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[INST] help me answer this question Question: 7 ≤ (3x + 11)/2 ≤ 11, think step by step [/INST] The given inequality is 7 ≤ (3x + 11)/2 ≤ 11. After solving this inequality, we get x ∈ [1, 11/3]. , The answer is The solutions of the given inequality are all real numbers x greater than or equal to 1 but less than or equal to 11/3.

[INST] help me answer this question N/A, think step by step [/INST] Class 11 Maths NCERT supplementary or miscellaneous exercise solutions PDFs are provided here. , The answer is N/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 i9 + i19, think step by step [/INST] i9 + i19 = (i2)4. i + (i2)9. i = (-1)4 . i + (-1)9 .i = 1 x i + -1 x i = i – i = 0 , The answer is 0 + i0

[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 2x + 19 ≤ 6x +47, think step by step [/INST] , The answer is x ≥ -7

[INST] help me answer this question Vertices (0, ± 5), foci (0, ± 8)., think step by step [/INST] Given vertices (0, ± 5) and foci (0, ± 8) are on the y-axis, so the equation of the hyperbola is of the form y2/a2 – x2/b2 = 1. Here, a = 5 and c = 8. We know that a2 + b2 = c2, so b2 = 39. , The answer is The equation of the hyperbola is y2/25 – x2/39 = 1.

[INST] help me answer this question Let A = {1, 2, 3, ... , 14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range., think step by step [/INST] The range of R is {3, 6, 9, 12} , The answer is The relation R from A to A is given as R = {(x, y): 3x – y = 0, where x, y ∈ A} = {(x, y): 3x = y, where x, y ∈ A} So, R = {(1, 3), (2, 6), (3, 9), (4, 12)} Now, The domain of R is the set of all first elements of the ordered pairs in the relation. Hence, Domain of R = {1, 2, 3, 4} The whole set A is the codomain of the relation R. Hence, Codomain of R = A = {1, 2, 3, ..., 14} The range of R is the set of all second elements of the ordered pairs in the relation.

[INST] help me answer this question What is the probability of getting exactly two tails?, think step by step [/INST] , The answer is The probability of getting exactly two tails is 3/8.

[INST] help me answer this question What is the number of subsets of the set A × B?, think step by step [/INST] , The answer is The set A × B has 16 subsets

[INST] help me answer this question Find the standard deviation for Y., think step by step [/INST] , The answer is The standard deviation for Y is 2.

[INST] help me answer this question If you do not have winter clothes, then you do not live in Delhi., think step by step [/INST] This is the contrapositive of the statement in (a). , The answer 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 `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 Describe two events which are mutually exclusive but not exhaustive., think step by step [/INST] , The answer is Let A be the event of getting at least two heads. So, A = (HHH, HHT, THH, HTH). Let B be the event of getting only head. So, B= (HHH). Therefore, A ∩ B = (HHH), and A and B are mutually exclusive but not exhaustive.

[INST] help me answer this question Find the distance between P (x1, y1) and Q (x2, y2) when: (i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis., think step by step [/INST] The distance formula is derived from the Pythagorean theorem and is used to calculate the distance between two points in a coordinate plane. , The answer is The distance between P and Q when PQ is parallel to y-axis is |y2 – y1|, and when PQ is parallel to x-axis, the distance between P and Q is |x2 – x1|.

[INST] help me answer this question What is the electron configuration of Sodium and Helium?, think step by step [/INST] The electron configuration of an element refers to the arrangement of electrons in an atom. It is derived by following the Aufbau principle, which states that electrons occupy the lowest energy level accessible to them. , The answer is The electron configuration of Sodium is 1s² 2s² 2p⁶ 3s¹ and the electron configuration of Helium is 1s².

[INST] help me answer this question For the given data, find the mean and standard deviation using the short-cut method., think step by step [/INST] Refer to the calculations in the answer. , The answer is The mean is 64 and the standard deviation is 1.691.

[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 x + 3y ≤ 30, what are the required points?, think step by step [/INST] , The answer is The required points are (0, 10) and (30, 0).

[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 `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 mean deviation about the mean for the data in Exercises 9 and 10., think step by step [/INST] Here, the user is expected to calculate the mean deviation about the mean for the given data in Exercises 9 and 10. , The answer is The user is being asked to find the mean deviation about the mean for the data given in Exercises 9 and 10.

[INST] help me answer this question Find the mean deviation about the mean for the data in Exercise 1 and 2., think step by step [/INST] In both cases, first, the mean of the given data is calculated and then the deviations from the mean are found. After that, the absolute values of the deviations are obtained, and the mean deviation is calculated as the sum of the deviations divided by the number of observations. , The answer is

[INST] help me answer this question 41n – 14n is a multiple of 27., think step by step [/INST] N/A , The answer is The given statement is true for all natural numbers.

[INST] help me answer this question What makes a collection a set?, think step by step [/INST] A set is a well-defined collection of objects, often referred to as elements or members. A collection is well-defined if the criteria for membership are clear. , The answer is A collection is a set if it is well-defined, and one can find a number which belongs to this collection. A collection is a set if it is well-defined, and one can find a book which belongs to this collection. A collection is a set if it is well-defined, and one can find an integer which belongs to this collection. A collection is a set if it is well-defined, and one can find a question which belongs to this chapter.

[INST] help me answer this question If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find (i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D (iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C) (vii) A ∩ D (viii) A ∩ (B ∪ D) (ix) (A ∩ B) ∩ (B ∪ C) (x) (A ∪ D) ∩ (B ∪ C), think step by step [/INST] , The answer is {'i': 'A ∩ B = {7, 9, 11}', 'ii': 'B ∩ C = {11, 13}', 'iii': 'A ∩ C ∩ D = Φ', 'iv': 'A ∩ C = {11}', 'v': 'B ∩ D = Φ', 'vi': 'A ∩ (B ∪ C) = {7, 9, 11}', 'vii': 'A ∩ D = Φ', 'viii': 'A ∩ (B ∪ D) = {7, 9, 11}', 'ix': '(A ∩ B) ∩ (B ∪ C) = {7, 9, 11}', 'x': '(A ∪ D) ∩ (B ∪ C) = {7, 9, 11, 15}'}

[INST] help me answer this question Find the derivative of the following functions. (vi) 5 sin x – 6 cos x + 7, think step by step [/INST] , The answer is The derivative of 5 sin x – 6 cos x + 7 is not provided.

[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 Find the discriminant of the quadratic equation x^2 + x + 1/√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 = 1/√2. Substituting these values into the formula, we get D = (1)^2 - 4*1*1/√2 = 2 - 2/√2 = 2*(√2 - 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 (i) {3, 4} ⊂ A is incorrect, think step by step [/INST] , The answer is Here 3 ∈ {3, 4}; where, 3∉A.

[INST] help me answer this question What is collinearity of points C (0, 1/3, 2), A, and B?, think step by step [/INST] , The answer is A, B, C are collinear.

[INST] help me answer this question Solve the inequality -12x > 30. Find the solution set when x is an integer., think step by step [/INST] N/A , The answer is The solution set is {…, -5, -4, -3}.

[INST] help me answer this question a^2 + 8a – 2a – 16 = 0, think step by step [/INST] , The answer is Adding -2a to both sides of the equation a^2 + 6a – 16 = 0, we get a^2 + 8a – 2a – 16 = 0.

[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 Are {2, 3, 4, 5} and {3, 6} disjoint?, think step by step [/INST] {2, 3, 4, 5} ∩ {3, 6} = {3} , The answer is Not disjoint

[INST] help me answer this question What is the standard deviation for the given data?, think step by step [/INST] , The answer is Standard Deviation = 5.553

[INST] help me answer this question 3(7 + i7) + i(7 + i7), think step by step [/INST] 3(7 + i7) + i(7 + i7) = 21 + i21 + i7 + i2 7 = 21 + i28 - 7 , The answer is 14 + i28

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

[INST] help me answer this question (i) {a, b} ⊄ {b, c, a}, think step by step [/INST] Here each element of {a, b} is an element of {b, c, a}. , The answer is False

[INST] help me answer this question Given, y + 8 ≥ 2x, think step by step [/INST] , The answer is The solution region of the given inequality is above the line y + 8 = 2x. (That is, the origin is included in the region.)

[INST] help me answer this question Is f a function from Z to Z?, think step by step [/INST] The same first element, 12, corresponds to two different images (8 and –8). , The answer is No, it is not.

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

[INST] help me answer this question How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?, think step by step [/INST] The word DAUGHTER has 3 vowels, A, E, U and 5 consonants D, G, H, T and R. The three vowels can be chosen in 3C2, as only two vowels are to be chosen. Similarly, the five consonants can be chosen in 5C3 ways. The number of choosing 2 vowels and 5 consonants would be 3C2 ×5C3. Each of these 5 letters can be arranged in 5 ways to form different words. , The answer is The total number of words formed would be = 30 × 120 = 3600

[INST] help me answer this question If the diagonals of a quadrilateral bisect each other, then it is a parallelogram., think step by step [/INST] This is the converse of the statement in (b). , The answer is

[INST] help me answer this question Calculate the distance PQ, QR, and PR using the given points: P(0, 7, 10), Q(–1, 6, 6), R(–4, 9, 6), think step by step [/INST] The distances are calculated using the distance formula, Distance = √[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2] , The answer is {'PQ': '√18', 'QR': '√18', 'PR': '√36'}

[INST] help me answer this question Which firm, A or B, shows greater variability in individual wages?, think step by step [/INST] , The answer is Firm B shows greater variability in individual wages with a variance of 121 compared to firm A's variance of 100.

[INST] help me answer this question A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment., think step by step [/INST] N/A , The answer is S = {WW, WR, RW}

[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 is the LCM of 12 and 18?, think step by step [/INST] , The answer is The LCM of 12 and 18 is 36.

[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) = (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 the point (2, 2) and cutting off intercepts on the axes whose sum is 9., think step by step [/INST] , The answer is The equation of the line is 2x + y – 6 = 0 or x + 2y – 6 = 0.

[INST] help me answer this question Write the first five terms of the sequence whose nth term is an = (-1)^n-1 5n+1, think step by step [/INST] , The answer is 25, -125, 625, -3125, 15625

[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 Find the equation for the ellipse that satisfies the given conditions: Vertices (± 5, 0) and foci (± 4, 0)., think step by step [/INST] , The answer is x^2/25 + y^2/16 = 1

[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 Vertex (0, 0) passes through (2, 3), and the axis is along the x-axis., think step by step [/INST] , The answer is The equation of the parabola is 2y2 = 9x

[INST] help me answer this question Checking for the origin (0, 0) 0 ≤ 6 this is true , think step by step [/INST] 0 ≤ 6 this is true , The answer is The origin lies in the solution area, and the area would be on the left of the line graph.

[INST] help me answer this question Write the reciprocal of the given fraction: 5/7, think step by step [/INST] , The answer is The reciprocal of the given fraction is 7/5.

[INST] help me answer this question a11 if an = (-1)^n-1 n, think step by step [/INST] , The answer is a11 = 11

[INST] help me answer this question (1 – i)4, think step by step [/INST] (1 – i)4 = [(1 – i)2]2 = [1 + i2 – 2i]2 = [1 – 1 – 2i]2 = (-2i)2 = 4(-1) = -4 + 0i , The answer is -4 + 0i

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

[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) = (x + 3)/(x - 4)`, think step by step [/INST] , The answer is The domain of the function is all real numbers except `x = 4`.

[INST] help me answer this question Is 102n – 1 + 1 divisible by 11 for all natural numbers, i.e., n?, think step by step [/INST] This can be proved using the principle of mathematical induction. For n=1, 102n – 1 + 1 = 11, which is divisible by 11. Assuming P (k) to be true for some positive integer k, it can be shown that P (k + 1) is also true, hence P (n) is true for all natural numbers, i.e., n. , The answer is Yes, 102n – 1 + 1 is divisible by 11 for all natural numbers.

[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] , The answer is R = {(2, 8), (3, 27), (5, 125), (7, 343)}

[INST] help me answer this question Is the statement “(a, a) ∈ R, for all a ∈ N” true?, think step by step [/INST] For (a, a) to be an element of R, a should be equal to the square of some natural number. However, this is not true for all a belonging to N. For example, 2 ∈ N, but √2 ∉ N. , The answer is No, it is not true.

[INST] help me answer this question Is 3 a member of set C?, think step by step [/INST] , The answer is Yes, 3 is a member of set C.

[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) = 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 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 `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 If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G., think step by step [/INST] Given, G = {7, 8} and H = {5, 4, 2} We know that The Cartesian product of two non-empty sets, P and Q, is given as P × Q = {(p, q): p ∈ P, q ∈ Q} So, G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)} H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)} , The answer is [(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)], [(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)]

[INST] help me answer this question If the sum of n terms of an A.P. is 3n^2 + 5n and its mth term is 164, 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 equating the sum of n terms from both the equations, simplifying the equation, and then solving for m. , The answer is The value of m is obtained by solving an equation in terms of m.

[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 relationship between set F and set G?, think step by step [/INST] , The answer is Set F is not equal to set G.

[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 x2/25 + y2/100 = 1, what is the length of the latus rectum?, think step by step [/INST] , The answer is Length of latus rectum = 2b2/a = (2×52)/10 = (2×25)/10 = 5

[INST] help me answer this question What is the probability of getting at least one tail when a coin is tossed twice?, think step by step [/INST] , The answer is 3/4

[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 If X and Y are two sets, such that X ∪Y has 18 elements, X has 8 elements, and Y has 15 elements, how many elements does X ∩ Y have?, think step by step [/INST] Using the formula n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y) , we find n(X ∩ Y) = 5 , The answer is n(X ∩ Y) = 5

[INST] help me answer this question If X and Y are two sets, such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩ Y)., think step by step [/INST] Using the formula n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y) , we find n(X ∩ Y) = 2 , The answer is n(X ∩ Y) = 2

[INST] help me answer this question Is the statement “(a, b) ∈ R, implies (b, a) ∈ R” true?, think step by step [/INST] Counterexample: (9, 3) ∈ R because 9, 3 ∈ N and 9 = 3^2. However, (3, 9) ∉ N because 3 ≠ 9^2 = 81. , The answer is No, it is not true.

[INST] help me answer this question Question: Find the value of n so that, think step by step [/INST] Explanation: To find the value of n that makes the square roots equal, we square both sides, perform cross-multiplication, and rearrange terms to solve for n. , The answer is Answer: n = (b^2 - a^2) / (2ab)

[INST] help me answer this question Given 3x + 4y ≤ 60, what are the required points?, think step by step [/INST] These points are obtained by putting values of x = 0 and y = 0 in the equation one by one. , The answer is The required points are (0, 15) and (20, 0).

[INST] help me answer this question Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + …, think step by step [/INST] , The answer is The sum of n terms of the series can be expressed as the sum of the first n terms of the sequence an, where an = n(n + 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 One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected randomly and rolled, and its colour and the number on its uppermost face are noted. Describe the sample space., think step by step [/INST] , The answer is Let 1, 2, 3, 4, 5 and 6 be the possible numbers when the die is thrown. Also, let the die of red colour be R, the die of white colour be W, and the die of blue colour be B. The total number of possible outcomes equals (6 × 3) = 18. Hence, the sample space of the event is S={(R,1),(R,2),(R,3),(R,4),(R,5),(R,6),(W,1),(W,2),(W,3),(W,4),(W,5),(W,6) (B,1),(B,2),(B,3),(B,4),(B,5),(B,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 ≥ 1`.

[INST] help me answer this question Find the equation of the tangent to the curve `y = x^2 - 3x + 2` at `x = 2`., think step by step [/INST] , The answer is The equation of the tangent is `y = 4x - 6`.

[INST] help me answer this question Show that the equation of the line is x/a + y/b = 2, if P (a, b) is the mid-point of a line segment between axes., think step by step [/INST] , The answer is Proved.

[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 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 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 The relation f is defined by f = {(x, y): y = x^2, 0 ≤ x < 3} and g is defined by g = {(x, y): y = x, 1 ≤ x ≤ 4}. Show that f is a function and g is not a function., think step by step [/INST] A relation is a function if for every value of x in its domain, there is only one value of y. In this case, f satisfies this property and hence is a function, while g does not and hence is not a function. , The answer is The relation f is a function as for every value of x in the given domain (0 ≤ x < 3), there is only one value of y such that (x, y) belongs to f. However, the relation g is not a function as for x = 2, there are two values of y (2 and 3) such that (x, y) belongs to g.