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[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 State whether each of the following statements is true or false. If the statement is false, rewrite the given statement correctly., think step by step [/INST] N/A , The answer is N/A

[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 A coin is tossed four times. Describe the sample space for this experiment., think step by step [/INST] When a coin is tossed four times, there are 2 possibilities (Head or Tail) for each toss. So, the total number of possible outcomes is 2*2*2*2 = 16. These outcomes form the sample space S. , The answer is The sample space S = {HHHH, THHH, HTHH, HHTH, HHHT, TTTT, HTTT, THTT, TTHT, TTTH, TTHH, HHTT, THTH, HTHT, THHT, HTTH}

[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 (0, ±13), the conjugate axis is of length 24., think step by step [/INST] Given foci (0, ±13) are on the y-axis, and the conjugate axis is of length 24, so the equation of the hyperbola is of the form y2/a2 – x2/b2 = 1. Here, a = 12 and c = 13. We know that a2 + b2 = c2, so b2 = 25. , The answer is The equation of the hyperbola is y2/25 – x2/144 = 1.

[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 (x – 4)/3 ≥ 1, think step by step [/INST] , The answer is x ∈ [7, ∞)

[INST] help me answer this question (2.1) What point is considered to check the inequality?, think step by step [/INST] N/A , The answer is The point (0, 0) is considered

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

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

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

[INST] help me answer this question Find the mean and variance for the data 6, 7, 10, 12, 13, 4, 8, 12., 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 9. The variance is calculated as the sum of (xi - x_mean)^2 divided by the number of observations, which comes out to be 9.25. , The answer is The mean of the data is 9 and the variance is 9.25.

[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 Write the relation R = {(x, x^3): x is a prime number less than 10} in roster form. , think step by step [/INST] , The answer is R = {(2, 8), (3, 27), (5, 125), (7, 343)}

[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 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 Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6). What is the equation of the ellipse?, think step by step [/INST] The major axis is on the y-axis and the ellipse passes through the points (3, 2) and (1, 6). Using these two points, we can calculate the values of a and b and hence, the equation of the ellipse. , The answer is The equation of the ellipse is 4x2 + y2 = 40.

[INST] help me answer this question What is the contrapositive of the given statement: "If two lines intersect in the same plane, then the two lines are not parallel.", 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 two lines are parallel, then they do not intersect in the same plane.

[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 (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 Ends of major axis (± 3, 0), ends of minor axis (0, ±2), think step by step [/INST] The major axis is along the x-axis, and the ends of the major and minor axes are given. So, the equation will be of the form x2/a2 + y2/b2 = 1. Here, a = 3, b = 2. So, the equation of the ellipse is x2/9 + y2/4 = 1. , The answer is The equation of the ellipse is x2/9 + y2/4 = 1

[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 θ = 15/75 radian = 1/5 radian

[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 Checking for the origin (0, 0) 0 ≥ 4, this is not true , think step by step [/INST] 0 ≥ 4, this is not true , The answer is The origin doesn’t lie in the solution area of the line graph. The solution area would be given by the right side of the line graph.

[INST] help me answer this question In which octant does the point (3, -1, 6) lie?, think step by step [/INST] Here, x is positive, y is negative, and z is positive. , The answer is It lies in the II octant.

[INST] help me answer this question Find the equation of the circle with Centre (1, 1) and radius √2., think step by step [/INST] , The answer is x2 + y2 - 2x - 2y + 1 = 0

[INST] help me answer this question Show that the points (–2, 3, 5), (1, 2, 3), and (7, 0, –1) are collinear., think step by step [/INST] , The answer is The points P, Q, and R are collinear.

[INST] help me answer this question Find the equation of the circle passing through (0,0) and making intercepts a and b on the coordinate axes., think step by step [/INST] , The answer is The equation of the required circle is x2 + y2 -ax - by = 0

[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 Write the first five terms of the sequence whose nth term is an = (-1)^n-1 5n+1, think step by step [/INST] , The answer is 25, -125, 625, -3125, 15625

[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 slope of the given line., think step by step [/INST] Given, m × (-2) = -1, so m = 1/2. , The answer is The slope of the given line is 1/2.

[INST] help me answer this question 38, 70, 48, 40, 42, 55, 63, 46, 54, 44, think step by step [/INST] , The answer is The mean deviation for the given data is 8.4.

[INST] help me answer this question Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3., think step by step [/INST] The given line has a slope of 1/7. Slope of a line perpendicular to a given line with slope m is -1/m. So the line having slope -7 and x intercept 3 has an equation of y = m(x – d) where d is the x intercept. Substituting the given slope and x intercept, we get the required equation. , The answer is 7x + y = 21

[INST] help me answer this question **Question**: If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets: (i) A = {a, b, c}, think step by step [/INST] , The answer is A' = {d, e, f, g, h}

[INST] help me answer this question Simplify (5x - 3y + 2)(5x - 3y - 2), think step by step [/INST] , The answer is (5x - 3y + 2)(5x - 3y - 2) = 25x² - 15xy - 3y² - 4

[INST] help me answer this question 2 ≤ 3x – 4 ≤ 5, think step by step [/INST] , The answer is x ∈ [2, 3]

[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 (ii) [6, 12], think step by step [/INST] , The answer is {x: x ∈ R, 6 ≤ x ≤ 12}

[INST] help me answer this question 2 boys and 2 girls are in Room X, and 1 boy and 3 girls are in Room Y. Specify the sample space for the experiment in which a room is selected and then a person., think step by step [/INST] , The answer is Let 2 boys and 2 girls in Room X be denoted by b1, b2 and, g1, g2, and 1 boy and 3 girls in Room Y by b3 and g3, g4, and g5. The overall sample space S is {(X,b1),(X,b2),(X,g1),(X,g2),(Y,b3),(Y,g3),(Y,g4),(Y,g5)}.

[INST] help me answer this question Using the section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and C (0, 1/3, 2) are collinear., think step by step [/INST] , The answer is Hence, A, B, C are collinear.

[INST] help me answer this question Question: Find the value of n so that, think step by step [/INST] Explanation: To find the value of n that makes the square roots equal, we square both sides, perform cross-multiplication, and rearrange terms to solve for n. , The answer is Answer: n = (b^2 - a^2) / (2ab)

[INST] help me answer this question Evaluate `((√3 + √2)⁶ - (√3 - √2)⁶) / (2 √6)`., think step by step [/INST] The given text provides a step-by-step solution for this calculation. , The answer is 396 √6

[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 The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find the values of m and c., think step by step [/INST] This is the solution for finding the values of m and c when the perpendicular from the origin to the line y = mx + c meets it at a given point. , The answer is The values of m and c are 1/2 and 5/2, respectively.

[INST] help me answer this question 3 (2 – x) ≥ 2 (1 – x), think step by step [/INST] The solutions of the given inequality are defined by all the real numbers greater than or equal to 4. Hence, the required solution set is [-∞, 4]. , The answer is x ≤ 4

[INST] help me answer this question Given 2x + y ≥ 4, Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = 4 and x =2 , think step by step [/INST] Putting values of x = 0 and y = 0 in the equation one by one, we get values of y = 4 and x =2 , The answer is The required points are (0, 4) and (2, 0)

[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 ≥ 1

[INST] help me answer this question Find the 4th term in the expansion of (x – 2y)¹², think step by step [/INST] In the binomial expansion (a + b)^n, the (r+1)th term is given by Tr+1 = nCran–rbr. Here, a = x, b = -2y, and n = 12. To find the 4th term, we need r = 3. Substituting these values, we get T4 = 12C3 (x)(-2y)³ (x⁻¹)(-2y) = -1344 x y² (2y)⁸. , The answer is The 4th term is -1344 x y² (2y)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 (ix) {x: 2x + 5 = 9}, think step by step [/INST] , The answer is {x: x ∈ N and x ≠ 2}

[INST] help me answer this question What is the relationship between set F and set G?, think step by step [/INST] , The answer is Set F is not equal to set G.

[INST] help me answer this question What can you say about lines AB and CA?, think step by step [/INST] , The answer is m1.m3 = -1.1 = -1, hence, the lines AB and CA are perpendicular to each other.

[INST] help me answer this question Expand the expression (3x2 – 2ax + 3a2)3, think step by step [/INST] This is the expansion of the given binomial expression using the binomial theorem. Each term is obtained by multiplying (3x2 – 2ax + 3a2) by itself for a certain number of times based on the binomial theorem formula. , The answer is 27x6 – 54ax5 + 117a2x4 – 116a3x3 + 117a4x2 – 54a5x + 27a6

[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) = (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 a17 and a24 if an = 4n – 3, think step by step [/INST] , The answer is a17 = 67, a24 = 99

[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 16x^2 + y^2 = 16., think step by step [/INST] , The answer is {'foci': ['(0, √15)', '(0, -√15)'], 'vertices': ['(0, 4)', '(0, -4)'], 'major axis length': 8, 'minor axis length': 2, 'eccentricity': '√15/4', 'latus rectum length': '½'}

[INST] help me answer this question What is the further simplified equation?, think step by step [/INST] , The answer is 4x ≤ 88

[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 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 What is the difference between a compound and a mixture and a physical change and a chemical change?, think step by step [/INST] , The answer is A compound is a substance composed of two or more elements in a fixed proportion by mass and is chemically combined, while a mixture is a combination of two or more substances in which each substance retains its chemical properties. A physical change is a change that does not alter the chemical identity of a substance, while a chemical change is a change that alters the chemical identity of a substance.

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

[INST] help me answer this question Find the equation of the circle with Centre (1/2, 1/4) and radius (1/12)., think step by step [/INST] , The answer is 36x2 + 36y2 – 36x – 18y + 11= 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 Question: The sum of two numbers is 6 times their geometric mean; show that numbers are in the ratio (a/b)^3 : (b/a)^3., think step by step [/INST] Explanation: Upon solving the expression and using the definition of a geometric mean, one can prove that the numbers are indeed in the given ratio. , The answer is Answer: The numbers are in the ratio (a/b)^3 : (b/a)^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 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 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 3(7 + i7) + i(7 + i7), think step by step [/INST] 3(7 + i7) + i(7 + i7) = 21 + i21 + i7 + i2 7 = 21 + i28 - 7 , The answer is 14 + i28

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

[INST] help me answer this question For 2x + y ≥ 8, x + 2y ≥ 10, find the region of the solution., think step by step [/INST] , The answer is The region of the solution for the given inequalities is the area to the right of the line 2x + y = 8 and the area to the left of the line x + 2y = 10.

[INST] help me answer this question How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?, think step by step [/INST] The word DAUGHTER has 3 vowels, A, E, U and 5 consonants D, G, H, T and R. The three vowels can be chosen in 3C2, as only two vowels are to be chosen. Similarly, the five consonants can be chosen in 5C3 ways. The number of choosing 2 vowels and 5 consonants would be 3C2 ×5C3. Each of these 5 letters can be arranged in 5 ways to form different words. , The answer is The total number of words formed would be = 30 × 120 = 3600

[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, n(A) = 3 and n(B) = 2, and (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements., think step by step [/INST] , The answer is Set A = {x, y, z}, Set B = {1, 2}

[INST] help me answer this question What is a simple event?, think step by step [/INST] , The answer is An event that has only one sample point of a sample space is called a simple (or elementary) event.

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

[INST] help me answer this question If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that a^(q-r) * b^(r-p) * c^(p-q) = 1, think step by step [/INST] Using the general term of a G.P., we express a, b, and c in terms of A (first term) and R (common ratio). Then, we substitute these expressions in a^(q-r) * b^(r-p) * c^(p-q) to get A^0 * R^0, which is equal to 1. , The answer is a^(q-r) * b^(r-p) * c^(p-q) = 1

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

[INST] help me answer this question Prove that cos (x + y) = cos x cos y - sin x sin y, think step by step [/INST] N/A , The answer is cos (x + y) = cos x cos y - sin x sin y

[INST] help me answer this question Solve the given equation: 2x - 3 = 0, think step by step [/INST] , The answer is The solution of the given equation is x = 3/2.

[INST] help me answer this question Given x2/100 + y2/400 = 1, what is the length of the major axis?, think step by step [/INST] , The answer is Length of major axis = 2a = 2 (20) = 40

[INST] help me answer this question Find C – A, think step by step [/INST] C – A represents the set of elements that are present in C but not in A. , The answer is {2, 4, 10, 14}

[INST] help me answer this question If A × B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B., think step by step [/INST] , The answer is Given, A × B = {(a, x), (a, y), (b, x), (b, y)} We know that the Cartesian product of two non-empty sets, P and Q, is given by P × Q = {(p, q): p ∈ P, q ∈ Q} Hence, A is the set of all first elements, and B is the set of all second elements. Therefore, A = {a, b} and B = {x, y}

[INST] help me answer this question Simplify the expression 2 sin 4x cos (– 2x) + 2 sin 4x, think step by step [/INST] Taking common terms, we get 2 sin 4x (cos 2x + 1). Using the formula sin(A + B) = sinA cosB + cosA sinB, we can write cos 2x + 1 as 2cos2 x. Thus, we get 2 sin 4x (2 cos2 x) or 4cos2 x sin 4x. , The answer is 4cos2 x sin 4x

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

[INST] help me answer this question Find the sum of odd integers from 1 to 2001., think step by step [/INST] , The answer is The sum of odd numbers from 1 to 2001 is 1002001.

[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 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 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 to n terms of the series can be expressed as the sum of n terms of the series 9n2 + 15n.

[INST] help me answer this question Is 100 divisible by 5?, think step by step [/INST] A number is divisible by 5 if its units digit is either 0 or 5. The units digit of 100 is 0. Therefore, 100 is divisible by 5. , The answer is Yes, 100 is divisible by 5.

[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 Which firm, A or B, shows greater variability in individual wages?, think step by step [/INST] To find out which firm shows greater variability, we need to compare the standard deviation of the wages of each firm. The standard deviation measures the spread of the data. The firm with the larger standard deviation shows greater variability. , The answer is Firm B shows greater variability in individual wages.

[INST] help me answer this question What are the values of y and x when we put x = 0 and y = 0 in the first inequality?, think step by step [/INST] , The answer is The values of y and x are 5 and 10 respectively.