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[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 Find `f''(x)` if `f(x) = x^3 - 2x^2 + 5`., think step by step [/INST] , The answer is `f''(x) = 6x - 4`.

[INST] help me answer this question Find the 20th and nth terms of the G.P. 5/2, 5/4, 5/8, ………, think step by step [/INST] We are given a geometric sequence with first term, a = 5/2 and common ratio, r = 1/2. We can find any term of the sequence using the formula a_n = a * r^(n-1). , The answer is The 20th term and nth term of the given G.P. are given by: a20 = ar19 = a * r^(20-1) = (5/2) * (1/2)^(20-1) an = ar(n-1) = a * r^(n-1) = (5/2) * (1/2)^(n-1)

[INST] help me answer this question Write the additive inverse of the given number: -15, think step by step [/INST] , The answer is The additive inverse of -15 is 15.

[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 value of x?, think step by step [/INST] , The answer is x ≤ 22

[INST] help me answer this question Question: Insert two numbers between 3 and 81 so that the resulting sequence is a G.P., think step by step [/INST] Explanation: Given the first term (3) and the fourth term (81), and the fact that the sequence is a G.P., we can find the common ratio (r) by solving for the cube root of the ratio between the first and fourth terms (r^3 = 81/3, hence r = 3). Using this ratio, we can find the two numbers, which are indeed the second and third terms of the sequence: 3 * 3 = 9 (second term) and 3 * 3^2 = 27 (third term). , The answer is Answer: 9 and 27

[INST] help me answer this question What is D – E?, think step by step [/INST] , The answer is The value of D – E is (1, 2, 3).

[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 Checking for x ≥ 1, The value of x would be unaffected by any value of y, The origin would not lie on the plane, think step by step [/INST] The required area to be included would be on the left of the graph x ≥1 , The answer is ⇒ 0 ≥ 1, which is not true

[INST] help me answer this question What is the value of the Hydrogen ion concentration if a solution has a pH of 4?, think step by step [/INST] The pH of a solution is a measure of its acidity or basicity, and is defined as the negative logarithm (base 10) of the Hydrogen ion concentration. Therefore, the Hydrogen ion concentration can be calculated by taking the inverse logarithm of the pH value. , The answer is The Hydrogen ion concentration is equal to 10 raised to the power of (-pH). So, if the pH is 4, the Hydrogen ion concentration would be equal to 10 raised to the power of -4.

[INST] help me answer this question 15x >= 2 (4x – 1), think step by step [/INST] (Blank) , The answer is x >= -2/7

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

[INST] help me answer this question (x + 1)/3 ≤ (x - 1)/2, think step by step [/INST] , The answer is x ∈ (-∞, -4] ∪ [1, ∞)

[INST] help me answer this question Factorize x² - x - 6, think step by step [/INST] , The answer is x² - x - 6 = (x + 2)(x - 3)

[INST] help me answer this question Is it a statement?, think step by step [/INST] All real numbers can be written as a × 1 + 0 × i. , The answer is Yes, it is a statement.

[INST] help me answer this question Checking for the origin (0, 0) 0 ≤ 30, this is true. Hence, the origin lies in the solution area, which is given by the left side of the line graph. , think step by step [/INST] 0 ≤ 30, this is true. , The answer is The origin lies in the solution area, which is given by the left side of the line graph.

[INST] help me answer this question Find the derivative of (v) x-4 (3 – 4x-5), think step by step [/INST] , The answer is The derivative of x-4 (3 – 4x-5) is not provided.

[INST] help me answer this question sin 2x + cos x = 0, think step by step [/INST] , The answer is The general solution of this equation is given by cos x = 0 or 2 sin x + 1 = 0, which implies x = nπ/2 or 2x = nπ + 3π/8 where n ∈ Z.

[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] This question is not a typical question with a definitive answer. Instead, it is a instruction for the reader to create a table by calculating mid-points, fixi, (xi – x̅), (xi – x̅)2, and fi(xi – x̅)2 for the given classes and their frequencies. , The answer is The answer is a set of instructions to be performed on the given 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 equation of the circle with the given condition: 144x2 – 144x + 144y2 – 72y + 44 = 0, think step by step [/INST] , The answer is The equation of the circle is 144x2 + 144y2 – 144x – 72y + 44 = 0

[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 In the expansion of (1 + a)m+n, prove that coefficients of am and an are equal., think step by step [/INST] , The answer is By substituting the values in the general form, Tr+1 = m+n Cr ar, we can prove that the coefficients of am and an are equal.

[INST] help me answer this question Write the equation of the ellipse., think step by step [/INST] , The answer is x²/52 + y²/13 = 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 The student can choose 3 questions from part I and 5 from part II, Or 4 questions from part I and 4 from part II, Or 5 questions from parts 1 and 3 from part II., think step by step [/INST] This is a combination problem. For each option given, we need to calculate the number of ways to choose the questions from part I and part II. For example, for the first option, there are 5C3 ways to choose 3 questions from 5 in part I and 7C5 ways to choose 5 questions from 7 in part II. To find these combinations, we need to use the formula mentioned in the previous question. , The answer is The student can choose the questions in 56, 175, or 273 ways, respectively

[INST] help me answer this question (a) What is the role of a catalyst in a chemical reaction?, think step by step [/INST] A catalyst works by providing an alternative reaction pathway with a lower energy barrier, allowing more reactant molecules to overcome the activation energy threshold and undergo a reaction. , The answer is A catalyst is a substance that increases the rate of a chemical reaction by lowering the activation energy required for the reaction to occur.

[INST] help me answer this question What is the final solution area for the given inequalities?, think step by step [/INST] , The answer is The final solution area for the given inequalities is the shaded area of the graph satisfying all the given inequalities.

[INST] help me answer this question Find the value of sin (1005°)., think step by step [/INST] We know that the values of sin x repeat after an interval of 2π or 360°. Therefore, sin (1005°) = sin (1005° - 900°) = sin 105° = sin (90° + 15°) = -cos 15°. Hence, sin (1005°) = -1/√2. , The answer is The value of sin (1005°) is -1/√2.

[INST] help me answer this question Convert the angle measure 120 degrees to radians., think step by step [/INST] To convert an angle measure from degrees to radians, we can use the formula: radians = (degrees * π) / 180. Therefore, 120 degrees is equivalent to (120 * π) / 180 = 2.0943951023932 radians. , The answer is The equivalent angle measure in radians is 2.0943951023932 radians.

[INST] help me answer this question The solution area of the line would not include the origin, as 0 ≥ 1 is not true. , think step by step [/INST] , The answer is N/A

[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 Given x2/49 + y2/36 = 1, what is the length of the minor axis?, think step by step [/INST] , The answer is Length of minor axis = 2b = 2 (6) = 12

[INST] help me answer this question b^2 = 12a/2, think step by step [/INST] , The answer is Simplifying 2b^2 = 12a, we get b^2 = 12a/2.

[INST] help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step [/INST] , The answer is The domain of the function is all real numbers.

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

[INST] help me answer this question If a, b, c and d are in G.P., show that (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2, think step by step [/INST] As a, b, c, and d are in G.P., we have b^2 = ac and c^2 = bd. We first express the given expression as a quadratic in ac, find its roots using the quadratic formula, and then show that these roots are indeed b^2 and d^2, thus verifying the identity. , The answer is (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2

[INST] help me answer this question What is the probability of getting no head?, think step by step [/INST] , The answer is The probability of getting no head is 1/8.

[INST] help me answer this question What is the difference between an Element and a Compound?, think step by step [/INST] Elements are the building blocks of matter, while compounds are formed by the chemical combination of elements. , The answer is An element is a pure substance that cannot be broken down into simpler substances by chemical means, while a compound is a substance composed of two or more elements chemically combined in a fixed proportion.

[INST] help me answer this question Given that P (3, 2, – 4), Q (5, 4, – 6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides PR., think step by step [/INST] , The answer is Hence, the ratio in which Q divides PR is 1: 2.

[INST] help me answer this question The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of (x + 1)n are in the ratio 1:3:5. Find n and r., think step by step [/INST] , The answer is By using the general term Tr+1 = nCr an-r br, we can find that n = 7 and r = 3.

[INST] help me answer this question Find the median of the following data: 36, 42, 45, 46, 46, 49, 51, 53, 60, 72., think step by step [/INST] First, we arrange the data in ascending order, then we calculate the median as the average of the 5th and 6th observations, which is (46 + 49)/2 = 47.5. , The answer is The median of the given data is 47.5.

[INST] help me answer this question The elements are odd natural numbers which are less than 10. Hence, (v) matches with (b)., think step by step [/INST] , The answer is The elements of set (iv) are odd natural numbers less than 10, which makes it match with set (b).

[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 (vii) {x: x is a perfect cube}, think step by step [/INST] , The answer is {x: x ∈ N and x is not a perfect cube}

[INST] help me answer this question Find the mean and variance for the first n natural numbers., think step by step [/INST] The mean is calculated using the formula for the sum of an arithmetic series. The variance is calculated using the formula for the variance of an arithmetic series. , The answer is The mean is (n + 1)/2 and the variance is (n^2 – 1)/12.

[INST] help me answer this question What is the perimeter of the triangle?, think step by step [/INST] , The answer is x + (x + 3) + 2x ≤ 91 cm

[INST] help me answer this question In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?, think step by step [/INST] Total number of letters in MISSISSIPPI = 11. Number of permutations = 11! / (4! * 4! * 2!) = 39,916,800 / (24 * 24 * 2) = 34,650. Number of permutations where the four I’s come together = 840. Therefore, the total number of permutations where the four I’s don't come together = 34,650 - 840 = 33,810. , The answer is The number of permutations where the four I’s do not come together is 33,810.

[INST] help me answer this question (v) {x: x is a triangle in a plane}…{x: x is a rectangle in the plane}, think step by step [/INST] Not every triangle in a plane is a rectangle in the same plane. , The answer is {x: x is a triangle in a plane} ⊄ {x: x is a rectangle in the plane}

[INST] help me answer this question What are the required points on the graph for the given inequalities of 5x + 4y ≤ 20?, think step by step [/INST] These points satisfy the given inequalities. , The answer is The required points would be (1, 0), (1, 2) and so on.

[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 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 `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 multiplicative inverse of each of the complex numbers given in Exercises 11 to 13., think step by step [/INST] , The answer is Not available in the text

[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 Find the equation of the line through the point (0, 2) making an angle 2π/3 with the positive x-axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin., think step by step [/INST] , The answer is The equation of the line is √3 x + y – 2 = 0 and the line parallel to it is √3 x + y + 2 = 0.

[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 Are the events BI and C mutually exclusive?, think step by step [/INST] BI ∩ C = {(1, 2), (3, 2), (4, 2), (6, 2)} , The answer is No, they are not mutually exclusive.

[INST] help me answer this question A die is thrown two times. Describe the sample space for this experiment., think step by step [/INST] When a die is thrown, there are 6 possible outcomes. Since the die is thrown twice, the total number of possible outcomes is 6*6 = 36. These outcomes form the sample space S. , The answer is The sample space S contains 36 elements and is given by: S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3)(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

[INST] help me answer this question Solve the inequality 3x – 7 > 5x – 1. , think step by step [/INST] N/A , The answer is x < -3.

[INST] help me answer this question Is it a statement?, think step by step [/INST] The product of (-1) and 8 is -8, but the sentence states the product to be -1. , The answer is No, it is not a statement.

[INST] help me answer this question Write the converse of the given statement: "If x is divisible by 4, then x is an even number.", think step by step [/INST] The converse statement is created by interchanging the hypothesis and conclusion of the original statement. , The answer is If x is not an even number, then x is not divisible by 4.

[INST] help me answer this question Given x2/16 + y2/9 = 1, what is the length of the major axis?, think step by step [/INST] , The answer is Length of major axis = 2a = 2 (4) = 8

[INST] help me answer this question In a class of 60 students, if 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS, find the probability that the student opted for NCC or NSS., think step by step [/INST] We know that, P(A∪B) = P(A) + P(B) – P(A∩B) = 30/60 + 32/60 – 24/60 = 19/30 , The answer is 19/30

[INST] help me answer this question Foci (±5, 0), the transverse axis is of length 8., think step by step [/INST] Given foci (±5, 0) are on the x-axis, and the transverse axis is of length 8, so the equation of the hyperbola is of the form x2/a2 – y2/b2 = 1. Here, a = 4 and c = 5. We know that a2 + b2 = c2, so b2 = 9. , The answer is The equation of the hyperbola is x2/16 – y2/9 = 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 Checking for the origin (0, 0) 0 ≤ 3, this is true , think step by step [/INST] 0 ≤ 3, this is true , The answer is The solution area would include the origin and hence would be on the left side of the line graph.

[INST] help me answer this question {2, 6, 10} ∩ {3, 7, 11} = Φ then is it true?, think step by step [/INST] , The answer is True

[INST] help me answer this question Find the equation of the parabola with equation x^2 = -9y., think step by step [/INST] On comparing the given equation with x^2 = -4ay, we get a = 9/4. The coordinates of the focus are (0,-a) = (0, -9/4). The given equation involves x^2, so the axis of the parabola is the y-axis. Therefore, the equation of directrix is y = a, which is y = 9/4. The length of latus rectum is 4a = 4(9/4) = 9. , The answer is x^2 = -9y.

[INST] help me answer this question Show that the products of the corresponding terms of the sequences a, ar, ar^2, …, a r^(n-1) and A, AR, AR^2, …, AR^(n-1) form a G.P., think step by step [/INST] None , The answer is To be proved: The sequence, aA, arAR, ar^2AR^2, …, a r^(n-1)AR^(n-1), forms a G.P.

[INST] help me answer this question 2(x + 1) + 3x ≤ 8x + 1, think step by step [/INST] , The answer is x ∈ (∞, -3.5)

[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 37 – (3x + 5) ≥ 9x – 8 (x – 3), think step by step [/INST] The solutions of the given inequality are defined by all the real numbers less than or equal to 3. Hence, the required solution set is (-∞, 3]. , The answer is x ≤ 3

[INST] help me answer this question (Miscellaneous Exercise on Chapter 12 Solutions), think step by step [/INST] The data contains a link to access the solutions for Miscellaneous Exercise on Chapter 12, but does not include the solutions directly. , The answer is Not provided in the data.

[INST] help me answer this question (iv) (2, –1, 3) and (–2, 1, 3), think step by step [/INST] , The answer is 2√11 units

[INST] help me answer this question Write the first five terms of the sequence whose nth term is an = (3n – 1)/2, think step by step [/INST] , The answer is 1, 5/2, 4, 11/2, 7

[INST] help me answer this question Write the first five terms of the sequence whose nth term is an = an-1 – 1, n > 2 and a1 = a2 = 2, think step by step [/INST] Given, a1 = a2 = 2, an = an-1 – 1, n > 2. Then, a3 = a2 – 1 = 2 – 1 = 1, a4 = a3 – 1 = 1 – 1 = 0, a5 = a4 – 1 = 0 – 1 = -1. , The answer is The first five terms of the sequence are 2, 2, 1, 0, and -1.

[INST] help me answer this question Show the ratio of the sum of the first n terms of a G.P. to the sum of terms from (n+1)th to (2n)th term is given by, think step by step [/INST] We first find the sum of the first n terms and the sum of terms from (n+1)th to (2n)th term using the formula for the sum of a G.P. Then, expressing the latter sum as a G.P., we show that the given ratio simplifies to r^(n^2). , The answer is The ratio of the sum of the first n terms of a G.P. to the sum of terms from (n+1)th to (2n)th term is given by

[INST] help me answer this question What is the length of the major axis?, think step by step [/INST] , The answer is Length of major axis = 2a = 2 (5) = 10

[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-39, think step by step [/INST] i-39 = 1/ i39 = 1/ i4 x 9 + 3 = 1/ (19 x i3) = 1/ i3 = 1/ (-i) [i4 = 1, i3 = -I and i2 = -1] Now, multiplying the numerator and denominator by i, we get i-39 = 1 x i / (-i x i) = i/ 1 = i , The answer is 0 + i

[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 The number of ways we can select a team of 11 players where 4 players are bowlers from 17 players is , think step by step [/INST] N/A , The answer is 3960

[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 coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem., think step by step [/INST] After expanding both expressions and multiplying them, the coefficient of x5 in the resulting expression is 171 , The answer is 171

[INST] help me answer this question (iii) Is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} a universal set for the sets A, B, and C?, think step by step [/INST] , The answer is Yes.

[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 minimum length of the shortest side., think step by step [/INST] , The answer is The minimum length of the shortest side will be 9 cm.

[INST] help me answer this question A card is selected from a pack of 52 cards. (a) How many points are there in the sample space?, think step by step [/INST] , The answer is The number of points in the sample space is 52.

[INST] help me answer this question Given P(A) = 0.75, P(A∩B) = 0.5, and P(AI∩BI) = 0.1, find the probability of passing the Hindi examination., think step by step [/INST] We have P(B) = P(A∩B) + P(AI∩BI) = 0.5 + 0.1 = 0.6, therefore, P(B) = 0.65 , The answer is 0.65

[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 Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x – x1) + B (y – y1) = 0., think step by step [/INST] The slope of the given line is found, then the formula of the line passing through a point and having slope m is used, which gives the required equation. , The answer is The equation is A(x – x1) + B(y – y1) = 0

[INST] help me answer this question Find A – B, think step by step [/INST] A – B represents the set of elements that are present in A but not in B. , The answer is {3, 6, 9, 15, 18, 21}

[INST] help me answer this question What is the relationship between set E and set G?, think step by step [/INST] , The answer is Set E is 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 The function ‘t’, which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = 9/5C + 32. Find, think step by step [/INST] , The answer is (i) t(0) = 32 (ii) t(28) = 82 (iii) t(–10) = 14 (iv) C when t(C) = 212 = 100

[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.