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<p>A. Aptitude:I.1.</p>
<p>Aptitude Questions and AnswersProblems on Trains 1. km/hr to m/s conversion: 5 a km/hr = ax m/s. 18 2. m/s to km/hr conversion: 18 a m/s = ax km/hr. 5</p>
<p>3. Formulas for finding Speed, Time and Distance1. Speed, Time and Distance: Distance Distance Speed = , Time = , Distance = (Speed x Time). Time Speed 2. km/hr to m/sec conversion: 5 x km/hr = xx m/sec. 18 3. m/sec to km/hr conversion: 18 x m/sec xx km/hr. = 5</p>
<p>4. If the ratio of the speeds of A and B is a : b, then the ratio of the1 : or b : a. a b 5. Suppose a man covers a certain distance at x km/hr and an equal distance at ykm/hr. Then, 2xy the average speed during the whole journey is km/hr. x+ y the times taken by then to cover the same distance is 1</p>
<p>4. Time taken by a train of length l metres to pass a pole or standing man or a signal post is equal to the timetaken by the train to cover l metres.</p>
<p>5. Time taken by a train of length l metres to pass a stationery object of length bmetres is the time taken bythe train to cover (l + b) metres.</p>
<p>6. Suppose two trains or two objects bodies are moving in the same direction at um/s and v m/s, where u > v,then their relative speed is = (u - v) m/s.</p>
<p>7. Suppose two trains or two objects bodies are moving in opposite directions at um/s and v m/s, then theirrelative speed is = (u + v) m/s.</p>
<p>8. If two trains of length a metres and b metres are moving in opposite directions atu m/s and v m/s, then:sec. (u + v) 9. If two trains of length a metres and b metres are moving in the same direction atu m/s and v m/s, then: The time taken by the faster train to cross the slower train (a + b) sec. = (u - v) 10. If two trains (or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then: (A's speed) : (B's speed) = (b : a) The time taken by the trains to cross each other = (a + b)</p>
<p>2.</p>
<p>Time and Distance</p>
<p>Time and Distance 1. Speed, Time and Distance: Distance Speed = , Time = Time 2. km/hr to m/sec conversion: 5 x km/hr = xx m/sec. 18 3. m/sec to km/hr conversion: 18 x m/sec xx km/hr. = 5</p>
<p>Distance Speed</p>
<p>, Distance = (Speed x Time).</p>
<p>4. If the ratio of the speeds of A and B is a : b, then the ratio of thethe times taken by then to cover the same distance is 1</p>
<p>5.</p>
<p>1 : or b : a. a b Suppose a man covers a certain distance at x km/hr and an equal distance at ykm/hr. Then, 2xy the average speed during the whole journey is km/hr. x+ y</p>
<p>3.</p>
<p>Height and Distance</p>
<p>4.</p>
<p>Time and Work1. Work from Days: If A can do a piece of work in n days, then A's 1 day's work = 2. Days from Work: If A's 1 day's work = 1 , then A can finish the work in n days. n 1 n</p>
<p>.</p>
<p>3. Ratio: If A is thrice as good a workman as B, then: Ratio of work done by A and B = 3 : 1. Ratio of times taken by A and B to finish a work = 1 : 3.</p>
<p>5.</p>
<p>Simple Interest</p>
<p>Simple Interest 1. Principal: The money borrowed or lent out for a certain period is called the principal or thesum. 2. Interest: Extra money paid for using other's money is called interest. 3. Simple Interest (S.I.): If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest. Let Principal = P, Rate = R% per annum (p.a.) and Time = T years. Then PxRx T (i). Simple Intereest = 100 100 x S.I. 100 x S.I. 100 x S.I. and T (ii). P = ;R= . = RxT PxT PxR</p>
<p>6.</p>
<p>Compound Interest</p>
<p>7.</p>
<p>Profit and Loss</p>
<p>Profit and Loss IMPORTANT FACTS Cost Price: The price, at which an article is purchased, is called its cost price, abbreviated as C.P. Selling Price: The price, at which an article is sold, is called its selling prices, abbreviated as S.P. Profit or Gain: If S.P. is greater than C.P., the seller is said to have a profit or gain. Loss: If S.P. is less than C.P., the seller is said to have incurred a loss. IMPORTANT FORMULAE 1. Gain = (S.P.) - (C.P.) 2. Loss = (C.P.) - (S.P.) 3. Loss or gain is always reckoned on C.P. 4. Gain Percentage: (Gain %) Gain x 100 Gain % = C.P. 5. Loss Percentage: (Loss %) Loss x 100 Loss % = C.P. Selling Price: (S.P.) (100 + Gain %) SP = x C.P 100 Selling Price: (S.P.) (100 - Loss %) SP = 100 Cost Price: (C.P.) C.P. = 9. 100 x C.P.</p>
<p>6.</p>
<p>7.</p>
<p>8.</p>
<p>(100 + Gain %) 100 (100 - Loss %)</p>
<p>x S.P.</p>
<p>Cost Price: (C.P.) C.P. = x S.P.</p>
<p>10. If an article is sold at a gain of say 35%, then S.P. = 135% of C.P. 11. If an article is sold at a loss of say, 35% then S.P. = 65% of C.P. 12. When a person sells two similar items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by: Common Loss and Gain % 2 x 2 Loss % = = . 10 10 13. If a trader professes to sell his goods at cost price, but uses false weights, then Error Gain % = x 100 (True Value) - (Error) %.</p>
<p>8.</p>
<p>Partnership</p>
<p>Partnership 1. Partnership: When two or more than two persons run a business jointly, they are calledpartners and the deal is known as partnership. 2. Ratio of Divisions of Gains:</p>
<p>When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments. Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year: (A's share of profit) : (B's share of profit) = x : y. II. When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital x number of units of time). Now gain or loss is divided in the ratio of these capitals. Suppose A invests Rs. x for p months and B invests Rs. y for q months then, (A's share of profit) : (B's share of profit)= xp : yq. 3. Working and Sleeping Partners: A partner who manages the the business is known as a working partner and the one who simply invests the money is a sleeping partner.</p>
<p>I.</p>
<p>9.</p>
<p>Percentage</p>
<p>Percentage 1. Concept of Percentage: By a certain percent, we mean that many hundredths. Thus, x percent means x hundredths, written as x%. x To express x% as a fraction: We have, x% = . 100 20 1 Thus, 20% = = . 100 5 a a a To as a percent: We have, = x 100 b b express b Thus, 1 4 = 1 4 x 100 = 25%.</p>
<p>%.</p>
<p>%</p>
<p>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 (100 + x 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: R n 1. Population after n years = 1+ P 100 P 2. Population n years ago = 1+ R 100n</p>
<p>4. Results on Depreciation: Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then: R n 1. Value of the machine after n years = P 1100 2. Value of the machine n years ago = P 1R 100n</p>
<p>R 3. If A is R% more than B, then B is less than A by (100 + R) R (100 - R) x 100 %.</p>
<p>4. If A is R% less than B, then B is more than A by</p>
<p>x 100</p>
<p>%.</p>
<p>10.</p>
<p>Problems on Ages</p>
<p>Problems on Ages 1. Odd Days: We are supposed to find the day of the week on a given date. For this, we use the concept of 'odd days'. In a given period, the number of days more than the complete weeks are calledodd days. 2. Leap Year: (i). Every year divisible by 4 is a leap year, if it is not a century. (ii). Every 4th century is a leap year and no other century is a leap year. Note: A leap year has 366 days. Examples: i. Each of the years 1948, 2004, 1676 etc. is a leap year. ii. Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year. iii. None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year. 3. Ordinary Year: The year which is not a leap year is called an ordinary years. An ordinary year has 365 days. 4. Counting of Odd Days: 1. 1 ordinary year = 365 days = (52 weeks + 1 day.) 1 ordinary year has 1 odd day. 2. 1 leap year = 366 days = (52 weeks + 2 days) 1 leap year has 2 odd days. 3. 100 years = 76 ordinary years + 24 leap years = (76 x 1 + 24 x 2) odd days = 124 odd days. = (17 weeks + days) 5 odd days. Number of odd days in 100 years = 5. Number of odd days in 200 years = (5 x 2) 3 odd days. Number of odd days in 300 years = (5 x 3) 1 odd day. Number of odd days in 400 years = (5 x 4 + 1) 0 odd day. Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days. Day of the Week Related to Odd Days:</p>
<p>2</p>
<p>No. of days: Day:</p>
<p>0 Su n.</p>
<p>1</p>
<p>2</p>
<p>3</p>
<p>4</p>
<p>5</p>
<p>6</p>
<p>Mon Tue We . s. d.</p>
<p>Thur Fri Sat s. . .</p>
<p>11.</p>
<p>Calendar</p>
<p>Calendar 1. Odd Days: We are supposed to find the day of the week on a given date. For this, we use the concept of 'odd days'.</p>
<p>2</p>
<p>In a given period, the number of days more than the complete weeks are calledodd days. 2. Leap Year: (i). Every year divisible by 4 is a leap year, if it is not a century. (ii). Every 4th century is a leap year and no other century is a leap year. Note: A leap year has 366 days. Examples: i. Each of the years 1948, 2004, 1676 etc. is a leap year. ii. Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year. iii. None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year. 3. Ordinary Year: The year which is not a leap year is called an ordinary years. An ordinary year has 365 days. 4. Counting of Odd Days: 1. 1 ordinary year = 365 days = (52 weeks + 1 day.) 1 ordinary year has 1 odd day. 2. 1 leap year = 366 days = (52 weeks + 2 days) 1 leap year has 2 odd days. 3. 100 years = 76 ordinary years + 24 leap years = (76 x 1 + 24 x 2) odd days = 124 odd days. = (17 weeks + days) 5 odd days. Number of odd days in 100 years = 5. Number of odd days in 200 years = (5 x 2) 3 odd days. Number of odd days in 300 years = (5 x 3) 1 odd day. Number of odd days in 400 years = (5 x 4 + 1) 0 odd day. Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days. Day of the Week Related to Odd Days:</p>
<p>No. of days: Day:</p>
<p>0 Su n.</p>
<p>1</p>
<p>2</p>
<p>3</p>
<p>4</p>
<p>5</p>
<p>6</p>
<p>Mon Tue We . s. d.</p>
<p>Thur Fri Sat s. . .</p>
<p>12.</p>
<p>Clock</p>
<p>Clock 1. Minute Spaces: The face or dial of watch is a circle whose circumference is divided into 60 equal parts, called minute spaces. Hour Hand and Minute Hand: A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called minute hand or long hand. 2. i. In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand. ii. In every hour, both the hands coincide once. iii. The hands are in the same straight line when they are coincident or opposite to each other. iv. When the two hands are at right angles, they are 15 minute spaces apart. v. When the hands are in opposite directions, they are 30 minute spaces apart. vi. Angle traced by hour hand in 12 hrs = 360 vii. Angle traced by minute hand in 60 min. = 360. viii. If a watch or a clock indicates 8.15, when the correct time is 8, it is said to be 15 minutes too fast. On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow.</p>
<p>13.</p>
<p>Average</p>
<p>Average 1. Average: Average =</p>
<p>Sum of observations Number of observations</p>
<p>2. Average Speed: Suppose a man covers a certain distance at x kmph and an equal distance at ykmph. 2x y Then, the average speed druing the whole journey is kmph</p>
<p>14.</p>
<p>Area</p>
<p>Area FUNDAMENTAL CONCEPTS 1. Results on Triangles: i. Sum of the angles of a triangle is 180. ii. The sum of any two sides of a triangle is greater than the third side. iii. Pythagoras Theorem: In a right-angled triangle, (Hypotenuse)2 = (Base)2 + (Height)2. iv. The line joining the mid-point of a side of a triangle to the positive vertex is called the median. v. The point where the three medians of a triangle meet, is called centroid.The centroid divided each of the medians in the ratio 2 : 1. vi. In an isosceles triangle, the altitude from the vertex bisects the base. vii. The median of a triangle divides it into two triangles of the same area. viii. The area of the triangle formed by joining the mid-points of the sides of a given triangle is onefourth of the area of the given triangle. 2. Results on Quadrilaterals: i. The diagonals of a parallelogram bisect each other. ii. Each diagonal of a parallelogram divides it into triangles of the same area. iii. The diagonals of a rectangle are equal and bisect each other. iv. The diagonals of a square are equal and bisect each other at right angles. v. The diagonals of a rhombus are unequal and bisect each other at right angles. vi. A parallelogram and a rectangle on the same base and between the same parallels are equal in area. vii. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area. IMPORTANT FORMULAE I. 1. Area of a rectangle = (Length x Breadth). Area Area Length = and Breadth = . Breadth Length II. 2. Perimeter of a rectangle = 2(Length + Breadth).</p>
<p>III.IV.</p>
<p>Area of a square = (side)2 = (diagonal)2. Area of 4 walls of a room = 2 (Length + Breadth) x Height. 1. 2. Area of a triangle = x Base x Height. Area of a triangle = s(s-a)(s-b)(s-c) where a, b, c are the sides of the triangle and s = (a + b + c). 3 Area of an equilateral triangle = x (side)2. 4 Radius of incircle of an equilateral triangle of side a = a . 2</p>
<p>V.</p>
<p>3. 4.</p>
<p>3 5. 6. VI. 1. 2. 3. 1. 2. 3. 4. Radius of circumcircle of an equilateral triangle of side a = Radius of incircle of a triangle of area Area of parallelogram = (Base x Height). Area of a rhombus = x (Product of diagonals).</p>
<p>a 3</p>
<p>. s</p>
<p>and semi-perimeter s =</p>
<p>VII.</p>
<p>Area of a trapezium = x (sum of parallel sides) x distance between them. Area of a circle = R2, where R is the radius. Circumference of a circle = 2 R. Length of an arc = Area of a sector = 2 R 360 , where is the central angle.</p>
<p>VIII.</p>
<p>1. 2.</p>
<p>1 R2 (arc x R) = . 2 360 Circumference of a semi-circle = R. Area of semi-circle = R2 2 .</p>
<p>15.</p>
<p>Volume and Surface Area</p>
<p>Volume and Surface Area 1. CUBOID Let length = l, breadth = b and height = h units. Then i. Volume = (l x b x h) cubic units.</p>
<p>ii. iii.</p>
<p>Surface area = 2(lb + bh + lh) sq. units. Diagonal = l2 + b2 + h2 units.</p>
<p>2. CUBE Let each edge of a cube be of length a. Then, i. Volume = a3 cubic units. Diagonal = 3a units. 3. CYLINDER Let radius of base = r and Height (or length) = h. Then, i. Volume = ( r2h) cubic units.</p>
<p>ii. iii.</p>
<p>Surface area = 6a2 sq. units.</p>
<p>ii. iii.</p>
<p>Curved surface area = (2 rh) sq. units. Total surface area = 2 r(h + r) sq. units.</p>
<p>4. CONE Let radius of base = r and Height = h. Then, i. Slant height, l = h2 + r2 units.</p>
<p>Total surface area = ( rl + r2) sq. units. 5. SPHERE Let the radius of the sphere be r. Then,</p>
<p>ii. iii. iv.</p>
<p>Volume =</p>
<p>r2h</p>
<p>cubic units.</p>
<p>Curved surface area = ( rl) sq. units.</p>
<p>Surface area = (4 r2) sq. units. 6. HEMISPHERE Let the radius of a hemisphere be r. Then,</p>
<p>i. ii.</p>
<p>Volume...</p>