MEASURES OF CENTRAL TENDENCY AND MEDIAN

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1 I. Solve the following: MEASURES OF CENTRAL TENDENCY AND MEDIAN 1. Find the value of p for the following distribution whose mean is 10. Variate Frequency 4 4 p Grade X Mathematics Practice Sheet Name 2. Find the mean for the following frequency distribution by i. direct method ii. short cut method iii. step-deviation method. Class-intervals Frequency The median of the following numbers, arranged in ascending order, is 25. Find x: 11, 13, 15, 19, x + 2, x + 4, 30, 35, 39, The marks scored by 16 students in a class test are: 3, 6, 8, 13, 15, 5, 21, 23, 17, 10, 9, 1, 20, 21, 18, 12. Find: i. the median ii. lower quartile iii. upper quartile 5. Find i. the median ii. lower quartile iii upper quartile Marks obtained No. of students II. 6. For the following distribution; find Monthly income (in Rs.) No. of employees i. The median ii. The lower and upper quartiles iii. The inter quartile range iv. Semi inter quartile range 7. Find i. the median ii. lower quartile iii. upper quartile Marks obtained No. of students Find mean by step deviation method for the above data. 9. Find the mean of squares of the first ten natural numbers. 10. Show that mean of first ten natural numbers is equal to the one less than one third of next ten natural numbers. SOLVE THE FOLLOWING: 1. The marks obtained by a set of students in an examination are given below: Marks No. of students x 4 2. Use graph paper for this question. The table given below shows the monthly wages of some factory workers. i. Using the table, calculate the cumulative frequencies of workers. ii. Draw the cumulative frequency curve. Draw the cumulative frequency curve. Use 2 cm=rs 500, starting the origin at Rs 6500 on x-axis, and 2 cm=10 workers on y-axis. iii. Use your graph to write down the median wage in Rs. Wages , , , , , , , Frequency 10, 18, 22, 25, 17, 10, 8,

2 3. The table below shows the distribution of the scores obtained by 120 shooters in a shooting competition. Using a graph sheet, draw an give for the distribution. Use your give to estimate: Scores obtained Number of shooters a. The median b. The inner quartile range c. The number of shooters who obtained more than 75% scores. 4. The daily wages of 160 workers in a building project are given below: Wages (in Rs.) No. of workers Using a graph paper, draw an Give for the above distribution. Use your Give to estimate: i. The median wage of the workers. ii. The upper quartile wage of the workers. iii. The lower quartile wage of the workers. iv. The percentage of workers who earn more than Rs. 45 a day. 5. Using a graph paper, draw an Give for the following distribution which shows a record of the weight in kilograms of 200 students. Use your Give to estimate the following: Wages (in Rs.) No. of workers i. The percentage of students weighing 55 kg or more. ii. The weight above which the heaviest 30% of the students fall. iii. The number of students who are: a. under-weight and b. over-weight, if kg is considered as standard weight. 6. For the following frequency distribution draw a histogram. Hence calculate the mode. Class Frequency The following table shows the distribution of the heights of a group of factory workers: Height (cm) No. of workers i. Determine the cumulative frequencies. ii. Draw the cumulative frequency curve on a graph paper. iii. From your graph, write down the median height in cm.

3 8. Find the mean of the following distribution: Class Interval Frequency The marks obtained by a set of students in an examination are given below: Marks No. of students x 4 Given that the mean mark of the set is 18, calculate the numerical value of x. 10. The mean of the following frequency distribution is 57.6 and the sum of the observation is 50. Find the missing frequency f 1 and f 2. Class Frequency 7 f 1 12 f RATIO AND PROPORTION 1. What should be added to each of the numbers 12, 22, 42 and 72 so that the resulting numbers may be in proportion? 2. Find two numbers such that the mean proportional between them is 18 and third proportional to them is If three quantities are in continued proportion, prove that the ratio of first to the third is the duplicate ratio of the first to the second. 4. If a b and a : b is the duplicate ratio of (a + c) and (b + c), prove that c is the mean proportional between a and b. 5. If b is the mean proportional between a and c, P.T a 2 b 2 + c 2 = b 4 6. If x = y = z, prove that a 2 x 2 + b 2 y 2 + c 2 z 2 = xyz a b c a 3 x + b 3 y + c 3 z abc a 2 b 2 + c 2 7. If a, b, c, d are in proportion, prove that a 4 + c 4 = ma 2 + nc 2 8. If a, b, c, d are in continued proportion, Prove that ab + bc cd = (a + b c)(b + c d). b 4 + d 4 mb 2 + nd 2 9. If x = y = z, Prove that ax + by + cz = 0. b c c a a b 10. If a : b : : c : d, Prove that 4a + 7b = 4a 7b 4c + 7d 4c 7d 11. If 2a + 2b 3c 3d = a + b 4c 4d, Prove that a : b : : c : d. 2a 2b 3c + 3d a b 4c + 4d 12. If p = 4xy, find the value of p + 2x + p + 2y p 2x p 2

4 13. If x = a + 3b + a 3b, Prove that 3bx 2 2ax + 3b = 0. a + 3b a 3b 14. If a 3 + 3ab 2 = x 3 + 3xy 2, Prove that x = y. 3a 2 b + b 3 3x 2 y + y 3 a b 15. What quantity must be added to each term of the ratio (a + 2b) : (a 2b) to make it equal to the duplicate ratio of (a + b) : (a b)? 16. If ay bx = cx az = bz cy, Prove that x = y = z. p q r a b c SIMILARITY AS SIZE TRANSFORMATION 1. A map of a square plot of land is drawn to a scale of 1: if the area of the plot in the map is 72 cm 2, find i. The actual area of the plot of land, and ii. The length of the diagonal in the actual plot of land. 2. The scale of map 1 : 2,00,000. A plot of land of area 25 km 2 is to be represented on the map. Find i. The number of kilometers on the ground which is represented by 1 cm on the map ii. The area in km 2 that can be represented by 1 cm 2 ii. The area on the map that represents the plot of land. 3. A model of a ship is made to a scale of 1 : 200. i. The length of the model is 5m. Calculate the length of the ship. ii. The area of the deck of the ship is 1,60,000m 2. Find the area of the deck of the model. iii. The volume of the model is 200 litres. Calculate the volume of the ship in m A model of a rectangular swimming pool was made to a scale of 1 : 500. i. The length and width of the actual swimming pool are 100m and 50m respectively. Calculate the length and width on the model in cm. ii. If the model has the capacity to hold 240cm 3 water, how much water can be stored in the actual swimming pool. 5. On a map drawn to a scale of 1 : 25000, a rectangular plot of land, ABCD has the following measurements. AB = 12cm and BC = 16cm. Angle A, B, C, and D are all 90 0 each. Calculate i. The diagonal distance of the plot in km ii. The are of the plot in sq. km. 6. The scale of map 1 : 2,00,000. A plot of land of area 25 km 2 is to be represented on the map. Find i. The number of kilometers on the ground which is represented by 1 cm on the map ii. The area in km 2 that can be represented by 1 cm 2 iii. The area on the map that represents the plot of land. 7. A model of a ship is made to a scale of 1 : 200. i. The length of the model is 5m. Calculate the length of the ship. ii. The area of the deck of the ship is 1,60,000m 2. Find the area of the deck of the model. iii. The volume of the model is 200 litres. Calculate the volume of the ship in m A model of a rectangular swimming pool was made to a scale of 1 : 500. i. The length and width of the actual swimming pool are 100m and 50m respectively. Calculate the length and width on the model in cm. ii. If the model has the capacity to hold 240cm 3 water, how much water can be stored in the actual swimming pool.

5 HEIGHT AND DISTANCES 1. Rita was observing a stationary balloon in the morning at an angle of elevation 45.The shadow of balloon which was on the western sky fell on the ground at a distance of meters from Rita. If the altitude of the sum be 60, find the height of the balloon from the ground and also the distance of the balloon from Rita. 2. The angle of elevation of a jet plane from a point A on the ground is 60. After a flight of 15 sec, the angle of elevation changes to 30. If the jet plane is flying at a constant height of meters, find the speed of the jet plane. 3. A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60º. When he moves 50 m, away from the bank, he finds the angle of elevation to be 30º. Calculate : i. The width of the river and ii. The height of the tree. 4. Two poles of equal heights are standing opposite to each other on either side of a road, which is 100 m wide. From a point between them on the road, the angle of elevation of their tops are 30º and 60º. Find the position of the point and also the height of the poles. [Use 3 = 1.732] 5. A man on the top of a vertical tower observes a car moving at uniform speed towards the tower. If it takes 12 minutes for the angle of depression to change from 30º to 45º, how soon, after this, will the car reach the tower? 6. The angle of elevation of the top of a hill at the foot of the tower is 60º and the angle of elevation of the top of the tower from the foot of the hill is 30º. If the tower is 50 m high, find the height of the hill. 7. From a window 15 m high above the ground in a street, the angle of elevation and depression of the top and foot of another house on the opposite side of the street are 30º and 45º respectively. Show that the height of the opposite house is m.(take 3 = 1.732). 8. The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is 30º. On advancing 150 m towards the foot of the tower, the angle of elevation becomes 60º. Show that the height of the tower is 129.9m. ( 3 = 1.732). 9. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 5m. From a point on the plane the angles of elevation of the bottom and top of the flagstaff are respectively 30º and 60º. Find the height of the tower. 10. As observed from the top of a light-house, 100 m high above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30º to 60º. Determine the distance traveled by the ship during the period of observation. (Use 3 = 1.732). PROBABILITY 1. An integer is chosen at random from 1 to 100. Find the probability that the number is: i. Is divisible by 5 ii. Is prime number iii. Is perfect cube 2. A dice is thrown once. What is the probability that the i. Number is even and ii. Number is greater than 2 3. Two different dice are thrown simultaneously. What is the probability that the sum of two numbers appearing on the top of disc is i. 9 ii. atleast10 iii A box contains 17 cards numbered 1, 2, and are mixed thoroughly. A card is drawn at random from the box. Find the probability that the number of card is i. Prime, ii. Divisible by 2 and 3 both, iii. Divisible by 2 or Ankita and Nagma are friends. They were both born in 1990.What is the probability that they have i. Same birthday? ii. Different birthdays? 6. A bag contains 1 balls out of which x are black. i. If a ball at random, what is the probability that it will be a black ball? ii. If 6 more balls are put in the bag, the probability of drawing a black ball will be double than that of Find the value of x.

6 1. If, MATRICES A, B and C show that AB AC. 2. Find x and y if 3. Find M Compute A (B + C) and (B + C) A. 7. find the value of x if A 2 = B 8. find the value of a, b and c. 9. find A 2 - A + BC 10. Compute (AB) C and (CB) A. Is (AB) C = (CB) A m n = find the values of k, n and m k 2

7 REFLECTION 1. Use graph paper for this question. The point P(2, 4)is reflected in the x = 0 to get the image P. i. Write down the co-ordinates of P. ii. Point P is reflected in the line y = 0, to get the image P. Write down the co-ordinates of P. iii. Name the figure PP P. iv. Find the area of the figure PP P. 2. A (3,2), B (4,0) and C ( 5,0) are the vertices of Δ ABC. Write down the co-ordinates of i. B 1 and C 1, the image of B and C under the reflection in y-axis. ii. B 2 and C 2, the image of B and C under the reflection in (0,0). Write down the co-ordinates of i. A 3 and C 3, the image of A and C by reflection in x-axis. ii. What type of figure is formed by Δ ABC and its image taken together. 3. The points (4,1), (4, 1), ( 4, 1) and ( 4, 1) are the vertices of a rectangle. If the rectangle is reflected in the line x = 5, find the co-ordinates of the reflected rectangle also find the area and perimeter of the reflected rectangle. 4. Use a graph paper for this question.(take 10 small division = 1 unit on both axes). P and Q have coordinates (0, 5) ( 2, 4). i. P is invariant when reflected in an axis. Name the axis. ii. Find the image of Q on reflection in the axis found in (i). iii. (0, k) on reflection in the origin is invariant. Write the value of k. BANKING 1. Amar opens a recurring deposit account in a bank. He deposits Rs.500 every month for one year. The interest paid by the bank is 5% per annum. Find the amount he receives at the time of maturity. 2. Shankar deposits Rs.300 per month in a recurring deposit account for 12 months. Find the amount he will receive at the time of maturity at the rate of 5% p.a. 3. Rajaram wants Rs at the end of 5 years by depositing a certain sum of money on a monthly basis in a bank paying 6% simple interest p.a. What should be the monthly instalment. 4. Rachna gets Rs.38, at the end of 6 years at the rate of 6.5% p.a. in a recurring deposit account. Find the monthly instalment. 5. Mrs. Mamun invests Rs.250 every month for 24 months in a bank and collects Rs at the end of the term. Find the rate of simple interest paid by the bank on this recurring deposit. 6. Which is better investment Rs in a saving deposit with bank for 3 years the interest being compounded half yearly at the of 6% or Rs.1200 p.m. in a recurring deposit with a bank paying simple interest at 6% p.a. for 3 years LINEAR INEQUATIONS 1. Solve the inequations and graph the solution set. 2. Solve : and draw graph of the solution set. 3. P is solution set of is the solution set of where. Find the set

8 4. Find the range of values of x which satisfies. Graph these values of x on the number line. 5. Find the range of values of x which satisfy Graph these values of x on the real number line. 6. Solve the following inequation, and graph the solution on the number line QUADRATIC EQUATIONS 1. 15x 2-28 = x 2. 3x 2-5x - 12 = 0 3. abx 2 + (b 2 - ac) x - bc = 0 4. x(2x + 1) = x 2-8x - 4 = (x + 4) (x + 5) = 3 (x + 1) (x + 2) + 2x Find the value of p for which the given equations has equal roots : i. 4x 2-5x + p = 0 ii. px 2-8x + 4 = 0 iii. 9x 2 + 3px + 4 = 0 iv. (p - 12) x (p - 12) x + 2 = Solve the equation 2x - 1/x = 7. Write your answer correct to two decimal places. 17. Find the values of k so that the equation kx(x-2) +6=0 gas two equal roots. Also find the root in each case. EQUATIONS OF STRIAGHT LINES 1. A line joining A(4, 5) and B(1, 2) is parallel to the joining C(1, -2) and D(0, k). Find the value of k. 2. Find the equation of the line passing through the intersection of 2x y = 1 and 3x + 2y + 9 = 0 having slope equal to T he co-ordinates of two points A and B are (0, 4) and (3, 7) respectively. Find i. The gradiant of AB. ii. The equation of AB. iii. The co-ordinates of the point where the line AB intersects the x-axis. 4. Write down the equation of the line parallel to x 2y + 8 = 0 and passing through the point (1, 2). 5. Find the equation of the perpendicular dropped from (-1, 2) on the line joined (1, 4) and (2, 3). 6. A (2, -4), B (3, 3) and C (-1, 5) are the vertices of the triangle ABC. Find the equation of i. The median of the triangle through A. ii. The attitude of the triangle through B.

7. Find the equation of a line, which has y-intercept 4 and which is parallel to the line 2x 3y = 7. Find the co-ordinates of the point, where it cuts the x-axis. 8.

A(1, 4), B(3, 2) and C(7, 5) are the vertices of a triangle ABC. Find i. The co-ordinates of the centroid G of ii. The equation of a line through G and parallel to AB 10.

9 7. Find the equation of a line, which has y-intercept 4 and which is parallel to the line 2x 3y = 7. Find the co-ordinates of the point, where it cuts the x-axis. 8. Find the equation of the straight line which passes through the point of intersection of the two lines 2x y + 5 = 0 ; 5x + 3y 4 = 0 and is perpendicular to the line x 3y + 21 = 0 9. A(1, 4), B(3, 2) and C(7, 5) are the vertices of a triangle ABC. Find i. The co-ordinates of the centroid G of ii. The equation of a line through G and parallel to AB 10. In the figure alongside, the lines are represented. Write down the angles that the lines make with the position direction of the x-axis. Hence determine angle. SECTION AND MID-POINT FORMULAE 1. The two vertices of a triangle are ( 1, 4) and (5, 2). If the centroid is (0, 3), find the third vertex. 2. Three consecutive vertices of a parallelogram ABCD are A (10,-6), B (2,-6) and C (-4,-2),findthe fourth vertex D. 3. A (10, 5), B (6, -3) and C (2, 1) are the vertices of a triangle ABC. L is the mid-point of AB and M is the mid-point of AC. Write down the co-ordinates of L and M. Show that LM = ½ BC. 4. Two vertices of a triangle are (2, 3), (1, 2) and the centroid is (-1, -2). Find the third vertex. 5. Find the co-ordinates of the centroid of a triangle whose vertices are A (-1, 3), B (1, -1) and C (5, 1). 6. Prove that the points A (-5, 4), B (-1, -2) and C (5, 2) are the vertices of an isosceles right angled triangle. Find the co-ordinates of D, so that ABCD is a square. 7. The line segment joining A(2, 3) and B(4, -5) is intersected by the x-axis at a point K. Write down the ordinate of the point. Hence find the ratio in which K divides AB. 8. Calculate the ratio in which the line segment joining (3, 4) and (-2, 1) is divided by the y-axis 9. The center O of a circle has the co-ordinates (4, 5) and one point on the circumference is (8, 10). Find the co-ordinates of the other and of the diameter of the circle through this point. 10. If the mid-point of the sides of a triangle are (-2, -5), (3, -2) and (3, -1); find its vertices 11. The co-ordinates of A and B are (2, a) and (-2, a+4). The mid-point of AB is (0, 1). Find the value of a. 12. find the points of trisection of ( - 2, 1) and (1, 4) 13. Find the circumcentre of the triangle formed by the points A (10, 5), B (6, -3) and C (2, 1) 14. Calculate the ratio in which the line joining. A (6, 5) and B (4, -3) is divided by the line y = 2. SIMILARITY OF TRIANGLES 1. In a triangle ABC, DE is parallel to BC. If AE = 2cm, EC = 4cm and BC = 12cm, find the length of DE 2. In a triangle ABC, DE is parallel to BC. If AD = 4cm, DB = 2cm and the area of i. Find the area of ii.

3. In ΔEF, GHǁF such that LP=2 cm and PM=6cm. Also MN=20 cm. 4. In the diagram alongside, DE is parallel to BC and Prove that i. ii. 5.

In the figure PQ RS is a parallelogram; PQ = 16cm; QR = 10cm, L is a point on PR such that RL : LP = 2 : 3, QL produced meets RS at M and PS produced at N. i. Prove that triangles RLQ and PLN.

RC is parallel to PQ and meets AB produced at C. Find QB : BR in two different ways and show that PR = 2RC TRIGONOMETRIC IDENTITIES 1.

Prove the identity: Sin 3 A - Cos 3 A = SinA CosA 1 SinA CosA 3. Prove the following identities : cos A/(1 tan A) + sin A/(1 cot A) = sin A + cos A 4.

(sina + seca) 2 + (cosa + coseca) 2 =(1 + seca coseca) 2 6. (1 + 1/tan 2 A)(1 + 1/cot 2 A)=1/(sin 2 A cos 2 A) 7.

10 3. In ΔEF, GHǁF such that LP=2 cm and PM=6cm. Also MN=20 cm. 4. In the diagram alongside, DE is parallel to BC and Prove that i. ii. 5. In triangle ABC, DE is parallel to BC. If AD : DB = 2 : 3 and area of is 8cm 2, find the area of BCED. 6. In the figure PQ RS is a parallelogram; PQ = 16cm; QR = 10cm, L is a point on PR such that RL : LP = 2 : 3, QL produced meets RS at M and PS produced at N. i. Prove that triangles RLQ and PLN. are similar. Find the length of PN ii. Find a triangle similar to RLM. Evaluate RM. 7. In triangle PQR, PB bisects and A is the mid-point of PQ. RC is parallel to PQ and meets AB produced at C. Find QB : BR in two different ways and show that PR = 2RC TRIGONOMETRIC IDENTITIES 1. Without using trigonometric table, evaluate : 7(sin 27 /cos 63 ) + 3(cos 21 /sin 69 ) 7(tan 36 /cot 54 ) 2. Prove the identity: Sin 3 A - Cos 3 A = SinA CosA 1 SinA CosA 3. Prove the following identities : cos A/(1 tan A) + sin A/(1 cot A) = sin A + cos A 4. Without using trigonometric tables, evaluate the following : 0 0 sec17 tan 68 cos 44 cos cos ec73 cot (sina + seca) 2 + (cosa + coseca) 2 =(1 + seca coseca) 2 6. (1 + 1/tan 2 A)(1 + 1/cot 2 A)=1/(sin 2 A cos 2 A) 7. coseca/(coseca 1) + coseca/(coseca + 1)=2 + 2tan 2 A. 8. tana/(1 cota ) + cota/(1 tana)=1 + seca coseca = 1 + tana + cota. 9. (sin 2 25º + sin 2 65º) + 3(tan5ºdtan15ºtan60ºtan75ºtan85º). 10. (cos 2 20º + cos 2 70º)/(sec 2 50º cot 2 40º) + 2cose 2 58º 2cot58ºtan32º 4tan13ºtan37ºtan45ºtan53ºtan77º. 11. Prove that sin ɵ +cos ɵ sin ɵ cos ɵ sin ɵ cos ɵ sin ɵ + cos ɵ = 2 1 2cos 2 ɵ

MATHEMATICS. (Two hours and a half) Answers to this Paper must be written on the paper provided separately.

MATHEMATICS. (Two hours and a half) Answers to this Paper must be written on the paper provided separately. MATHEMATICS (Two hours and a half) Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading