Permutation & Com- Com-bination
COORDINATE GEOMETRY Introduction
Coordinate geometry is a branch of mathematics that uses the principles of algebra to study geometry.
Figures like lines and circles can be represented using algebraic equations and their properties can be studied using these equations. Coordinate geometry also helps us understand the behaviour of functions.
A one-dimensional coordinate system, used to represent points, is the number line similar to the one that we have studied in Number Systems.
A two-dimensional coordinate system is used to represent two-dimensional figures like lines, circles and other curves.
The Coordinate System
Two perpendicular number lines, XOX' and YOY', intersecting at O form the coordinate system.
Quadrants, Axes and Origin
The line XOX' is called the x-axis and the line YOY' is called the y-axis. The x-axis and the y-axis are called the coordinate axes. The point O is called the origin. The axes divide the plane on which they are drawn into four parts known as quadrants and the plane is called the x-y plane.
Coordinates of a Point
Any point on the x-y plane can be identified by coordinates. The x-coordinate is the distance of the point from the y-axis and the y-coordinate is the distance of the point from the x-axis. The x-coordinate is also known as the abscissa and the y-coordinate
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is also known as the ordinate. Any point can be represented by an ordered pair of x and y coordinates as (x, y). For example, a point A with x-coordinate 4 and y-coordinate 3 can be represented as A (4, 3). The distance of this point A is 4 units from the y-axis and 3 units from the x-axis.
The signs of the x and y-coordinates of a point change depending on the quadrant in which it lies.
Quadrant x-coordinate y-coordinate I Positive Positive II Negative Positive III Negative Negative IV Positive Negative Remember:
The points lying on the axes are not considered to be in any quadrant.
Distance Formula and Section Formula The Distance Formula
The distance between any two points lying on a line parallel to the x-axis and having x-coordinates
Similarly, the distance between any two points lying on a line parallel to the y-axis and having y-coordinates
Distance between two points
is given by
Lines Positive and Negative Angles
The angles measured anti-clockwise from the positive direction of the x-axis are considered to be positive. In the following figure, the first two angles are positive.
The angles measured clockwise from the positive direction of the x-axis are considered to be negative.
The third angle is a negative angle.
Slope of a Line
The slope of a line, generally denoted by m, is the slant of a line. The slope of a horizontal line is 0. Its magnitude goes on increasing as we gradually make the line vertical as shown.
Clearly, the slope depends on the angle that it makes with the horizontal. It can be calculated in two ways.
1. If the angle made by the line in the positive direction of the x-axis is known, and is say , the slope is given by:
is called the inclination of the line . Slopes can be positive or negative depending on This is illustrated in the following table.
2. If are any two points on a line, the slope is given by:
Slopes of Parallel Lines
Parallel lines make equal angles with the positive direction of the x-axis. Hence, the slopes of parallel lines are always equal.
Slopes of Perpendicular Lines
If there are two perpendicular lines, then the product of their slopes will be -1.
Remember: Slope of the x-axis is 0 and the slope of the y-axis is .
Equation of a Line
As stated earlier, coordinate geometry helps us study geometry using algebraic principles. We can describe a line completely using an algebraic equation. Now let us first try to understand how a line is described.
We have seen the concept of slope. It denotes the slant of a line. If we consider a line with slope 1, we know that it makes an angle of with the positive direction of the x-axis.
All the above lines have slope 1. But all the lines are different. If we know any one point on the line in addition to the slope, we will know exactly where the line lies. The equation of a line can be stated in various forms as described below.
Slope-point Form
The equation of a line having slope m and passing through the point
Remember: Any point on a line always satisfies the equation of the line.
Two Point Form
As stated earlier, we need a point on a line and the slope to write the equation of the line. When we have two points, we can find the slope. Using this slope and any one of the given points, the equation of the line can be found.
The equation of a line passing through points is given by
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1. What is half of ?
a. b.
c. d. None of the above
2. If , then, what is the value of a?
a. 9/10 b. 5/7
c. 2/3 d. Cannot be determined 3. Which of the given numbers is largest?
a. b.
c. d.
4. Which of the given values are negative?
a.
b.
c.
d. None of the above
5. Which of the given numbers is larger:
a. b.
c. Both are equal d. None of the above 6. If , then what is the value
of ?
a. 0 b. 1
c. abc d. none of the above 7. What is the value of ?
a. 2 b. 5
c. 1 d. 0
8. Find the number of digits in if log 3= 0.477
a. 12 b. 14
c. 15 d. 16
9. If then what is the value of
?
a. b.
c. d.
10. If is the arithmetic mean of , then what is the value of x
a. 2 b. 3
c. 4 d. Either 2 or 3
11. Find the value of x if .
a. 10 b. 1
c. d. None of the above
12. If x, y and z are in GP, then are in
a. AP b. GP
c. HP d. None of the above
13. If , then what is the value
of x?
a. 2 b. 3
c. 5 d. Either 2 or 5
14. Which of the given value is higher:
a.
b.
c. Both of these numbers are equal d. Cannot be determined
15. Find the percentage increase in income of a person who decided to sell his Rs 100 shares being sold at a premium of 20% and paying 12% dividend and reinvest his money in another Rs 100 share sold at a discount of 20% paying a 10% dividend.
a. 12 % b. 25 %
c. 20% d. 22%
16. Find the return on investment for a person who has invested all his savings in Rs 10 share whose market value was Rs 40 paying a dividend of 25%.
a. 25% b. 12.5%
c. 6.25% d. 5%
17. Find the annual income of a person who has invested Rs 1,00,000 in a Rs 10 share selling at Rs 12.5 if the dividend percentage on the share
is 12 %.
a. Rs 9000 b. Rs 9,200 c. Rs. 9,600 d. Rs 9,800
18. A person earns Rs 1,200 through his investments on a share paying Rs 6 per share and the market price for the share is Rs 120. How much did he invest in the shares?
a. Rs 12,000 b. Rs 10,000 c. Rs 8,000 d. Rs 5,000
19. Ramu sold all his Rs 100 shares at Rs 120 and invested all his money in another Rs 10 share selling at Rs 12. Find the percentage change in his income if initially he was earning Rs 1000 and both the shares are paying a dividend of 10%.
a. His income will increase b. His income will decrease c. His income will remain same d. Depend upon the investment
20. Yadav ji had to raise funds for his tabela. He decided to float shares for his tabela at Rs 100 but sold these shares at Rs 120. He needed exactly Rs 96000 for his tabela. Find how much he needs to pay per annum as dividend if he raised exactly the same amount of money which he needed and has already declared a payout of 12 %.
a. Rs 9,200 b. Rs 9,600 c. Rs 9,800 d. Rs 10,000
21. Sheela and Munni invested Rs 1,00,000 and Rs 80,000 respectively in two shares. Ratio of dividend percentages was a:b and ratio of their market price was b:a. Find the ratio of the face value of the shares if both of them were earning same amount of money from their respective investments.
a. b.
c. d.
22. Calculate
a. b.
c. d. None of these
23. A particular angle is given in degree. Which of the given factors should be multiplied to the given angle to convert them into radians ?
a. b.
c. d.
24. Angle of elevation of the top of a building from a 100 m far point was 60°. Find the height of the building.
a. 100 b.
c. d. 50
25. After moving 100 m angle of elevation of top of building decreased from 45 to 30. What could be the height of the building? Assume that initially person was moving towards the building and after reaching building he started moving away from the building and he moved in a straight line only.
a. b.
c. d. None of these
26. There are six circles of equal radius inscribed in a triangle as shown in the figure. If radius of the circle is 1 cm then find the length of the side of the triangle?
a. b.
c. d.
27. Find the length of the thread required to tie three cylinders of radius 10 m.
a. b.
c. d.
28. Find the value of
a. 3/5 b. 4/3
c. 5/3 d. 5/4
29. Which of the given relationship is true
a. b.
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c. d.
30. Which of the following is an odd function
a. b.
c. d. None of the above
31. If
a. 11/3 b. -11/3
c. 3/11 d. -3/11
32. Shadow of a 200 m high building on a sunny day was 150 m. What will be the length of the shadow of a 15m tall pole?
a. 10 m b. 12m
c. 13.5 m d. 11.25 m
33. In the given figure if AB = 150 m, then what is the length of CD?
a. b.
c. d. 100
34. Refer to the figure given in question number 33, if CD = 150 m then what is the value of AB ?
a. b.
c. d. 100
35. Refer to the figure given in question number 33, if BC = 150 m then what is the length of CD?
a. b.
c. 100 d. 300
36. In the given figure, if AD = 100 m, then what is the length of BC?
a. b.
c. 300 d. 250
37. Refer to figure given in question number 36, if BC = 100 m, then what is the length of AD?
a. b.
c. d. 150
38. Refer to figure given in question number 36, if BD = 100 m, then what is the length of DC?
a. 100 b. 150
c. d.
39. Find the equation of a line passing though point (3, 4) which makes an angle of 60° with the x-axis.
a. b.
c. d.
40. Find the equation of a line which passes through points (3, 5) and (4, 4)?
a. 3x+4y = 5 b. x+y = 5 c. x+y = 8
d. 3x+5y = 16
41. Find the point which divides the line joining point (6, 8) and (-3, -7) in a ratio 2 : 1.
a. (0, -2) b. (2, 0) c. (2, 5) d. (1, -2)
42. Find the equation of a circle with center as (3,4) and radius of the circle as 3 units.
a.
b.
c.
d.
43. Find the value of a + b if (6, 9) is the mid-point of (4, 7) and (a, b)
a. 17 b. 19
c. 21 d. 23
44. Which of the following can be the equation of the circle given in the figure if the coordinates of the center of the circle are (5, 4)?
a.
b.
c.
d.
45. Line will not pass through which of the given points?
a. (11, 8) b. (-11, 8) c. (8, 11) d. (-8, 11)
46. Which of the following statements are true regarding the line whose equation is
i. If value of x increases, then value of y also increases.
ii. If value of x increases, then value of y decreases.
iii. If value of x decreases, then value of y also increases.
iv. If value of x decreases, then value of y also decreases.
a. (i) and (iv) b. (ii) and (iii) c. (i) and (ii) d. (iii) and (iv) 47. is equation of a_______
a. Circle b. Parabola
c. Ellipse d. Hyperbola
48. Find the area of a triangle whose vertices are .
a. b. 6
c. d. Cannot be determined
49. What will be the coordinates of the circumcenter of a triangle whose vertices are (0, 0), (0, 8) and (10, 0)?
a. (5, 6) b. (2, 3)
c. (5, 4) d. (3, 5)
50. Does ellipse has a constant radius?
a. Yes b. No
c. Depends on the axis d. None of the above