J. K. SHAH CLASSES ?

Size: px

Start display at page:

Download "J. K. SHAH CLASSES ?"

Valentine Wilkerson
6 years ago
Views:

1 J. K. SHAH CLASSES BRANCH : BORIVALI &ANDHERI(A1, B1 + B) SUB :MATHS - SET 1 MEASURES OF DISPERSION 1. For the following frequency distribution the value of Q 1 is 33. Find the missing frequency and then calculate quartile deviation. ( Marks) Wages (` per hour) No of workers ? Let x be the missing frequency. Now we shall form 1.c.f. table as follows : Class Interval Frequency l.c.f 30-3 x x x x x x x x Given that, Q 1 33, 3-3 is the first quartile class. L 3, c.f. x, h, f 18 Q 1 L + h ff NN cc. ff (7+xx) xx 9 x (33-3) (7+xx) xx 9 + x (7+xx) 36 + x 7 + x 3x 36 x 1 Now, N After getting missing frequency, we can form l.c.f. table as follows :

2 Class Interval Frequency l.c.f For Q 3 consider 3NN 63 Q 3 lies in the class 38-0 L 38, c.f. 60, h, f 1 Q 3 L + h ff 3NN cc. ff (63 60) Q Q.D. QQ 3 QQ Q.d..75. Compute the mean deviation about (i) the mean and (ii) the median for the price of share of particular company in a week 0,7,38,1,9. First we find the mean and the median to be given data. Mean xx 15 3 nn 5 Now we shall find median of the observation. For this we arrange the observation in ascending order as 38,0,1,7,9. By definition, Median Me 1 x i lx i Meanl lx i Mel lx i 3l lx i 1l Total 0 18 From table

3 nn nn lxx ii Meanl 0, lxx Mel 18 ii1 ii 1 M.D. about Mean llxx MMMMMMMMMM nn 55 M.D. abut Median llxx MMMMMMMMMMMMMM nn Prices of a particular commodity in five years in two cities are as follows : ( Marks) Price in city A Price in city B Determine which city had more stability in prices. Prepare following table for calculating mean and S.D. of prices in both cities A and B. Price in (xx ii xx ) (xx ii xx ) Price in (yy ii yy ) (xx ii yy ) city a xi city B yy ii From table xx ii 110, (xx ii xx ) 30 yy ii 75, (yy ii yy ) 68 Mean xx 1 nn xx ii Mean yy 1 nn yy ii Var (X) 1 nn (xx ii xx ) Var (Y) 1 nn (yy ii yy) S.D. σσ xx VVVVVV (XX) 6.5, S.D. σσ YY VVVVVV (YY) C.V. (X) σσ xx xx C.V. σσ xx yy C.V. (X).5 x 100% C.V. (Y) C.V. (X) 11.1% C.V. (Y).6% Since the C.V. corresponding to city A is less than that corresponding to city B, the prices show more stability in city A. MOMENTS. The first four raw moments of a distribution are,0,0 and 50 respectively. Find the first four central moments of the distribution. The first four raw moments of the distribution are μμ 1, μμ 0, μμ 3 0 and μμ 50 We want to find first four central moments μμ 1, μμ,μμ 3 and μμ. We know that μμ 1 0. Using relations between raw moments and central moments, We get

4 μμ μμ - μμ μμ 3 μμ 3-3μμ μμ 1 + μμ (0)() + (8) μμ μμ - μμ 3 μμ 1 + 6μμ μμ 1 3μμ 1 50 (0)() + 6(0)() 3(16) Thus, the first four central moments are μμ 1 0, μμ 16, μμ 3-6 and μμ 16 SKEWNESS AND KURTOSIS 5. For a distribution.μμ 16 and μμ Find the coefficient of skewness γγ 11 ( Marks) ββ 1 μμ 3 3 ( 0) μμ (16) γγ 1 ββ (-ve sign is selected since μμ 3 is ve) Thus, γγ (Using log) γγ 1 < 0, the distribution is negatively skewed. 6. Compute Bowley s coefficient of skewnwsssk b for the following set of observation. 5,,1,3,7,9,8,11,9,6 For computation of Sk b, we need to find the three quartiles Q 1, Q,Q 3. For computation of quartiles, we first arrange the observation in ascending order. 1,,3,5,6,7,8,9,9,11 n 10 Q 1 value of nn+1 tth observation n 10 value of.75 th observation value of nd observation (value of 3 rd observation value of nd observation) (3-).75 Q value of nn+1 tth observation value of 5.5 th observation vvvvvvvvvv oooo 5 tth oooooooooooooooooooooo + vvvvvvvvvv oooo 6 tth oooooo eeeeeeeeeeeeeeee Q 3 value of 3 nn+1 tth

5 value of 8.5 th observation value of 8 th observation (value of 9 th observation value of 8 th observation) (9-9) 9 Bowley s coefficient of skewness is given by QQ Skb 3 + QQ 1 QQ QQ 3 + QQ (6.5) Skb -0. < 0, the distribution is negatively skewed. ANGLE AND ITS MEASUREMENT 7. One angle of a quadrilateral has measure radians and the measures of the other three angles are 99 in the ratio 3 :5 : 8, find the measures in radians. The sum of angles of a quadrilateral is 360 o. One of the angles is given to be ππ ππ x ππ 0o Sum of the remaining three angles is 360 o 0 o 30 o Since these three angles are in the ratio 3 : 5 :8, degree measures of these angles are 3k, 5k, 8k, where k is constant. 3k + 5k + 8k 30 o 16k 30 o K 0 o The measures of three angles are (3k) o (3 x 0) o 60 o (5k) o (5 x 0) o 100 o and (8k) o (8 x 0) o 160 o Three angles are 30o, 100o, 160o. These three angles in radians are 60 o 60 xx 100 o 100 xx 160 o 160 xx ππ 180 c πc 3 ππ 180 c 5πc 9 ππ 180 c 8πc 9 TRIGONOMETRY

6 8. Find the trigonometric function of - 55ππcc 66 Let m XOB 5ππcc 6 Draw a unit circle with centre at the origin. Let ray OB meet the standard unit circle in P (x,y). ll(op) 1 Draw segment PM perpendicular to X axis In POM, m POM ππcc 6 30o In POM, m POM ππcc 6 30 o ll(oooo) ll(op). 1 Since P lies in third quadrant, P (x,y) 3, 1 by definition sin 5ππ 6 cc y 1 cosec 5ππ 6 cc - cos 5ππ 3 x 6 sec 5ππ 6 cc 3 tan 5ππ 6 cc yy xx ( 1/) 1 3/ 3 cot 5ππ 6 cc 3 9. Eliminateθθ, if x asecθθ + b tan θθ ; y asec θθ- b tanθθ ( Marks) x a sec θθ + b tan θθ (I) Y a sec θθ - b tan θθ.(ii) On adding (I) and (II), on subtracting (II) from (I) x + y asec θθ, x y b tan θθ

7 secθθ xx+yy xx yy ; tan θθ aa bb We have,ssssss θθ tan θθ 1 xx+yy aa - xx yy bb 1 (xx+yy) aa - (xx yy) bb 1. LOCUS 10. A (,5) and B (9,-1), are the vertices of ABC. The third vertex C lies on the locus whose equation is 3x + y Find the locus of the centroid of ABC. Let G (x,y) be the centroid of ABC. Let C(h,k) be any point on the locus 3x + y h + k (i) G(x,y) is the centroid of ABC. G(x,y) G +9+h 3, 5 1+kk 3 (By centroid formula) x 11+h 9+kk, and y 3 3 h 3x 11 and k 3y + 9 Putting the values of h and k in equation (i), we get 3(3x 11) + (3y + 9) x + 1y This is the required equation of the locus of centroid of ABC.

MTP_FOUNDATION_Syllabus 2012_Dec2016 SET - I. Paper 4-Fundamentals of Business Mathematics and Statistics

MTP_FOUNDATION_Syllabus 2012_Dec2016 SET - I. Paper 4-Fundamentals of Business Mathematics and Statistics SET - I Paper 4-Fundamentals of Business Mathematics and Statistics Full Marks: 00 Time allowed: 3 Hours Section A (Fundamentals of Business Mathematics) I. Answer any two questions. Each question carries