Answer: Arrange the numbers in ascending order. Median is the middle number of all the observation. 34, 38, 43, 44, 47, 48, 53, 55, 63, 70 Here the Number of observations are Even then Median =...
Calculate the mean deviation from the mean for the following data : (i) 4, 7, 8, 9, 10, 12, 13, 17 (ii) 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Answers: (i)Β 4, 7, 8, 9, 10, 12, 13, 17 |di| = |xiΒ β x| βxβ be the mean of the given observation. x = [4 + 7 + 8 + 9 + 10 + 12 + 13 + 17]/8 = 80/8 = 10 Number of observations, βnβ = 8 xi |di|...
Calculate the mean deviation from the mean for the following data : (i) 38, 70, 48, 40, 42, 55, 63, 46, 54, 44 (ii) 36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Answers: (i) 38, 70, 48, 40, 42, 55, 63, 46, 54, 44 |di| = |xiΒ β x| βxβ be the mean of the given observation. x = [38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44]/10 = 500/10 = 50 Number of...
Calculate the mean deviation from the mean for the following data : 57, 64, 43, 67, 49, 59, 44, 47, 61, 59
Answer: 57, 64, 43, 67, 49, 59, 44, 47, 61, 59 |di| = |xiΒ β x| βxβ be the mean of the given observation. x = [57 + 64 + 43 + 67 + 49 + 59 + 44 + 47 + 61 + 59]/10 = 550/10 = 55 Number of...
Calculate the mean deviation of the following income groups of five and seven members from their medians:
I Income in βΉ II Income in βΉ 4000 3800 4200 4000 4400 4200 4600 4400 4800 4600 4800 5800 Answer: Data is arranged in ascending order, 4000, 4200, 4400, 4600, 4800 Median = 4400 Total...
The lengths (in cm) of 10 rods in a shop are given below: 40.0, 52.3, 55.2, 72.9, 52.8, 79.0, 32.5, 15.2, 27.9, 30.2 (i) Find the mean deviation from the median. (ii) Find the mean deviation from the mean also.
Answers: (i)Β Arrange the data in ascending order, 15.2, 27.9, 30.2, 32.5, 40.0, 52.3, 52.8, 55.2, 72.9, 79.0 |di| = |xiΒ β M| The number of observations are Even then MedianΒ = (40+52.3)/2 =...
In 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 find the number of observations lying between β M . D and + M . D , where M.D. is the mean deviation from the mean.
Answer: |di| = |xiΒ β x| βxβ be the mean of the given observation. x = [34 + 66 + 30 + 38 + 44 + 50 + 40 + 60 + 42 + 51]/10 = 455/10 = 45.5 Number of observations, βnβ = 10 xi |di| = |xiΒ β...
In 22, 24, 30, 27, 29, 31, 25, 28, 41, 42 find the number of observations lying between β M . D and + M . D , where M.D. is the mean deviation from the mean.
Answer: 22, 24, 30, 27, 29, 31, 25, 28, 41, 42 |di| = |xiΒ β x| βxβ be the mean of the given observation. x = [22 + 24 + 30 + 27 + 29 + 31 + 25 + 28 + 41 + 42]/10 = 299/10 = 29.9 Number of...
In 38, 70, 48, 34, 63, 42, 55, 44, 53, 47 find the number of observations lying between β M . D and + M . D , where M.D. is the mean deviation from the mean.
Answer: 38, 70, 48, 34, 63, 42, 55, 44, 53, 47 We know that, |di| = |xiΒ β x| βxβ be the mean of the given observation. x = [38 + 70 + 48 + 34 + 63 + 42 + 55 + 44 + 53 + 47]/10 = 494/10 = 49.4...
Find the slopes of a line (i) which bisects the first quadrant angle (ii) which makes an angle of 300 with the positive direction of y β axis measured anticlockwise.
(i)Β Which bisects the first quadrant angle? Given: Line bisects the first quadrant We know that, if the line bisects in the first quadrant, then the angle must be between line and the positive...
State whether the two lines in each of the following are parallel, perpendicular or neither: (i) Through (5, 6) and (2, 3); through (9, β2) and (6, β5) (ii) Through (9, 5) and (β 1, 1); through (3, β5) and (8, β3)
(i)Β Through \[\left( 5,\text{ }6 \right)\text{ }and\text{ }\left( 2,\text{ }3 \right)\] Through \[\left( 9,-\text{ }2 \right)\text{ }and\text{ }\left( 6,-\text{ }5 \right)\] By using the formula,...
Find the slopes of a line passing through the following points : (i) (β3, 2) and (1, 4) (ii) (at21, 2at1) and (at22, 2at2)
(i) \[\left( -3,\text{ }2 \right)\text{ }and\text{ }\left( 1,\text{ }4 \right)\] By using the formula, β΄ The slope of the line is \[{\scriptscriptstyle 1\!/\!{ }_2}\] ...
Find the slopes of the lines which make the following angles with the positive direction of x β axis: (i) β Ο/4 (ii) 2Ο/3
(i) \[-\text{ }\pi /4\] Let the slope of the line be \[m\] Where, \[m\text{ }=\text{ }tan\text{ }\theta \] So, the slope of Line is \[m\text{ }=\text{ }tan~\left( -\text{ }\pi /4 \right)~\]...
Find the A.M. between: (i) 7 and 13 (ii) 12 and β 8
Answers: (i) A be the Arithmetic mean 7, A, 13 are in AP A-7 = 13-A 2A = 13 + 7 A = 10 (ii) A be the Arithmetic mean 12, A, β 8 are in AP A β 12 = β 8 β A 2A = 12 + 8 A =...
Find the A.M. between: (x β y) and (x + y)
Answer: A be the Arithmetic mean x β y, A, x + y are in AP A β (x β y) = (x + y) β A 2A = x + y + x β y A = x
Insert 4 A.M.s between 4 and 19.
Answer: A1, A2, A3, A4 - 4 AM Between 4 and 19 4, A1, A2, A3, A4, 19 are in AP. By using the formula, d = (b-a) / (n+1) d = (19 β 4) / (4 + 1) d = 15/5 d = 3 A1Β = a + d = 4 + 3 = 7 A2Β = A1 + d = 7 +...
Insert 7 A.M.s between 2 and 17.
Answer: A1, A2, A3, A4, A5, A6, A7 - 7 AMs between 2 and 17 2, A1, A2, A3, A4, A5, A6, A7, 17 are in AP By using the formula, anΒ = a + (n β 1)d anΒ = 17, a = 2, n = 9 17 = 2 + (9 β 1)d 17 = 2 + 9d β...
Insert six A.M.s between 15 and β 13.
Answer: A1, A2, A3, A4, A5, A6 - 7 AM between 15 and β 13 15, A1, A2, A3, A4, A5, A6, β 13 are in AP anΒ = a + (n β 1)d anΒ = -13, a = 15, n = 8 -13 = 15 + (8 β 1)d -13 = 15 + 7d 7d = -13 β 15 7d =...
There are n A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3: 1. Find the value of n
Answer: Let the series be 3, A1, A2, A3, β¦β¦.., An, 17 Given, an/a1Β = 3/1 Total terms in AP are n + 2 17 is the (n + 2)th term By using the formula, AnΒ = a + (n β 1)d AnΒ = 17, a = 3 So, 17 = 3 + (n +...
Insert A.M.s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.s.
Answer: Let the series be 7, A1, A2, A3, β¦β¦.., An, 71 Total terms in AP are n + 2 71 is the (n + 2)th term By using the formula, AnΒ = a + (n β 1)d AnΒ = 71, n = 6 A6Β = a + (6 β 1)d a + 5d = 27...
If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.
Answer: Let a and b be the first and last terms The series be a, A1, A2, A3, β¦β¦.., An, b Mean = (a+b)/2 Mean of A1Β and AnΒ = (A1Β + An)/2 A1Β = a+d AnΒ = a β d AM = (a+d+b-d)/2 AM = (a+b)/2 AM between...
If x, y, z are in A.P. and A1is the A.M. of x and y, and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
Answer: Given, A1Β = AM of x and y A2Β = AM of y and z A1Β = (x+y)/2 A2Β = (y+x)/2 AM of A1Β and A2Β = (A1Β + A2)/2 => [(x+y)/2 + (y+z)/2]/2 => [x+y+y+z]/2 => [x+2y+z]/2 x, y, z are in AP, y =...
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P
Answer: A1, A2, A3, A4, A5 - 5 numbers between 8 and 26 8, A1, A2, A3, A4, A5, 26 are in AP By using the formula, AnΒ = a + (n β 1)d AnΒ = 26, a = 8, n = 7 26 = 8 + (7 β 1)d 26 = 8 + 6d 6d = 26 β 8 6d...
If are in AP., prove that a/(b+c), b/(c+a), c/(a+b) are in AP.
Answer: If a2, b2, c2Β are in AP then, b2Β β a2Β = c2Β β b2 If a/(b+c), b/(c+a), c/(a+b) are in AP then, b/(c+a) β a/(b+c) = c/(a+b) β b/(c+a) Let us take LCM on both the sides, b2Β β a2Β = c2Β β b2...
If a, b, c are in A.P., then show that: are also in A.P. (ii) b + c β a, c + a β b, a + b β c are in A.P.
Answers: (i)Β If b2(c + a) β a2(b + c) = c2(a + b) β b2(c + a) b2c + b2a β a2b β a2c = c2a + c2b β b2a β b2c Given, b β a = c β b a, b, c are in AP, c(b2Β β a2Β ) + ab(b β a) = a(c2Β β b2Β ) + bc(c β b)...
If a, b, c are in A.P., then show that:
Answer: If (ca β b2) β (bc β a2) = (ab β c2) β (ca β b2) bc β a2, ca β b2, ab β c2Β are in A.P. Consider LHS and RHS, (ca β b2) β (bc β a2) = (ab β c2) β (ca β b2) (a β b2Β β bc + a2) = (ab β c2Β β ca...
If (b+c)/a, (c+a)/b, (a+b)/c are in AP., prove that: (i) 1/a, 1/b, 1/c are in AP (ii) bc, ca, ab are in AP
Answer: (i)Β If 1/a, 1/b, 1/c are in AP 1/b β 1/a = 1/c β 1/b Consider LHS, 1/b β 1/a = (a-b)/ab = c(a-b)/abc Consider RHS, 1/c β 1/b = (b-c)/bc = a(b-c)/bc [by multiplying with βaβ on both the...
If a, b, c are in A.P., prove that:
Answer: (i)Β Expanding, a2Β + c2Β β 2ac = 4(ab β ac β b2Β + bc) a2Β + 4c2b2Β + 2ac β 4ab β 4bc = 0 (a + c β 2b)2Β = 0 a + c β 2b = 0 a, b, c are in AP b β a = c β b a + c β 2b = 0 a + c = 2b (a β c)2Β = 4...
If a, b, c are in A.P., prove that:
Answer: Expanding, a3Β + c3Β + 6abc = 8b3 a3Β + c3Β β (2b)3Β + 6abc = 0 a3Β + (-2b)3Β + c3Β + 3a(-2b)c = 0 if a + b + c = 0, a3Β + b3Β + c3Β = 3abc (a β 2b + c)3Β = 0 a β 2b + c = 0 a + c = 2b b = (a+c)/2 a, b,...
If a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP., prove that a, b, c are in AP.
Answer: Given, a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP Also, a(1/b + 1/c) + 1, b(1/c + 1/a) + 1, c(1/a + 1/b) + 1 are in AP Let us take LCM for each expression, (ac+ab+bc)/bc ,...
Show that are in consecutive terms of an A.P., if x, y and z are in A.P.
Answer: Given, x2Β + xy + y2, z2Β + zx + x2Β and y2Β + yz + z2Β are in AP (z2Β + zx + x2) β (x2Β + xy + y2) = (y2Β + yz + z2) βΒ (z2Β + zx + x2) d = common difference, Y = x + d and x = x + 2d Consider the...
If 1/a, 1/b, 1/c are in A.P., prove that: (i) (b+c)/a, (c+a)/b, (a+b)/c are in A.P. (ii) a(b + c), b(c + a), c(a + b) are in A.P.
Answer: (i)Β If a, b, c are in AP b β a = c β b 1/a, 1/b, 1/c are in AP 1/b β 1/a = 1/c β 1/b If (b+c)/a, (c+a)/b, (a+b)/c are in AP (c+a)/b β (b+c)/a = (a+b)/c β (c+a)/b Let us take LCM, 1/a, 1/b,...
Find the sum of the following arithmetic progressions: (i) 50, 46, 42, β¦. to 10 terms (ii) 1, 3, 5, 7, β¦ to 12 terms
Answers: (i)Β n = 10 First term, a = a1Β = 50 Common difference, d = a2Β β a1Β = 46 β 50 = -4 By using the formula, S = n/2 (2a + (n β 1) d) Substitute the values of βaβ and βdβ, we get S = 10/2 (100 +...
Find the sum of the following arithmetic progressions: (i) 3, 9/2, 6, 15/2, β¦ to 25 terms (ii) 41, 36, 31, β¦ to 12 terms
Answers: (i)Β n = 25 First term, a = a1Β = 3 Common difference, d = a2Β β a1Β = 9/2 β 3 = (9 β 6)/2 = 3/2 By using the formula, S = n/2 (2a + (n β 1) d) Substitute the values of βaβ and βdβ, we get S =...
Find the sum of the following arithmetic progressions: (i) a+b, a-b, a-3b, β¦ to 22 terms to n terms
Answers: (i)Β n = 22 First term, a = a1Β = a+b Common difference, d = a2Β β a1Β = (a-b) β (a+b) = a-b-a-b = -2b By using the formula, S = n/2 (2a + (n β 1) d) Substitute the values of βaβ and βdβ, we...
Find the sum of the following arithmetic progression: (x β y)/(x + y), (3x β 2y)/(x + y), (5x β 3y)/(x + y), β¦ to n terms
Answer: n = n First term, a = a1Β = (x-y)/(x+y) Common difference, d = a2Β β a1Β = (3x β 2y)/(x + y) β (x-y)/(x+y) = (2x β y)/(x+y) By using the formula, S = n/2 (2a + (n β 1) d) Substitute the values...
Find the sum of the following series: (i) 2 + 5 + 8 + β¦ + 182 (ii) 101 + 99 + 97 + β¦ + 47
Answers: (i) First term, a = a1Β = 2 Common difference, d = a2Β β a1Β = 5 β 2 = 3 anΒ term of given AP is 182 anΒ = a + (n-1) d 182 = 2 + (n-1) 3 182 = 2 + 3n β 3 182 = 3n β 1 3n = 182 + 1 n = 183/3 n =...
Find the sum of the following series:
Answer: First term, a = a1Β = (a-b)2 Common difference, d = a2Β β a1Β = (a2Β + b2) β (a β b)2Β = 2ab anΒ term of given AP is [(a + b)2Β + 6ab] anΒ = a + (n-1) d [(a + b)2Β + 6ab] = (a-b)2Β + (n-1)2ab ...
Find the sum of first n natural numbers.
Answer: First term, a = a1Β = 1 Common difference, d = a2Β β a1Β = 2 β 1 = 1 l = n Sum of n terms = S S = n/2 [2a + (n-1) d] S = n/2 [2(1) + (n-1) 1] S = n/2 [2 + n β 1] S = n/2 [n β 1] β΄ The sum of...
Find the sum of all β natural numbers between 1 and 100, which are divisible by 2 or 5
Answer: The natural numbers which are divisible by 2 or 5 are: 2 + 4 + 5 + 6 + 8 + 10 + β¦ + 100 = (2 + 4 + 6 +β¦+ 100) + (5 + 15 + 25 +β¦+95) (2 + 4 + 6 +β¦+ 100) + (5 + 15 + 25 +β¦+95) are AP with...
Find the sum of first n odd natural numbers.
Answer: Given, AP of first n odd natural numbers whose first term a is 1 Common difference d = 3 The sequence is 1, 3, 5, 7β¦β¦n a = 1, d = 3-1 = 2, n = n By using the formula, S = n/2 [2a + (n-1)d] S...
Find the sum of all odd numbers between 100 and 200
Answer: The series is 101, 103, 105, β¦, 199 Number of terms be n a = 101, d = 103 β 101 = 2, anΒ = 199 anΒ = a + (n-1)d 199 = 101 + (n-1)2 199 = 101 + 2n β 2 2n = 199 β 101 + 2 2n = 100 n = 100/2 n =...
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667
Answer: The odd numbers between 1 and 1000 divisible by 3 are 3, 9, 15,β¦,999 Number of terms be βnβ, so the nth term is 999 a = 3, d = 9-3 = 6, anΒ = 999 anΒ = a + (n-1)d 999 = 3 + (n-1)6 999 = 3 + 6n...
Find the sum of all integers between 84 and 719, which are multiples of 5
Answer: The series is 85, 90, 95, β¦, 715 βnβ terms in the AP a = 85, d = 90-85 = 5, anΒ = 715 anΒ = a + (n-1)d 715 = 85 + (n-1)5 715 = 85 + 5n β 5 5n = 715 β 85 + 5 5n = 635 n = 635/5 n = 127 By using...
Find the sum of all integers between 50 and 500 which are divisible by 7
Answer: The series of integers divisible by 7 between 50 and 500 are 56, 63, 70, β¦, 497 Number of terms be βnβ a = 56, d = 63-56 = 7, anΒ = 497 anΒ = a + (n-1)d 497 = 56 + (n-1)7 497 = 56 + 7n β 7 7n...
Find the sum of all even integers between 101 and 999
Answer: All even integers will have a common difference of 2. AP is 102, 104, 106, β¦, 998 a = 102, d = 104 β 102 = 2, anΒ = 998 By using the formula, anΒ = a + (n-1)d 998 = 102 + (n-1)2 998 = 102 + 2n...
The Sum of the three terms of an A.P. is 21 and the product of the first, and the third terms exceed the second term by 6, find three terms.
Answer: Given. The sum of first three terms is 21 Assume, the first three terms as a β d, a, a + d [where a is the first term and d is the common difference] Sum of first three terms is a β d + a +...
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers
Answer: Given, Sum of first three terms is 27 Assume, the first three terms as a β d, a, a + d [where a is the first term and d is the common difference] Sum of first three terms is a β d + a + a +...
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Answer: Given, Sum of four terms is 50. Assume these four terms as a β 3d, a β d, a + d, a + 3d Sum of these terms - 4a = 50 a = 50/4 = 25/2 β¦ (i) It is also given that the greatest number is 4 time...
The sum of three numbers in A.P. is 12, and the sum of their cubes is 288. Find the numbers.
Answer: Given, Sum of three numbers = 12 Assume the numbers in AP are a β d, a, a + d 3a = 12 a = 4 It is also given that the sum of their cube is 288 (a β d)3Β + a3Β + (a + d)3Β = 288 a3Β β d3Β β 3ad(a...
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Answer: Given, Sum of first three terms is 24 Assume the first three terms are a β d, a, a + d [where a is the first term and d is the common difference] Sum of first three terms is a β d + a + a +...
The angles of a quadrilateral are in A.P. whose common difference is 10. Find the angles
Answer: Given, d = 10 Sum of all angles in a quadrilateral = 360 Assume the angles are a β 3d, a β d, a + d, a + 3d a β 2d + a β d + a + d + a + 2d = 360 4a = 360 a = 90β¦ (i) (a β d) β (a β...
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
Answer: Given, 24thΒ term is twice the 10thΒ term a24Β = 2a10 an = a + (n β 1) d When n = 10, a10Β = a + (10 β 1)d = a + 9d When n = 24, a24Β = a + (24 β 1)d = a + 23d When n = 34, a34Β = a + (34 β 1)d =...
The 10th and 18th term of an A.P. are 41 and 73 respectively, find 26th term.
Answer: Given, 10thΒ term of an A.P is 41, and 18thΒ terms of an A.P. is 73 a10 = 41 a18Β = 73 an = a + (n β 1) d When n = 10, a10Β = a + (10 β 1)d = a + 9d When n = 18, a18Β = a + (18 β 1)d = a + 17d...
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that the 25th term of the A.P. is Zero.
Answer: Given, 10 times the 10thΒ term of an A.P. is equal to 15 times the 15thΒ term 10a10Β = 15a15 an = a + (n β 1) d When n = 10, a10Β = a + (10 β 1)d = a + 9d When n = 15, a15Β = a + (15 β 1)d = a +...
If 9th term of an A.P. is Zero, prove that its 29th term is double the 19th term.
Answer: Given, 9thΒ term of an A.P is 0 a9Β = 0 an = a + (n β 1) d [w When n = 9, a9Β = a + (9 β 1)d = a + 8d a9Β = 0 a + 8d = 0 a = -8d When n = 19, a19Β = a + (19 β 1)d = a + 18d = -8d + 18d =...
The 6th and 17th terms of an A.P. are 19 and 41 respectively. Find the 40th term.
Answer: Given, 6thΒ term of an A.P is 19 and 17thΒ terms of an A.P. is 41 a6 = 19 a17Β = 41 an = a + (n β 1) d When n = 6, a6Β = a + (6 β 1) d = a + 5d When n = 17, a17Β = a + (17 β 1)d = a + 16d a6Β = 19...
The first term of an A.P. is 5, the common difference is 3, and the last term is 80; find the number of terms.
Answer: Given, First term, a = 5; last term, l = anΒ = 80 Common difference, d = 3 an = a + (n β 1) d anΒ = 5 + (n β 1)3 = 5 + 3n β 3 = 3n + 2 Put anΒ = 80 as 80 is last term of A.P. 3n + 2 = 80 3n =...
How many terms are there in the A.P. -1, -5/6, -2/3, -1/2, β¦, 10/3 ?
Answer: Given, AP of -1, -5/6, -2/3, -1/2, β¦ a1 = a = -1 a2Β = -5/6 Common difference, d = a2Β β a1 = -5/6 β (-1) = -5/6 + 1 = (-5+6)/6 = 1/6 an = a + (n β 1) d anΒ = -1 + (n β 1) 1/6 = -1 + 1/6n β 1/6...
How many terms are in A.P. 7, 10, 13,β¦43?
Answer: Given, AP of 7, 10, 13,β¦ a1 = a = 7 a2Β = 10 Common difference, d = a2Β β a1 = 10 β 7 = 3 an = a + (n β 1) d anΒ = 7 + (n β 1)3 = 7 + 3n β 3 = 3n + 4 Put, an = 43 3n + 4 = 43 3n = 43 β 4 3n =...
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, β¦ is (a) purely real (b) purely imaginary ?
Answer: Given, AP of 12 + 8i, 11 + 6i, 10 + 4i, β¦ a1 = a = 12 + 8i a2Β = 11 + 6i Common difference, d = a2Β β a1 = 11 + 6i β (12 + 8i) = 11 β 12 + 6i β 8i = -1 β 2i an = a + (n β 1) d anΒ = 12 + 8i +...
Which term of the sequence 24, 23 ΒΌ, 22 Β½, 21 ΒΎ is the first negative term?
Answer: Given, AP of 24, 23 ΒΌ, 22 Β½, 21 ΒΎ, β¦ = 24, 93/4, 45/2, 87/4, β¦ a1 = a = 24 a2Β = 93/4 Common difference, d = a2Β β a1Β = 93/4 β 24 = (93 β 96)/4 = β 3/4 an = a + (n β 1) d anΒ = a + (n β 1) d...
(i) Is 68 a term of the A.P. 7, 10, 13,β¦? (ii) Is 302 a term of the A.P. 3, 8, 13,β¦?
Answers: (i)Β Given, A.P is 7, 10, 13,β¦ a1 = a = 7 a2Β = 10 Common difference, d = a2Β β a1Β = 10 β 7 = 3 an = a + (n β 1)d anΒ = 7 + (n β 1)3 = 7 + 3n β 3 = 3n + 4 Put, anΒ = 68 3n + 4 = 68 3n = 68 β 4...
Which term of the A.P. 4, 9, 14,β¦ is 254 ?
Answer: Given, A.P is 4, 9, 14,β¦ a1 = a = 4 a2Β = 9 Common difference, d = a2Β β a1Β = 9 β 4 = 5 anΒ = a + (n β 1)d anΒ = 4 + (n β 1)5 = 4 + 5n β 5 = 5n β 1 Put, anΒ = 254 5n β 1 = 254 5n = 254 + 1 5n =...
(i) Which term of the A.P. 3, 8, 13,β¦ is 248 ? (ii) Which term of the A.P. 84, 80, 76,β¦ is 0 ?
Answer: (i)Β Given, A.P is 3, 8, 13,β¦ a1 = a = 3 a2Β = 8 Common difference, d = a2Β β a1Β = 8 β 3 = 5 anΒ = a + (n β 1)d anΒ = 3 + (n β 1)5 = 3 + 5n β 5 = 5n β 2 (Put, an = 248) β΄Β 5n β 2 = 248 = 248 + 2...
In an A.P., show that am+n + amβn = 2am.
Answer: Using the formula, anΒ = a + (n β 1)d LHS: am+nΒ + am-n am+nΒ + am-nΒ = a + (m + n β 1)d + a + (m β n β 1)d = a + md + nd β d + a + md β nd β d = 2a + 2md β 2d = 2(a + md β d) = 2[a + d(m β 1)]...
Find: nth term of the A.P 13, 8, 3, -2, β¦.
Answer: nth term of the A.P 13, 8, 3, -2, β¦. Arithmetic Progression (AP) whose common difference is = anΒ β an-1Β where n > 0 Consider, a = a1Β = 13, a2Β = 8 β¦ Common difference, d = a2Β β a1Β = 8 β 13...
Find: (i) 10th term of the A.P. 1, 4, 7, 10, β¦.. (ii) 18th term of the A.P. β2, 3β2, 5β2, β¦
Answer: (i)Β Arithmetic Progression (AP) whose common difference is = anΒ β an-1Β where n > 0 Consider, a = a1Β = 1, a2Β = 4 β¦ Common difference, d = a2Β β a1Β = 4 β 1 = 3 Finding an anΒ = a + (n-1) d =...
If the nth term of a sequence is given by an = n2 β n+1, write down its first five terms.
Answer: Using n = 1, 2, 3, 4, 5, the first five terms can be calculated. If n = 1, a1Β = (1)2Β β 1 + 1 a1Β = 1 β 1 + 1 a1Β = 1 If n = 2, a2Β = (2)2Β β 2 + 1 a2Β = 4 β 2 + 1 a2Β = 3 If n = 3, a3Β = (3)2Β β 3 +...
A sequence is defined by an = n3 β 6n2 + 11n β 6, n β N. Show that the first three terms of the sequence are zero and all other terms are positive.
Answer: Using n = 1, 2, 3, the first three terms can be calculated. If n = 1, a1Β = (1)3Β β 6(1)2Β + 11(1) β 6 a1 = 1 β 6 + 11 β 6 a1 = 12 β 12 a1 = 0 If n = 2, a2Β = (2)3Β β 6(2)2Β + 11(2) β 6 a2Β = 8 β...
Find the first four terms of the sequence defined by a1 = 3 and an = 3anβ1 + 2, for all n > 1.
Answer: Using n = 1, 2, 3, 4, the first four terms can be calculated. If n = 1, a1Β = 3 If n = 2, a2Β = 3a2β1Β + 2 a2Β = 3a1Β + 2 a2Β = 3(3) + 2 a2Β = 9 + 2 a2Β = 11 If n = 3, a3Β = 3a3β1Β + 2 a3Β = 3a2Β + 2...
Write the first five terms in each of the following sequences: (i) a1 = 1, an = anβ1 + 2, n > 1 (ii) a1 = 1 = a2, an = anβ1 + anβ2, n > 2
Answer: (i) Using n = 1, 2, 3, 4, 5, the first five terms can be calculated. If n = 1, a1Β = 1 If n = 2, a2Β = a2β1Β + 2 a2Β = a1Β + 2 a2Β = 1 + 2 a2Β = 3 If n = 3, a3Β = a3β1Β + 2 a3Β = a2Β + 2 a3Β = 3 + 2...
Write the first five terms in each of the following sequence: a1 = a2 =2, an = anβ1 β 1, n > 2
Answer: Using n = 1, 2, 3, 4, 5, the first five terms can be calculated. If n = 1, a1Β = 2 If n = 2, a2Β = 2 If n = 3, a3Β = a3β1Β β 1 = a2Β β 1 = 2 β 1 = 1 If n = 4, a4Β = a4β1Β β 1 = a3Β β 1 = 1 β...
The Fibonacci sequence is defined by a1 = 1 = a2, an = anβ1 + anβ2 for n > 2. Find (an+1)/an for n = 1, 2, 3, 4, 5.
3Answer: anΒ = anβ1Β + anβ2 If n = 1, (an+1)/anΒ = (a1+1)/a1 = a2/a1 = 1/1 = 1 a3Β = a3β1Β + a3β2 = a2Β + a1 = 1 + 1 = 2 If n = 2, (an+1)/anΒ = (a2+1)/a2 = a3/a2 = 2/1 = 2 a4Β = a4β1Β + a4β2 = a3Β + a2 = 2 +...
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
Answer: Given, Coefficients of (r + 1)th term in (1 + x)n+1 =Β n+1Cr Sum of the coefficients of the rth and (r + 1)th terms in (1 + x)n , (1 + x)nΒ =Β nCr-1Β +Β nCr...
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
Answer: Coefficient of the r term in the expansion of (1 + x)n - nCr-1 Coefficients of the (2r + 1) and (r + 2) terms in the given expansion are 43C2r+1-1Β andΒ 43Cr+2-1 Equaling two coefficients,...
If the coefficients of (2r + 4)th and (r β 2)th terms in the expansion of (1 + x)18 are equal, find r.
Answer: Coefficient of the r term in the expansion of (1 + x)nΒ isΒ nCr-1 Coefficients of the (2r + 4) and (r β 2) terms in the given expansion are 18C2r+4-1Β andΒ 18Cr-2-1 Equaling two coefficients,...
Find the term independent of x in the expansion of the following expressions:
Answers: (ii) If (rΒ + 1)th term in the given expression is independent of x. Tr+1Β =Β nCrΒ xn-rΒ ar For this term to be independent of r, (18-r)/3 β r/3 = 0 (18 β r β r)/3 = 0 18 β 2r = 0 2r = 18 r =...
Find the term independent of x in the expansion of the following expressions:
Answers: (i)Β If (rΒ + 1)th term in the given expression is independent of x. Tr+1Β =Β nCrΒ xn-rΒ ar For this term to be independent of x, (8-r)/3 β r/5 = 0 (40 β 5r β 3r)/15 = 0 40 β 5r β 3r = 0 40 β 8r...
Find the term independent of x in the expansion of the following expressions:
Answers: (i) If (rΒ + 1)th term in the given expression is independent of x. Tr+1Β =Β nCrΒ xn-rΒ ar For this term to be independent of x, (10-r)/2 β 2r = 0 10 β 5r = 0 5r = 10 r = 10/5 r = 2 The term is...
Find the term independent of x in the expansion of the following expressions:
Answers: (i) If (rΒ + 1)th term in the given expression is independent of x. Tr+1Β =Β nCrΒ xn-rΒ ar =Β 25CrΒ (2x2)25-rΒ (-3/x3)r = (-1)rΒ 25CrΒ Γ 225-rΒ Γ 3rΒ x50-2r-3r For this term to be independent of x, 50...
Find the term independent of x in the expansion of the following expressions:
Answers: (i)Β If (rΒ + 1)th term in the given expression is independent of x. Tr+1Β =Β nCrΒ xn-rΒ ar For this term to be independent of x, 18 β 3r = 0 3r = 18 r = 18/3 r = 6 The term is 7thΒ term. T7Β =...
Find the middle terms in the expansion of:
Answers: (i) Given, (p/x + x/p)9Β [n = 9] Middle terms - ((n+1)/2) = ((9+1)/2) = 10/2 = 5 and ((n+1)/2 + 1) = ((9+1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6 The terms are 5thΒ and 6th. T5Β = T4+1 T6Β = T5+1...
Find the middle terms in the expansion of:
Answers: (i) Given, (3 β x3/6)7Β [n = 7] Middle terms are ((n+1)/2) = ((7+1)/2) = 8/2 = 4 and ((n+1)/2 + 1) = ((7+1)/2 + 1) = (8/2 + 1) = (4 + 1) = 5 The terms are 4thΒ and 5th. T4Β = T3+1...
Find the middle terms in the expansion of:
Answers: (v) Given, (x β 1/x)2n+1Β [n = (2n + 1)] Middle terms - ((n+1)/2) = ((2n+1+1)/2) = (2n+2)/2 = (n + 1) and ((n+1)/2 + 1) = ((2n+1+1)/2 + 1) = ((2n+2)/2 + 1) = (n + 1 + 1) = (n + 2) The terms...
Find the middle terms in the expansion of:
Answers: (iii) Given, (1 + 3x + 3x2Β + x3)2nΒ = (1 + x)6nΒ [n is an even number] Middle term - (n/2 + 1) = (6n/2 + 1) = (3n + 1)th term. T2nΒ = T3n+1 =Β 6nC3nΒ x3n = (6n)!/(3n!)2Β x3n Middle term is...
Find the middle terms in the expansion of:
Answers: (i) Given, (x β 1/x)10Β [n = 10] Middle term - (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6. The term is 6thΒ term T6Β = T5+1 Middle term is -252. (ii)Β Given, (1 β 2x + x2)nΒ = (1 β x)2nΒ [n is an...
Find the 11th term from the beginning and the 11th term from the end in the expansion of (2x β 1/x2)25.
Answer: This expression contains 26 terms. 11thΒ term from the end is the (26 β 11 + 1)Β thΒ term from the beginning. T16Β = T15+1Β =Β 25C15Β (2x)25-15Β (-1/x2)15 T16 = 25C15Β (210) (x)10Β (-1/x30) T16Β =...
Find the 7th term in the expansion of (3×2 β 1/x3)10.
Answer: Consider, 7thΒ term as T7 T7Β = T6+1 T7 =Β 10C6Β (3x2)10-6Β (-1/x3)6 T7 =Β 10C6Β (3)4Β (x)8Β (1/x18) T7 = [10Γ9Γ8Γ7Γ81] / [4Γ3Γ2Γx10] T7 = 17010 / x10 β΄ 7thΒ term of the expression (3x2Β β 1/x3)10 =...
Find the 5th term in the expansion of (3x β 1/x2)10.
Answer: 5thΒ term from the end is the (11 β 5 + 1)th, is., 7thΒ term from the beginning. T7Β = T6+1 T7 =Β 10C6Β (3x)10-6Β (-1/x2)6 T7 =Β 10C6Β (3)4Β (x)4Β (1/x12) T7 = [10Γ9Γ8Γ7Γ81] / [4Γ3Γ2Γx8] T7 = 17010 /...
Find the 8th term in the expansion of (x3/2 y1/2 β x1/2 y3/2)10.
Answer: Consider, 8thΒ term as T8 T8Β = T7+1 T8 =Β 10C7Β (x3/2Β y1/2)10-7Β (-x1/2Β y3/2)7 T8 = -[10Γ9Γ8]/[3Γ2] x9/2Β y3/2Β (x7/2Β y21/2) T8 = -120 x8y12 β΄ 8thΒ term of the expression (x3/2Β y1/2Β β x1/2Β y3/2)10...
Find the 7th term in the expansion of (4x/5 + 5/2x) 8.
Answer: Consider, 7thΒ term as T7 T7Β = T6+1 β΄Β 7thΒ term of the expression (4x/5 + 5/2x)Β 8 = 4375/x4.
Find the 4th term from the beginning and 4th term from the end in the expansion of (x + 2/x) 9.
Answer: Consider, Tr+1Β be the 4th term from the end. Tr+1 is (10 β 4 + 1)th is the 7th term from the beginning.
Find the 4th term from the end in the expansion of (4x/5 β 5/2x) 9.
Answer: Consider, Tr+1Β be theΒ 4th term from the end of the given expression. Tr+1Β is (10 β 4 + 1)th term The term is the 7th term from the beginning. T7Β = T6+1 β΄ 4th term from the end =...
Find the 7th term from the end in the expansion of (2×2 β 3/2x) 8.
Answer: Consider, Tr+1Β be theΒ 4th term from the end of the expression. Tr+1Β is (9 β 7 + 1)th term, The term is 3rd term from the beginning. T3Β = T2+1 β΄ 7th term from the end = 4032...
Find the coefficient of: (i) x10 in the expansion of (2×2 β 1/x)20 (ii) x7 in the expansion of (x β 1/x2)40
Answers: (i) Β x10Β is in the (r + 1)th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar (ii) If x7Β is at the (r + 1) th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar 40 β 3rΒ =7 3r = 40 β 7 3r = 33 r = 33/3 r...
Find the coefficient of: (i) x-15 in the expansion of (3×2 β a/3×3)10 (ii) x9 in the expansion of (x2 β 1/3x)9
Answers: (i) IfΒ xβ15 is at the (r + 1)th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar (ii) IfΒ x9 occurs at the (r + 1)th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar ForΒ thisΒ termΒ toΒ containΒ x9, 18 β...
Find the coefficient of: (i) xm in the expansion of (x + 1/x)n (ii) x in the expansion of (1 β 2×3 + 3×5) (1 + 1/x)8
Answers: (i)Β IfΒ xm is at the (r + 1)th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar (ii) If x is at the (r + 1)th term in the expression. (1 β 2x3Β + 3x5) (1 + 1/x)8Β =Β (1 β 2x3Β + 3x5) (8C0Β +Β 8C1Β (1/x)...
Find the coefficient of: (i) a5b7 in the expansion of (a β 2b)12 (ii) x in the expansion of (1 β 3x + 7×2) (1 β x)16
Answers: (i) IfΒ a5b7 is at the (r + 1)th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar (ii) If x is at the (r + 1)th term in the expression. (1 β 3x + 7x2) (1 β x)16Β = (1 β 3x + 7x2) (16C0Β +Β 16C1Β (-x)...
Which term in the expansion of contains x and y to one and the same power.
Answer: Consider, Tr+1Β th term in the given expansion contains x and y to one and the same power. Tr+1Β =Β nCrΒ xn-rΒ ar
Does the expansion of (2×2 β 1/x) contain any term involving x9?
Answer: IfΒ x9 is at the (r + 1)th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar For this term to contain x9, 40 β 3r = 9 3r = 40 β 9 3r = 31 r = 31/3 It is not possible.Β [r is not an integer] Thus,...
Show that the expansion of (x2 + 1/x)12 does not contain any term involving x-1.
Answer: IfΒ x-1 is at the (r + 1)th term in the expression. Tr+1Β =Β nCrΒ xn-rΒ ar For this term to contain x-1, 24 β 3r = -1 3r = 24 + 1 3r = 25 r = 25/3 It is not possible.Β Β [r is not an integer]...
Find the middle term in the expansion of: (i) (2/3x β 3/2x)20 (ii) (a/x + bx)12
Answers: (i)Β Given, (2/3x β 3/2x)20Β [n = 20] Middle term - (n/2 + 1) = (20/2 + 1) = (10 + 1) = 11. The term is 11thΒ term T11Β = T10+1 T11Β =Β 20C10Β (2/3x)20-10Β (3/2x)10 T11Β =Β 20C10Β 210/310Β Γ...
Find the middle term in the expansion of: (i) (x2 β 2/x)10 (ii) (x/a β a/x)10
Answers: (i) Given, (x2Β β 2/x)10Β [n = 10] Middle term - (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6. The term is 6thΒ term T6Β = T5+1 Middle term is -8064x5. (iI) Given, (x/a β a/x)Β 10Β [n = 10] Middle term...
Find the middle terms in the expansion of: (i) (3x β x3/6)9 (ii) (2×2 β 1/x)7
Answers: (i) Given, (3x β x3/6)9Β [n = 9] Middle terms - ((n+1)/2) = ((9+1)/2) = 10/2 = 5 and ((n+1)/2 + 1) = ((9+1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6 The terms are 5thΒ and 6th. T5Β = T4+1 Middle...
Find the middle terms in the expansion of: (i) (3x β 2/x2)15 (ii) (x4 β 1/x3)11
Answers: (i) Given, (3x β 2/x2)15Β [n = 15] Middle terms - ((n+1)/2) = ((15+1)/2) = 16/2 = 8 and ((n+1)/2 + 1) = ((15+1)/2 + 1) = (16/2 + 1) = (8 + 1) = 9 The terms are 8thΒ and 9th. Middle term are...
Using binomial theorem, write down the expressions of the following:
Answers: (i) Solving, (2x + 3y)Β 5Β =Β 5C0Β (2x)5Β (3y)0Β +Β 5C1Β (2x)4Β (3y)1Β +Β 5C2Β (2x)3Β (3y)2Β +Β 5C3Β (2x)2Β (3y)3Β +Β 5C4Β (2x)1Β (3y)4Β +Β 5C5Β (2x)0Β (3y)5 = 32x5Β + 5 (16x4) (3y) + 10 (8x3) (9y)2Β + 10...
Using binomial theorem, write down the expressions of the following:
Answers: (i) Solving, (ii) Solving, (1 β 3x)Β 7Β =Β 7C0Β (3x)0Β βΒ 7C1Β (3x)1Β +Β 7C2Β (3x)2Β βΒ 7C3Β (3x)3Β +Β 7C4Β (3x)4Β βΒ 7C5Β (3x)5Β +7C6Β (3x)6Β βΒ 7C7Β (3x)7 = 1 β 7 (3x) + 21...
Evaluate the following:
Answers: (i) Solving, (ii) Solving,
Evaluate the following:
Answers: (i) Solving, = 2 [5C0Β (2βx)0Β +Β 5C2Β (2βx)2Β +Β 5C4Β (2βx)4] = 2 [1 + 10 (4x) + 5 (16x2)] = 2 [1 + 40x + 80x2] (ii) Solving, = 2 [6C0Β (β2)6Β +Β 6C2Β (β2)4Β +Β 6C4Β (β2)2Β +Β 6C6Β (β2)0] = 2 [8 + 15 (4) +...
Evaluate the following:
Answers: (i) Solving, = 2 [5C1Β (34) (β2)1Β +Β 5C3Β (32) (β2)3Β +Β 5C5Β (30) (β2)5] = 2 [5 (81) (β2) + 10 (9) (2β2) + 4β2] = 2β2 (405 + 180 + 4) = 1178β2 (ii) Solving, = 2 [7C0Β (27) (β3)0Β +Β 7C2Β (25)...
Evaluate the following:
Answers: (i) Solving, = 2 [5C1Β (β3)4Β +Β 5C3Β (β3)2Β +Β 5C5Β (β3)0] = 2 [5 (9) + 10 (3) + 1] = 2 [76] = 152 (ii) Solving, = (1 β 0.01)5Β + (1 + 0.01)5 = 2 [5C0Β (0.01)0Β +Β 5C2Β (0.01)2Β +Β 5C4Β (0.01)4] = 2 [1 +...
Find (x + 1) 6 + (x β 1) 6. Hence, or otherwise evaluate (β2 + 1)6 + (β2 β 1)6.
Answer: To solve, (x + 1)Β 6Β + (x β 1)Β 6 (x + 1)Β 6Β + (x β 1)Β 6Β = 2 [6C0Β x6Β +Β 6C2Β x4Β +Β 6C4Β x2Β +Β 6C6Β x0] = 2 [x6Β + 15x4Β + 15x2Β + 1] Evaluating, (β2 + 1)6Β + (β2 β 1)6 Let us consider, x = β2, (β2 +...
Using binomial theorem evaluate each of the following: (i) (96)3 (ii) (102)5
Answers: (i) Split the given expression into two different entities, Applying binomial theorem. (96)3Β = (100 β 4)3 => 3C0Β (100)3Β (4)0Β βΒ 3C1Β (100)2Β (4)1Β +Β 3C2Β (100)1Β (4)2Β βΒ 3C3Β (100)0Β (4)3 =>...
Using binomial theorem evaluate each of the following: (i) (101)4 (ii) (98)5
Answers: (i)Β Split the given expression into two different entities, Applying binomial theorem. (101)4Β = (100 + 1)4 => 4C0Β (100)4Β +Β 4C1Β (100)3Β +Β 4C2Β (100)2Β +Β 4C3Β (100)1Β +Β 4C4Β (100)0 =>...
Find (a + b) 4 β (a β b) 4. Hence, evaluate (β3 + β2)4 β (β3 β β2)4.
Answer: To solve, (a + b)Β 4Β β (a β b)Β 4 (a + b)Β 4Β β (a β b)Β 4Β = 2 [4C1Β a3b1Β +Β 4C3Β a1b3] = 2 [4a3b + 4ab3] = 8 (a3b + ab3) Evaluating, (β3 + β2)4Β β (β3 -β2)4 Consider, a = β3 and b = β2 (β3 + β2)4Β β...
Evaluate the following:
Answers: (i) Solving, = 2 [6C1Β (β3)5Β (β2)1Β +Β 6C3Β (β3)3Β (β2)3Β +Β 6C5Β (β3)1Β (β2)5] = 2 [6 (9β3) (β2) + 20 (3β3) (2β2) + 6 (β3) (4β2)] = 2 [β6 (54 + 120 + 24)] = 396Β β6 (ii) Solving, = 2...
Using binomial theorem, write down the expressions of the following:
Answers: (i) Solving, (ii) Solving,
Using binomial theorem, write down the expressions of the following:
Answers: (i) Solving, (ii) Consider, (1 + 2x) and 3x2 - Different entities Applying binomial theorem, (1 + 2x β 3x2)5Β =Β 5C0Β (1 + 2x)5Β (3x2)0Β βΒ 5C1Β (1 + 2x)4Β (3x2)1Β +Β 5C2Β (1 +...
Using binomial theorem, write down the expressions of the following:
Answers: (i) Solving, (ii) Solving,
How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?
Given: The word \[INVOLUTE\] Total number of letters \[=\text{ }8\] Total vowels are \[=\text{ }I,\text{ }O,\text{ }U,\text{ }E\] Total consonants \[=\text{ }N,\text{ }V,\text{ }L,\text{ }T\] So...
Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
Here, it is clear that \[3\]things are already selected and we need to choose \[\left( r\text{ }-\text{ }3 \right)\] things from the remaining \[\left( n\text{ }-\text{ }3 \right)\] things. Let us...
How many words, with or without meaning can be formed from the letters of the word βMONDAYβ, assuming that no letter is repeated, if all letters are used but first letter is a vowel ?
Given: The word \[MONDAY\] Total letters \[=\text{ }6\] All letters are used but first letter is a vowel ? In the word \[MONDAY\] the vowels are \[O\text{ }and\text{ }A\] We need to choose one vowel...
How many words, with or without meaning can be formed from the letters of the word βMONDAYβ, assuming that no letter is repeated, if (i) 4 letters are used at a time (ii) all letters are used at a time
Given: The word \[MONDAY\] Total letters \[=\text{ }6\] (i)Β \[4\]letters are used at a time Number of ways = (No. of ways of choosing 4 letters from MONDAY) \[=\text{ }{{(}^{6}}{{C}_{4}})\] By using...
There are 10 persons named P1, P2, P3 β¦, P10. Out of 10 persons, 5 persons are to be arranged in a line such that is each arrangement P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.
Given: Total persons \[=\text{ }10\] Number of persons to be selected \[=\text{ }5\text{ }from\text{ }10\]persons \[({{P}_{1}},\text{ }{{P}_{2}},\text{ }{{P}_{3}}~\ldots \text{ }{{P}_{10}})\] It is...
How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?
Given: Total number of vowels \[=\text{ }5\] Total number of consonants \[=\text{ }17\] Number of ways = (No. of ways of choosing 2 vowels from 5 vowels) Γ (No. of ways of choosing 3 consonants from...
How many triangles can be obtained by joining 12 points, five of which are collinear?
We know that \[3\] points are required toΒ draw a triangle and the collinear points will lie on the same line. Number of triangles formed = (total no. of triangles formed by all 12 points) β (no. of...
Find the number of diagonals of (i) a hexagon (ii) a polygon of 16 sides
(i)Β a hexagon We know that a hexagon has 6 angular points. By joining those any two angular points we get a line which is either a side or a diagonal. So number of lines formed...
There are 10 points in a plane of which 4 are collinear. How many different straight lines can be drawn by joining these points.
Given: Total number of points \[=\text{ }10\] Number of collinear points \[=\text{ }4\] Number of lines formed = (total no. of lines formed by all 10 points) β (no. of lines formed by collinear...
A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?
Given: Total number of questions \[=\text{ }12\] Total number of questions to be answered \[=\text{ }7\] Number of ways = (No. of ways of answering 5 questions from group 1 and 2 from group 2) +...
In an examination, a student to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make a choice.
Given: Total number of questions \[=\text{ }5\] Total number of questions to be answered \[=\text{ }4\] Number of ways = we need to answer 2 questions out of the remaining 3 questions as 1 and 2 are...
A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions?
Given: Total number of questions \[=\text{ }10\] Questions in part \[A\text{ }=\text{ }6\] Questions in part \[B\text{ }=\text{ }7\] Number of ways = (No. of ways of answering 4 questions from part...
A sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII. If there are 20 students in each of these classes, in how many ways can the teams be constituted?
Given: Total number of students in XI \[=\text{ }20\] And, Total number of students in XII \[=\text{ }20\] Total number of students to be selected in a team = 11 (with atleast 5 from class XI and 5...
From 4 officers and 8 jawans in how many ways can 6 be chosen (i) to include exactly one officer (ii) to include at least one officer?
Given: Total number of officers \[=\text{ }4\] Total number of jawans \[=\text{ }8\] Total number of selection to be made is \[6\] (i)Β to include exactly one officer Number of ways = (no. of ways of...
How many different selections of 4 books can be made from 10 different books, if two particular books are never selected
Given: Total number of books \[=\text{ }10\] Total books to be selected \[=\text{ }4\] Two particular books are never selected Number of ways = select 4 books out of remaining 8 books as 2 books are...
How many different selections of 4 books can be made from 10 different books, if (i) there is no restriction (ii) two particular books are always selected
Given: Total number of books \[=\text{ }10\] Total books to be selected \[=\text{ }4\] (i)Β there is no restriction Number of ways = choosing 4 books out of 10 books \[={{~}^{10}}{{C}_{4}}\] By using...
From a class of 12 boys and 10 girls, 10 students are to be chosen for the competition, at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?
Given: Total number of boys \[=\text{ }12\] Total number of girls \[=\text{ }10\] Total number of girls for the competition \[=\text{ }10\text{ }+\text{ }2\text{ }=\text{ }12\] Number of ways = (no....
How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)?
Given that we need to find the no. of ways of obtaining a product by multiplying two or more from the numbers \[3,\text{ }5,\text{ }7,\text{ }11\] Number of ways = (no. of ways of multiplying two...
There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further, find in how many of these committees: a particular student is excluded.
As per the given question, Since, Total number of professor \[=\text{ }10\] And, Total number of students \[=\text{ }20\] And, Number of ways = (choosing 2 professors out of 10 professors) Γ...
There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further, find in how many of these committees: (i) a particular professor is included. (ii) a particular student is included.
As per the given question, Since, Total number of professor \[=\text{ }10\] And, Total number of students \[=\text{ }20\] And, Number of ways = (choosing 2 professors out of 10 professors) Γ...
In how many ways can a football team of 11 players be selected from 16 players? How many of these will (i) Include 2 particular players? (ii) Exclude 2 particular players?
Given: Total number of players \[=\text{ }16\] Number of players to be selected \[=\text{ }11\] So, the combination is \[^{16}{{C}_{11}}\] By using the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left(...
In how many ways can a student choose 5 courses out of 9 courses if 2 courses are compulsory for every student?
Given: Total number of courses is \[9\] So out of \[9\text{ }courses\text{ }2\text{ }courses\] are compulsory. Student can choose from \[7\left( i.e.,\text{ }5+2 \right)\] courses only. That too out...
How many different boat parties of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls?
Given: Total boys are \[=\text{ }25\] Total girls are \[=\text{ }10\] Boat party of \[8\]to be made from \[25\] boys and \[10\]girls, by selecting \[5\text{ }boys\text{ }and\text{ }3\text{ }girls\]...
From a group of 15 cricket players, a team of 11 players is to be chosen. In how many ways can this be done?
Given: Number of players \[=\text{ }15\] Number of players to be selected \[=\text{ }11\] By using the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left( n\text{ }-\text{ }r \right)!\]...
If n + 2C8: n β 2P4 = 57: 16, find n
Given: \[^{n\text{ }+\text{ }2}{{C}_{8}}:{{~}^{n\text{ }-\text{ }2}}{{P}_{4}}~=\text{ }57:\text{ }16\] \[^{n\text{ }+\text{ }2}{{C}_{8}}~/{{~}^{n\text{ }-\text{ }2}}{{P}_{4}}~=\text{ }57\text{...
If 15Cr: 15Cr β 1 = 11: 5, find r
Given: \[^{15}{{C}_{r}}:{{~}^{15}}{{C}_{r\text{ }-\text{ }1}}~=\text{ }11:\text{ }5\] \[^{15}{{C}_{r}}~/{{~}^{15}}{{C}_{r\text{ }-\text{ }1}}~=\text{ }11\text{ }/\text{ }5\] Let us use the formula,...
If 8Cr β 7C3 = 7C2, find r
To find \[r\] let us consider the given expression, \[^{8}{{C}_{r}}-{{~}^{7}}{{C}_{3}}~={{~}^{7}}{{C}_{2}}\] \[^{8}{{C}_{r}}~={{~}^{7}}{{C}_{2}}~+{{~}^{7}}{{C}_{3}}\] We know...
If 15C3r = 15Cr + 3, find r
We know that ifΒ \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
If 18Cx = 18Cx + 2, find x
We know that ifΒ \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
If 24Cx = 24C2x + 3, find x
We know that ifΒ \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
If nC10 = nC12, find 23Cn
We know that ifΒ \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
If nC4 = nC6, find 12Cn.
We know that ifΒ \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
If nC12 = nC5, find the value of n.
We know that ifΒ \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
Evaluate the following:
Solution: As per the given question
Evaluate the following: (i) 35C35 (ii) n+1Cn
(i)Β \[^{35}{{C}_{35}}\] Let us use the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left( n\text{ }-\text{ }r \right)!\] So now, value of \[n\text{ }=\text{ }35\text{ }and\text{ }r\text{ }=\text{ }35\]...
Evaluate the following: (i) 14C3 (ii) 12C10
(i)Β \[^{14}{{C}_{3}}\] Let us use the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left( n\text{ }-\text{ }r \right)!\] So now, value of \[n\text{ }=\text{ }14\text{ }and\text{ }r\text{ }=\text{ }3\]...
Find the number of numbers, greater than a million that can be formed with the digit 2, 3, 0, 3, 4, 2, 3.
Given: The digits \[2,\text{ }3,\text{ }0,\text{ }3,\text{ }4,\text{ }2,\text{ }3\] Total number of digits \[=\text{ }7\] We know, zero cannot be the first digit of the \[7\] digit numbers. Number...
How many permutations of the letters of the word βMADHUBANIβ do not begin with M but end with I?
Given: The word \[MADHUBANI\] Total number of letters \[=\text{ }9\] A totalΒ number of arrangements of word \[MADHUBANI\] excluding \[I\] Total letters \[8\] Repeating letter \[A\] Β repeating twice....
How many words can be formed from the letters of the word βSERIESβ which start with S and end with S?
Given: The word \[SERIES\] There are \[6\] letters in the word \[SERIES\] out of which \[2\] are \[Ss,\text{ }2\text{ }are\text{ }Es\] and the rest all are distinct. Now, Let us fix \[5\] letters at...
How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.
Given: The digits \[0,\text{ }1,\text{ }1,\text{ }5,\text{ }9\] Total number of digits \[=\text{ }5\] So now, number greater than \[50000\] will have either \[5\text{ }or\text{ }9\] in the first...
In how many ways can the letters of the word βARRANGEβ be arranged so that the two Rβs are never together?
There are \[7\] letters in the word \[ARRANGE\]out of which \[2\text{ }are\text{ }As,\text{ }2\text{ }are\text{ }Rs\] and the rest all are distinct. So by using the formula, \[n!/\text{ }\left(...
How many numbers of four digits can be formed with the digits 1, 3, 3, 0?
Given: The digits \[1,\text{ }3,\text{ }3,\text{ }0\] Total number of digits \[=\text{ }4\] Digits of the same type \[=\text{ }2\] Total number of \[4\]digit numbers \[=\text{ }4!\text{ }/\text{...
How many different signals can be made from 4 red, 2 white, and 3 green flags by arranging all of them vertically on a flagstaff?
Given: Number of red flags \[=\text{ }4\] Number of white flags \[=\text{ }2\] Number of green flags \[=\text{ }3\] So there are total \[9\] flags, out of which \[4\] are red, \[2\] are white,...
How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?
Given: The digits \[1,\text{ }2,\text{ }3,\text{ }4,\text{ }3,\text{ }2,\text{ }1\] The total number of digits are \[7\] There are \[4\] odd digits \[1,1,3,3\text{ }and\text{ }4\]odd places \[\left(...
How many words can be formed by arranging the letters of the word βMUMBAIβ so that all Mβs come together?
Given: The word \[MUMBAI\] There are \[6\] letters in the word \[MUMBAI\]out of which \[2\] are \[Ms\] and the rest all are distinct. So let us consider both \[Ms\]together as one letter, the...
How many words can be formed with the letters of the word βPARALLELβ so that all Lβs do not come together?
Given: The word \[PARALLEL\] There are \[8\] letters in the word \[PARALLEL\] out of which \[2\text{ }are\text{ }As,\text{ }3\text{ }are\text{ }Ls\] and the rest all are distinct. So by using the...
Find the total number of arrangements of the letters in the expression a^3 b^2 c^4 when written at full length.
There are \[9\text{ }\left( i.e\text{ }powers\text{ }3\text{ }+\text{ }2\text{ }+\text{ }4\text{ }=\text{ }9 \right)\]objects in the expression \[{{a}^{3}}~{{b}^{2}}~{{c}^{4}}\]Β and there are...
How many words can be formed with the letters of the word βUNIVERSITY,β the vowels remaining together?
Given: The word \[UNIVERSITY\] There are \[10\] letters in the word \[UNIVERSITY\] out of which \[2\text{ }are\text{ }Is\] There are \[4\]vowels in the word \[UNIVERSITY\] out of which \[2\text{...
In how many ways can the letters of the word βALGEBRAβ be arranged without changing the relative order of the vowels and consonants?
Given: The word \[ALGEBRA\] There are \[4\]consonants in the word \[ALGEBRA\] The number of ways to arrange these consonants isΒ \[^{4}{{P}_{4}}~=\text{ }4!\] There are \[3\] vowels in the word...
Find the number of words formed by permuting all the letters of the following words : CONSTANTINOPLE
\[CONSTANTINOPLE\] There are \[14\] letters in the word \[CONSTANTINOPLE\]out of which \[2\text{ }are\text{ }Os,\text{ }3\text{ }are\text{ }Ns,\text{ }2\text{ }are\text{ }Ts\]and the rest all are...
Find the number of words formed by permuting all the letters of the following words : (i) SERIES (ii) EXERCISES
(i)Β \[SERIES\] There are \[6\]letters in the word \[SERIES\]out of which \[2\text{ }are\text{ }Ss,\text{ }2\text{ }are\text{ }Es\]and the rest all are distinct. So by using the formula, \[n!/\text{...
Find the number of words formed by permuting all the letters of the following words : (i) PAKISTAN (ii) RUSSIA
(i)Β \[PAKISTAN\] There are \[8\]letters in the word \[PAKISTAN\]out of which \[2\text{ }are\text{ }As\]and the rest all are distinct. So by using the formula, \[n!/\text{ }\left( p!\text{ }\times...
Find the number of words formed by permuting all the letters of the following words : (i)ARRANGE (ii) INDIA
(i)Β \[ARRANGE\] There are \[7\] letters in the word \[ARRANGE\] out of which\[2\text{ }are\text{ }As,\text{ }2\text{ }are\text{ }Rs\] and the rest all are distinct. So by using the formula,...
Find the number of words formed by permuting all the letters of the following words : (i) INDEPENDENCE (ii) INTERMEDIATE
(i)Β \[INDEPENDENCE\] There are \[12\] letters in the word \[INDEPENDENCE\] out of which \[2\text{ }are\text{ }Ds,\text{ }3\text{ }are\text{ }Ns,\text{ }4\text{ }are\text{ }Es\]and the rest all are...
By using the method of completing the square, show that the equation has no real roots.
$2 x^{2}+x+4=0$ $\Rightarrow 4 x^{2}+2 x+8=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}+2 x=-8$ $\Rightarrow(2 x)^{2}+2 \times 2 x \times...
The following are some particulars of the distribution of weights of boys and girls in a class:
Which of the distributions is more variable? Solution:
An analysis of the weekly wages paid to workers in two firms A and B, belonging to the same industry gives the following results:
(i) Which firm A or B pays out the larger amount as weekly wages? (ii) Which firm A or B has greater variability in individual wages? Solution:
Calculate coefficient of variation from the following data:
Solution:
The coefficient of variation of two distribution are 60% and 70%, and their standard deviations are 21 and 16 respectively. What is their arithmetic means?
Here, the Coefficient of variation for the first distribution is 60 And, Coefficient of variation for the first distribution is 70 $\mathrm{SD}\left(\sigma_{1}\right)=21$ and...
The means and standard deviations of heights and weights of 50 students in a class are as follows:
Which shows more variability, heights or weights? Solution: Given: The mean and SD is given of 50 students. Let us find which shows more variability, height and weight. By using the formulas,...
Two plants A and B of a factory show the following results about the number of workers and the wages paid to them
In which plant A or B is there greater variability in individual wages? Solution: Variation of the distribution of wages in plant $\mathrm{A}\left(\sigma^{2}=18\right)$ So, Standard deviation of the...
Calculate the mean, median and standard deviation of the following distribution
Solution:
A student obtained the mean and standard deviation of 100 observations as 40 and 5.1 respectively. It was later found that one observation was wrongly copied as 50, the correct figure is 40. Find the correct mean and S.D.
Calculate the A.M. and S.D. for the following distribution:
Solution:
Calculate the standard deviation for the following data:
Solution:
Calculate the mean and S.D. for the following data:
Solution:
Find the standard deviation for the following data:
(i) Solution: (ii) Solution:
Find the mean, and standard deviation for the following data:
(i) Solution: (ii) Solution:
Table below shows the frequency f with which βxβ alpha particles were radiated from a diskette
Calculate the mean and variance. Solution:
Find the standard deviation for the following distribution:
Solution: By using the formula for standard deviation: $\begin{array}{l} \mathrm{SD}=\sqrt{\operatorname{Var}(\mathrm{X})} \\ \text { Mean }=\sum \frac{\mathrm{f}_{\mathrm{i}}...
One number is chosen from numbers 1 to 100. Find the probability that it is divisible by 4 or 6?
Sample space is the set of first 100 natural numbers. n (S) = 100 Let βAβ be the event of choosing the number such that it is divisible by 4 n (A) = [100/4] = [25] = 25 {where [.] represents...
The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75. What is the probability of passing the Hindi examination?
Let βEβ denotes the event that student passes in English examination. And βHβ be the event that student passes in Hindi exam. It is given that, P (E) = 0.75 P (passing both) = P (E β© H) = 0.5 P...
A card is drawn from a deck of 52 cards. Find the probability of getting an ace or a spade card.
A card is drawn from a deck of 52 cards. Let βSβ denotes the event of card being a spade and βKβ denote the event of card being ace. As we know that a deck of 52 cards contains 4 suits (Heart,...
A die is thrown twice. What is the probability that at least one of the two throws come up with the number 3?
If a dice is thrown twice, it has a total of 36 possible outcomes. If S represents the sample space then, n (S) = 36 Let βAβ represent events the event such that 3 comes in the first throw. A =...
A natural number is chosen at random from amongst first 500. What is the probability that the number so chosen is divisible by 3 or 5?
Sample space is the set of first 500 natural numbers. n (S) = 500 Let βAβ be the event of choosing the number such that it is divisible by 3 n (A) = [500/3] = [166.67] = 166 {where [.] represents...
In a single throw of two dice, find the probability that neither a doublet nor a total of 9 will appear.
In a single throw of 2 die, we have total 36 outcomes possible. Say, n (S) = 36 Where, βSβ represents sample space Let βAβ denotes the event of getting a double. So, A = {(1,1), (2,2), (3,3), (4,4),...
A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability of its being a spade or a king.
A card is drawn from a deck of 52 cards is given. Let βSβ denotes the event of card being a spade and βKβ denote the event of card being King. As we know that a deck of 52 cards contains 4 suits...
One of the two events must happen. Given that the chance of one is two-third of the other, find the odds in favour of the other.
Let A and B be two events. As, out of 2 events, only one can happen at a time which means no event have anything common. β΄ We can say that A and B are mutually exclusive events. So, by definition of...
There are three events A, B, C one of which must and only one can happen, the odds are 8 to 3 against A, 5 to 2 against B, fins the odds against C.
As, out of 3 events, only one can happen at a time which means no event have anything common. So, A, B and C are mutually exclusive events. Now, by definition of mutually exclusive events we know, P...
Given two mutually exclusive events A and B such that P (A) = 1/2 and P (B) = 1/3, find P (A or B).
A and B are two mutually exclusive events is given P (A) = 1/2 and P (B) = 1/3 We need to find P (A βorβ B). P (A or B) = P (A βͺ B) So by definition of mutually exclusive events we know, P (A βͺ B) =...