Datasets:

nwadgeri

/

cs541_dataset

like 0

[INST] help me answer this question Expand (1 – 2x)5, think step by step [/INST] , The answer is 1 – 10x + 40x2 – 80x3 + 80x4 – 32x5

[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 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}

[INST] help me answer this question Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, – 4) and B (8, 0)., think step by step [/INST] The slope of a line is the change in the y-coordinate divided by the change in the x-coordinate. , The answer is The slope of the line is -1/2.

[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.

[INST] help me answer this question Find the sum to n terms of the A.P., whose kth term is 5k + 1., think step by step [/INST] We found the expressions for a1 and an using the kth term given in the problem and then used the formula for the sum of an A.P. , The answer is The sum formula is n(a1 + an) / 2.

[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.

[INST] help me answer this question What is the fifth inequality given in the problem?, think step by step [/INST] , The answer is y ≥ 0

[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}

[INST] help me answer this question Are the events A and B mutually exclusive?, think step by step [/INST] A ∩ B = {(1, 1), (1, 2), (1, 3), (3, 1)} eq φ , The answer is No, they are not mutually exclusive.

[INST] help me answer this question Write the equations for the x-and y-axes., think step by step [/INST] , The answer is The y-coordinate of every point on x-axis is 0. ∴ Equation of x-axis is y = 0. The x-coordinate of every point on y-axis is 0. ∴ Equation of y-axis is x = 0.

[INST] help me answer this question Prove that cos (x + y) = cos x cos y - sin x sin y, think step by step [/INST] N/A , The answer is cos (x + y) = cos x cos y - sin x sin y

[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)}

[INST] help me answer this question Let us consider the equation of the required circle be (x – h)2 + (y – k)2 = r2. Find the equation of the circle., think step by step [/INST] We know that the radius of the circle is 5 and its centre lies on the x-axis, so k = 0 and r = 5. , The answer is The equation of the circle is (x – h)2 + y2 = 25.

[INST] help me answer this question The shaded area in the graph is the solution area for the given inequalities x + 2y ≤ 10, x + y ≥ 1, x - y ≤ 0, x ≥ 0, y ≥ 0, think step by step [/INST] , The answer is The solution area for the given inequalities is the shaded area in the graph, which satisfies all the given inequalities simultaneously. This area is to the left of the line graph of x + 2y = 10, above the line graph of x + y = 1, below the line graph of x - y = 0, and in the first quadrant. The required points are (0, 5) and (10, 0).

[INST] help me answer this question Find the value of x for which the points (x, – 1), (2, 1) and (4, 5) are collinear., think step by step [/INST] , The answer is The required value of x is 1.

[INST] help me answer this question Which is more varying, the length or weight?, think step by step [/INST] Variance is a measure of the spread of a data set. It measures how far each number in the set is from the mean. A larger variance indicates that the numbers are more spread out from the mean. , The answer is To find out which is more varying, we need to compare the variance of the length and weight measurements.

[INST] help me answer this question 120 ≥ x, think step by step [/INST] The solutions of the given inequality are defined by all the real numbers less than or equal to 120. Hence, the required solution set is (-∞, 120]. , The answer is x ≤ 120

[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 Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = -2 and x = 3 , think step by step [/INST] Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = -2 and x = 3 , The answer is The required points are (0, -2), (3, 0)

[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 Calculate the standard deviation for the group A data., think step by step [/INST] , The answer is The standard deviation for group A is 15.09.

[INST] help me answer this question What is the value of x for the relationship between the third piece and the second piece?, think step by step [/INST] , The answer is x ≥ 8

[INST] help me answer this question Ends of major axis (0, ±√5), ends of minor axis (±1, 0), think step by step [/INST] The major axis is along the y-axis, and the ends of the major and minor axes are given. So, the equation will be of the form x2/b2 + y2/a2 = 1. Here, a = √5, b = 1. So, the equation of the ellipse is x2/1 + y2/5 = 1. , The answer is The equation of the ellipse is x2/1 + y2/5 = 1

[INST] help me answer this question Given x2/16 + y2/9 = 1, what is the length of the latus rectum?, think step by step [/INST] , The answer is Length of latus rectum = 2b2/a = (2×32)/4 = (2×9)/4 = 9/2

[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 Given x2/49 + y2/36 = 1, what is the length of the latus rectum?, think step by step [/INST] , The answer is Length of latus rectum = 2b2/a = (2×62)/7 = (2×36)/7 = 72/7

[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 Which group, A or B, has a larger standard deviation?, think step by step [/INST] , The answer is Group B has a larger standard deviation (15.61) compared to group A (15.09).

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

[INST] help me answer this question D ∩ E, think step by step [/INST] , The answer is D ∩ E = (1, 2, 3) ∩ (6) = (φ)

[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 Does the origin (0, 0) lie in the line graph of y > x?, think step by step [/INST] Solving for y = x, we get x= 0, y = 0, which are the coordinates of the origin. , The answer is Yes, the origin lies on the line graph.

[INST] help me answer this question Find the equation of the circle with Centre (-2, 3) and radius 4., think step by step [/INST] , The answer is x2 + y2 + 4x – 6y – 3 = 0

[INST] help me answer this question Find the equation of the circle with Centre (1, 1) and radius √2., think step by step [/INST] , The answer is x2 + y2 - 2x - 2y + 1 = 0

[INST] help me answer this question Find the equation of the circle with the given condition: 36x2 + 36x + 36y2 – 18y + 11 = 0, think step by step [/INST] , The answer is The equation of the circle is 36x2 + 36y2 + 36x – 18y + 11 = 0

[INST] help me answer this question From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman, assuming one person cannot hold more than one position?, think step by step [/INST] Total number of people in committee = 8. Number of positions to be filled = 2. Number of permutations = 8P2 = 56. , The answer is The number of ways to choose a chairman and a vice chairman from a committee of 8 persons, with no repetition, is 56.

[INST] help me answer this question Find the centre and radius of the circle with the given condition: (x + 5)2 + (y – 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.

[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)

[INST] help me answer this question Find the 4th term in the expansion of (x – 2y)¹², think step by step [/INST] In the binomial expansion (a + b)^n, the (r+1)th term is given by Tr+1 = nCran–rbr. Here, a = x, b = -2y, and n = 12. To find the 4th term, we need r = 3. Substituting these values, we get T4 = 12C3 (x)(-2y)³ (x⁻¹)(-2y) = -1344 x y² (2y)⁸. , The answer is The 4th term is -1344 x y² (2y)⁸

[INST] help me answer this question If A is a set with m elements, think step by step [/INST] , The answer is n (A) = m then n [P (A)] = 2m

[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) = 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 Are set B and set D mutually exclusive?, think step by step [/INST] , The answer is No, they are not mutually exclusive as there are common elements in B & D.

[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 sum to n terms of the series 3 × 8 + 6 × 11 + 9 × 14 +…, think step by step [/INST] , The answer is The sum to n terms of the series can be expressed as the sum of n terms of the series 9n2 + 15n.

[INST] help me answer this question Find the coordinates of the foci, vertices, length of major axis, length of minor axis, eccentricity, and length of latus rectum for the equation 4x^2 + 9y^2 = 36., think step by step [/INST] , The answer is {'foci': ['(√5, 0)', '(-√5, 0)'], 'vertices': ['(3, 0)', '(-3, 0)'], 'major axis length': 6, 'minor axis length': 4, 'eccentricity': '√5/3', 'latus rectum length': '8/3'}

[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.

[INST] help me answer this question Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6). What is the equation of the ellipse?, think step by step [/INST] The major axis is on the y-axis and the ellipse passes through the points (3, 2) and (1, 6). Using these two points, we can calculate the values of a and b and hence, the equation of the ellipse. , The answer is The equation of the ellipse is 4x2 + y2 = 40.

[INST] help me answer this question Find the region of the solution for x + y ≤ 9, y > x, x ≥ 0., think step by step [/INST] , The answer is The region of the solution for the given inequalities is the area below the line y = x, above the line x + y = 9 and to the right of the line x = 0.

[INST] help me answer this question Given x2/49 + y2/36 = 1, what is the eccentricity?, think step by step [/INST] , The answer is Eccentricity, e = c/a = √13/7

[INST] help me answer this question Simplify the given expression: (2x)^3 * (3x)^2, think step by step [/INST] , The answer is The simplified form of the given expression is 216x^5.

[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) = 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 are the coordinates of the foci?, think step by step [/INST] , The answer is The coordinates of the foci are (0, √21) and (0, -√21).

[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 X – Y, think step by step [/INST] X – Y represents the set of elements that are present in X but not in Y. , The answer is {a, c}

[INST] help me answer this question Find four numbers forming a geometric progression in which the third term is greater than the first term by 9, and the second term is greater than the 4th by 18., think step by step [/INST] The given conditions lead us to a system of equations, which we solve to find the first term (a) and the common ratio (r). Substituting these values, we find the first four terms of the G.P. , The answer is The first four numbers of the G.P. are 3, 3(-2), 3(-2)^2, and 3(-2)^3 i.e., 3, -6, 12, and -24.

[INST] help me answer this question Does the equation of the circle becomes when h = 6?, think step by step [/INST] , The answer is (x – 6)2 + y2 = 25

[INST] help me answer this question Write the product of the given numbers in standard form: 2 × 10⁴ × 3 × 10³, think step by step [/INST] , The answer is The product of the given numbers in standard form is 6 × 10⁷.

[INST] help me answer this question Question: 5x + 1 > -24, 5x – 1 < 24, think step by step [/INST] The given inequalities are 5x + 1 > -24 and 5x – 1 < 24. After solving these inequalities, we get the solution as (-5, 5). , The answer is The solutions of the given inequalities are all real numbers x such that -5 < x < 5.

[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) = 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) = (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 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.

[INST] help me answer this question What is the length of the shortest board?, think step by step [/INST] , The answer is The length of the shortest board is greater than or equal to 8 cm and less than or equal to 22 cm.

[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 The set of isosceles triangles, think step by step [/INST] , The answer is Universal set (s) would be the set of all triangles.

[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.

[INST] help me answer this question 7^2 + b^2 = (28/3)^2, think step by step [/INST] , The answer is Squaring both sides of the equation c/a = 4/3, we get a^2 + b^2 = c^2. Substituting the values of a and c, we get 7^2 + b^2 = (28/3)^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 Which of the following pairs of sets are disjoint?, think step by step [/INST] See subparts , The answer is Not disjoint

[INST] help me answer this question Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 if no digit is repeated. How many of these will be even?, think step by step [/INST] The number will be even when 2 and 4 are in one’s place. The possibility of (2, 4) at one’s place = 2/5 = 0.4. The total number of even numbers = 120 × 0.4 = 48. , The answer is The total number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repetition is 120. The number of even numbers among these is 48.

[INST] help me answer this question Find the equation of the circle with the given condition: 144x2 – 144x + 36y2 – 72y + 36 + 144y2 – 72y + 9 – 1 = 0, think step by step [/INST] , The answer is The equation of the circle is 36x2 + 36y2 – 36x – 18y + 44 = 0

[INST] help me answer this question Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non-defective (N). Write the sample space of this experiment., think step by step [/INST] N/A , The answer is S = {DDD, DDN, DND, NDD, DNN, NDN, NND, NNN}

[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 sin 4x = 4 cos2 x sin 4x - 8 cos4 x sin4 x + 4 cos6 x sin4 x, think step by step [/INST] The given expression is incorrect, and it should be cos 4x = 1 - 8 cos2 x sin2 x. , The answer is False

[INST] help me answer this question Prove that (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0, think step by step [/INST] This involves the use of trigonometric formulas and simplification. , The answer is LHS = sin 3x sin x + sin^2 x + cos 3x cos x – cos^2 x = cos 2x – cos^2 x = 0 = RHS

[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

[INST] help me answer this question Find the equation of the line passing through the points (-1, 1) and (2, -4)., think step by step [/INST] , The answer is The equation of the line is 5x + 3y + 2 = 0.

[INST] help me answer this question What is the negation of the statement `for every real number x, x is less than x + 1`?, think step by step [/INST] , The answer is The negation of this statement is: `There exist a real number x, such that x is not less than x + 1.`

[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 equation of the line passing through point (x1, y1) and having slope m = -A/B is, think step by step [/INST] This is the formula for finding the equation of a line when the slope and a point through which the line passes is given. , The answer is The equation of the line passing through point (x1, y1) and having slope m = -A/B is y – y1 = -A/B (x – x1) or B (y – y1) = -A (x – x1)

[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.

[INST] help me answer this question 2x + y ≥ 4, x + y ≤ 3, 2x – 3y ≤ 6 , think step by step [/INST] N/A , The answer is N/A

[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 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.

[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 Identify the following angle as acute, obtuse, straight, reflex or a full rotation., think step by step [/INST] A reflex angle measures more than 180 degrees but less than 360 degrees. , The answer is The given angle is a reflex angle.

[INST] help me answer this question Is f a relation from A to B?, think step by step [/INST] A relation from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. Here, f is a subset of A × B. , The answer is Yes, it is.

[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 A coin is tossed 3 times, and the outcomes are recorded. How many possible outcomes are there?, think step by step [/INST] Since a coin has two outcomes (head or tail) and it is tossed three times, so the total number of possible outcomes is 2*2*2 = 8. , The answer is 8

[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 Exercise 1.2 Solutions, think step by step [/INST] None , The answer is Not applicable, as this is a section header, not a question.

[INST] help me answer this question (i) Write the relation R = {(x, y): y = x - 2; x ∈ P} in set-builder form. (ii) Write the relation R in roster form. What are their domain and range? , think step by step [/INST] , The answer is (i) Set-builder form: R = {(x, y): y = x - 2; x ∈ P} (ii) Roster form: R = {(3, 1), (4, 2), (5, 3)}

[INST] help me answer this question Find the value of `^5√((0.99)`., think step by step [/INST] The given text provides a detailed explanation for this calculation. , The answer is An approximation of (0.99)⁵ using the first three terms of its expansion is 0.951.

[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.