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[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 Find the centre and radius of the circle with the given condition: x2 + y2 -8x + 10y -12 = 0, think step by step [/INST] , The answer is The centre of the given circle is (4, -5), and its radius is √53.
[INST] help me answer this question The 4th term of a G.P. is the square of its second term, and the first term is -3. Determine its 7th term., think step by step [/INST] We are given that the 4th term of the sequence is the square of its second term and the first term is -3. We first find the second term using the given relationship and then find the 7th term using the formula a_n = a * r^(n-1). , The answer is a7 = a * r^(7-1) = -3*1^6 = -3
[INST] help me answer this question What is the probability of getting 3 tails?, think step by step [/INST] , The answer is The probability of getting 3 tails is 1/8.
[INST] help me answer this question How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5, assuming that Repetition of the digits is not allowed?, think step by step [/INST] N/A , The answer is 60
[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 `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 How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?, think step by step [/INST] Total number of different letters in EQUATION = 8. Number of letters to be used to form a word = 8. Number of permutations = 8! = 40,320. , The answer is The total number of arrangements of the word EQUATION is 40,320.
[INST] help me answer this question What is (iii) (6, 12] in interval notation?, think step by step [/INST] (6, 12] represents all real numbers greater than 6 and less than or equal to 12. , The answer is {x: x ∈ R, 6 < x ≀ 12}
[INST] help me answer this question Find the equation of the parabola that satisfies the given conditions: Focus (6,0); directrix x = – 6, think step by step [/INST] The focus lies on the x-axis, so the equation of the parabola is either of the form y^2 = 4ax or y^2 = -4ax. The directrix, x = -6, is to the left of the y-axis, and the focus (6, 0) is to the right of the y-axis. Hence, the parabola is of the form y^2 = 4ax. Here, a = 6. , The answer is y^2 = 24x.
[INST] help me answer this question What is the negation of the statement `in India, there exists a state which does not have a capital`?, think step by step [/INST] , The answer is The negation of this statement is: `For every state in India, it has a capital.`
[INST] help me answer this question Solve the inequality 4x + 3 < 5x + 7. , think step by step [/INST] N/A , The answer is The solutions of the given inequality are defined by all the real numbers greater than -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 Simplify the expression: (x^2 - 1) / (x - 1), think step by step [/INST] To simplify the expression, we recognize that it is a difference of squares. Therefore, we can rewrite the expression as (x - 1)(x + 1) / (x - 1). Since the denominator and the first term in the numerator are the same, we can cancel them out to get x + 1. However, since x cannot equal 1 (because we are dividing by zero), we must exclude this value from the domain of the expression. , The answer is The simplified expression is x.
[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 Evaluate (x + 1)6 + (x - 1)6. Hence, or otherwise, evaluate (√2 + 1)6 + (√2 - 1)6., think step by step [/INST] The binomial theorem is used to express the given expressions and then the values of x are substituted. , The answer is 198
[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 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 In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of the first pq terms is 1/2 (pq + 1) where p β‰  q., think step by step [/INST] Using the pth and qth term expressions, this problem required us to prove the sum of the first pq terms using the A.P. definition. , The answer is The sum of the first pq terms is 1/2 (pq + 1).
[INST] help me answer this question What is the second inequality given in the problem?, think step by step [/INST] , The answer is x + y β‰₯ 1
[INST] help me answer this question Length of the latus rectum is 12, think step by step [/INST] , The answer is The length of the latus rectum is given by the formula 2b^2/a. Here, it is 12.
[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 indicated terms in the sequence whose nth term is an = 4n – 3; a17, a24, think step by step [/INST] On substituting n = 17, we get a17 = 4(17) – 3 = 65. Next, on substituting n = 24, we get a24 = 4(24) – 3 = 93. , The answer is a17 = 65, a24 = 93
[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 an+1/an, for n = 1, 2, 3, 4, 5 in the Fibonacci sequence defined by 1 = a1 = a2 and an = an-1 + an-2, n > 2, think step by step [/INST] Given, 1 = a1 = a2 and an = an-1 + an-2, n > 2. Then, a3/a2 = 2/1 = 1, a4/a3 = 3/2, a5/a4 = 5/3, a6/a5 = 8/5. , The answer is a3/a2 = 1, a4/a3 = 3/2, a5/a4 = 2, a6/a5 = 5/3
[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 12th term of a G.P. whose 8th term is 192, and the common ratio is 2., think step by step [/INST] We are given that the 8th term of the sequence is 192 and the common ratio is 2. We first find the first term, a, using the equation a8 = a * r^7 = 192 and then find the 12th term using the formula a_n = a * r^(n-1). , The answer is The 12th term of the G.P. is given by: a12 = a * r^(12-8) = a * r^4 = 192 * 2^4 = 12288
[INST] help me answer this question What is the probability of getting at least 2 heads?, think step by step [/INST] , The answer is The probability of getting at least 2 heads is 1/2.
[INST] help me answer this question Given that the centre of the circle is (h, k) = (2,2), and the radius (r) of the circle is the distance between points (2,2) and (4,5). Find the required circle., think step by step [/INST] , The answer is The equation of the required circle is x2 + y2 – 4x – 4y = 5
[INST] help me answer this question Find the 13th term in the expansion of , think step by step [/INST] , The answer is Not provided in the data
[INST] help me answer this question In the given word ASSASSINATION, there are 4 β€˜S’. Since all the 4 β€˜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 Find the equation of the line passing through the point (2, 3) and cutting off intercepts on the axes whose sum is 5., think step by step [/INST] , The answer is The equation of the line is x + y – 5 = 0.
[INST] help me answer this question Calculate the mean deviation about the median for the data: 36, 42, 45, 46, 46, 49, 51, 53, 60, 72., think step by step [/INST] First, we find the median (which is 47.5), 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 7.
[INST] help me answer this question Prove that cos(-x) = cos x, think step by step [/INST] N/A , The answer is cos(-x) = cos x
[INST] help me answer this question (ii) If the length of a pendulum is 75 cm and the tip describes an arc of length 15 cm, what angle does the pendulum swing in radians?, think step by step [/INST] , The answer is ΞΈ = 15/75 radian = 1/5 radian
[INST] help me answer this question b^2 = 6a, think step by step [/INST] , The answer is Further simplifying b^2 = 12a/2, we get b^2 = 6a.
[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 Hence, the solution to the inequalities is given by the shaded region in the graph. , think step by step [/INST] , The answer is N/A
[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) = 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 What is the probability of getting 3 heads?, think step by step [/INST] , The answer is The probability of getting 3 heads is 1/8.
[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 : 2x + y >= 6, think step by step [/INST] N/A , The answer is The solution region of the given inequality is above the line 2x + y = 6. (Away from the origin).
[INST] help me answer this question Women occupy even places, which means they will be sitting in 2nd, 4th, 6th and 8th places, whereas men will be sitting in 1st, 3rd, 5th,7th and 9th places. 4 women can sit in four places and ways they can be seated = 4P4, think step by step [/INST] We have 4 women to be seated in 4 even places. The number of permutations of n objects is nPn = n!. So, the number of ways to seat 4 women in 4 even places is 4P4 = 4! = 4 * 3 * 2 * 1 = 24. , The answer is Ways women can be seated = 24
[INST] help me answer this question Find the variance of the following data: 2.25, 21, 24, 27, 30, 16.5., think step by step [/INST] The variance can be calculated by first finding the mean (average) of the data, then finding the deviation of each number from the mean, squaring those deviations, adding them together, and dividing by the number of data points less one. The mean is 16.5 and the variance is 74.25. , The answer is The variance is 74.25.
[INST] help me answer this question If a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d} then {a, e, i, o, u} ∩ {a, b, c, d} = {a}, think step by step [/INST] , The answer is True
[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 Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour., think step by step [/INST] None. , The answer is 6C3 * 5C3 * 5C3
[INST] help me answer this question Evaluate the given limit:, think step by step [/INST] Given, substitute x = 3, we get (3 + 3) = 6 , The answer is 6
[INST] help me answer this question What is the converse of the given statement: "If two lines do not intersect in the same plane, then they are parallel.", think step by step [/INST] The converse statement is created by interchanging the hypothesis and conclusion of the original statement. , The answer is If two lines intersect in the same plane, then they are not parallel.
[INST] help me answer this question Find the derivative of (iii) x-3 (5 + 3x), think step by step [/INST] , The answer is The derivative of x-3 (5 + 3x) is not provided.
[INST] help me answer this question Does the point (-2.5, 3.5) lie inside, outside or on the circle x2 + y2 = 25?, think step by step [/INST] When we substitute x = -2.5 and y = 3.5 in the equation x2 + y2 = 25, we get x2 + y2 > 25, which implies that the point lies outside the circle. , The answer is The point (-2.5, 3.5) lies outside the circle x2 + y2 = 25.
[INST] help me answer this question Given x – 2y ≀ 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 β‰₯ 1`.
[INST] help me answer this question An experiment consists of recording the boy-girl composition of families with 2 children. (i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births?, think step by step [/INST] , The answer is The sample space if we are interested in knowing whether it is a boy or girl in the order of their births, S = {GG, BB, GB, BG}.
[INST] help me answer this question Find the slope of the given line., think step by step [/INST] Given, m Γ— (-2) = -1, so m = 1/2. , The answer is The slope of the given line is 1/2.
[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) = 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 length of the third piece?, think step by step [/INST] , The answer is The length of the third piece = 2x cm.
[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 coordinates of the foci, vertices, length of major axis, length of minor axis, eccentricity, and length of latus rectum for the equation 36x^2 + 4y^2 = 144., think step by step [/INST] , The answer is {'foci': ['(0, 4√2)', '(0, -4√2)'], 'vertices': ['(0, 6)', '(0, -6)'], 'major axis length': 12, 'minor axis length': 4, 'eccentricity': '2√2/3', 'latus rectum length': '4/3'}
[INST] help me answer this question Explain the roster form of a set., think step by step [/INST] , The answer is The roster form of a set lists all its elements between braces {}.
[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 Prove that (cos x – cos y)^2 + (sin x – sin y)^2 = 2 - 2cos(x – y), think step by step [/INST] This involves the use of trigonometric formulas and simplification. , The answer is LHS = (cos^2 x + cos^2 y - 2cos x cos y) + (sin^2 x + sin^2 y - 2sin x sin y) = 2 - 2cos(x – y) = RHS
[INST] help me answer this question Find the slope of the line CA., think step by step [/INST] , The answer is The slope of the line CA (m3) = (4+1)/(4+1) = 5/5 = 1
[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) = (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 Is it a statement?, think step by step [/INST] The given sentence is correct and evident. , The answer is Yes, it is a statement.
[INST] help me answer this question (iv) [U’ ∩ A], think step by step [/INST] , The answer is Ξ¦
[INST] help me answer this question None, think step by step [/INST] None , The answer is The given data is converted into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval.
[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 Prove that sin2 6x – sin2 4x = sin 2x sin 10x, think step by step [/INST] Using sum and difference formulas, sin 6x can be written as sin (2x + 4x) and sin 4x can be written as sin (2x + 2x). Then applying the formula for sin (a + b) and sin (a - b) and using some trigonometric identities, the result can be obtained. , The answer is sin2 6x – sin2 4x = sin 2x sin 10x
[INST] help me answer this question Expand (102)5, think step by step [/INST] Solution: Given, (102)5 102 can be expressed as the sum or difference of two numbers, and then the binomial theorem can be applied. The given question can be written as 102 = 100 + 2 (102)5 = (100 + 2)5 = 5C0 (100)5 + 5C1 (100)4 (2) + 5C2 (100)3 (2)2 + 5C3 (100)2 (2)3 + 5C4 (100) (2)4 + 5C5 (2)5 = (100)5 + 5 (100)4 (2) + 10 (100)3 (2)2 + 5 (100) (2)3 + 5 (100) (2)4 + (2)5 , The answer is 11040808032
[INST] help me answer this question Find the domain of the function f(x) = (x - 1)/(x - 2)., think step by step [/INST] , The answer is The domain of the function f(x) is R - {2}, where R is the set of all real numbers.
[INST] help me answer this question Is 100 divisible by 3?, think step by step [/INST] A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 100 is 1+0+0=1, which is not divisible by 3. Therefore, 100 is not divisible by 3. , The answer is No, 100 is not divisible by 3.
[INST] help me answer this question How many chords can be drawn through 21 points on a circle?, think step by step [/INST] We know that we require two points on the circle to draw a chord.<br>The number of chords is 21C2 = 21! / (21 - 2)! * 2! = 210. , The answer is 210
[INST] help me answer this question If x1 = 4, y1 = – 7, z1 = 8, x2 = 2, y2 = – 3, z2 = 4, find the distance CD., think step by step [/INST] The distance CD is calculated using the formula: √[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]. Substituting the given values, we get: √[4 + 16 + 16] = √36 = 6. , The answer is 6
[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 The solution area would be on the left side of the line graph. , think step by step [/INST] , The answer is N/A
[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 Which of the following is the HCF of 16 and 24?, think step by step [/INST] , The answer is The HCF of 16 and 24 is 8.
[INST] help me answer this question If A = Ξ¦ we get n (A) = 0, think step by step [/INST] Therefore, P (A) has one element. , The answer is n [P(A)] = 1
[INST] help me answer this question Which class interval contains Nth/2 or 50th item for the given continuous frequency distribution?, think step by step [/INST] None , The answer is 35.5 – 40.5
[INST] help me answer this question Write each of the following statements in the form 'if-then'. (i) You get a job implies that your credentials are good., think step by step [/INST] , The answer is If you get a job, then your credentials are good.
[INST] help me answer this question cos 4x = cos 2x, think step by step [/INST] , The answer is The general solution of this equation is given by 4x = 2x + 2Ο€n or 4x = -2Ο€n, where n ∈ Z.
[INST] help me answer this question What are the required points for the third inequality?, think step by step [/INST] , The answer is The required points are (0, 0) and (0, 0).
[INST] help me answer this question Prove that p^2 + 4q^2 = k^2, if p and q are the lengths of perpendiculars from the origin to the lines x cos ΞΈ βˆ’ y sin ΞΈ = k cos 2ΞΈ and x sec ΞΈ + y cosec ΞΈ = k, respectively., think step by step [/INST] The solution involves deriving the formula for perpendicular distance from a line to the origin, squaring both sides and then adding them. , The answer is The proof is provided in the solution.
[INST] help me answer this question Find the domain and range of the following real function., think step by step [/INST] , The answer is Given, f(x) = –|x|, x ∈ R The domain of f is R because f(x) is defined for x ∈ R. The range of f(x) = –|x| is all real numbers except positive real numbers. Therefore, the range of f is given by (β€“βˆž, 0].
[INST] help me answer this question Find the derivative of the following functions. (ii) sec x, think step by step [/INST] , The answer is The derivative of sec x is not provided.
[INST] help me answer this question Vertex (0, 0), passing through (5, 2) and symmetric with respect to the y-axis., think step by step [/INST] , The answer is The equation of the parabola is 2x2 = 25y
[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 D ∩ E?, think step by step [/INST] , The answer is The value of D ∩ E is the empty set (Ο†).
[INST] help me answer this question Evaluate: , think step by step [/INST] , The answer is The value of the expression.
[INST] help me answer this question The remaining 10 places can be filled with 10 remaining letters. Find the number of permutations., think step by step [/INST] , The answer is No. of permutations = 14 Γ— 10!/4! = 25401600
[INST] help me answer this question Is it a statement?, think step by step [/INST] In this question, it is not evident to whom 'she' is referred. , The answer is No, it is not a statement.
[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.