Datasets:

mwz

/

11th-maths

like 0

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

### Instruction: help me answer this question What is √(25-4)?, think step by step ### Assistant: , The answer is √21

### Instruction: 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 ### Assistant: , The answer is The equation of the circle is 144x2 + 144y2 – 144x – 72y + 44 = 0

### Instruction: help me answer this question Prove that cos 2x = 2cos^2 x - 1, think step by step ### Assistant: This is a standard identity in trigonometry. , The answer is LHS = cos 2x = 2cos^2 x - 1 = RHS

### Instruction: help me answer this question cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x), think step by step ### Assistant: Consider LHS = cot 4x (sin 5x + sin 3x). Using the formula sin(A + B) = sinA cosB + cosA sinB, it can be written as 2 cos 4x cos x. Thus, LHS = RHS. , The answer is True

### Instruction: help me answer this question For all values of x, what is y in the given inequalities of 5x + 4y ≤ 20?, think step by step ### Assistant: , The answer is For all the values of x, y would be 2.

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

### Instruction: help me answer this question Find the derivative of, think step by step ### Assistant: , The answer is The derivative of the given expression is not provided.

### Instruction: help me answer this question What are mRNA and rRNA? Define them., think step by step ### Assistant: Both mRNA and rRNA are essential for protein synthesis. mRNA carries the genetic information from DNA to the ribosomes, while rRNA plays a structural role in ribosomes and facilitates the protein synthesis process. , The answer is mRNA, or messenger RNA, is a type of RNA (ribonucleic acid) that plays a crucial role in biology. The primary function of mRNA is to carry genetic information from the DNA in the cell nucleus to the ribosomes in the cytoplasm, where protein synthesis occurs.<br><br>rRNA, or ribosomal RNA, is another type of RNA that is a major component of ribosomes. Ribosomes are cellular structures where protein synthesis takes place. rRNA is vital for the assembly and function of ribosomes, helping to decode the mRNA and facilitating the formation of peptide bonds during protein synthesis.

### Instruction: help me answer this question sin x = 1/4, x in quadrant II, think step by step ### Assistant: N/A , The answer is cos x/2 = ± √[(1 + cos x)/2]

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question Which class interval contains Nth/2 or 50th item for the given continuous frequency distribution?, think step by step ### Assistant: None , The answer is 35.5 – 40.5

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

### Instruction: help me answer this question Calculate the standard deviation and mean diameter of the circles., think step by step ### Assistant: To calculate the mean and standard deviation, the reader must first convert the given classes to a continuous form, then calculate the midpoints, Yi, Yi2, fiyi, and fiyi2, and finally use those values to find the mean and standard deviation using the formulae provided in the question. , The answer is The mean diameter of the circles is 43.5 mm and the standard deviation is 5.553 mm.

### Instruction: help me answer this question a^2 + 6a – 16 = 0, think step by step ### Assistant: , The answer is Subtracting 16 from both sides of the equation a^2 + 6a = 16, we get a^2 + 6a – 16 = 0.

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question Given x2/16 + y2/9 = 1, what is the length of the latus rectum?, think step by step ### Assistant: , The answer is Length of latus rectum = 2b2/a = (2×32)/4 = (2×9)/4 = 9/2

### Instruction: 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 ### Assistant: , The answer is The equation of the hyperbola has the general form x^2/a^2 – y^2/b^2 = 1.

### Instruction: help me answer this question Foci (±5, 0), the transverse axis is of length 8., think step by step ### Assistant: 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.

### Instruction: help me answer this question What is D ∩ E?, think step by step ### Assistant: , The answer is The value of D ∩ E is the empty set (φ).

### Instruction: help me answer this question Solve the given inequality: (x - 2)(x - 3) > 0, think step by step ### Assistant: , The answer is The solution of the given inequality is x < 2 or x > 3.

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

### Instruction: help me answer this question (i) Write the relation R = {(x, y): y = x - 2; x ∈ P} in set-builder form. (ii) Write the relation R in roster form. What are their domain and range? , think step by step ### Assistant: , The answer is (i) Set-builder form: R = {(x, y): y = x - 2; x ∈ P} (ii) Roster form: R = {(3, 1), (4, 2), (5, 3)}

### Instruction: help me answer this question Is it a statement?, think step by step ### Assistant: In this question, it is not evident to whom 'she' is referred. , The answer is No, it is not a statement.

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

### Instruction: help me answer this question P(A∪B) is the probability of passing at least one of the examinations. If P(A) = 0.8, P(B) = 0.7, and P(A∪B) = 0.95, find P(A∩B)., think step by step ### Assistant: We know that, P(A∪B) = P(A) + P(B) – P(A∩B), therefore, P(A∩B) = P(A) + P(B) – P(A∪B) = 0.8 + 0.7 - 0.95 = 0.55 , The answer is 0.55

### Instruction: help me answer this question Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = -2 and x = 3 , think step by step ### Assistant: Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = -2 and x = 3 , The answer is The required points are (0, -2), (3, 0)

### Instruction: help me answer this question Find the distance of the point (-1, 1) from the line 12(x + 6) = 5(y – 2)., think step by step ### Assistant: N/A , The answer is The distance is 5 units.

### Instruction: help me answer this question Find the discriminant of the quadratic equation x^2 + 3x + 5 = 0., think step by step ### Assistant: , 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 = 3, and c = 5. Substituting these values into the formula, we get D = (3)^2 - 4*1*5 = -11.

### Instruction: help me answer this question (ii) {3, 4} ∈A is correct, think step by step ### Assistant: , The answer is {3, 4} is an element of A.

### Instruction: help me answer this question Find the equation for the ellipse that satisfies the given conditions: Vertices (± 5, 0) and foci (± 4, 0)., think step by step ### Assistant: , The answer is x^2/25 + y^2/16 = 1

### Instruction: help me answer this question Find the equation of the median through the vertex R., think step by step ### Assistant: This is the equation of the median through the vertex R of the triangle. , The answer is 3x – 4y + 8 = 0

### Instruction: help me answer this question Evaluate the given limit:, think step by step ### Assistant: Given limit, substitute x = 1, we get (a + b + c) / (a + b + c) = 1 , The answer is 1

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

### Instruction: help me answer this question 2x + y ≥ 4, x + y ≤ 3, 2x – 3y ≤ 6 , think step by step ### Assistant: N/A , The answer is N/A

### Instruction: help me answer this question Find the derivative of the following functions. (iii) 5 sec x + 4 cos x, think step by step ### Assistant: , The answer is The derivative of 5 sec x + 4 cos x is not provided.

### Instruction: help me answer this question Find the sum to n terms of the series 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + …, think step by step ### Assistant: , The answer is The sum of n terms of the series can be expressed as the sum of the first n terms of the sequence an, where an = n(n + 1)(n + 2).

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: help me answer this question Find the 4th term in the expansion of (x – 2y)12., think step by step ### Assistant: 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

### Instruction: 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 ### Assistant: 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.

### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step ### Assistant: , The answer is The domain of the function is `x ≥ 1`.

### Instruction: 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 ### Assistant: , The answer is The centre of the given circle is (4, -5), and its radius is √53.

### Instruction: help me answer this question 32n + 2 – 8n – 9 is divisible by 8., think step by step ### Assistant: N/A , The answer is The given statement is true for all natural numbers.

### Instruction: help me answer this question Find the value of tan (1050°)., think step by step ### Assistant: We know that the values of tan x repeat after an interval of π or 180°. Therefore, tan (1050°) = tan (900° + 150°) = tan 150° = tan (180° - 30°) = -tan 30°. Hence, tan (1050°) = -1/√3 = √3. , The answer is The value of tan (1050°) is √3.

### Instruction: help me answer this question What is E ∩ F’?, think step by step ### Assistant: , The answer is The value of E ∩ F’ is the empty set (φ).

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

### Instruction: 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 ### Assistant: , The answer is The equation of the required circle is x2 + y2 – 4x – 4y = 5

### Instruction: help me answer this question Solve 24x < 100, when - x is a natural number. - x is an integer. , think step by step ### Assistant: N/A , The answer is {'naturalNumber': '[1, 2, 3, 4]', 'integer': '...,-1, 0, 1, 2, 3, 4'}

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

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

### Instruction: help me answer this question Find the sum to n terms in the geometric progression √7, √21, 3√7, …., think step by step ### Assistant: , The answer is An expression to find the sum to n terms in the geometric progression √7, √21, 3√7, ….

### Instruction: help me answer this question (ii) [Φ′ ∩ A], think step by step ### Assistant: , The answer is A

### Instruction: help me answer this question Write each of the following statements in the form 'if-then'. (iv) To get an A+ in the class, it is necessary that you do the exercises in the book., think step by step ### Assistant: , The answer is If you want to score an A+ in the class, then you do all the exercises in the book.

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

### Instruction: help me answer this question Find A – B, think step by step ### Assistant: 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}

### Instruction: help me answer this question 3x + 4y ≥ 12, , think step by step ### Assistant: , The answer is N/A

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

### Instruction: help me answer this question cos 4x = cos 2x, think step by step ### Assistant: , The answer is The general solution of this equation is given by 4x = 2x + 2πn or 4x = -2πn, where n ∈ Z.

### Instruction: help me answer this question Expand (1 – 2x)5, think step by step ### Assistant: , The answer is 1 – 10x + 40x2 – 80x3 + 80x4 – 32x5

### Instruction: help me answer this question Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6)., think step by step ### Assistant: This is the equation of the line that passes through the given points and is perpendicular to the line through the points (2, 5) and (–3, 6). , The answer is 5x – y + 20 = 0

### Instruction: help me answer this question Solve the inequality 4x + 3 < 5x + 7. , think step by step ### Assistant: N/A , The answer is The solutions of the given inequality are defined by all the real numbers greater than -4.

### Instruction: help me answer this question Does the origin lie in the solution area for the given inequalities of 5x + 4y ≤ 20?, think step by step ### Assistant: The origin satisfies the inequality 5x + 4y ≤ 20. , The answer is Yes, the origin would lie in the solution area.

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

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

### Instruction: help me answer this question For the following frequency distributions, find the mean and standard deviation., think step by step ### Assistant: , The answer is The mean is 90 and the standard deviation is 42.16.

### Instruction: help me answer this question What is the fourth inequality given in the problem?, think step by step ### Assistant: , The answer is x ≥ 0

### Instruction: help me answer this question How many 4-digit numbers are there with no digit repeated?, think step by step ### Assistant: , The answer is 9 × 9 × 8 × 7 = 4536

### Instruction: help me answer this question What is the probability of getting 2 heads?, think step by step ### Assistant: , The answer is The probability of getting 2 heads is 3/8.

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

### Instruction: help me answer this question Find the discriminant of the quadratic equation -x^2 + x - 2 = 0., think step by step ### Assistant: , 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) = -7.

### Instruction: help me answer this question Question: P (k + 1) is true whenever P (k) is true., think step by step ### Assistant: 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.

### Instruction: help me answer this question What is the probability of getting at most 2 heads?, think step by step ### Assistant: , The answer is The probability of getting at most 2 heads is 7/8.

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

### Instruction: help me answer this question Find the mean for the data 4, 7, 10., think step by step ### Assistant: 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.

### Instruction: help me answer this question One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected randomly and rolled, and its colour and the number on its uppermost face are noted. Describe the sample space., think step by step ### Assistant: , The answer is Let 1, 2, 3, 4, 5 and 6 be the possible numbers when the die is thrown. Also, let the die of red colour be R, the die of white colour be W, and the die of blue colour be B. The total number of possible outcomes equals (6 × 3) = 18. Hence, the sample space of the event is S={(R,1),(R,2),(R,3),(R,4),(R,5),(R,6),(W,1),(W,2),(W,3),(W,4),(W,5),(W,6) (B,1),(B,2),(B,3),(B,4),(B,5),(B,6)}.

### Instruction: help me answer this question By using the concept of equation of a line, prove that the three points (3, 0), (– 2, – 2) and (8, 2) are collinear., think step by step ### Assistant: Collinearity of points can be proved by showing that the line passing through any two points also passes through the third point. Here, we have used the formula to find the equation of a line passing through two points and then checked if the third point also satisfies the equation of the line. , The answer is According to the question, if we have to prove that the given three points (3, 0), (– 2, – 2) and (8, 2) are collinear, then we have to also prove that the line passing through the points (3, 0) and (– 2, – 2) also passes through the point (8, 2). By using the formula, the equation of the line passing through the points (x1, y1) and (x2, y2) is given by -5y = -2(x – 3). Further simplifying the equation, we get 2x – 5y = 6. If 2x – 5y = 6 passes through (8, 2), then 2x – 5y = 2(8) – 5(2) = 16 – 10 = 6 = RHS. Hence, the line passing through the points (3, 0) and (– 2, – 2) also passes through the point (8, 2).

### Instruction: help me answer this question What is the fifth inequality given in the problem?, think step by step ### Assistant: , The answer is y ≥ 0

### Instruction: help me answer this question A solution is to be kept between 68° F and 77° F. What is the range in temperature in degree Celsius (C) if the Celsius / Fahrenheit (F) conversion formula is given by F = (9/5) C + 32?, think step by step ### Assistant: , The answer is The range of temperature in degree Celsius is between 20° C to 25° C.

### Instruction: help me answer this question (xi) {Φ} ⊂ A is incorrect, think step by step ### Assistant: , The answer is Φ∈ {Φ}; where, Φ ∈ A.

### Instruction: help me answer this question Ends of major axis (0, ±√5), ends of minor axis (±1, 0), think step by step ### Assistant: The major axis is along the y-axis, and the ends of the major and minor axes are given. So, the equation will be of the form x2/b2 + y2/a2 = 1. Here, a = √5, b = 1. So, the equation of the ellipse is x2/1 + y2/5 = 1. , The answer is The equation of the ellipse is x2/1 + y2/5 = 1

### Instruction: help me answer this question Checking for the origin (0, 0) 0 ≤ 3, this is true. , think step by step ### Assistant: 0 ≤ 3, this is true. , The answer is The solution area would be on the left of the line graph

### Instruction: help me answer this question What is the relationship between set G and set H?, think step by step ### Assistant: , The answer is Set G is not equal to set H.

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

### Instruction: help me answer this question What is the length of the third piece?, think step by step ### Assistant: , The answer is The length of the third piece = 2x cm.

### Instruction: help me answer this question (Miscellaneous Exercise on Chapter 12 Solutions), think step by step ### Assistant: 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.

### Instruction: help me answer this question Write the reciprocal of the given fraction: 5/7, think step by step ### Assistant: , The answer is The reciprocal of the given fraction is 7/5.

### Instruction: help me answer this question Find the equation of the circle with radius 5 whose centre lies on the x-axis and passes through the point (2, 3)., think step by step ### Assistant: The solution involves using the given conditions, the general form of the equation of a circle (x – h)2 + (y – k)2 = r2, and the fact that the center lies on the x-axis (k = 0). , The answer is x2 + y2 – 7x + 5y – 14 = 0

### Instruction: 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 ### Assistant: 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.

### Instruction: help me answer this question How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?, think step by step ### Assistant: N/A , The answer is 90

### Instruction: help me answer this question What are the required points for the first inequality?, think step by step ### Assistant: , The answer is The required points are (0, 5) and (10, 0).

### Instruction: help me answer this question Solve the quadratic equation x2 + 3x + 9 = 0, think step by step ### Assistant: Comparing the given equation with ax2 + bx + c = 0, we have a = 1, b = 3, and c = 9. The discriminant (D) is calculated as D = b2 - 4ac = 32 - 4 × 1 × 9 = -27. Since D < 0, the roots of the equation are given by x = [-b ± √(b2 - 4ac)]/2a = [-3 ± √(-27)]/2 = -1.5 ± 1.5√3. , The answer is x = -1.5 ± 1.5√3

### Instruction: help me answer this question Does the equation of the circle becomes when h = 6?, think step by step ### Assistant: , The answer is (x – 6)2 + y2 = 25

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

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

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

### Instruction: help me answer this question Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5., think step by step ### Assistant: , The answer is The sum of such numbers is 99500.

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

### Instruction: help me answer this question In a group of 65 people, 40 like cricket, and 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?, think step by step ### Assistant: This is derived from n(C ∪ T) = n(C) + n(T) - n(C ∩ T) and n (T) = n (T - C) + n (T ∩ C). , The answer is 25 people like only tennis, and 35 people like tennis.

### Instruction: help me answer this question Evaluate: log₀ 10000, think step by step ### Assistant: , The answer is log₀ 10000 = 4