-axis = 120° Perpendicular distance from origin = 2, Angle with positive x-axis = 90° Perpendicular distance from origin = 4/√2, Angle with positive x-axis = 135°

[INST] help me answer this question 3(x – 2)/5 ≤ 5 (2 – x)/3, think step by step [/INST] The solutions of the given inequality are defined by all the real numbers less than or equal to 2. Hence, the required solution set is (-∞, 2]. , The answer is x ≤ 2

[INST] help me answer this question 5x + 3 > 17, think step by step [/INST] , The answer is x ∈ (-∞, 2.6)

[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`.

[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`.

[INST] help me answer this question (ii) [6, 12], think step by step [/INST] , The answer is {x: x ∈ R, 6 ≤ x ≤ 12}

[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

[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`.

[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.

[INST] help me answer this question Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1), think step by step [/INST] The distance is calculated using the formula Distance PQ = √[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]. The values of x1, y1, z1, x2, y2, z2 are substituted in the formula for the first pair of points, and the square roots of the sums are calculated. , The answer is 2√5 units

[INST] help me answer this question What is the value of the expression ((5!)/(3!)) * (1/2!) when simplified?, think step by step [/INST] The value of the expression can be simplified by cancelling out the common factors in the numerator and denominator. The expression reduces to (5!/(3!*2!)) = (5*4)/(2*1) = 10. , The answer is The value of the expression is 10.

[INST] help me answer this question Given, x > – 3, think step by step [/INST] , The answer is The solution region of the given inequality is to the right of the line x = – 3. (That is, away from the origin.)

[INST] help me answer this question (1 – i)4, think step by step [/INST] (1 – i)4 = [(1 – i)2]2 = [1 + i2 – 2i]2 = [1 - 1 - 2i]2 = (-2i)2 = -4 + 8i , The answer is -4 + 8i

[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`.

[INST] help me answer this question Foci (± 3√5, 0), the latus rectum is of length 8., think step by step [/INST] Given foci (± 3√5, 0) are on the x-axis, and the latus rectum is of length 8, so the equation of the hyperbola is of the form x2/a2 – y2/b2 = 1. Here, a = 5 and c = 3√5. We know that a2 + b2 = c2, so b2 = 20. , The answer is The equation of the hyperbola is x2/25 – y2/20 = 1.

[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

[INST] help me answer this question How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5, assuming that Repetition of the digits is allowed?, think step by step [/INST] N/A , The answer is 125

[INST] help me answer this question Show that the given points are the vertices of a parallelogram., think step by step [/INST] , The answer is ABCD is a parallelogram.

[INST] help me answer this question x/3 > x/2 + 1, think step by step [/INST] The solutions of the given inequality are defined by all the real numbers less than -6. Hence, the required solution set is (-∞, -6). , The answer is x < -6

[INST] help me answer this question Expand the expression (3x2 – 2ax + 3a2)3, think step by step [/INST] This is the expansion of the given binomial expression using the binomial theorem. Each term is obtained by multiplying (3x2 – 2ax + 3a2) by itself for a certain number of times based on the binomial theorem formula. , The answer is 27x6 – 54ax5 + 117a2x4 – 116a3x3 + 117a4x2 – 54a5x + 27a6

[INST] help me answer this question What are the possible outcomes when a coin is tossed?, think step by step [/INST] , The answer is The possible outcomes when a coin is tossed are Head (H) or Tail (T).

[INST] help me answer this question What is the Cartesian product of sets A = {1, 2} and B = {3, 4}?, think step by step [/INST] The Cartesian product of sets A and B is the set of all ordered pairs (a, b), where a is an element of set A and b is an element of set B. , The answer is A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}

[INST] help me answer this question Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R., think step by step [/INST] For question 3, since the difference between any two integers is always an integer, therefore, the domain and range of R is Z. , The answer is Domain of R = Z and Range of R = Z.

[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).

[INST] help me answer this question The number of ways a student can choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student is , think step by step [/INST] The problem doesn't provide enough information to calculate the answer. We need to know how many courses the student can choose freely from the 9 available courses. , The answer is N/A

[INST] help me answer this question Calculate the standard deviation for the group B data., think step by step [/INST] , The answer is The standard deviation for group B is 15.61.

[INST] help me answer this question Find the coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6)., think step by step [/INST] , The answer is The coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6) are R(6.67, -5.33, -4) and S(7.33, -10.67, 4).

[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`.

[INST] help me answer this question Expand (1 - 3x)⁷ using the binomial theorem., think step by step [/INST] The given text provides a detailed explanation for this calculation. , The answer is The expanded expression of (1 - 3x)⁷ is 1 - 21x + 210x² - 1260x³ + 4725x⁴ - 11340x⁵ + 17010x⁶ - 15120x⁷ + 6237x⁸.

[INST] help me answer this question a^2 + 6a – 16 = 0, think step by step [/INST] , The answer is Subtracting 16 from both sides of the equation a^2 + 6a = 16, we get a^2 + 6a – 16 = 0.

[INST] help me answer this question a^2 + b^2 = c^2, think step by step [/INST] , The answer is This equation represents one of the properties of a right-angled triangle, where a, b, and c are the lengths of the sides.

[INST] help me answer this question Find the equation of the circle with radius 5 whose centre lies on the x-axis and passes through the point (2, 3)., think step by step [/INST] The solution involves using the given conditions, the general form of the equation of a circle (x – h)2 + (y – k)2 = r2, and the fact that the center lies on the x-axis (k = 0). , The answer is x2 + y2 – 7x + 5y – 14 = 0

[INST] help me answer this question For 2x + y ≥ 8, x + 2y ≥ 10, find the region of the solution., think step by step [/INST] , The answer is The region of the solution for the given inequalities is the area to the right of the line 2x + y = 8 and the area to the left of the line x + 2y = 10.

[INST] help me answer this question What is the sample space for the experiment where a die is thrown repeatedly until a six comes up?, think step by step [/INST] , The answer is The sample space, denoted as S, is infinitely defined and includes all the possible combinations of throws that result in a six, such as {(6), (1,6), (2,6), (3,6), (4,6), (5,6), (1,1,6), (1,2,6)...}.

[INST] help me answer this question Without using Pythagoras’ theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right-angled triangle., think step by step [/INST] We can find the slope of the lines formed by joining the given points and check if the product of the slopes of the lines, which form the angles of the triangle, is -1. , The answer is The points are the vertices of a right-angled triangle.

[INST] help me answer this question (iii) [A ∩ A'], think step by step [/INST] , The answer is Φ

[INST] help me answer this question How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?, think step by step [/INST] N/A , The answer is 90

[INST] help me answer this question (iv) 1∈A is correct, think step by step [/INST] , The answer is 1 is an element of A.

[INST] help me answer this question What can you say about lines AB and CA?, think step by step [/INST] , The answer is m1.m3 = -1.1 = -1, hence, the lines AB and CA are perpendicular to each other.

[INST] help me answer this question (i) [A U A'], think step by step [/INST] , The answer is U

[INST] help me answer this question 6) Where, A = 92.5, h = 5, think step by step [/INST] This is not a question but a part of the instructions for calculating the mean, variance, and standard deviation. , The answer is Not applicable.

[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.

[INST] help me answer this question Evaluate the following: ∫(x^2 + 2x + 1) dx., think step by step [/INST] We integrate each term separately, using the power rule for integration. , The answer is The integral of (x^2 + 2x + 1) with respect to x is (1/3)x^3 + x^2 + x + C, where C is the constant of integration.

[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`.

[INST] help me answer this question Is number 3 odd?, think step by step [/INST] An odd number is a number that cannot be divided evenly by 2. Number 3 cannot be divided evenly by 2, therefore, 3 is an odd number. , The answer is Yes, number 3 is odd.

[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.

[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

[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`.

[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.

[INST] help me answer this question Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2)., think step by step [/INST] , The answer is The angle between the x-axis and the line joining the points (3, –1) and (4, –2) is 135°.

[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.

[INST] help me answer this question (iii) {x: x is a student of Class XI of your school} … {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} ⊂ {x: x student of your school}

[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 – 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.

[INST] help me answer this question What is the eccentricity?, think step by step [/INST] , The answer is Eccentricity, e = c/a = √21/5

[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}

[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`.

[INST] help me answer this question (iv) {x: x is a circle in the plane} … {x: x is a circle in the same plane with radius 1 unit}, think step by step [/INST] Not every circle in the plane is a circle with radius 1 unit in the same plane. , The answer is {x: x is a circle in the plane} ⊄ {x: x is a circle in the same plane with radius 1 unit}

[INST] help me answer this question Number of vowels in PERMUTATIONS = 5 (E, U, A, I, O), Now, we consider all the vowels together as one. Number of permutations of vowels = 120, Now total number of letters = 12 – 5 + 1= 8. Find the number of permutations., think step by step [/INST] , The answer is Number of permutations = 8!/4! = 1680

[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`.

[INST] help me answer this question If A = {x: x is a natural number}, B ={x: x is an even natural number} C = {x: x is an odd natural number} and D = {x: x is a prime number}, find (i) A ∩ B (ii) A ∩ C (iii) A ∩ D (iv) B ∩ C (v) B ∩ D (vi) C ∩ D, think step by step [/INST] , The answer is {'i': 'A ∩ B = {2, 4, 6, 8, ...}', 'ii': 'A ∩ C = {1, 3, 5, 7, ...}', 'iii': 'A ∩ D = {2, 3, 5, 7, ...}', 'iv': 'B ∩ C = Φ', 'v': 'B ∩ D = {2}', 'vi': 'C ∩ D = Φ'}

[INST] help me answer this question What is E ∩ F’?, think step by step [/INST] , The answer is The value of E ∩ F’ is the empty set (φ).

[INST] help me answer this question cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1, think step by step [/INST] cos 6x = cos 3(2x), and using the formula cos 3A = 4 cos3 A – 3 cos A, it can be written as 4 cos3 2x – 3 cos 2x. Further simplification yields the given expression. , The answer is True

[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.

[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`.

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

[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.