HEIGHT AND DISTANCE ALL CONCEPT IN ONE GO

Trigonometry:

I.   sin  =

Perpendicular

=

AB

;

Hypotenuse

OB

ii.   cos  =	Base	=	OA	;
	Hypotenuse		OB

iii.  tan  =	Perpendicular	=	AB	;
	Base		OA

iv.  cosec  =	1	=	OB	;
	sin		AB

v.   sec  =	1	=	OB	;
	cos		OA

vi.  cot  =	1	=	OA	;
	tan		AB

Trigonometrical Identities:

i.        sin²  + cos²  = 1.

ii.        1 + tan²  = sec² .

iii.        1 + cot²  = cosec² .

Values of T-ratios:

	0°	(/6) 30°	(/4) 45°	(/3) 60°	(/2) 90°
sin	0		1 √2	√3 2	1
cos	1	√3 2	1 √2		0
tan	0	1 √3	1	√3	not defined

     Angle of Elevation:

Suppose a man from a point O looks up at an object P, placed above the level of his eye. Then, the angle which the line of sight makes with the horizontal through O, is called the angle of elevation of P as seen from O.

 Angle of elevation of P from O = AOP.

Angle of Depression:

Suppose a man from a point O looks down at an object P, placed below the level of his eye, then the angle which the line of sight makes with the horizontal through O, is called the angle of depression of P as seen from O.

Q1-Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is:

A. 173 m B. 200 m C. 273 m D. 300 m Explanation: Let AB be the lighthouse and C and D be the positions of the ships. Then, AB = 100 m, ACB = 30° and ADB = 45°. AB = tan 30° = 1          AC = AB x 3 = 100ROOT3 m. AC 3 AB = tan 45° = 1          AD = AB = 100 m. AD  CD = (AC + AD) = (100√3 + 100) m = 100(√3 + 1) = (100 x 2.73) m = 273 m.

Q2-The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:

A. 2.3 m B. 4.6 m C. 7.8 m D. 9.2 m Explanation: Let AB be the wall and BC be the ladder. Then, ACB = 60º and AC = 4.6 m. AC = cos 60º = 1 BC 2  BC = 2 x AC = (2 x 4.6) m = 9.2 m.

ALL CONCEPT OF PERCENTAGE IN ONE GO

1.    Concept of Percentage:

By a certain percent, we mean that many hundredths.

Thus, x percent means x hundredths, written as x%.

To express x% as a fraction: We have, x% =	x	.
	100

Thus, 20% =	20	=	1	.
	100		5

To express	a	as a percent: We have,	a	=		a	x 100	%.
	b		b			b

Thus,	1	=		1	x 100	%	= 25%.
	4			4

2.    Percentage Increase/Decrease:

If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is:

	R	x 100	%
	(100 + R)

If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure is:

	R	x 100	%
	(100 - R)

3.    Results on Population:

Let the population of a town be P now and suppose it increases at the rate of R% per annum, then:

1. Population after n years = P		1 +	R		n
			100

2. Population n years ago =	P
	1 + R n 100

4.    Results on Depreciation:

Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then:

1. Value of the machine after n years = P		1 -	R		n
			100

2. Value of the machine n years ago =	P
	1 - R n 100

3. If A is R% more than B, then B is less than A by		R	x 100	%.
		(100 + R)

4. If A is R% less than B, then B is more than A by		R	x 100	%.
		(100 - R)

Q1:- A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total score did he make by running between the wickets?

A. 45% B. 45 5 % 11 C. 54 6 % 11 D. 55% Explanation: Number of runs made by running = 110 - (3 x 4 + 8 x 6) = 110 - (60) = 50.  Required percentage = 50 x 100 % = 45 5 % 110 11

Q2-Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are:

A. 39, 30 B. 41, 32 C. 42, 33 D. 43, 34 Explanation: Let their marks be (x + 9) and x. Then, x + 9 = 56 (x + 9 + x) 100  25(x + 9) = 14(2x + 9)  3x = 99  x = 33 So, their marks are 42 and 33. Q3- In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is  of the number of students of 8 years of age which is 48. What is the total number of students in the school? A. 72 B. 80 C. 120 D. 150 E. 100 Explanation: Let the number of students be x. Then, Number of students above 8 years of age = (100 - 20) % of x = 80% of x. 80% of x = 48 + 2 of 48 3 80 x = 80 100  x = 100.

Q4-In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was:

A. 2700 B. 2900 C. 3000 D. 3100 Explanation: Number of valid votes = 80% of 7500 = 6000. Valid votes polled by other candidate = 45% of 6000 = 45 x 6000 = 2700. 100

Q5-Two tailors X and Y are paid a total of Rs. 550 per week by their employer. If X is paid 120 percent of the sum paid to Y, how much is Y paid per week?

A. Rs. 200 B. Rs. 250 C. Rs. 300 D. None of these Explanation: Let the sum paid to Y per week be Rs. z. Then, z + 120% of z = 550.  z + 120 z = 550 100 11 z = 550 5  z = 550 x 5   = 250. 11