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[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 What is the difference between a physical change and a chemical change?, think step by step [/INST] , The answer is A physical change is a change in which matter does not undergo a chemical transformation, while a chemical change is a change in which matter undergoes a chemical transformation.

[INST] help me answer this question Write each of the following statements in the form 'if-then'. (ii) The Banana trees will bloom if it stays warm for a month., think step by step [/INST] , The answer is If it stays warm for a month, then the Banana trees will bloom.

[INST] help me answer this question We have a deck of cards that has 4 kings. The numbers of remaining cards are 52. Ways of selecting a king from the deck = 4C1, think step by step [/INST] We have 4 kings in a deck of 52 cards. We need to find the number of ways to select 1 king from 4. Since there are 4 kings, there is only 1 way to choose 1 king, which is 4C1 = 4. , The answer is Ways of selecting a king from the deck = 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 `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 a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3)., think step by step [/INST] This is the equation of the line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3). Here, 'a' is the length of the intercept. , The answer is x/a + y/a = 1

[INST] help me answer this question What are the possible values of D?, think step by step [/INST] , The answer is The possible values of D are 1, 2, and 3.

[INST] help me answer this question What is the eccentricity?, think step by step [/INST] , The answer is Eccentricity, e = c/a = √21/5

[INST] help me answer this question The shaded area in the graph shows the solution area for the given inequalities 4x + 3y ≤ 60, y ≥ 2x, x ≥ 3, x, y ≥ 0, think step by step [/INST] , The answer is The solution area for the given inequalities is the shaded area in the graph, which satisfies all the given inequalities simultaneously. This area is to the left of the line graph of y = 2x and 4x + 3y = 60, on the left of the line x = 3, and in the first quadrant. The required points are (0, 20) and (15, 0).

[INST] help me answer this question `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 A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of at least 3 girls?, think step by step [/INST] , The answer is The total number of ways of making the committee are 588.

[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 What is the mean (x̅) for the given data?, think step by step [/INST] , The answer is x̅ = 43.5

[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 `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 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 A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find the probability that the sum of numbers that turn up is (i) 3. (ii) 12., think step by step [/INST] , The answer is The sample space S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)} n(S) = 12 (i) Let 'P' be the event having the sum of numbers as 3. n(P) = 1 P(P) = n(P)/n(S) = 1/12 (ii) Let 'Q' be the event having the sum of the number as 12. n(Q) = 1 P(Q) = n(Q)/n(S) = 1/12

[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 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 `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 Evaluate<br>(i) 8!<br>(ii) 4! – 3!, think step by step [/INST] {'part1': '8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320', 'part2': '4! - 3! = (4 × 3!) - 3! = 3! (4-1) = 3 × 2 × 1 × 3 = 18'} , The answer is {'part1': '8! = 40320', 'part2': '4! - 3! = 18'}

[INST] help me answer this question Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0); focus (3, 0), think step by step [/INST] The vertex of the parabola is (0, 0), and the focus lies on the positive x-axis. The equation of the parabola is of the form y^2 = 4ax. Since the focus is (3, 0), a = 3. , The answer is y^2 = 12x.

[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 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 If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?, think step by step [/INST] Using the formula n(S ∪ T) = n(S) + n(T) - n(S ∩ T) , we find n(S ∪ T) = 42 , The answer is n(S ∪ T) = 42

[INST] help me answer this question What is the length of the minor arc of the chord in a circle of radius 20 cm, if the arc subtends an angle of θ = 60° = π/3 radian at the centre?, think step by step [/INST] , The answer is The length of the minor arc of the chord is 20π/3 cm.

[INST] help me answer this question The set of isosceles triangles, think step by step [/INST] , The answer is Universal set (s) would be the set of all triangles.

[INST] help me answer this question Find the discriminant of the quadratic equation x^2 - x + 2 = 0., think step by step [/INST] , The answer is The discriminant (D) of the quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac. For the given equation, a = 1, b = -1, and c = 2. Substituting these values into the formula, we get D = (-1)^2 - 4*1*2 = -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 Given x2/49 + y2/36 = 1, what is the length of the major axis?, think step by step [/INST] , The answer is Length of major axis = 2a = 2 (7) = 14

[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 **Question**: Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) {x: x is an even natural number}, think step by step [/INST] , The answer is {x: x is an odd natural number}

[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 Evaluate the given limit:, think step by step [/INST] Given limit, substitute r = 1, we get π(1)^2 = π , The answer is π

[INST] help me answer this question Is f a relation from A to B?, think step by step [/INST] A relation from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. Here, f is a subset of A × B. , The answer is Yes, it is.

[INST] help me answer this question The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon., think step by step [/INST] It's understood from the question that the angles of the polygon will form an A.P. with a common difference d = 5° and first term a = 120°. And, we know that the sum of all angles of a polygon with n sides is 180°(n – 2). Thus, we can say a + (a + d) + (a + 2d) + ... + [a + (n – 2)d] = 180(n – 2) Simplifying the above equation, we get: n * [2a + (n – 2)d] / 2 = 180(n – 2) Since a = 120° and d = 5°, substituting these values in the equation, we get: n * [240 + 5(n – 2)] = 360(n – 2) Further simplification leads us to: n^2 + 47n - 456 = 0 Upon solving the above quadratic equation, we get: n = 9 or n = 16 Therefore, a polygon having 9 or 16 sides will satisfy the condition in the question. , The answer is A polygon having 9 and 16 sides will satisfy the condition in the question.

[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 variance for X., think step by step [/INST] , The answer is The variance for X is 35.

[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 If x is the A.M. between a and b, then find the value of n., think step by step [/INST] Explanation provided in the markdown text outlines the steps to derive the answer. It includes equating the arithmetic mean between two numbers, a and b, and then solving for n. , The answer is The value of n is 1.

[INST] help me answer this question If sin x + sin 3x + sin 5x = 0, find the value of x., think step by step [/INST] We can write sin x + sin 3x + sin 5x = 0 as (sin x + sin 5x) + sin 3x = 0. Using the formula 2 sin A cos B = sin (A + B) + sin (A - B), we can simplify it to sin 3x (2 cos 2x + 1) = 0. Therefore, sin 3x = 0 or 2 cos 2x + 1 = 0. If sin 3x = 0, then x = nπ/3, where n ∈ Z. If 2 cos 2x + 1 = 0, then cos 2x = – 1/2 = – cos π/3 = cos (π – π/3), so 2x = 2π/3 or 4π/3, and x = π/3 or 2π/3. However, x = 2π/3 is not a solution because sin 3x is negative when x = 2π/3. , The answer is x = nπ/3, where n ∈ Z

[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 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 If A (1, 2, 3) and B (3, 2, –1), what is the set of points P (x, y, z) such that PA = PB?, think step by step [/INST] By equating the distances PA and PB, we get an equation in x, y, and z, which simplifies to x – 2z = 0. , The answer is x – 2z = 0

[INST] help me answer this question Is it a statement?, think step by step [/INST] It is not evident to whom 'he' is referred. , The answer is No, it is not a statement.

[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) = 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) Is {1, 2, 3, 4, 5, 6, 7, 8} a universal set for the sets A, B, and C?, think step by step [/INST] , The answer is No.

[INST] help me answer this question Evaluate the given limit:, think step by step [/INST] The limit has a form of 0/0. Apply L'Hopital's rule or factorization, but not possible to evaluate further. , The answer is Not possible to evaluate

[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 Question: 5x + 1 > -24, 5x – 1 < 24, think step by step [/INST] The given inequalities are 5x + 1 > -24 and 5x – 1 < 24. After solving these inequalities, we get the solution as (-5, 5). , The answer is The solutions of the given inequalities are all real numbers x such that -5 < x < 5.

[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 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 (iv) [U’ ∩ A], think step by step [/INST] , The answer is Φ

[INST] help me answer this question Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n – 1., think step by step [/INST] , The answer is The coefficient of xn in (1+x)2n is 2nCn and the coefficient of xn in (1+x)2n-1 is 2n-1Cn. We need to prove that 2nCn = 2 * 2n-1Cn.

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

[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 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 `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 a⁵b⁷ in (a – 2b)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 = a, b = -2b & n =12. Substituting the values, we get Tr+1 = 12Cr a12-r (-2b)r. To find a⁵, we have to equate a12-r =a5 => r = 7. Putting r = 7 in (i), we get T8 = 12C7 a5 (-2b)7 = -101376 a5 b7. Hence, the coefficient of a⁵b⁷ = -101376. , The answer is The coefficient of a⁵b⁷ = -101376

[INST] help me answer this question For the data 6, 10, 14, 18, 24, 28, 30, find the deviations from the mean and (xi – x̅)2., think step by step [/INST] , The answer is Refer to the table in the answer.

[INST] help me answer this question Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R., think step by step [/INST] For question 3, since the difference between any two integers is always an integer, therefore, the domain and range of R is Z. , The answer is Domain of R = Z and Range of R = Z.

[INST] help me answer this question Calculate the mean deviation about the mean for the data: 5, 10, 15, 20, 25., think step by step [/INST] First, we find the mean of the data, then we calculate the absolute deviations from the mean, and finally, we find the mean of these deviations. , The answer is The mean deviation about the mean for the given data is 7.5.

[INST] help me answer this question If the coin shows Tail, what is the sample space?, think step by step [/INST] , The answer is The sample space, denoted as ST, is {(T,R1), (T,R2), (T,B1), (T,B2), (T,B3)}.

[INST] help me answer this question Is 100 divisible by 11?, think step by step [/INST] A number is divisible by 11 if the difference between the sum of its digits in the odd places and the sum of its digits in the even places is divisible by 11. In 100, the sum of the digits in the odd places is 0 and the sum of the digits in the even places is 10. The difference is 10-0=10, which is not divisible by 11. Therefore, 100 is not divisible by 11. , The answer is No, 100 is not divisible by 11.

[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 Foci (± 3√5, 0), the latus rectum is of length 8., think step by step [/INST] Given foci (± 3√5, 0) are on the x-axis, and the latus rectum is of length 8, so the equation of the hyperbola is of the form x2/a2 – y2/b2 = 1. Here, a = 5 and c = 3√5. We know that a2 + b2 = c2, so b2 = 20. , The answer is The equation of the hyperbola is x2/25 – y2/20 = 1.

[INST] help me answer this question 1.(a) Passing through the point (– 4, 3) with slope 1/2, think step by step [/INST] , The answer is Given: Point (-4, 3) and slope, m = 1/2 We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0), if and only if, its coordinates satisfy the equation y – y0 = m (x – x0) So, y – 3 = 1/2 (x – (-4)) y – 3 = 1/2 (x + 4) 2(y – 3) = x + 4 2y – 6 = x + 4 x + 4 – (2y – 6) = 0 x + 4 – 2y + 6 = 0 x – 2y + 10 = 0 ∴ The equation of the line is x – 2y + 10 = 0.

[INST] help me answer this question What is the mean for X?, think step by step [/INST] , The answer is The mean (x̄) for X is 51.

[INST] help me answer this question If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation., think step by step [/INST] Let the roots of the quadratic equation be a and b. Given the arithmetic mean (A.M.) and geometric mean (G.M.) of the roots, we can use the properties of A.M. and G.M. to determine the values of (a + b) and ab. Subsequently, we can form a quadratic equation using the formula x² - x(a + b) + (ab) = 0. , The answer is The quadratic equation is x² - 16x + 25 = 0.

[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 Are set B and set D mutually exclusive?, think step by step [/INST] , The answer is No, they are not mutually exclusive as there are common elements in B & D.

[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 (iii) [A ∩ A'], think step by step [/INST] , The answer is Φ

[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 variance for the given data?, think step by step [/INST] , The answer is Variance = 30.84

[INST] help me answer this question What is the distance AB, BC, CD, and DA?, think step by step [/INST] , The answer is {'AB': 6, 'BC': '√43', 'CD': 6, 'DA': '√30'}

[INST] help me answer this question Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. , think step by step [/INST] The process involves comparing the given equation with the standard form of the ellipse, i.e., x2/a2 + y2/b2 = 1, to find the values of a, b, and c. Then, using these values, the coordinates of the foci, vertices, and other parameters are calculated. , The answer is {'1.1': {'coordinates_foci': '(2√5, 0) and (-2√5, 0)', 'coordinates_vertices': '(6, 0) and (-6, 0)', 'length_major_axis': 12, 'length_minor_axis': 8, 'eccentricity': '√5/3', 'length_latus_rectum': '16/3'}, '1.2': {'coordinates_foci': '(0, √21) and (0, -√21)', 'coordinates_vertices': '(0, 5) and (0, -5)', 'length_major_axis': 10, 'length_minor_axis': 4, 'eccentricity': '√21/5'}}

[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 `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 centre and radius of the circle with the given condition: 2x2 + 2y2 – x = 0, think step by step [/INST] , The answer is The centre of the given circle is (1/4, 0), and its radius is 1/4.

[INST] help me answer this question What is the length of the minor axis?, think step by step [/INST] , The answer is Length of minor axis = 2b = 2 (2) = 4

[INST] help me answer this question Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2)., think step by step [/INST] , The answer is The angle between the x-axis and the line joining the points (3, –1) and (4, –2) is 135°.

[INST] help me answer this question What is the value of 5C5?, think step by step [/INST] , The answer is 5C5 = 1

[INST] help me answer this question 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 Divide the given numbers and express the result in standard form: 5 × 10⁵ ÷ 2 × 10³, think step by step [/INST] , The answer is The result of the given division in standard form is 2.5 × 10².

[INST] help me answer this question Evaluate<br>(i) 6C2<br>(ii) 9C5, think step by step [/INST] nCr = n! / {(n-r)! × r!} , The answer is {'part1': '15', 'part2': '126'}

[INST] help me answer this question 8. Find sin x/2, cos x/2 and tan x/2 in each of the following., think step by step [/INST] N/A , The answer is N/A

[INST] help me answer this question Write the product of the given numbers in standard form: 2 × 10⁴ × 3 × 10³, think step by step [/INST] , The answer is The product of the given numbers in standard form is 6 × 10⁷.

[INST] help me answer this question Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 if no digit is repeated. How many of these will be even?, think step by step [/INST] The number will be even when 2 and 4 are in one’s place. The possibility of (2, 4) at one’s place = 2/5 = 0.4. The total number of even numbers = 120 × 0.4 = 48. , The answer is The total number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repetition is 120. The number of even numbers among these is 48.

[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 Find the derivative of the following functions. (i) sin x cos x, think step by step [/INST] , The answer is The derivative of sin x cos x is not provided.