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[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 b^2 = 6a, think step by step [/INST] , The answer is Substituting the value of a from the equation a = 2 into the previously derived equation b^2 = 6a, we get b^2 = 6a.
[INST] help me answer this question Given x2/25 + y2/100 = 1, what is the eccentricity?, think step by step [/INST] , The answer is Eccentricity, e = c/a = √3/2
[INST] help me answer this question (x – 4)/3 β‰₯ 1, think step by step [/INST] , The answer is x ∈ [7, ∞)
[INST] help me answer this question Foci (Β±3, 0), a = 4. What is the equation of the ellipse?, think step by step [/INST] The foci are on the x-axis, so the major axis is along the x-axis. The equation of the ellipse will be of the form x2/a2 + y2/b2 = 1. Since the foci are (Β±3, 0), c = 3 and a = 4. Using the formula a2 = b2 + c2, we get b2 = 7. Therefore, the equation of the ellipse is x2/16 + y2/7 = 1. , The answer is The equation of the ellipse is x2/16 + y2/7 = 1.
[INST] help me answer this question What is the relationship between set G and set H?, think step by step [/INST] , The answer is Set G is not equal to set H.
[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 By using the concept of equation of a line, prove that the three points (3, 0), (– 2, – 2) and (8, 2) are collinear., think step by step [/INST] Collinearity of points can be proved by showing that the line passing through any two points also passes through the third point. Here, we have used the formula to find the equation of a line passing through two points and then checked if the third point also satisfies the equation of the line. , The answer is According to the question, if we have to prove that the given three points (3, 0), (– 2, – 2) and (8, 2) are collinear, then we have to also prove that the line passing through the points (3, 0) and (– 2, – 2) also passes through the point (8, 2). By using the formula, the equation of the line passing through the points (x1, y1) and (x2, y2) is given by -5y = -2(x – 3). Further simplifying the equation, we get 2x – 5y = 6. If 2x – 5y = 6 passes through (8, 2), then 2x – 5y = 2(8) – 5(2) = 16 – 10 = 6 = RHS. Hence, the line passing through the points (3, 0) and (– 2, – 2) also passes through the point (8, 2).
[INST] help me answer this question Evaluate the given limit:, think step by step [/INST] Apply L'Hopital's rule or factorization , The answer is a / c
[INST] help me answer this question (iv) Only subset of Ξ¦ is Ξ¦., think step by step [/INST] , The answer is N/A
[INST] help me answer this question cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x), think step by step [/INST] Consider LHS = cot 4x (sin 5x + sin 3x). Using the formula sin(A + B) = sinA cosB + cosA sinB, it can be written as 2 cos 4x cos x. Thus, LHS = RHS. , The answer is True
[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 the following pair of statements a negation of each other, give reasons for the answer. (i) x + y = y + x is true for every real number x and y. (ii) There exists real numbers x and y for which x + y = y + x., think step by step [/INST] , The answer is The negation of the (i) statement is: `There exists real numbers x and y for which x + y β‰  y + x.` This statement is not the same as the (ii) statement, hence the given statements are not a negation of each other.
[INST] help me answer this question If the ratio of the fifth term from the beginning to the fifth term from the end of the expansion of (a + b)ⁿ is √6:1, find n., think step by step [/INST] The given text provides a detailed explanation for this calculation. , The answer is The value of n is 10.
[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 Length of major axis 26, foci (Β±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 `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 In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?, think step by step [/INST] This is derived from n(C βˆͺ T) = n(C) + n(T) - n(C ∩ T) by substituting the given values and solving for n(C ∩ T). , The answer is 19 people like both coffee and tea.
[INST] help me answer this question Exercise 1.4 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 What is the coefficient of variation (C.V) for group A?, think step by step [/INST] , The answer is The coefficient of variation (C.V) for group A is 33.83.
[INST] help me answer this question Given, – 3x + 2y β‰₯ – 6, think step by step [/INST] , The answer is The solution region of the given inequality is above the line – 3x + 2y β‰₯ – 6. (That is, the origin is included in the region)
[INST] help me answer this question (v) {a} ∈ (a, b, c), think step by step [/INST] Here {a} is a set and (a, b, c) is an ordered triple, so a set cannot be an element of an ordered triple. , The answer is False
[INST] help me answer this question (5.1) A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?, think step by step [/INST] , The answer is The number of litres of 2% of boric acid solution that has to be added will be more than 320 litres but less than 1280 litres.
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[INST] help me answer this question P (k + 1) is true whenever P (k) is true., think step by step [/INST] Principle of Mathematical Induction , The answer is Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.
[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 shaded area in the graph shows the solution area for the given inequalities 4x + 3y ≀ 60, y β‰₯ 2x, x β‰₯ 3, x, y β‰₯ 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 y = 2x and 4x + 3y = 60, on the left of the line x = 3, and in the first quadrant. The required points are (0, 20) and (15, 0).
[INST] help me answer this question What is the formula used to calculate the standard deviation?, think step by step [/INST] , The answer is Οƒ = √Variance
[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 Prove that a – b is a factor of an – bn, whenever n is a positive integer., think step by step [/INST] The given text provides a detailed proof for this statement. , The answer is In order to prove that (a – b) is a factor of (an – bn), it has to be proved that an – bn = k(a – b), where k is some natural number.
[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 β‰₯ 1`.
[INST] help me answer this question 2 boys and 2 girls are in Room X, and 1 boy and 3 girls are in Room Y. Specify the sample space for the experiment in which a room is selected and then a person., think step by step [/INST] , The answer is Let 2 boys and 2 girls in Room X be denoted by b1, b2 and, g1, g2, and 1 boy and 3 girls in Room Y by b3 and g3, g4, and g5. The overall sample space S is {(X,b1),(X,b2),(X,g1),(X,g2),(Y,b3),(Y,g3),(Y,g4),(Y,g5)}.
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[INST] help me answer this question Vertices (0, Β± 13), foci (0, Β± 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 `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 What is the contrapositive of the given statement: "If two lines intersect in the same plane, then the two lines are not parallel.", think step by step [/INST] The contrapositive statement is created by negating both the hypothesis and conclusion of the original statement and then interchanging them. , The answer is If two lines are parallel, then they do not intersect in the same plane.
[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 20 (2x – 1) >= 3 (19x – 18), think step by step [/INST] All the real numbers of x, which are less than or equal to 2, are the solutions to the given inequality. , The answer is -∞, 2]
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[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 β‰₯ 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 β‰₯ 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 β‰₯ 1`.
[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 cos 3x + cos x – cos 2x = 0, think step by step [/INST] , The answer is The general solution of this equation is given by cos x (2 cos x - 1) = 0, which implies cos x = 0 or 2 cos x - 1 = 0.
[INST] help me answer this question The base of an equilateral triangle with side 2a lies along the y-axis, such that the mid-point of the base is at the origin. Find the vertices of the triangle., think step by step [/INST] , The answer is Let the triangle be ABC with base BC lying along the y-axis. O (0, 0) is the mid-point of the base BC. Let the third vertex be A (x, y). Side of the triangle is 2a units. Applying distance formula, we get x^2 + (y-a)^2 = a^2. Therefore, the coordinates of A are (a, 0) and the coordinates of B and C are (0, a) and (0, -a) respectively.
[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 required equation in the data., think step by step [/INST] , The answer is The required equation is 9xΒ² + 25yΒ² + 25zΒ² – 225 = 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 Is event A a simple event?, think step by step [/INST] , The answer is Yes, A = (HHH) is a simple event.
[INST] help me answer this question (1 – i) – (–1 + i6), think step by step [/INST] , The answer is 2 – i7
[INST] help me answer this question 1/x > 1/(x + 1), think step by step [/INST] , The answer is x ∈ (-1, 0) βˆͺ (0, ∞)
[INST] help me answer this question Evaluate: logβ‚€ 0.1, think step by step [/INST] , The answer is logβ‚€ 0.1 cannot be evaluated because the base (0.1) is less than 1.
[INST] help me answer this question If A (1, 2, 3) and B (3, 2, –1), what is the set of points P (x, y, z) such that PA = PB?, think step by step [/INST] By equating the distances PA and PB, we get an equation in x, y, and z, which simplifies to x – 2z = 0. , The answer is x – 2z = 0
[INST] help me answer this question False., think step by step [/INST] , The answer is 2 ∈ {1, 2, 3} where, 2βˆ‰ {1, 3, 5}
[INST] help me answer this question (viii) {x: x + 5 = 8}, think step by step [/INST] , The answer is {x: x ∈ N and x β‰  3}
[INST] help me answer this question (iii) {{3, 4}} βŠ‚ A is correct, think step by step [/INST] , The answer is {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.
[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 value of 5C5?, think step by step [/INST] , The answer is 5C5 = 1
[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 (12.1 Exercises Solutions), think step by step [/INST] The data contains a link to access the solutions for Exercise 12.1, but does not include the solutions directly. , The answer is Not provided in the data.
[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 Evaluate the given limit:, think step by step [/INST] Given limit, substitute x = 1, we get (a*1^2 + b*1 + c) / (c*1^2 + b*1 + a) = 1, given a + b + c = 1 , The answer is 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 Sum of the first p, q and r terms of an A.P. are a, b and c, respectively. Prove that (p - q)a + (q - r)b + (r - p)c = 0., think step by step [/INST] Explanation provided in the markdown text outlines the steps to derive the answer. It includes solving the first term a1 and common difference d equations simultaneously, and then substituting d in the terms (p - q)a, (q - r)b, and (r - p)c and adding them to get the desired result. , The answer is (p - q)a + (q - r)b + (r - p)c = 0.
[INST] help me answer this question Substitute the value of 1/aΒ² in equation (3), 36/aΒ² + 4/bΒ² = 1., think step by step [/INST] , The answer is 9/4 – 81/4bΒ² + 4/bΒ² = 1 – 9/4
[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 β‰₯ 1`.
[INST] help me answer this question Find the discriminant of the quadratic equation -x^2 + x - 2 = 0., think step by step [/INST] , The answer is The discriminant (D) of the quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac. For the given equation, a = -1, b = 1, and c = -2. Substituting these values into the formula, we get D = (1)^2 - 4*(-1)*(-2) = -7.
[INST] help me answer this question Find the domain and the range of the real function f defined by f(x) = √(x - 1)., think step by step [/INST] , The answer is The domain of f is [1, ∞) and its range is [0, ∞).
[INST] help me answer this question Find the equation of the line intersecting the y-axis at a distance of 2 units above the origin and making an angle of 30o with positive direction of the x-axis., think step by step [/INST] , The answer is The equation of the line is x – √3 y + 2√3 = 0.
[INST] help me answer this question We know that there are 5 vowels and 21 consonants in the English alphabet. Choosing two vowels out of 5 would be done in 5C2 ways. Choosing 2 consonants out of 21 can be done in 21C2 ways., think step by step [/INST] This is a combination problem because the order of choosing the vowels and consonants does not matter. We need to find the number of ways to select 2 vowels from 5 and 2 consonants from 21. This can be calculated using the combination formula: nCr = n! / (r! Γ— (n-r)!), where n is the total number of objects, r is the number of objects to choose, and '!' denotes factorial. , The answer is The total number of ways to select 2 vowels and 2 consonants = 5C2 Γ— 21C2
[INST] help me answer this question Find the sum to n terms in the geometric progression x^3, x^5, x^7, … (if x β‰  Β±1 ), think step by step [/INST] , The answer is An expression to find the sum to n terms in the geometric progression x^3, x^5, x^7, … (if x β‰  Β±1 )
[INST] help me answer this question True., think step by step [/INST] , The answer is Each element of {a} is also an element of {a, b, c}.
[INST] help me answer this question Calculate (a + b)4 - (a - b)4. Hence, evaluate (√3 + √2)4 - (√3 - √2)4., think step by step [/INST] The binomial theorem is used to expand the expressions, and then the values of a and b are substituted. , The answer is 40 √6
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[INST] help me answer this question 9. cos x = -1/3, x in quadrant III, think step by step [/INST] N/A , The answer is tan x/2 = sin x/2 / cos x/2
[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 (v) 1βŠ‚ A is incorrect, think step by step [/INST] , The answer is An element of a set can never be a subset of itself.
[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 β‰₯ 0, y β‰₯ 0?, think step by step [/INST] , The answer is Yes, the origin lies in the solution area.
[INST] help me answer this question Find the sum to n terms of the series whose nth term is given by n2 + 2n., think step by step [/INST] , The answer is The sum to n terms of the series can be expressed as (1/3)n(n+1)(n+2).
[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 Are {2, 6, 10} and {3, 7, 11} disjoint?, think step by step [/INST] {2, 6, 10} ∩ {3, 7, 11} = Φ , The answer is Disjoint
[INST] help me answer this question What is the formula used to calculate the mean (xΜ…)?, think step by step [/INST] , The answer is xΜ… = A + (h/N) * i
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[INST] help me answer this question The sum of the first three terms of a G.P. is 39/10, and their product is 1. Find the common ratio and the terms., think step by step [/INST] , The answer is The common ratio and the terms of the G.P.
[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 1.(b) Passing through (0, 0) with slope m., think step by step [/INST] , The answer is Given: Point (0, 0) and slope, m = m We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0), if and only if, its coordinates satisfy the equation y – y0 = m (x – x0) So, y – 0 = m (x – 0) y = mx y – mx = 0 ∴ The equation of the line is y – mx = 0.
[INST] help me answer this question Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m - 1)th numbers is 5:9. Find the value of m., think step by step [/INST] Explanation provided in the markdown text outlines the steps to derive the answer. It includes finding the common difference, deriving an equation on the basis of given ratio, and then solving for m. , The answer is The value of m is 11.
[INST] help me answer this question Given 2x – y >1, think step by step [/INST] Checking for the origin, putting (0, 0) 0 >1, which is false , The answer is The points are (0,-1) and (0.5, 0)
[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 What is √(25-4)?, think step by step [/INST] , The answer is √21
[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x β‰₯ 1`.
[INST] help me answer this question Find the slope of the line BC., think step by step [/INST] , The answer is The slope of the line BC (m2) = (-1-5)/(-1-3) = -6/-4 = 3/2
[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 Solve the quadratic equation x2 + 3 = 0, think step by step [/INST] Comparing the given equation with ax2 + bx + c = 0, we have a = 1, b = 0, and c = 3. The discriminant (D) is calculated as D = b2 - 4ac = 02 - 4 Γ— 1 Γ— 3 = -12. Since D < 0, the roots of the equation are given by x = [-b Β± √(b2 - 4ac)]/2a = [-0 Β± √(-12)]/2 = ±√(-3) or x = Β±1.732i. , The answer is x = ±√(-3) or x = Β±1.732i
[INST] help me answer this question Exercise 1.3 Solutions, think step by step [/INST] None , The answer is Not applicable, as this is a section header, not a question.