Points:
A quadratic equation in the variable x is of the form ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
Different methods to findout the solutions of second degree equations
Completing the square method
Quadratic Formula (Shreedharacharya’s rule)method
√ 4
2
is the solutions of second degree equation ax2+bx+c=0,a≠0
b2‐4ac is the discriminant of ax2+bx+c=0
If b2‐4ac=0, then the equation has only one solution and the solution is –b/2a.
If b2‐4ac <0 (a negative number),then the equation has no solution
If b2‐4ac >0 (a positive number) then the equation has two different solutions
Sample Questions
PROBLEM
25. Find the roots of the second degree equation 2x2 - 7x +3 = 0 by the method of completing the square.
Answer
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25
of their perimeters is 40 m, find the sides of the two squares.Answer
Let the side of the squares be x and y meters.
According to the condition, x2 + y2 = 500 ….(1)
4x - 4y = 40 (x – y) = 10
y = x - 10
Substituting the value of y in (1), we get, x2 + (x - 10)2 = 500
26
Side of the second square, y = (20 - 10) = 10 m
PROBLEM
27. Three consecutive positive integers are taken such that the sum of the square of the first and the product of the other two is 232. Find the integers.
Answer
Let the three consecutive positive integers be x, x + 1, x + 2.
x2 + (x + 1) (x + 2) = 232 x2 + (x2 + 3x + 2) = 232 2x2 + 3x - 230 = 0
√ 4
2 3 √1849 x = 10 or -11.5 4
But, x is a positive integer, so, x = 10.
Thus, the numbers are 10, 11, and 12.
PROBLEM
28. The sum of the squares of two consecutive even numbers is 164. Find the numbers.
Answer
Let the consecutive numbers be x, x 2.
x2 x 2 2 164 x2 x2 4x 4 164 2x2 4x ‐ 160 0 x2 2x ‐ 80 0
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27
The numbers are 8 and 10.
PROBLEM
29. 250 Rupees is divided equally among a certain number of children. If there were 25 children more, each would have received 50 paise less. Find the number of children.
Answer
Let the number of children be x.
It is given that Rs 250 is divided amongst x children.
So, money received by each child 250/x If there were 25 children more, then
Money received by each child 250/ x 25 From the given information,
250 250
Since, the number of children cannot be negative, so, x 100.
Hence, the number of children is 100.
PROBLEM
30. By increasing the speed of a bus by 10 km/hr, it takes one and half hours less to cover a journey of 450 km. Find the original speed of the bus.
Answer
Let speed of the bus be x km/hr Time t = 450/x
If speed is x + 10, then time T = 450/(x + 10)
28
31. A person has a rectangular garden whose area is 100 sq m.
He fences three sides of the garden with 30 m barbed wire.
On the fourth side, the wall of his of his house is constructed;
find the dimensions of the garden.
Answer
Let the length and breadth of garden be x m and y m respectively.
Area of the garden = 100 sq m xy = 100 m2 or y =100/x
Suppose the person builds his house along the breadth of the garden.
Then, we have:
32. The hypotenuse of a right triangle is 20m. If the difference between the length of the other sides is 4m. Find the sides.
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29
Sides are 12cm and 16cm
PRACTICE EXERCISE
36) Perimeter of a rectangle is 40cm. If area is 96cm2, find the sides.
[12, 8]
37) If from a number, twice its reciprocal is subtracted we get 1. What is the number?
[2 or -1]
38) The sum of the squares of two consecutive odd positive integers is
290. Find them.
[11, 13]
39) Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.
[10, 6]
40) The speed of a boat in still water is 8 km/hr. It can go 15 km upstream 22 km downstream in 5 hours. Find the speed of the
stream.
[3 km/hr]
41) Find two consecutive numbers whose squares have the sum 85.
[6, 7]
42) Sum of 2 numbers is 12. If the sum of their squares is 90, find the numbers.
[9, 3]
30
43) If the sum of the squares of 2 consecutive even natural numbers is 244. Find the numbers.
[10, 12]
44) Square of a number is 60 more than 7times the number. Find the number.
[12 or -5]
45) The sum of squares of 2 consecutive odd numbers is 74. Which is
the smaller of the numbers?
[5 or 7]
EXERCISE
46) Anu is 4 years older than Vinu. If 4 is added to the product of their ages, the result is 169. What are their ages?
47) To the square of a natural number, four times the next natural number is added and the result is 36. What is number?
48) The difference of two numbers is 6 and their product is 16. What are the numbers
49) If from the squre of a number, six time the number is subtracted, we get 40. What is the number?
50) How many terms of the arithmetic progression 3, 7, 11… must be added to get 253?
51) If the product of a number with 6 more than the number is 160.
What is the number?
52) If the product of a number with 8 less than the number is 65, what is the number?
53) How many terms of the arithmetic progression 4, 10, 16… starting from the first, are to be added to get 252?
54) The width of a rectangle is 7metre more than its height and its area is 60 squre metres. Find the dimensions of rectangle.
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31
Trigonometry
Points:
If the angles of a triangle are 600,600,600; then its sides will be in the ratio 1:1:1
If the angles of a triangle are 450,450,900; then its sides will be in the ratio 1:1:√2
If the angles of a triangle are 300,600,900; then its sides will be in the ratio 1:√3:2
In triangle ABC Sin A=BC/AC Cos A= AB/AC Tan A=BC/AB
The angle of elevation of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level
The angle of depression of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level
Sample Questions
PROBLEM
33. One angle of a right triangle is 300 and its hypotenuse is 4cm.What is its area?
Answer
Triangle side ratio is ratio 1:√3:2
32
Altitude = hypotenuse × √3/2 = 4 × √3/2
= 3.46 cm
PROBLEM
34. One angle of a triangle 600 and the length of its opposite side is 4cm.What is its circumradius.
Answer
From figure, sin60 = 4/BD 0.8660 = BC/BD BD = 4.62 cm
Therefore radius = 2.31 cm
PROBLEM
35. Two sides of a triangle are 7 and 6 centimeters and the angle between them is 1200. Find length of third side?
Answer
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33
= 6×0.8660 = 5.2 AD = AC cos60
= 6×0.5 = 3
BD = BA + AD = 3+7=10 In triangle BDC
BC2 = BD2+CD2
= 102+5.22 = 100+27.04 = 127.04 BC = 11.3
Third side is 11.3cm
PROBLEM
36. In the figure, ∠B = 90◦; also, AB = 10 cm and ∠C = 60◦
(a) What is the measure of ∠A
(b) What are the lengths of AC and BC
Answer (a) ∠A=300
(b) sin 60 = AB/AC = 10/AC AC= 10/0.8660=11.55 cm tan 60 = AB/BC
BC = 10/1.73 =5.78 cm
PROBLEM
37. In the figure, ∠BAC = 90◦, AD=6cm, CD=9cm, ∠ACD = x
(a) What is tan x?
(b) How much is ∠BAD?
(c) What is the length of BD?
34
Answer
(a) tan x= AD/DC=6/9=2/3 (b) x
(c) tan x = BD/AD=BD/6 2/3=BD/6
BD=(2/3)× 6=4 cm
PROBLEM
38. In the figure, AQB is an arc of a circle centred at O. Also,
∠AOB = 120◦, ∠AOQ = 60◦, PQ = 3 cm What is the radius of the circle?
Answer
cos 60 = OP/AO=(r-3)/r 0.5 r = r - 3
r - 0.5 r = 3 r = 3/0.5 = 6cm
PROBLEM
39. The shadow of a tower standing on a level ground is found to be 45 m longer when the sun’s altitude is 30° than when it
Page
35
Answer
In Δ ABD, AB/BD = tan 30 h /(45+x)=1/√3
x = (√3h - 45) …..(1) In ΔABC,
AB/BC = tan 60 h / x = √3
x = h/√3 …….(2)
From equation (1) and (2), we get (√3h - 45) = h/√3
h = 38.97m
PROBLEM
40. The length of shadow of a tower is 24 m, when the sun is at an angle of elevation of 550. Find height of tower.
36
Answer
In the figure AB is tower.
tan 50 = AB/BC AB = CB tan 55 AB = 24 × 1.4281 = 34.2744
Height of the tower = 34.3 m
PROBLEM
41. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.
Answer
In figure PR = 28.5 In ΔPAR,
PR/AR = tan 30 28.5/AR = 0.5773
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37
PR/BR = tan 60 28.5/BR = √3
BR = 16.4545 ST = AR – BR
= 49.3634 – 16. 3634 = 32.9089 m
PROBLEM
42. The angle of elevation of the top of the building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building
In ΔABC, AB/BC=tan 60°
BC=50/√3 …..(1) In ΔDCB,
DC/BC = tan 30 h /BC = 1/√3 h= BC/√3 h = 16.67m
38
long. What is its circumradius?[2.13cm]
57) The angle between the radius and slant height of a cone is 600. Find the radius of the cone, if its slant height is 14 cm
[7cm]
58) A man standing on the deck of a ship, which is 10 m above water level. He observes the angle of elevation of the top of a hill as 60°
and the angle of depression of the base of the hill as 30°.
(a) Calculate the distance of the hill from the ship (b) Find the height of the hill.
[10√3m, 40m]
59) An observer in a lighthouse 100 m above the sea-level is watching the ship sailing towards the lighthouse. The angle of depression of the ship from the observer is 30°. How far is the ship from the lighthouse?
[100√3m]
60) A ladder is placed along a wall such that its upper end is touching the top of the wall. The foot of the ladder is 2m away from the wall and the ladder is making an angle of 60° with the level ground.
Find the height of the wall.
[3.46m]
61) The top of a tower is seen at an angle of elevation of 400 from a point 30m away from the base of the tower. What is the height of the tower?
[Sin 40 = 0.64; Cos 40 = 0.77; tan 40 = 0.84]
[25.20m]
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39
m towards the foot of the tower, its angle of elevation becomes 60°.
Find the height of the tower.
[94.64m]
63) Two men are on the opposite sides of a tower. They measure the angles of elevation of the top of the towers as 30° and 45°. If the height of the tower is 60 m, find the distance between them.
[163.92 m]
64) From the top of a building 60m high the angles of depression of the top and bottom are observed to be 30º and 60º.Find the height of the tower.
[40m]
65) The horizontal distance between 2 towers is 70m.The angle of depression of the top of first tower when seen from the top of second tower is 30º. If the height of the second tower is 120m, find the height of the first tower.
[79.6m]
66) An aero plane when 3000m high passes vertically above another aero plane at an instant when the angle of elevation of the two aero planes from the same point on the ground are 60 and 45
respectively. Find the vertical distance between the aero planes.
[1268m]
EXERCISE
67) A long pole leans against a short wall, making a 400 angle with the ground. The foot of the pole is 2metres away from the bottom of the wall. What is the height of the wall?
68) A man 1.7metres tall standing 10 metres away from a tree sees the top of the tree at an angle of elevation 500. What is the height of the tree?
69) When the sun is at an angle of elevation 480, the shadow of a tree is 18metres long. What is the height of the tree?
70) The difference in the lengths of the shadow of a tower when the sun is at angles of elevations 300 and 600 is 45 metres. Compute the height of the tower.
40
Solids
Points:
Square Pyramid:
Lateral surface area = 4×½(Base edge × Slant height) L.S.A 2
Total Surface Area = Base Area × Lateral surface area
T.S.A 2
Relations connecting base edge a, lateral edge e, slant height l, height h and base diagonal d:
1 41 41
4
Volume = Base area × height
Volume Cones
The radius of the sector becomes the slant height of the cone;
the arc length of the sector becomes the base circumference of the cone.
Suppose that a cone of base radius r and slant height l, radius of the sector l and the central angle x ,then
L.S.A of Cone 360
T.S.A of Cone
Volume
Page
41
Spheres
T.S.A 4
Volume = Hemispheres
L.S.A 2
T.S.A 3
Volume =
Sample Questions
PROBLEM
43. A toy in the shape of a square pyramid has base edge 16cm and slant height 10cm.
(a) Find lateral surface area.
(b) Find Total surface area.
(c) Calculate its volume.
Answer
L.S.A = 2 =2×16×10=320 cm2
T.S.A= 2 = 16×16 + 320=576 cm2
=100 – 64 6
Volume = = 256 × 6=1536 cm3
PROBLEM
44. Height of a cone is 40cm. Slant height is 41cm.
(a) Find diameter of its base.
(b) Find Volume
41 40 =81 9
∴ Diameter = 18cm
42
(a) What is the least area of leather required to make 50 such footballs?
(b) Also find volume of air inside 50 such footballs Answer
Diameter of a football is 30cm.
Surface area of a football 4
4 15 900 The least area of leather required to make 50 footballs
50 900 45000 Volume of air inside 50 such footballs
= 50 50 15 = 225000 cm3
PROBLEM
46. A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 19cm and diameter of the cylinder is 7cm.
(a) Find volume.
(b) Total Surface Area of solid.
Answer
Volume of solid 2
3.5 12 3.5 641.67 cm3
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43
PROBLEM
47. A circus tent is made of canvas and is in the form of a right circular cylinder and a right circular cone above it. The diameter and height of the cylindrical part of the tent are 126m and 5m respectively. The total height of the tent is 21m. Find the total cost of the tent if the canvas used costs Rs.12 per sq.m
Answer
Diameter and height of the cylindrical part of the tent are 126m and 5m. Total height of the tent is 21m.
16 63
256 3969 = 4225 65m
The total surface area 2
2 63 5 63 65
48. A cylindrical jar of radius 6cm contains oil. Iron spheres each of radius 1.5cm are immersed in the oil. How many spheres are necessary to raise the oil by 2cm?
Answer
Volume of cylinder with height 2cm = n Volume of iron spheres
Number of spheres are 16.
44
PROBLEM
49. A Gulab Jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found is 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm.
Answer
Volume of one gulab jamun = volume of cylindrical part + 2 × (volume of hemispherical part)
2 2
3
1.4 2.2 1.4 Volume of such 45 gulab jamun
7.97
Volume of Syrup 30% Volume of such 45 gulab jamun 338 cm3
PRACTICE EXERCISE
71) A solid is hemispherical at the bottom and conical above. If the curved surface area of the two parts are equal, then from the ratio of the radius and height of the conical part.
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45
cm. The total height of the toy is 15.5 cm.
(a) Find slant height.
(b) Find its total surface area.
[12.5 cm, 214.5 cm2]
73) A tent is of the shape of a right circular cylinder upto a height of 3 metres and conical above it. The total height of the tent is 13.5 metres above the ground. Calculate the cost of painting the inner side of the tent at the rate of Rs. 2 per square metre, if the radius of the base is 14 metres.
[2068]
74) A solid sphere of radius 3cm is melted and then cast into small spherical balls each of diameter 0.6cm.Find the number of small balls thus obtained.
[1000]
75) A cylinder of radius 12 cm contains water to a depth of 20cm, a spherical iron ball is dropped into the cylinder and thus the level
of water is raised to 6.75cm.Find the radius of the ball.
[9cm]
76) A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
[572 cm2]
77) A glass cylinder with diameter 20 cm has water to the height of 9 cm. A metal cube of 8 cm edge is immersed in it completely. Find the height by which the water will rise in the cylinder.
[1.6 cm]
78) The diameters of the internal and external surfaces of a hollow spherical shell are 6 cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm, find the height of the cylinder.
[8/3 cm]
46
79) If the number of square centimetres on the surface of a sphere is equal to the number of cubic centimetres in its volume, what is the diameter of the sphere?
[12 cm]
80) The diameter of a metallic solid sphere is 12 cm. It is melted and drawn into a wire having diameter of the cross-section 0.2 cm. Find the length of the wire.
[1188 cm2]
81) A semi-circular thin sheet of metal of diameter 28 cm, is bent to make an open conical cap. Find the capacity of the cap.
[622.38 cm3]
82) An ice-cream cone has a hemispherical top. If the height of the cone is 9 cm and base radius is 2.5 cm, find the volume of ice cream
cone. [91.7 cm3]
EXERCISE
83) The volume of a square pyramid is 720 cubic centimetres and its base edge is12 centimetres. Find its height.
84) Two square pyramids are of equal volume and the base edge of the first is double that of the second. What fraction of the height of the second is the height of the first?
85) Two square pyramids are of equal volume and the height of the first is double that of the second. What fraction of the base edge of the second is the base edge of the first?
86) Compute the curved surface area of a cone of base radius 12 cm and slant height 25cm.
87) What is the surface area of a cone with the diameter of the base 30cm and height 40cm?
88) If the volumes of two spheres are 27 cubic centimetres and 64 cubic centimetres, what is the ratio of their radii?
89) The ratio of the surface areas of two spheres is 3: 5. What is the ratio of their volumes?
90) Compute the surface area of the largest sphere that can be cut out
Page
47
their heights are in the ratio 1: 3. The volume of the first pyramid is 180 cubic centimetres. Compute the volume of the second.
92) A cylindrical vessel of base radius 10 centimetres and height 85 centimetres is completely filled with water. Spheres of radius 10 centimetres, as many as can be completely immersed in water, are put into the vessel. Find the volume of water remaining in the vessel.
93) A cylindrical rod of length 4 centimetres and diameter 4 centimetres is melted and recast into spheres of radius 2 centimetres. How many such spheres can be made?
94) A metal sphere of diameter 24 centimetres is melted and recast into cones of base radius and height 6 centimetres. How many such cones are made?
95) A metal sphere of diameter 24 centimetres is melted and recast into cones of base radius and height 6 centimetres. How many such cones are made?
96) The cost of painting a hemispherical paper weight was 80 rupees.
What will be the cost to paint a hemisphere of triple the radius at the same rate?
97) If the surface area of a solid hemisphere is 432π square centimetres, what is its radius?
98) If the inner radius of a hemispherical bowl is 60 centimetres, how many litres of water can it contain?
99) A petrol tank is in the shape of a cylinder with hemispheres of the same radius attached to both ends. If the total length of the tank is 6 metres and the radius is 1 metre, what is the capacity of the tank in litres?
100) A rocket is in the shape of a cylinder with a cone attached to one end and a hemisphere attached to the other. All of them of the same radius of 1.5 metre. The total length of the rocket is 7 metres and the height of the cone is 2 metres. Compute the volume of the rocket.
48 Coordinates
Points
The coordinates of a point are the distances of the point from x-axis and y-axis.
The coordinates of a point on the x-axis are of the form (x, 0) and of a point on the y-axis are of the form (0, y).
Sample Questions
PROBLEM
50. Find the coordinates of the other three vertices of the rectangle in the figure below.
Answer
T The coordinates of rectangle are (0,0),(5,0),(5,4) and (0,4)
PROBLEM
51. A circle is drawn with centre at (0, 0) and radius 6 units
51. A circle is drawn with centre at (0, 0) and radius 6 units