Class name Subject Book title Chapter number Content Created at
Class XI Mathematics Mathematics 12 = sin x. Solution We use the Leibnitz product rule to evaluate this. df (x) d = (sinxsin x) dx dx ′ ′ =(sin x)sin x+sin x(sin x) =(cosx )inx +sinx ...
Class XI Mathematics Mathematics 12 have 2(x +h )3 2x +3 f x+ h − f x − f ( ) lim ( ) ( ) = lim x +h−2 x −2 h→ 0 h h→0 h Reprint 2026-27 250 MATHEMATICS (2x+ 2h+3 x)( − 2) (3 x+ h−)( ...
Class XI Mathematics Mathematics 12 (cos xsin x) +limsin x +limcosx h→0 h h→0 h h→0 h = cosx −sinx f (+ h ) f x( ) (x+ h ) ( x+ h −)xsin x (ii) f ( ) = h→0 h = h→0 h (x + )(sin x cosh...
Class XI Mathematics Mathematics 12 function wherever sin x itisdefined. (x −cosx) sin x−(x −cosx)(sin x) ′ h (x) = 2 (sin x) Reprint 2026-27 LIMITS AND DERIVATIVES 253 4 5 = (5x +sin...
Class XI Mathematics Mathematics 12 at a point is the common value of the left and right hand limits,iftheycoincide. fi For a function f and a real number a,im f(x) and f (a) may not b...
Class XI Mathematics Mathematics 12 of sinα ∆ y f (x i) f (x) for α = 0. He wrote = , and called the limit for α ∆x i i →0, the “function derive’e, y′ for f ′(x)”. Before1900,itwastho...
Class XI Mathematics Mathematics 13 Chapter 13 STATISTICS v “Statistics may be rightly called the science of averages and their estimates.” – A.L.BOWLEY & A.L. BODDINGTON v 13.1 Intro...
Class XI Mathematics Mathematics 13 (maximum). Whereas, the range of the runs scored by batsman B is from 46 to 60. Let us now plot the above scores as dots on a number line.We find t...
Class XI Mathematics Mathematics 13 or scatter but does not tell about the dispersion of the data from a measure of central tendency. For this purpose, we need some other measure of v...
Class XI Mathematics Mathematics 13 of deviations from 'a' M.D.( a) = . Number of observations Remark Mean deviation may be obtained from any measure of central tendency. However, mea...
Class XI Mathematics Mathematics 13 –3, –2, 1, 3, 4, –5, –1, 3 Step 3 The absolute values of the deviations, i.i.,x are 3, 2, 1, 3, 4, 5, 1, 3 Step 4 The required mean deviation about...
Class XI Mathematics Mathematics 13 n distinct values x1, x2, ...,nx occurring with frequencies f ,1f 2 ..., n respectively. This data can be represented in the tabular form as given ...
Class XI Mathematics Mathematics 13 60 1.5 15 8 7 56 0.5 3.5 10 8 80 2.5 20 12 5 60 4.5 22.5 40 300 92 6 6 6 N = ∑ fi=40 , ∑ fi i=300 , ∑ fixi− x =92 i=1 i=1 i=1 x = 1 f x = 1 ×300 =7...
Class XI Mathematics Mathematics 13 students are presented in a continuous frequencydistributionasfollows: Marksobtained 0-10 10-20 20-30 30-40 40-50 50-60 Number of Students 12 18 27...
Class XI Mathematics Mathematics 13 , where ‘a’is the assumed mean and h is the common factor. Then, the i h mean xby step-deviation method is given by n ∑ fi i x = a +i=1 ×h N Let us...
Class XI Mathematics Mathematics 13 Then M.D. (M) = N i=1i i − M Theprocessisillustratedinthefollowingexample: Example 7 Calculate the mean deviation about median for the following da...
Class XI Mathematics Mathematics 13 mean deviation about the mean for the data in Exercises 9 and 10. 9. Incomeper 0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 day in...
Class XI Mathematics Mathematics 13 Ifthissumis zero,theneach (x i x) hastobezero.Thisimpliesthatthereisno dispersion at all as all observations are equal to the mean x . n 2 If ∑ (xi...
Class XI Mathematics Mathematics 13 n 2 If ∑ (x i x) is simply our measure of dispersion or scatter about mean, we i=1 willtendtosaythatthesetAofsixobservationshasalesserdispersionabo...
Class XI Mathematics Mathematics 13 square-root of the variance and is called standard deviation. Therefore, the standard deviation,usuallydenotedbyσ ,isgivenby 1 n σ = ∑ (xi− x)2 ......
Class XI Mathematics Mathematics 13 x – (x −x) 2 f(x − x)2 i i i i i x i i i 4 3 12 –10 100 300 8 5 40 –6 36 180 11 9 99 –3 9 81 17 5 85 3 9 45 20 4 80 6 36 144 24 3 72 10 100 300 32 ...
Class XI Mathematics Mathematics 13 2  1 n n  = i i + x N 2 .N x Here ∑ xi i= x or ∑ xi i= Nx  N i=1  N i=1 i=1  1 n 2 1 n 2 2 = ∑ fi i + x 2 − 2x2 = ∑ fi i − x N i=1 N i=1 n 2 ...
Class XI Mathematics Mathematics 13 10 6 SolutionLet us form the followingTable 13.10: Table13.10 x f f x x 2 f x2 i i i i i i i 3 7 21 9 63 8 10 80 64 640 13 15 195 169 2535 18 10 18...
Class XI Mathematics Mathematics 3 results in this connection are called trigonometric identities. We have seen that 1. sin (– x) = – sin x 2. cos (– x) = cos x We shall now prove so...
Class XI Mathematics Mathematics 13 (A + hy − A − h y) 2 = N ∑ i i (Using (1) and (3)) i=1 Reprint 2026-27 280 MATHEMATICS n 1 f h (y − y)2 = N ∑ i i i=1 2 n h f (y − y)2 2 = N ∑ i i ...