nwadgeri/maths-1k · Datasets at Hugging Face

entry stringlengths 86 1.26k	text stringclasses 1 value
[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) = (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 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 Vertices (0, ± 13), foci (0, ± 5), think step by step [/INST] The vertices and foci given are on the y-axis, so the equation will be of the form x2/b2 + y2/a2 = 1. Here, a = 13, c = 5. Using a2 = b2 + c2, b2 = 169 – 15 = 144. So, the equation of the ellipse is x2/144 + y2/169 = 1. , The answer is The equation of the ellipse is x2/144 + y2/169 = 1
[INST] help me answer this question D – E, think step by step [/INST] , The answer is D – E = (1, 2, 3) - (6) = (1, 2, 3)
[INST] help me answer this question In which direction is the solution area for the first inequality?, think step by step [/INST] , The answer is The solution area for the first inequality is towards the origin, including the same.
[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 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 Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola x2/16 – y2/9 = 1, think step by step [/INST] Given the equation x2/16 – y2/9 = 1, we can compare it with the standard equation of a hyperbola x2/a2 – y2/b2 = 1. By doing so, we find that a = 4 and b = 3. We then calculate c using a2 + b2 = c2, which gives us c = 5. The coordinates of the foci are (±c, 0), so they are (±5, 0). The coordinates of the vertices are (±a, 0), so they are (±4, 0). The eccentricity is e = c/a = 5/4. The length of the latus rectum is given by 2b2/a = (2×9)/4 = 9/2. , The answer is The coordinates of the foci are (±5, 0), the coordinates of the vertices are (±4, 0), the eccentricity is e = c/a = 5/4, and the length of the latus rectum is 9/2.
[INST] help me answer this question Find the equation of the circle with Centre (0, 2) and radius 2., think step by step [/INST] , The answer is x2 + y2 – 4y = 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 `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 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 `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 Given x2/16 + y2/9 = 1, what is the eccentricity?, think step by step [/INST] , The answer is Eccentricity, e = c/a = √7/4
[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 A ∩ B, think step by step [/INST] , The answer is A ∩ B = (1, 2, 3, 4, 5, 6) ∩ (φ) = (φ)
[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 mean deviation about the median for the data: 10, 30, 50, 70, 90., think step by step [/INST] First, we find the median (which is 50), then we calculate the absolute deviations from the median, and finally, we find the mean of these deviations. , The answer is The mean deviation about the median for the given data is 24.
[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.
[INST] help me answer this question (x) Φ ⊂ A is correct, think step by step [/INST] , The answer is Φ is a subset of every set.
[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 Perimeter of a square?, think step by step [/INST] The Perimeter is the distance around a shape. For a square, it is found by adding up all the side lengths, which are equal. So, 4 times the length of one side gives the perimeter. , The answer is The Perimeter of a square is calculated by the formula 4 multiplied by the side length.
[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 For the data 92, 93, 97, 98, 98, 102, 104, 109, find the deviations from the mean and (xi – x̅)2., think step by step [/INST] , The answer is Refer to the table in the answer.
[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 Given, y < – 2, think step by step [/INST] , The answer is The solution region of the given inequality is below the line y < – 2. (That is, away from the origin.)
[INST] help me answer this question The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q^2 = ps., think step by step [/INST] We are given the 5th, 8th and 11th terms of a G.P. and asked to prove a relationship between them. We first express each term using the formula a_n = a * r^(n-1) and then prove the relationship. , The answer is Given the 5th, 8th and 11th terms of a G.P. as p, q and s respectively, we can say that: q^2 = (a * r^7)^2 = a^2 * r^14 = a * r^3 * a * r^10 = a^2 * r^14 = ps
[INST] help me answer this question How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?, think step by step [/INST] , The answer is 3 × 20 = 60
[INST] help me answer this question An experiment consists of recording the boy-girl composition of families with 2 children. (ii) What is the sample space if we are interested in the number of girls in the family?, think step by step [/INST] , The answer is If we are interested in the number of girls in the family, the sample space contains only two outcomes: S = {0 girls, 2 girls}.
[INST] help me answer this question Does the origin lie in the solution area for the given inequalities of 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0?, think step by step [/INST] , The answer is Yes, the origin lies in the solution area.
[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 `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 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 `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 Solve 3x + 8 >2. , think step by step [/INST] N/A , The answer is The solution set is {-1, 0, 1, 2, …}.
[INST] help me answer this question What is the coefficient of variation (C.V) for Y?, think step by step [/INST] , The answer is The coefficient of variation (C.V) for Y is 1.904.
[INST] help me answer this question Find the points on the x-axis whose distances from the line x/3 + y/4 = 1 are 4 units., think step by step [/INST] N/A , The answer is The required points on the x-axis are (-2, 0) and (8, 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 3 (1 – x) < 2 (x + 4), think step by step [/INST] (Blank) , The answer is x > -1
[INST] help me answer this question , think step by step [/INST] , The answer is
[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 False., think step by step [/INST] , The answer is 2 ∈ {1, 2, 3} where, 2∉ {1, 3, 5}
[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 x + y ≤ 3, , think step by step [/INST] , 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 `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 `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 Divide both sides by √(02 + 12) = √1 = 1 0 (x) + 1 (y) = 2, think step by step [/INST] N/A , The answer is N/A
[INST] help me answer this question What is the universal set for the set of isosceles triangles?, think step by step [/INST] , The answer is The set of triangles or the set of polygons or the set of two-dimensional figures.
[INST] help me answer this question We have x ≥ 0, , think step by step [/INST] , The answer is N/A
[INST] help me answer this question Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range. , think step by step [/INST] , The answer is The relation R is given by R = {(1, 6), (2, 7), (3, 8)}
[INST] help me answer this question From the given data, find out which is more stable in value, X or Y., think step by step [/INST] , The answer is Y is more stable than X.
[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 Length of major axis 26, foci (±5, 0), think step by step [/INST] The major axis is along the x-axis, the length of the major axis is given as 26, and the foci are given. So, the equation will be of the form x2/a2 + y2/b2 = 1. Here, 2a = 26, a = 13, and c = 5. Using a2 = b2 + c2, b2 = 169 – 25 = 144. So, the equation of the ellipse is x2/169 + y2/144 = 1. , The answer is The equation of the ellipse is x2/169 + y2/144 = 1
[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 To receive a Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in the first four examinations are 87, 92, 94 and 95, find the minimum marks that Sunita must obtain in the fifth examination to get a Grade ‘A’ in the course., think step by step [/INST] To find the minimum marks, we need to first find the total marks of five examinations and divide it by 5. After substituting Sunita’s marks in the first four examinations and simplification, we get x ≥ 82. , The answer is Sunita must obtain 82 or more marks in her fifth examination.
[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 Expand (101)4 using the binomial theorem., think step by step [/INST] The binomial theorem is applied with 101 expressed as 100 + 1. , The answer is 104060401
[INST] help me answer this question From the data given below, the state which group is more variable, A or B?, think step by step [/INST] To find the coefficient of variation (C.V.) for each group, we first need to calculate the standard deviation (σ) and the mean (x̄) for each group. The formula for C.V. is: C.V. = (σ/ x̄) × 100 For Group A, the standard deviation (σ) is 15.09 and the mean (x̄) is 44.6. So, C.V. for Group A is: C.V. = (15.09/44.6) × 100 = 33.83 For Group B, the standard deviation (σ) and the mean (x̄) need to be calculated following a similar process as for Group A. After calculating the C.V. for both groups, we can compare them to determine which group is more variable. , The answer is To compare the variability or dispersion of two series, we calculate the coefficient of variance for each series. The series having greater C.V. is said to be more variable than the other. The series having lesser C.V. is said to be more consistent than the other.
[INST] help me answer this question Checking for y ≥ 2, The value of y will be unaffected by any value of x in the given equality. Also, the origin doesn’t satisfy the given inequality., think step by step [/INST] The region to be included in the solution would be towards the left of the equality y≥ 2 , The answer is ⇒ 0 ≥ 2, which is not true
[INST] help me answer this question Is the origin included in the solution area for the first inequality?, think step by step [/INST] , The answer is No, the origin is not included in the solution area for the first inequality.
[INST] help me answer this question If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term., think step by step [/INST] We found the sum formula for N terms, solved for N, and then found the 8th term using the nth term formula for the given A.P. , The answer is The last term is the 8th term, 4.
[INST] help me answer this question Solve – 12x > 30, when - x is a natural number. - x is an integer. , think step by step [/INST] N/A , The answer is {'naturalNumber': 'No solution', 'integer': '...,-5, -4, -3'}
[INST] help me answer this question Are PQ, QR, and PR sides of an isosceles triangle?, think step by step [/INST] PQ = QR, and PR is not equal to PQ or QR. Hence, PQR is an isosceles triangle. , The answer is Yes
[INST] help me answer this question (ii) Find the distance between parallel lines l(x + y) + p = 0 and l (x + y) – r = 0, think step by step [/INST] N/A , The answer is The distance between parallel lines is \|p+r\|/l√2
[INST] help me answer this question Find the distance between the points (2, –1, 3) and (–2, 1, 3)., think step by step [/INST] , The answer is The required distance is 2\sqrt5 units.
[INST] help me answer this question In ΔOAB, if OA = OB = 20 cm and AB = 20 cm, what type of triangle is ΔOAB?, think step by step [/INST] , The answer is ΔOAB is an equilateral triangle.
[INST] help me answer this question Find the region of the solution for x – 2y < – 1., think step by step [/INST] Putting values of x = 0 and y = 0 in the equation, we get points (0, 0.5) and (-1, 0). Checking for the origin, putting (0, 0), we get 0 < -1, which is false. Hence, the origin does not lie in the solution area. , The answer is The region of the solution for x – 2y < – 1 is the area on the left side of the line graph.
[INST] help me answer this question Find the equation of the line having normal distance p = 5 from the origin and angle ω = 30o which the normal makes with the positive x-axis., think step by step [/INST] , The answer is The equation of the line is √3 x + y – 10 = 0.
[INST] help me answer this question Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?, think step by step [/INST] Since there are 5 flags of different colours, the number of flags possible for the upper flag is 5. Now, as one of the flags is taken, the number of flags remaining for the lower flag in the signal is 4. The number of ways in which signal can be given = 5 × 4 = 20. , The answer is 20
[INST] help me answer this question If , find the values of x and y., think step by step [/INST] Given, As the ordered pairs are equal, the corresponding elements should also be equal. Thus, x/3 + 1 = 5/3 and y – 2/3 = 1/3 Solving, we get x + 3 = 5 and 3y – 2 = 1 [Taking L.C.M and adding] x = 2 and 3y = 3 Therefore, x = 2 and y = 1 , The answer is x = 2 and y = 1
[INST] help me answer this question Consider x ≥ 0, y ≥ 0,, think step by step [/INST] , The answer is The solution lies in the first quadrant only.
[INST] help me answer this question Find the sum to n terms of the series 1 + 3 + 5 + … + (2n - 1), think step by step [/INST] , The answer is The sum of n terms of the series can be expressed as n2.
[INST] help me answer this question Find the middle terms in the expansions of , think step by step [/INST] , The answer is Not provided in the data
[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 (6, -4, -2) and (8, -10, 2).
[INST] help me answer this question IQ of a person is given by the formula, IQ = MA/CA, where MA is mental age and CA is chronological age. If 80 ≤ IQ ≤ 140 for a group of 12 years old children, find the range of their mental age., think step by step [/INST] According to the question, Chronological age = CA = 12 years IQ for the age group of 12 is 80 ≤ IQ ≤ 140. We get that 80 ≤ IQ ≤ 140 Substituting, IQ = MA/CA 80 ≤ MA/12 ≤ 140 Further solving, 96 ≤ MA ≤ 168 9.6 ≤ MA ≤ 16.8 , The answer is 9.6 ≤ MA ≤ 16.8
[INST] help me answer this question For some constants a and b, find the derivative of (i) (x − a) (x − b), think step by step [/INST] , The answer is The derivative of (x − a) (x − b) is 2x - a - b.
[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 In which direction is the solution area for the third inequality?, think step by step [/INST] , The answer is The solution area for the third inequality is towards the left side of the line graph.
[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 Which area of the line graph is the required solution for the given inequalities of y > x?, think step by step [/INST] , The answer is The shaded area on the graph shows the required solution for the given inequalities of y > x.
[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 universal set for the set of right triangles?, think step by step [/INST] , The answer is The set of triangles or the set of polygons.
[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 Download All Questions of NCERT Solutions for Class 11 Maths Chapter 1 – Sets [here.](http://ncerthelp.com/ncert-solutions-for-class-11-maths-chapter-1-sets/), think step by step [/INST] None , The answer is Not applicable, as this is a link, not a question.
[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 What is the law of conservation of mass?, think step by step [/INST] This means that the total mass of all the reactants in a chemical reaction must be equal to the total mass of all the products. , The answer is The law of conservation of mass states that mass can neither be created nor destroyed in a chemical reaction.
[INST] help me answer this question Write the following sets in the set-builder form. (i) (3, 6, 9, 12) (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6 … } (v) {1, 4, 9 … 100}, think step by step [/INST] In each case, I have described the sets using the set-builder notation, where x is the element of the set defined by a rule or condition. , The answer is (i) {x: x = 3n, n ∈ N and 1 ≤ n ≤ 4} (ii) {x: x = 2n, n ∈ N and 1 ≤ n ≤ 5} (iii) {x: x = 5n, n ∈ N and 1 ≤ n ≤ 4} (iv) {x: x is an even natural number} (v) {x: x = n^2, n ∈ N and 1 ≤ n ≤ 10}
[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 `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 In the given word ASSASSINATION, there are 4 ‘S’. Since all the 4 ‘S’ have to be arranged together, let us take them as one unit., think step by step [/INST] The problem deals with permutations, as the order in which the letters are arranged matters. Here, the task is to find the number of distinguishable arrangements of given letters. Some of the letters are repeated, thus requiring the use of a specific formula for this purpose. , The answer is The number of ways to arrange the letters is equal to 181440
[INST] help me answer this question (2.2) Does the point (0, 0) satisfy the inequality?, think step by step [/INST] N/A , The answer is Yes, it does satisfy the inequality
[INST] help me answer this question Write the relation R = {(x, x^3): x is a prime number less than 10} in roster form., think step by step [/INST] For question 1, the prime numbers less than 10 are 2, 3, 5, and 7. Therefore, R = {(2, 8), (3, 27), (5, 125), (7, 343)}. , The answer is R = {(2, 8), (3, 27), (5, 125), (7, 343)}

[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 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 Vertices (0, ± 13), foci (0, ± 5), think step by step [/INST] The vertices and foci given are on the y-axis, so the equation will be of the form x2/b2 + y2/a2 = 1. Here, a = 13, c = 5. Using a2 = b2 + c2, b2 = 169 – 15 = 144. So, the equation of the ellipse is x2/144 + y2/169 = 1. , The answer is The equation of the ellipse is x2/144 + y2/169 = 1

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

[INST] help me answer this question In which direction is the solution area for the first inequality?, think step by step [/INST] , The answer is The solution area for the first inequality is towards the origin, including the same.

[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 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 Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola x2/16 – y2/9 = 1, think step by step [/INST] Given the equation x2/16 – y2/9 = 1, we can compare it with the standard equation of a hyperbola x2/a2 – y2/b2 = 1. By doing so, we find that a = 4 and b = 3. We then calculate c using a2 + b2 = c2, which gives us c = 5. The coordinates of the foci are (±c, 0), so they are (±5, 0). The coordinates of the vertices are (±a, 0), so they are (±4, 0). The eccentricity is e = c/a = 5/4. The length of the latus rectum is given by 2b2/a = (2×9)/4 = 9/2. , The answer is The coordinates of the foci are (±5, 0), the coordinates of the vertices are (±4, 0), the eccentricity is e = c/a = 5/4, and the length of the latus rectum is 9/2.

[INST] help me answer this question Find the equation of the circle with Centre (0, 2) and radius 2., think step by step [/INST] , The answer is x2 + y2 – 4y = 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 `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 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 `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 Given x2/16 + y2/9 = 1, what is the eccentricity?, think step by step [/INST] , The answer is Eccentricity, e = c/a = √7/4

[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 A ∩ B, think step by step [/INST] , The answer is A ∩ B = (1, 2, 3, 4, 5, 6) ∩ (φ) = (φ)

[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 mean deviation about the median for the data: 10, 30, 50, 70, 90., think step by step [/INST] First, we find the median (which is 50), then we calculate the absolute deviations from the median, and finally, we find the mean of these deviations. , The answer is The mean deviation about the median for the given data is 24.

[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.

[INST] help me answer this question (x) Φ ⊂ A is correct, think step by step [/INST] , The answer is Φ is a subset of every set.

[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 Perimeter of a square?, think step by step [/INST] The Perimeter is the distance around a shape. For a square, it is found by adding up all the side lengths, which are equal. So, 4 times the length of one side gives the perimeter. , The answer is The Perimeter of a square is calculated by the formula 4 multiplied by the side length.

[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 For the data 92, 93, 97, 98, 98, 102, 104, 109, find the deviations from the mean and (xi – x̅)2., think step by step [/INST] , The answer is Refer to the table in the answer.

[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 Given, y < – 2, think step by step [/INST] , The answer is The solution region of the given inequality is below the line y < – 2. (That is, away from the origin.)

[INST] help me answer this question The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q^2 = ps., think step by step [/INST] We are given the 5th, 8th and 11th terms of a G.P. and asked to prove a relationship between them. We first express each term using the formula a_n = a * r^(n-1) and then prove the relationship. , The answer is Given the 5th, 8th and 11th terms of a G.P. as p, q and s respectively, we can say that: q^2 = (a * r^7)^2 = a^2 * r^14 = a * r^3 * a * r^10 = a^2 * r^14 = ps

[INST] help me answer this question How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?, think step by step [/INST] , The answer is 3 × 20 = 60

[INST] help me answer this question An experiment consists of recording the boy-girl composition of families with 2 children. (ii) What is the sample space if we are interested in the number of girls in the family?, think step by step [/INST] , The answer is If we are interested in the number of girls in the family, the sample space contains only two outcomes: S = {0 girls, 2 girls}.

[INST] help me answer this question Does the origin lie in the solution area for the given inequalities of 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0?, think step by step [/INST] , The answer is Yes, the origin lies in the solution area.

[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 `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 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 `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 Solve 3x + 8 >2. , think step by step [/INST] N/A , The answer is The solution set is {-1, 0, 1, 2, …}.

[INST] help me answer this question What is the coefficient of variation (C.V) for Y?, think step by step [/INST] , The answer is The coefficient of variation (C.V) for Y is 1.904.

[INST] help me answer this question Find the points on the x-axis whose distances from the line x/3 + y/4 = 1 are 4 units., think step by step [/INST] N/A , The answer is The required points on the x-axis are (-2, 0) and (8, 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 3 (1 – x) < 2 (x + 4), think step by step [/INST] (Blank) , The answer is x > -1

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

[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 False., think step by step [/INST] , The answer is 2 ∈ {1, 2, 3} where, 2∉ {1, 3, 5}

[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 x + y ≤ 3, , think step by step [/INST] , 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 `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 `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 Divide both sides by √(02 + 12) = √1 = 1 0 (x) + 1 (y) = 2, think step by step [/INST] N/A , The answer is N/A

[INST] help me answer this question What is the universal set for the set of isosceles triangles?, think step by step [/INST] , The answer is The set of triangles or the set of polygons or the set of two-dimensional figures.

[INST] help me answer this question We have x ≥ 0, , think step by step [/INST] , The answer is N/A

[INST] help me answer this question Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range. , think step by step [/INST] , The answer is The relation R is given by R = {(1, 6), (2, 7), (3, 8)}

[INST] help me answer this question From the given data, find out which is more stable in value, X or Y., think step by step [/INST] , The answer is Y is more stable than X.

[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 Length of major axis 26, foci (±5, 0), think step by step [/INST] The major axis is along the x-axis, the length of the major axis is given as 26, and the foci are given. So, the equation will be of the form x2/a2 + y2/b2 = 1. Here, 2a = 26, a = 13, and c = 5. Using a2 = b2 + c2, b2 = 169 – 25 = 144. So, the equation of the ellipse is x2/169 + y2/144 = 1. , The answer is The equation of the ellipse is x2/169 + y2/144 = 1

[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 To receive a Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in the first four examinations are 87, 92, 94 and 95, find the minimum marks that Sunita must obtain in the fifth examination to get a Grade ‘A’ in the course., think step by step [/INST] To find the minimum marks, we need to first find the total marks of five examinations and divide it by 5. After substituting Sunita’s marks in the first four examinations and simplification, we get x ≥ 82. , The answer is Sunita must obtain 82 or more marks in her fifth examination.

[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 Expand (101)4 using the binomial theorem., think step by step [/INST] The binomial theorem is applied with 101 expressed as 100 + 1. , The answer is 104060401

[INST] help me answer this question From the data given below, the state which group is more variable, A or B?, think step by step [/INST] To find the coefficient of variation (C.V.) for each group, we first need to calculate the standard deviation (σ) and the mean (x̄) for each group. The formula for C.V. is: C.V. = (σ/ x̄) × 100 For Group A, the standard deviation (σ) is 15.09 and the mean (x̄) is 44.6. So, C.V. for Group A is: C.V. = (15.09/44.6) × 100 = 33.83 For Group B, the standard deviation (σ) and the mean (x̄) need to be calculated following a similar process as for Group A. After calculating the C.V. for both groups, we can compare them to determine which group is more variable. , The answer is To compare the variability or dispersion of two series, we calculate the coefficient of variance for each series. The series having greater C.V. is said to be more variable than the other. The series having lesser C.V. is said to be more consistent than the other.

[INST] help me answer this question Checking for y ≥ 2, The value of y will be unaffected by any value of x in the given equality. Also, the origin doesn’t satisfy the given inequality., think step by step [/INST] The region to be included in the solution would be towards the left of the equality y≥ 2 , The answer is ⇒ 0 ≥ 2, which is not true

[INST] help me answer this question Is the origin included in the solution area for the first inequality?, think step by step [/INST] , The answer is No, the origin is not included in the solution area for the first inequality.

[INST] help me answer this question If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term., think step by step [/INST] We found the sum formula for N terms, solved for N, and then found the 8th term using the nth term formula for the given A.P. , The answer is The last term is the 8th term, 4.

[INST] help me answer this question Solve – 12x > 30, when - x is a natural number. - x is an integer. , think step by step [/INST] N/A , The answer is {'naturalNumber': 'No solution', 'integer': '...,-5, -4, -3'}

[INST] help me answer this question Are PQ, QR, and PR sides of an isosceles triangle?, think step by step [/INST] PQ = QR, and PR is not equal to PQ or QR. Hence, PQR is an isosceles triangle. , The answer is Yes

[INST] help me answer this question (ii) Find the distance between parallel lines l(x + y) + p = 0 and l (x + y) – r = 0, think step by step [/INST] N/A , The answer is The distance between parallel lines is |p+r|/l√2

[INST] help me answer this question Find the distance between the points (2, –1, 3) and (–2, 1, 3)., think step by step [/INST] , The answer is The required distance is 2\sqrt5 units.

[INST] help me answer this question In ΔOAB, if OA = OB = 20 cm and AB = 20 cm, what type of triangle is ΔOAB?, think step by step [/INST] , The answer is ΔOAB is an equilateral triangle.

[INST] help me answer this question Find the region of the solution for x – 2y < – 1., think step by step [/INST] Putting values of x = 0 and y = 0 in the equation, we get points (0, 0.5) and (-1, 0). Checking for the origin, putting (0, 0), we get 0 < -1, which is false. Hence, the origin does not lie in the solution area. , The answer is The region of the solution for x – 2y < – 1 is the area on the left side of the line graph.

[INST] help me answer this question Find the equation of the line having normal distance p = 5 from the origin and angle ω = 30o which the normal makes with the positive x-axis., think step by step [/INST] , The answer is The equation of the line is √3 x + y – 10 = 0.

[INST] help me answer this question Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?, think step by step [/INST] Since there are 5 flags of different colours, the number of flags possible for the upper flag is 5. Now, as one of the flags is taken, the number of flags remaining for the lower flag in the signal is 4. The number of ways in which signal can be given = 5 × 4 = 20. , The answer is 20

[INST] help me answer this question If , find the values of x and y., think step by step [/INST] Given, As the ordered pairs are equal, the corresponding elements should also be equal. Thus, x/3 + 1 = 5/3 and y – 2/3 = 1/3 Solving, we get x + 3 = 5 and 3y – 2 = 1 [Taking L.C.M and adding] x = 2 and 3y = 3 Therefore, x = 2 and y = 1 , The answer is x = 2 and y = 1

[INST] help me answer this question Consider x ≥ 0, y ≥ 0,, think step by step [/INST] , The answer is The solution lies in the first quadrant only.

[INST] help me answer this question Find the sum to n terms of the series 1 + 3 + 5 + … + (2n - 1), think step by step [/INST] , The answer is The sum of n terms of the series can be expressed as n2.

[INST] help me answer this question Find the middle terms in the expansions of , think step by step [/INST] , The answer is Not provided in the data

[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 (6, -4, -2) and (8, -10, 2).

[INST] help me answer this question IQ of a person is given by the formula, IQ = MA/CA, where MA is mental age and CA is chronological age. If 80 ≤ IQ ≤ 140 for a group of 12 years old children, find the range of their mental age., think step by step [/INST] According to the question, Chronological age = CA = 12 years IQ for the age group of 12 is 80 ≤ IQ ≤ 140. We get that 80 ≤ IQ ≤ 140 Substituting, IQ = MA/CA 80 ≤ MA/12 ≤ 140 Further solving, 96 ≤ MA ≤ 168 9.6 ≤ MA ≤ 16.8 , The answer is 9.6 ≤ MA ≤ 16.8

[INST] help me answer this question For some constants a and b, find the derivative of (i) (x − a) (x − b), think step by step [/INST] , The answer is The derivative of (x − a) (x − b) is 2x - a - b.

[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 In which direction is the solution area for the third inequality?, think step by step [/INST] , The answer is The solution area for the third inequality is towards the left side of the line graph.

[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 Which area of the line graph is the required solution for the given inequalities of y > x?, think step by step [/INST] , The answer is The shaded area on the graph shows the required solution for the given inequalities of y > x.

[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 universal set for the set of right triangles?, think step by step [/INST] , The answer is The set of triangles or the set of polygons.

[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 Download All Questions of NCERT Solutions for Class 11 Maths Chapter 1 – Sets [here.](http://ncerthelp.com/ncert-solutions-for-class-11-maths-chapter-1-sets/), think step by step [/INST] None , The answer is Not applicable, as this is a link, not a question.

[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 What is the law of conservation of mass?, think step by step [/INST] This means that the total mass of all the reactants in a chemical reaction must be equal to the total mass of all the products. , The answer is The law of conservation of mass states that mass can neither be created nor destroyed in a chemical reaction.

[INST] help me answer this question Write the following sets in the set-builder form. (i) (3, 6, 9, 12) (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6 … } (v) {1, 4, 9 … 100}, think step by step [/INST] In each case, I have described the sets using the set-builder notation, where x is the element of the set defined by a rule or condition. , The answer is (i) {x: x = 3n, n ∈ N and 1 ≤ n ≤ 4} (ii) {x: x = 2n, n ∈ N and 1 ≤ n ≤ 5} (iii) {x: x = 5n, n ∈ N and 1 ≤ n ≤ 4} (iv) {x: x is an even natural number} (v) {x: x = n^2, n ∈ N and 1 ≤ n ≤ 10}

[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 `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 In the given word ASSASSINATION, there are 4 ‘S’. Since all the 4 ‘S’ have to be arranged together, let us take them as one unit., think step by step [/INST] The problem deals with permutations, as the order in which the letters are arranged matters. Here, the task is to find the number of distinguishable arrangements of given letters. Some of the letters are repeated, thus requiring the use of a specific formula for this purpose. , The answer is The number of ways to arrange the letters is equal to 181440

[INST] help me answer this question (2.2) Does the point (0, 0) satisfy the inequality?, think step by step [/INST] N/A , The answer is Yes, it does satisfy the inequality

[INST] help me answer this question Write the relation R = {(x, x^3): x is a prime number less than 10} in roster form., think step by step [/INST] For question 1, the prime numbers less than 10 are 2, 3, 5, and 7. Therefore, R = {(2, 8), (3, 27), (5, 125), (7, 343)}. , The answer is R = {(2, 8), (3, 27), (5, 125), (7, 343)}