SECTION FORMULA
1) Determine the coordinates of the middle points of the sides of the triangle whose vertices have coordinates are (3,2), (-1, -2) and (-5,-4). (1,0),(-3,-3),(-1,-1)
2) find the coordinates of the centroid of the triangles whose vertices are
A) (-4,5),(8,2),(2,-1). (2,2)
B) (3,4),(- 1,7),(10,10). (4,7)
3) find the co-ordinates of the point which divides the line segment PQ joining the points P(6,4) and Q(7,-5) in the ratio 3:2. 33/5, -7/5
4) Find the co-ordinate of the point which divide the line segment joining the points (9,5) and (-7,-3) in the ratio 5:3. (-1,0)
5) Write down the coordinates of the point which divides the augment joining the points (1,-2) and (6,8) in the ratio 2:3. (3,2)
6) Find the coordinates of the point of trisection of the line segment joining the points P(-2,3) and Q(3,-1) that is nearer to P. (-/3, 5/3)
7) A line segment directed from (-3,2) to (1,-4) is trebled. Find the coordinates of the terminal point. (9,-16)
8) The line segment joining the point (2,-2) and (4,6) is extended each way a distance equal to half its own length; find the coordinates of its terminal points. (5,10),(1,-6)
9) Show that the points (0,2),(4,1) and (16,-2) lie in a straight line.
10) Show that the points (-4,0),(6,3) and (36,12) lie in a straight line.
11) if (19,8),(15,-6),(-11,- 12) are the three vertices of a parallelogram and The Fourth vertex lies in the second quadrant, find the coordinates of the fourth vertex. (-7,2)
12) Show that the points A(8,12), B(-2,7) and C(2,9) lie on a straight line. Also, find the ratio in which the line segment AB is divided at C. (3:2)
13) If the point (9,2) divides the line segment joining the points P(6,8) and (x,y) in the ratio 3:7, find the coordinates of Q. (16,-12)
14) If the point (6,3) divides the segment of the line from (4,5) to (x, y) in the ratio 2:5, find the co-ordinates (x, y) of Q. What are the coordinates of the mid point of PQ. (11,-2),(15/2,3/2)
15) Find the ratio in which the points (-2,2) divides the line segment joining the points (-4,6) and (1/2, -3). 4:5
16) Find the ratio in which point (-5, -20) divides the line joining the points (4,7) and (1,-2). 3:2
17) find the ratio in which the point (-1,0) divides the line segment joining the points (- 7,-3) and (9,5). 3:5
18) Prove that the points (2,1),(0,0),(-1,2) and (1,3) form a square.
Ratio And Proportion
1) Find the duplicate ratio of 3:4 and the sub duplicate ratio of 169: 256. 9:16, 13:16
2) Find the ratio compounded of the ratios 6a: 5b, 2ab: 3c² and c: a. 4a: 5c
3) Find the ratio compound of the Triplicate ratio of 2a: 3b and sub duplicate ratio of 9:64. a³: 9b³
4) If the ratio (x+7):2(x+14) be equal to the duplicate ratio of 5:8, find the value of x. 18
5) Show that the ratio x : y is the duplicate of the ratio (x+z):(y+z) if z² - xy= 0.
6) If x: y = 3: 4, find the value of (7x - 4y):(3x+y). 5:13
7) If (x - y):(x+ y)= 7:11, find the value of x: y. 9:2
8) If (a+ b):(a - b)= 5:2, find the value of b: a . 3:7
9) If (2x+ 5y):(3x +5y)= 9: 10. Find x : y. 5:7
10) For what value of x will the ratio (23 +x): (19+ x) be equal to 2. -15
11) If the ratio of (5+x) and (37+x) be equal to the ratio 1, and 3. Find x. 11
11) What number must be added to each term of the ratio 5: 37 to make it equal to the ratio 1:3. 11
12) Two numbers are in the ratio 3: 4 and if 7 is added to each term of the ratio, then the new ratio is 4: 5. Find the numbers. 21, 28
13) Two numbers are in the ratio 7: 9 and if 10 be subtracted from each term of the ratio, then the new ratio is 8:11. Find the numbers. 42, 54
14) What number must be added to each term of the ratio a : b to make it equal to c: d. (ad - bc)/(c - d)
15) If x/a = y/b = z/c,
Prove:
A) (x² - yz)/(a² - bc) = (y² - zx)/(b² - ac) = (z² - xy)/(c² - ab).
B) (x² + a²)/(x+ a) + (y² +b²)/(y+ b) + (z² + c²)/(z+ c( = {(x+ y+ z)² +(a+ b + c)²}/{(x+ y+ z) + (a+ b+ c)}.
C) (3x² + 5y² + 4z²)/(3a²+ 5b²+ 4c²) = (x²+y² +z²)/(a²+ b²+ c²).
16) If the work done by (x -1) men in (x - 1) days is to the work done by (x+2) men in (x -1) days be in the ratio 9:10, find x. 8
17) The monthly income of two person are the ratio 4:5 and their monthly expenditures are in the ratio 7:9. If each saves ₹50 per month, find their monthly incomes. ₹400, ₹500
18) The monthly salary of two persons are in the ratio 3:5. If each receives an increase of ₹20 in the monthly salary, the ratio is altered to 13 :21, find their salaries. 240, 400
19) find in what ratio will the total wages of the workers of a factory be increased or decreased if there be a reduction in the number of workers in the ratio 15 :11 and an increament in the wages in the ratio 22: 25. 6:5
20) An employer raises hourly rate of wages in the ratio 5:6, but reduces the working hours per week in the ratio 7:8. What will be the increase or decrease in the weekly wages bill ? if it was Rs 2400 previously, what it will be now ? 21:20 increase, ₹2520
21) The railway fare in a certain year increases in the ratio 22: 25 but the number of passengers decreases in the ratio 13 :11. Find in what ratio will the total income from Passengers' fare increases or decrease. 26:25, decrease
22) The Railway fare in a certain year increases in the ratio 14:25 but the number of passengers decreases in the ratio 15:7. Find in what ratio will the total income increase or decrease. 6:5, decrease
23) The ratio of prices of two articles was 16:23. Two years later when the price of the first had risen by 10% and that of second by ₹477, the ratio of their prices becomes 11:20. Find the original prices of the two articles. 848, 1219
24) The profit of a trader increases every year in the ratio 4:5 for the first 5 years but from the sixth year in the ratio 4:3 in the 3 subsequent years. If his profit in the first year was ₹2400, find out the change in his profit in the eighth year. 71.92
25) The prime cost of an article was three times the value of the raw materials used. The cost of raw materials increases in the ratio 3:7 and productive wages increases in the ratio 4:9. Find the present Prime cost of an article which could formerly be made for ₹18. 41
26) Factory on cost of the articles produced by a factory is taken as the sum of the cost of material, cost of labour and factory overhead charges. These three costs are in the ratio 15: 6 :5. Later the cost of material is decreased in the ratio 10:9 and the cost of labour is increased in the ratio 2:3, find the subsequent altered ratio in the factory overhead charges so as to keep the factory on-Cost unaltered. 10:7 , decreased.
27) A mixture of 30 litres contains milk and water in the ratio 7 to 3; how much water must be added to the mixture so that the ratio of milk to water may be 3: 7. 40 litres
28) A mixture of 30 litres contains milk and water in the ratio of 7:3. How much mixture must be replaced by water to make the ratio 3:7 ? 17:14
29) water and milk are mixed in the ratio 2:7 in one vessel and in the ratio 2:9 in the another vessel. Determine the volume of the mixture of the second vessel that should be mixed with 225c.c. of the mixture in the first vessel to form a new mixture containing milk and water in the ratio 4:1. 275cc
30) Two mixture contains milk and honey with ratio of 7:2 and 5 : 1. In what ratio these two mixture should we mixed so that the resulting mixture may contain milk honey in the ratio 9:2 ? 3:8
31) Find the third proportional
A) 16 and 20. 25
B) a²b and ab. b
32) Find the fourth proportional to
A) 2, ,3, 4. 6
B) 2, 5 , 22. 55
C) 4a², 7ab, 8ab². 14b³
33) Find the mean proportional between
A) 3 and 12. 6
B) 30a²b a8120a²b³. 60a²b²
34) If x be the mean proportional between find (x-3) and (x -6), find x. 2
35) If m/n = p/q = √(7/15) find the value of
A) (n+ m)/(n - m). (11+√105)/4
B) (q²+ p²)/(q² - p²). 11/4
C) (m²+ n²)/(n²+ q²). 7/15
36) what number must be added to each of the four numbers 9,11, 15, 18 so that the sums will be proportional. 3
37) What number must be subtracted from each of the four numbers 15,18, 21, 27 so that the difference will be proportional. 9
38) if 3, x, 1083 are in continued proportion, find x. ±57
39) If 2,x , 18 are in continued proportion, find x. ±6
40) Divide 345 into 4 parts proportional to 3,5, 6, 9. 45, 75, 90, 135
41) if x :y :z = 2:7 :11 and 4x- 5y + 3z= 60, find x, y, z. 20, 70, 110
42) If a: b= c: d prove:
A) a²+ c² : b² + d²= ac: bd.
B) ac(a+ c): bd(b +d)= (a+c)³: (b+d)³
C) (a+ c)³ : (b + d)³= a(a - c)²: b(b - d)².
D) pa²+ qb² : pa²² - qb²= pc²+ qd²: pc² - qd².
43) If a, b, c are in continued proportion, prove that:
A) (a+ b)²: (b+ c)² =(a²+ b²):(b²+ c²)
B) (a²+ ab+b²) : (b² + bc+ c²)= a: c.
44) If a, b, c, d are in continued proportion, prove that:
A) a : (b+ d)= c³: (c²a +d³).
B) (a+ b) : (c+ d)= (a²+ b²+ c²): (b² + c² +d²).
C) (2a+ 3d) : (3a - 4d)= (2a³ + 3b³): (3a³ - 4b³).
D) (a²+ b²+ c²)(b²+ c²+ d²) = (ab+ bc+ cd)².
45) If x: y: z = a: b : c, prove that
A) (x²+ y²+ z²)(a²+ b²+ c²) = ax+ by + cz)².
B) (x²+ 5y²+ 4z²) : (3a² + 5b²+ 4c²) = (x²+ y² +z²): (a²+b² + c²).
46) If x/{(b+c)(b+ c - 2a)} = y/{(c - a)(c+ a - 2b)} = z/{(a- b)(a+ b - 2c)}, Prove that x+ y+ z = 0.
47) If x/(q+ r - p) = y/(r+ p - q) = z/(p+ q - r), Prove that (q-r)x+ (r - p)y+ (p -q)z = 0.
48) If (a+b+c+ d)(a- b - c + d) = (a- b+ c - d)(a+ b - c - d), = Prove that a, b, c ,d are proportional.
49) If y be the mean proportional between x and z, prove that x²y²z²{1/x³ +1/y³+ 1/z³) x³+ y³+ z³.
TRIGONOMETRIC RATIOS
1) If x = 30°, verify the following:
A) cos 2x = 2 cos²x - 1 = 1 - 2 sin²x = cos²x - sin²x.
B) sin 2x = 2 sinx cosx.
C) sin3x = 3 sinx - 4 sin³x.
2) Find the value of the following:
A) sec30 tan 60+ sin45 cosec45+ cos30 cot60. 7/2
B) cot²60+ 3 cos² 60- 3/2 sec²45 - 8 sun²60. - 311/12
C) 4/3 cot²30+ 3 sin²60 - 2 cosec²60 - 3/4 tan²30 10/3
3) If x= 60 and y= 30 verify
A) sin(x+y)= sinx cosy + cosx siny
B) tan(x+y)= (tanx + tany)/(1- tanx tany)
C) sin(x+y). Sin(x - y)= sin²x - sin²y = cos²y - cos²x.
4) Show that
A) √{(1+ cos30)/(1- cos30){= sec 60 + tan 60.
B) √{(1+ cos30)/(1- cos30)} - cosec 30 = cosec 30 - √{(1- cos30)/(1+ cos30).
C) (√3+1)(3- cot 30)= tan³60 - 2 sin 60.
D) sec²45. Tan²30 - tan²45. sec²30 = tan²30 - tan²45.
E) (tan 60+ tan 30)(1- cot60. Cot30)+ (cot60+ cot30)(1- tan60 tan30)= 0.
5) If x and y are two positive acute angles such that cos(x - y)= 1 and cot(x + y)= 0, find x and y. 45
6) The angle C of a triangle ABC is obtuse and sin(B+ 2A)= tan(B - A) = 1. Find the angles A, B, C. 15,60,105
7) Solve:
A) sinx = cosx. 45
B) 2 cosx+ 5 tanx =4 secx. 30
C) tanx +√3 cotx =√3+ 1. 45, 60
D) sinx+ cosx = √2. 45
E) 7 sin²x + 3cos²x = 4. 30
F) 3 sec⁴x - 10 sec²x + 8 = 0. 30, 45
MEASUREMENT OF ANGLES
1) Express the following angles in both centesimal and circular measure:
A) 30°
B) 15°15'15"
C) 40°29"
2) The angles A of a parallelogram ABCD is 45°. Find the other angles in sexagesimial
3) The angles of a triangle in the ratio 1:4:5. Find them in centesimal. 20, 80, 100
4) Find in sexagesimial measure of each interior angle of a regular pentagon. 108°
5) The sum of two angles is equal to 112°. If the number of grades in one is twice the number of degrees in the other, find the angles in grades. 80, 44.44
Quadratic Equation
1) x² - x - 12= 0. 4, -3
2) 4x² - 16x +15= 0. 3/2, 5/2
3) 4x² - 4x - 3= 0. -1/2,3/2
4) x² - 0.5x + 0.06= 0. 0.2, 0.3
5) 2x² - 3x +1= 0. 1, 1/2
6) 2x² - 3x - 1= 0. 1/4 (3±√17)
7) x² - 2√3x - 3x- 13= 0. √3±4
8) x² +3x - (a-1)(a+2)= 0. 1-a, a+2
9) x² - (p+ 1/p) x + 1= 0. a, 1/a
10) x² - {a/(a+ b) + (a+b)/a} x + 1= 0. -a/(a+ b), -(a+b)/a
11) (x+2)/(x-2) + (x -2)/(x+2)= 11/2. ±2 √(15/7)
12) 1/(x+1) + 1/(x +3) = 1/(x +4) + 1/(x+6). (-7± √5)/2
13) 2x² - 3x -5= 0. 5/2, -1
14) √(x+2) + √(x -3) = 5. 7
15) 2√(x+5) - √(2x +8) = 2 ±4
16) √(2x+1) + √(3x +4) = 7 4, 480
17) 4x²+6x + √(2x²+ 3x +4) = 13. 1, -5/2
18) √(2x²- x+10) - √(2x²- x +3) =1. 2, -3/2
19) √{(x+1)/(x -1)} + √{(x -1)/(x+1)} = 5/√6. ±5
20) √{(1-x)/x} + √{x/(1- x)} = 13/6. 4/13, 9/13
21) √(x+5) + √(x +12)= √(2x +41) -21, 4
22) √(x² 8x+15) + √(x²+ 2x -15)= √(4x² - 18x +18) 3, 17/3
23) {√(x²+4) + √(x +1)}/{√(x² +4)- √(x +1)}= 3. 0, 4
24) 1/(x+ a+ b) = 1/x + 1/a + 1/b. - a, - b
Height and Distance
Type -1
1) What is the ratio between the height of a vertical pole and length of its shadow, when the elevation of the sun is
A) 30°. 1:√3
B) 45°. 1:1
C) 60°. √3:1
2) The length of the shadow of x metres high vertical tower is x√3. What is the elevation of the sun ? 30°
3) What is the angle of the elevation of the sun, when the length of shadow of a vertical Pole is equal to its height ? 45°
4) The height of a tree is √3 times, the length of its Shadow. Find the angle of the elevation the sun. 60
5) The angle of the elevation of the top of a tower from a point on the ground and at a distance of 160 m from its foot, is found to be 60°. Find the height of the tower. 277.12m
6) A kite is attached to a string. Find the length of the string, when the height of the kite 60°m and the string makes an angle of 30° with the ground. 120m
7) A boy, 1.6 m tall, is 20m away from a tower and observes the angle of the elevation of the top of the tower to be
A) 45°
B) 60°
Find the height of the tower in each case. 21.6m, 36.24m
8) A vertical flagstaff stands on a horizontal plane. From a point 80 m from its foot, the angle of elevation of its top is found to be 30°. Find the height of the flagstaff. 46.19m
9) The upper part of a tree, broken over by the wind, makes on angle of 45° with ground; and the distance from the root to the point where the top touches the ground, is 15m. What was the height of the tree before it was broken. 36.21m
10) The angle of Bangalore elevation of the top of an unfinished tower at a point distance 80m from its base is 30°. How much higher must the Tower be raised so that its angle of elevation at the same point may be 60°. 92.37m
11) The angle of elevation of the top of a tower, which is incomplete 45° from a point on the level ground and at a distance of 150 m from the base of the tower. How much higher should it be raised so that the elevation of the top of the tower may become 60° at the same point? 109.8m
12) At a particular time, when the sun's altitude is 30°, the length of the shadow of a vertical Tower is 45°m. Calculate
A) the height of the tower.
B) the length of the shadow of the same Tower, when the sun's altitude is
A) 45°. 25.98m
A) 60°. 25.98m
Type -2
1) The length of the shadow of a vertical tower on level ground increases by 10m, when the altitude of the sun changes from 45° to 30°. Calculate the height of the tower. 13.66m
2) An observer on the top of a cliff, 200 m above the sea level, observes the angle of a depression of the two ships to be 45° and 30° respectively. Find the distance between the ships, if the ships are
A) on the same side of the cliff.
B) on the opposite sides of the cliff. 146.4m, 546.4m
3) A man on the top of a vertical observation Tower of a car moving at uniform speed coming directly towards it, if it takes 12 minutes for the angle of depression to change from 30° to 45°. How soon after this will be the car reach the observation Tower ? 16.39 minutes
4) Find the height of a tree when it is found that walking away from it 20 m, in a horizontal line through its base, the elevation of its top changes from 60° to 30°. 17.32m
5) A person standing on the bank of a river observes the angle of elevation of the top of a tree, on a opposite bank, to be 60°. When he retires 30m from the bank, he finds the angle of the elevation to be 30°. Find the height of the tree and the breadth of the river. 25.98m, 15m
6) Find the height of a building, when it is found that on walking towards it 40 m in a horizontal line through its base the angular elevation of its top changes from 30° to 45°. 54.64m
7) The shadow of a tower standing on a level ground is found to be 40m longer, when Sun's altitude decreases from 45° to 30°. Find the height of the tower. 54.64m
8) Two pillars of equal heights stands on either side of a roadway, which is 150m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are 60° and 30°, find the height of the pillars and the position of the point. 64.95m, 37.5m
9) The angle of the elevation of the tower of the top of a tower is observed to be 60°. At a point 30m vertically above the first point of the observation, the elevation is found to be 45°. Find
A) the height of the tower. 70.98m
B) its horizontal distance from the points from the points of observation. 40.98m
10) From the top of a cliff, 60m high the angles of depression of the top and bottom of a tower are observed to be 30° and 60°. Find the height of the tower. 40m
11) The angle of elevation of the top P of a vertical Tower PQ from a point X is 60°, at a point Y, 40 m vertically above X, the angle of elevation is 45°.
A) Find the height of the tower PQ.
B) Find the distance XQ. 95, 55
(Give your answers to the nearest metre)
12) A man on a cliff observes a boat at an angle of depression 30° which is sailing towards the shore to the pointing immediately beneath him. 3 minutes later the angle of depression of the boat is found to be 60° assuming that the boat sail at a uniform speed, determine:
A) how much more time it will take to reach the shore. 1.5min
B) the speed of the boat in metres per second. If the height of the cliff is 500m. 3.21 m/s
13) A man in a Boat rowing away from a Lighthouse 150m high, takes 2 minutes to change the angle of elevation of the top of the Lighthouse from 60° to 45°. Find the speed of the boat. 0.53 m/sec
14) A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle evelvation to be 30°. Find
A) the height of the tree, correct to 2 decimal places. 34.64m
B) the width of the River. 20m
15) The horizontal distance between the two Towers is 75m and the angular depression of the top of the first tower as seen from the top of the second, which is 160m high, is 45°. Find the height of the first tower. 85m
16) The length of the shadow of a tower standing on a level plane is found to be 2y metre longer when the sun's altitude is 30° than when it was 45°. Prove that the height of the tower is y(√3+1) m.
17) An aeroplane flying horizontally 1 km above the ground is observed at an elevation at 60°. After 10 seconds its elevation is observed to the 30°, find the uniform speed of the Aeroplane in km per hour. 415.67 km/hr
1) cos x/(secx - tanx)= 1 + sinx
2) (1+ cos x - sin²x)/{sinx(1+ cos x)})= cotx
3) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1
4) sinx/(1+ cos x) + (1+ cosx)/sin x = 2 cosec x
5) (1+cos x)/(1 - cosx)= (cosec x + cot x)²
6) (1+sin x)/(1 - sinx)= (sec x + tan x)²
7) √{(1- sin x)/(1 + sinx)= sec x - tan x
8) (1+cos x + sinx)/(1 - cosx + sinx)= sin x/(1- cos x) = (1 + cos x)/sinx
9) (secx+tan x)/(cosec c + cotx) - (sec x - tanx )/(cosecx - cotx) = 2(secx - cosecx)
10) (secx- cos x)(cosecx - sinx)= tanx/ (1+ tan²x)
11) (1+secx + tanx)(1 - secx + tanx)= 2 tan x
12) (1+ cosecx + cotx)(1 - cosecx + cotx)= 2 cot x
13) sec⁴x + tan⁴x = 1+ 2 sec²x tan²x.
14) sinx/(1- cot x) + cos x/(1 - tanx) = sinx + cos x
15) (sin x - secx)² + (cosx - cosec x)² = (1- secx cosecx)²
16) (1+ tanx)/(1- tan x) - (1 - tanx)/(1 + tanx)= 4 sinx cosx/(1- 2 sin²x)
17) tanx/(1+tan²x)² + cotx/(1 + cot²x)²= sin x cosx
18) 1/(cosecx + cotx) - 1/sinx = 1/sinx - 1/(cosecx - cotx).
19) √{(1+ cos)/(1+ cosx)} = cosecx - cot x)
20) (1 + secx + tanx)= 2/(1- cosecx + cot x)
21) (secx - tanx)/(secx + tanx)= 1 - 2 secx tan x + 2 tan²x.
22) (1 + tan²x)/(1 + cot²x)= {(1- tan x)/(1- cotx)}²
23) {(1+sin x)/(1- sinx)} - sec x = secx- √{(1 - sinx)/(1 + sinx).
24) sin⁶x + cos⁶x = 1 - 3sin²x cos²x.
25) cosec⁶x - cot⁶x = 1 + 3 cosec²x cot²x.
26) (cosx + cosy)/(sinx - sin y) = (sinx + siny)/(cosy - cos x).
27) sec²x tan²y - tan²x sec²y = 3tan²y - tan²x.
28) If sinx = (sinx+ sin y)/(1+ sin x sin y), Prove cos x = = cos x cos y/( 1 + sinx sin y).
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