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[INST] help me answer this question We know that n(S) = 21, n(T) = 32, and n(S ∩ T) = 11. Find n(S ∪ T)., think step by step [/INST] This is based on the formula n(S ∪ T) = n(S) + n(T) - n(S ∩ T) , The answer is n(S ∪ T) = 21 + 32 - 11 = 42

[INST] help me answer this question What are the required points for the second inequality?, think step by step [/INST] , The answer is The required points are (0, 1) and (1, 0).

[INST] help me answer this question If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference., think step by step [/INST] Using the sum formula and coefficients of n and n^2, we solved for the common difference. , The answer is The common difference is 2q.

[INST] help me answer this question What is a compound event?, think step by step [/INST] , The answer is An event that has more than one sample point is called a Compound event.

[INST] help me answer this question Find equation of the line parallel to the line 3x − 4y + 2 = 0 and passing through the point (–2, 3)., think step by step [/INST] The slope of the given line is 3/4. Equation of line having slope m and passing through (x1, y1) is given by y – y1 = m (x – x1). Substituting the given slope and point we get the required equation. , The answer is 3x – 4y = 18

[INST] help me answer this question Determine which number is larger, (1.1)10000 or 1000, using the binomial theorem., think step by step [/INST] The binomial theorem is applied to (1.1)10000 to compare. , The answer is (1.1)10000 > 1000

[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 Calculate the mean deviation about median age for the age distribution of 100 persons given below., think step by step [/INST] Here, the user is expected to calculate the mean deviation about the median age for the given age distribution of 100 persons. , The answer is The user is being asked to calculate the mean deviation about the median age for the given age distribution of 100 persons.

[INST] help me answer this question 1/3 ≥ 1/x - 1/2, think step by step [/INST] , The answer is x ∈ (0, 3)

[INST] help me answer this question The equation of the hyperbola is of form x^2/a^2 – y^2/b^2 = 1, think step by step [/INST] , The answer is The equation of the hyperbola has the general form x^2/a^2 – y^2/b^2 = 1.

[INST] help me answer this question A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form. , think step by step [/INST] , The answer is The relation R is given by R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}

[INST] help me answer this question (i) (–3, 0), think step by step [/INST] , The answer is {x: x ∈ R, –3 < x < 0}

[INST] help me answer this question Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively., think step by step [/INST] We have used the formula (r + 1)th term, (T(r+1)) in the binomial expansion of (a + b)n is given by T(r+1) = nCr an-t br. After equating the three given terms with the first three terms of the expansion and solving the equations, we got a = 3, b = 5 and n = 76 , The answer is a = 3, b = 5 and n = 76

[INST] help me answer this question B ∪ C, think step by step [/INST] , The answer is B ∪ C = (φ) ∪ (3, 6) = (3, 6)

[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 `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 (ii) (–3, 7, 2) and (2, 4, –1), think step by step [/INST] , The answer is √43 units

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

[INST] help me answer this question 2 and 3 are prime numbers which are divisors of 6. Hence, (ii) matches with (a)., think step by step [/INST] , The answer is Set (ii) contains prime numbers that are divisors of 6, which makes it match with set (a).

[INST] help me answer this question Find the derivative of (iv) x5 (3 – 6x-9), think step by step [/INST] , The answer is The derivative of x5 (3 – 6x-9) is not provided.

[INST] help me answer this question sin x + sin 3x + sin 5x = 0, think step by step [/INST] , The answer is The general solution of this equation is given by sin 3x (2 cos 2x + 1) = 0, which implies sin 3x = 0 or 2 cos 2x + 1 = 0.

[INST] help me answer this question Find the derivative of the following functions. (iii) 5 sec x + 4 cos x, think step by step [/INST] , The answer is The derivative of 5 sec x + 4 cos x is not provided.

[INST] help me answer this question Find the equation of the line if point R (h, k) divides a line segment between the axes in the ratio 1:2., think step by step [/INST] , The answer is The equation of the line is 2x/3h + y/3k = 1.

[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 Write the contrapositive of the given statement: "If x is not divisible by 4, then x is not an even number.", 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 x is an even number, then x is divisible by 4.

[INST] help me answer this question Which term of the following sequences: (a) 2, 2√2, 4,… is 128 ? (b) √3, 3, 3√3,… is 729 ? (c) 1/3, 1/9, 1/27, … is 1/19683 ?, think step by step [/INST] , The answer is {'part_a': 'The 13th term of the given sequence is 128.', 'part_b': 'The 12th term of the given sequence is 729.', 'part_c': 'The 9th term of the given sequence is 1/19683.'}

[INST] help me answer this question Find the value of: (i) sin 75o, (ii) tan 15o, think step by step [/INST] N/A , The answer is {'part1': {'subpart': '(i)', 'content': 'sin 75o = sin (45o + 30o) = sin 45o cos 30o + cos 45o sin 30o = √3 + 1 / 2√2'}, 'part2': {'subpart': '(ii)', 'content': 'tan 15o can be written as tan (45o – 30o) = (tan 45o – tan 30o) / (1 + (tan 45o * tan 30o)) = √3 / (√3 + 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 (iii) If the length of a pendulum is 75 cm and the tip describes an arc of length 21 cm, what angle does the pendulum swing in radians?, think step by step [/INST] , The answer is θ = 21/75 radian = 7/25 radian

[INST] help me answer this question A fair coin is tossed four times, and a person wins Rs 1 for each head and loses Rs 1.50 for each tail that turns up., think step by step [/INST] , The answer is Not provided in the data.

[INST] help me answer this question Find the sum to n terms of the series 52 + 62 + 72 + … + 202, 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 = n2 + 8n + 16.

[INST] help me answer this question Evaluate the given limit:, think step by step [/INST] Given limit, substitute x = 0, we get (a*0 + b) / (c*0 + 1) = b , The answer is b

[INST] help me answer this question If slope of one line is 2, find equation of the other line., think step by step [/INST] , The answer is If the slope of one line is 2 and it intersects another line at an angle of 60°, then the slope of the other line can be found using the formula m2 = m1 - tan(60°) and the equation of the other line can be found using the slope-point form of a line.

[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 (iv) [–23, 5), think step by step [/INST] , The answer is {x: x ∈ R, –23 ≤ x < 5}

[INST] help me answer this question What is (iv) [–23, 5) in interval notation?, think step by step [/INST] [–23, 5) represents all real numbers greater than or equal to -23 and less than 5. , The answer is {x: x ∈ R, –23 ≤ x < 5}

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

[INST] help me answer this question Expand (1 + x)7, think step by step [/INST] , The answer is From the binomial theorem, the given equation can be expanded as Solution: Co (1)7 + 7C1 (1)6 (x) + 21C2 (1)5 (x)2 + 35C3 (1)4 (x)3 + 35C4 (1)3 (x)4 + 21C5 (1)2 (x)5 + 7C6 (1) (x)6 + (x)7

[INST] help me answer this question Vertices (±7, 0), e = 4/3, think step by step [/INST] , The answer is The vertices of the hyperbola are (±7, 0) and the eccentricity is e = 4/3.

[INST] help me answer this question Draw a quadrilateral in the Cartesian plane whose vertices are (– 4, 5), (0, 7), (5, – 5) and (– 4, –2). Also, find its area., think step by step [/INST] , The answer is Let ABCD be the given quadrilateral with vertices A (-4,5) , B (0,7), C (5,-5) and D (-4,-2). To find the area, draw the diagonal AC. Area (ABCD) = area (∆ABC) + area (∆ADC). Area of ∆ ABC = ½ [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)] = 29 square units. Area of ∆ ACD = ½ [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)] = 63/2 square units. Area (ABCD) = 121/2 square units.

[INST] help me answer this question 5. sin x + sin 3x + sin 5x + sin 7x, think step by step [/INST] N/A , The answer is N/A

[INST] help me answer this question The 6-digit number is to be formed out of which unit place is fixed as 0. The remaining 5 places can be filled by 1, 3, 5, 7 and 9. Here, n = 5 and the numbers of choice available are 5. So, the total ways in which the rest of the places can be filled are 5P5, think step by step [/INST] We have 5 places and 5 numbers to fill them. This is a permutation problem because the order matters. The number of ways to arrange n objects is nPn. So, the total ways to fill 5 places with 5 numbers is 5P5 = 120. , The answer is The total ways in which the rest of the places can be filled are 120

[INST] help me answer this question Multiply by “-1” and change the sign., think step by step [/INST] The solutions of the given inequality are defined by all the real numbers greater than 4. , The answer is 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 Find the sum to n terms of the series 3 × 8 + 6 × 11 + 9 × 14 + …, 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 = 3n + 3.

[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 32n + 2 – 8n – 9 is divisible by 8., 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 Is n (n + 1) (n + 5) a multiple of 3 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, n (n + 1) (n + 5) = 12, which is divisible by 3. 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, n (n + 1) (n + 5) is a multiple of 3 for all natural 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 Find the coefficient of x⁵ in the expression `(1+2x)⁶(1-x)⁷`., think step by step [/INST] The given text does not provide an explanation for this answer. , The answer is 171

[INST] help me answer this question Given x2/100 + y2/400 = 1, what is the length of the latus rectum?, think step by step [/INST] , The answer is Length of latus rectum = 2b2/a = (2×102)/20 = (2×100)/20 = 10

[INST] help me answer this question 4, 7, 8, 9, 10, 12, 13, 17, think step by step [/INST] , The answer is The mean deviation about the mean for the given data is 3.

[INST] help me answer this question How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?, think step by step [/INST] To find the total number of 5-digit telephone numbers, we need to find the number of possible digits for each place (excluding the ones already taken) and multiply them together. Here, the number of possible digits for the third place is 8, for the fourth place is 7 and for the fifth place is 6. , The answer is 336

[INST] help me answer this question Rewrite the following statement with 'if-then' in five different ways conveying the same meaning. , think step by step [/INST] N/A , The answer is N/A

[INST] help me answer this question Expand (101)4, think step by step [/INST] Solution: Given, (101)4 101 can be expressed as the sum or difference of two numbers, and then the binomial theorem can be applied. , The answer is 1030401

[INST] help me answer this question Expand (96)3, think step by step [/INST] Solution: The given question can be written as 96 = 100 – 4 (96)3 = (100 – 4)3 = 3C0 (100)3 – 3C1 (100)2 (4) – 3C2 (100) (4)2– 3C3 (4)3 = (100)3 – 3 (100)2 (4) + 3 (100) (4)2 – (4)3 = 1000000 – 120000 + 4800 – 64 = 884736 , The answer is 884736

[INST] help me answer this question A point is on the x-axis. What are its y-coordinate and z-coordinates?, think step by step [/INST] , The answer is If a point is on the x-axis, then the coordinates of y and z are 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 Find the mean for the data 4, 7, 10., think step by step [/INST] The mean is calculated as the sum of all observations divided by the number of observations, which comes out to be 7. , The answer is The mean of the data is 7.

[INST] help me answer this question - 3 ≤ 4 – 7x/2 ≤ 18, think step by step [/INST] , The answer is x ∈ [-4, 2]

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

[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 4th term in the expansion of (x – 2y)12., think step by step [/INST] The general term Tr+1 in the binomial expansion is given by Tr+1 = n C r an-r br. Here, a= x, n =12, r= 3 and b = -2y. By substituting the values, we get T4 = 12C3 x9 (-2y)3 = -1760 x9 y3. , The answer is T4 = 12C3 x9 (-2y)3 = -1760 x9 y3

[INST] help me answer this question Question : y + 8 >= 2x, think step by step [/INST] The question is incomplete and missing the solution region information. , The answer is N/A

[INST] help me answer this question Find the distance between parallel lines 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0, think step by step [/INST] N/A , The answer is The distance between parallel lines is 65/17.

[INST] help me answer this question FI, think step by step [/INST] I is not defined in the question. , The answer is FI = Not provided

[INST] help me answer this question Find the discriminant of the quadratic equation √2x^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 = √2, b = 1, and c = √2. Substituting these values into the formula, we get D = (1)^2 - 4*√2*√2 = -7.

[INST] help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step [/INST] Explaination: N/A , The answer is Answer: Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers, i.e., n.

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

[INST] help me answer this question `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 In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that the 20th term is –112., think step by step [/INST] Given the first term and the relation between the sums, the 20th term was found after calculating the common difference. , The answer is The 20th term of the A.P. is -112.

[INST] help me answer this question Let the 3-digit number be ABC, where C is at the unit's place, B at the tens place and A at the hundreds place. As the number has to be even, the digits possible at C are 2 or 4 or 6. That is, the number of possible digits at C is 3. Now, as repetition is allowed, the digits possible at B is 6. Similarly, at A, also, the number of digits possible is 6. Therefore, the total number of possible 3-digit numbers = 6 × 6 × 3 = 108 , think step by step [/INST] To find the total number of 3-digit numbers, we need to find the number of possible digits for each place and multiply them together. Here, the number of possible digits for C is 3, for B is 6 and for A is also 6. , The answer is 108

[INST] help me answer this question What is the inequality for the relationship between the third piece and the second piece?, think step by step [/INST] , The answer is 2x ≥ (x+3) + 5

[INST] help me answer this question Hence, the shaded area in the graph is the required solution area for the given inequalities. , think step by step [/INST] , The answer is N/A

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

[INST] help me answer this question Given 5x + 4y ≤ 20, 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, 5) and (4, 0).

[INST] help me answer this question (vi) {x: x is a perfect square}, think step by step [/INST] , The answer is {x: x ∈ N and x is not a perfect square}

[INST] help me answer this question What is the formula used to calculate the variance?, think step by step [/INST] , The answer is σ2 = (1/N) [N * Σ(xi^2) - (Σxi)^2]

[INST] help me answer this question Find the standard deviation for X., think step by step [/INST] , The answer is The standard deviation for X is √35, which is approximately 5.91.

[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 Simplify the expression: (2x + 3)(x - 1) + 2., think step by step [/INST] , The answer is The simplified expression is 2x^2 + x + 1.

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

[INST] help me answer this question Find the slope of the line that makes an angle of 30° with the positive direction of the y-axis measured anticlockwise., think step by step [/INST] , The answer is The slope of the given line is tan 120° = tan (180° – 60°) = – tan 60° = –√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 `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 Checking for y ≥ 2, The value of y will be unaffected by any value of x in the given equality. Also, the origin doesn’t satisfy the given inequality., think step by step [/INST] The region to be included in the solution would be towards the left of the equality y≥ 2 , The answer is ⇒ 0 ≥ 2, which is not true

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

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

[INST] help me answer this question `f(x) = sqrt(x - 1)`, think step by step [/INST] , The answer is The domain of the function is `x ≥ 1`.

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

[INST] help me answer this question For all the values of y, the value of x would be the same in the given inequality, which would be the region above the x-axis on the graph. , think step by step [/INST] , 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 How many questions are there in this exercise?, think step by step [/INST] There are 9 questions in the exercise, and each question (and its corresponding subparts) represents a unique problem. , The answer is There are 9 questions in this exercise.

[INST] help me answer this question 3(x – 1) ≤ 2 (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 A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line., think step by step [/INST] This is the equation of the line that is perpendicular to the line segment joining the points (1, 0) and (2, 3) and divides it in the ratio 1: n. , The answer is (1 + n) x + 3(1 + n) y – n – 11 = 0