Icse Questions and Answers

8/2/2019 Icse Questions and Answers

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EXERCISE 21(A) ...- ----_.- .. _ - ~

1. The circumference of a circular field is440 m. Find:(i) its radius;(ii) its area;(iii) the cost of levelling the field at the rate

of ~ 17.50 per sq. m.2. The area of a circular plot is 9856 sq. m.Calculate the cost of fencing the plot at therate of ~ 6 per metre.

3. Find the circumference of the circle whosearea is 16 times the area of the circle withdiameter 1-4 m.

4. Calculate the circumference of a circle whosearea is equal to the sum of the areas of thecircles with diameters 24 em, 32 em and96 ern.

5. The ratio between the areas of two circles is9 : 25. Find the ratio between :(i) their radii; (ii) their circumferences.

6. The ratio between the circumferences of twocircles is 4 : 9, find the ratio between theirareas.

7. A racing track is bounded by two concentriccircles with circumferences 220 m and 286 m.Calculate the area of the racing track.

8, The circumference of a circular garden is572 m. Outside the garden a road, 35 m wide,runs around it. Calculate the cost of repairingthe road at the rate of ~ 375 per 100 sq. m.9. The diameter of the wheel of a car is 56 em,Calculate:(i) the number of times the wheel will rotate

in travelling through a distance of1056 km.

(ii) the speed of the car, in km per hour, ifits wheel makes 750 revolutions perminute.

A bucket is raised from a well by means of arope which is wound round a wheel ofdiameter 77 em. Given that the bucket ascentsin 1 minute 28 seconds with a uniform speedof 1.1 mis, calculate the number of completerevolutions ,the wheel makes in raising thebucket. I< )9 -: I11Ie wheel of a cart is making 5 revolutions::u second. If the diameter of the wheel is

em, find its speed in kmlhr. Give yourrer, correct to the nearest km. "

12. The area enclosed between the circumferencesof two concentric circles is 2464 sq. cm.If their radii are in the ratio 5 : 3; calculate:(i) the area of the outer circle;(ii) the circumference of the inner circle.If a third circle is drawn, outside the largercircle, so that the area enclosed between thiscircle and the larger circle is twice the areaenclosed between the given circles; calculatethe area of this circle.

13. A circular running track is the portion boundedby two concentric circles. If the cost of fixingthe fence along the outer circumference of thetrack is ~ 3,696 at ~ 24 per metre and the costof levelling the track is ~ 6,006 at~ 12 per sq. m; calculate the width of thetrack.

14. The given figure shows a rectangle ABCD anda circle in it.Given AB = = 105 em and BC = = 7 em.

D ,..".-~ "".f'F'.:~~r ............ --" ~ . .... C.1 .:~(~~~; .:~j5r/:_ ::';.:!" '. n b . : . ' ~ ; . 8

A ~i.~rik-~~)~~~:~~:..-~:..~: B(i) If the area of shaded portion is 4886 sq.

cm; calculate the radius of the circle.(ii) If the circumference of the circle is

15-4 em; find the area of the shadedportion.15. The circumference of a circle exceeds the

diameter by 168 em. Find the area of thecircle.

16. The sum of the radii of two circles is140 ern and the difference of theircircumferences is 88 em. Find the ratiobetween their areas.

17. Find the area of a circle whose circumferenceis equal to the sum of the circumferences withdiameters 36 em and 20 em.

18. A toothed wheel of radius 40 cm is attached toa smaller toothed wheel of diameter 24 cm. Howmany revolutions wilI the smaller wheel makewhen the larger one makes 150 revolutions., _ . _ - ._---_._-~. --'--'. Req. no. of rev. = I

1 Distance moved by bigger wheel in150 rev. :_________~~ lDistance moved by smaller wheel in 1rev. j

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--~ --- ------- --_ -----_ - - --~~=-- - - -

21. A sheet is 30 cm long and 10 em wide.Circular pieces, all of equal diameters, arecut from the sheet to prepare discs.Calculate the number of discs prepared ifthe diameter of each disc is :(i) 1 em (ii) 12 em

19. The side of a square is 20 em. Find theareas of the circumcribed and inscribedcircles.

20. Find the area of the region bounded bytwo concentric circles, if the length of thechord of the outer circle, touching theinner circle, is 28 cm. (iii) 15 em (iv) 25 em

_ ..._--_. cpn= -_---.-.

Each diameter of a circle divides the circle into twocongruent parts and each part is called a semi-circle. o:" 0 is'- . .:". -:In the adjoining figure, AB is a diameter of a circleand so APB is a semi-circle.

(i) n ,,'".. (circular part)= ~ x circumference of the circle.

1= 2 x 21tr = .r(ii) - II 0\;'::1' ':; ,,; t I~ ':I:i-,;ir.:'

= length of arc + diameter AB= ;".-.... r r.diameter = 2 x r]

(iii) 11 ~= 2 " x area of circle = ~?:. .... ..--.....-.. --.-..----.-------- .--.----.------.-----------J

~ Fi~d the ar~.~n~ :~~~~r ~ ! _ ~s~~.~~.~~~ ..~l.at~ o:_d~~~: 14 cm. ..Jflllt

~< - . .- - . . .. _ - - . ~14cm1t,-2= 21 22= '2 x "'7 X (7)2 sq. em

.. _ ...._ - _ ...._--_ .. .--,!S' 14 7 Imce; r = "2 ern = cm I ..;__ .__ ._---l

\ns.,,., "I ~I

1= 2 x (circumference of circle) + diameter= ~ x 21tr + diameter = 272 x 7 + 14 = \ ~ ern ADS.~n the adjoining figure; PS is a diameter of a circle and

is of length 6 ern, Q and R are points on the diameter suchthat PQ, QR and RS are equal. Semicircles are drawn withPQ and QS as diameters. Find the perimeter of the shadedportion.Also, find the area of the shaded portion.

A

p s I. . _ . J al..

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8 = > 2x2 - 5x - 3 = 0= > (x - 3) (2x + 1) = 0= > x - 3 = 0, or 2x + 1 = 0

1= > x = 3, or x = -- ..2_____ ... ~___ ... __ a _oa _ ... _ _ . . . . ........

111.

eDivide 8 into two parts such that the sum of their reciprocals is 1~.----_. -- - .. _ .... _ - -- _ . . . . ---_ . . . . - .... .. ... ... . -_.., .\" 1/11,) Let the two parts be x and 8 - x.1 1 8 8- x + x 8- + = = =::} =. 8- x 15 x(8 - x) 15

= > 120 = 8(8x - x 2 )= > x 2 - 8x + 15 = 0 r--...--'---.= > x = 5, or x = 3 On solving.

x = = 5 = > One part = 5 and other part = 8 - 5 = 3x = 3 = > One part = 3 and other part = 8 - 3 = = 5

'! II:8j!ie For the same amount of work, A takes 6 hours less than B. If together they Jcomplete th e work in 13 hours 20 minutes; find how much time will B alone take tocomplete the work .

:, .J .. .

If B alone takes x hours then A alone takes (x - 6) hours for the same work.r-'--' ._---- ....---..----.;. . (20) 40.: 13 hrs. 20 mm. = 13+ 60 hrs. = "3hrs .1 1.. - -+ -= =x -6 x 403

= > x+x-6(x-6)x340

= > 3x2 - 18x = = 80x - 240 i.e. 3x2 - 98x + 240 = = 0= > 3x2 - 90x - 8x + 240 = 0

8i.e. (x - 30) (3x - 8) = 0

x = 30> x = 30, or x = 3 i.e._ _ _ _ _ _ . ~ _ , . . . 0 r 4 - ~ ._ . . . . . . . . . _ . .. .

L The product of two consecutive integers is 56.Find the integers.

~ The sum of the squares of two consecutivenatural numbers is 41. Find the numbers.

3. Find the two natural numbers which differby 5 and the sum of whose squares is 97.

4. The sum of a number and its reciprocal is4.25. Find the number.1Q5


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5. Two natural numbers differ by 3. Find the Inumbers, if the sum of their reciprocals is 1~ .

6. Divide 15 into two parts such that the sum of I7 ; : : : : : ~ : h :::U~,"s of two positive Iintegers is 208. If the square of the largernumber is 18 times the smaller number, findthe numbers.

8. The sum of the squares of two consecutivepositive even numbers is 52. Find thenumbers.

9. Find two consecutive positive odd numbers,the sum of whose squares is 74.

10. The denominator of a fraction is one morethan twice the numerator. If the sum of thefraction and its reciprocal is 2.9; find thefraction.

11. Three positive numbers are in the ratio1 1 1 . th if h f thei2 : : 3 : 4 Fmd e numbers t e sum 0 theirsquares is 244.

12. Divide 20 into two parts such that three timesthe square of one part exceeds the other partby 10.

13. Three consecutive natural numbers are suchthat the square of the middle number exceedsthe difference of the squares of the other twoby 60.Assume the middle number to be x and forma quadratic equation satisfying the abovestatement. Hence; find the three numbers.

14. Out of three consecutive positive integers, themiddle number is p. If three times the squareof the largest is greater than the sum of thesquares of the other two numbers by 67;calculate the value of p.

15. A can do a piece of work in 'x' days and Bcan do the same work in (x + 16) days. If bothworking together can do it in 15 days;calculate 'x'.

16. One pipe can fill a cistern in 3 hours less thanthe other. The two pipes together can fill thecistern in 6 hours 40 minutes. Find the timethat each pipe will take to fill the cistern.

17. A positive number is divided into two parts suchthat the sum of the squares of the two parts I S20. The square of the larger part is 8 times thesmaller part. Taking x as the smaller part of thetwo parts, find the number. : _ : UJ

_________ ,~ ~ .,w_~ __

8.4 . . ,. :- _ ..... _ . - . - .. -- _ . - _ ._.- - - _ . - - - _ . _ . _ .._ .... - - ..- _ . . . _ _ .._------o The hypotenuse of a right triangle is 13 em and the difference between the othertwo sides is 7 em.

Taking 'x' as the length of the shorter of the two sides, write an equation in 'x' thatrepresents the above statement and also solve the equation to find the two unknownsides of the triangle.

Since, the shorter side = x em.:. Longer side = (x + 7) em.

Using Pythagoras Theorem, we get:x2 + (x + 7)2 = 132~ x2 + x2 + l4x + 49 = 169~ 2 x2 + 14x - 1 2 0 = 0

x2 + 7x - 60 = 0

x

G

S i

all

.. .... ---- . . . . . . . . . _,I : Dividing each term by 2 :....__ .---. _ . . . . . . . _.'

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are suchexceedsther two

and forme above

thee squarem of theby 67;

ys and Bs. If both15 days;

less thanfill thethe time

parts sucho parts istimes thepart of the

I,'...-..

On solving, it givesx = -12, or x = 5

Since the side of triangle cannot be negative, therefore, x = 5.

I) = (x + 7) em = (5 + 7) em = U P. Ans,and II I= x em ='. ,.t

- ----------,o The length of a verandah is 3 m more than its breadth. The numerical value of Iits area is equal to the numerical value of its perimeter. ' ;1(i) Taking 'x' as the breadth of the verandah, write an equation in 'x' that I

represents the above statement.(ii) Solve the equation obtained in (i) above and hence find the dimensions of :

the verandah.

==~-.--._..- "EXERCISE 8(B) ,-. . . . - . . . . ... _ . ._.". T h e sides of a right -angled triangle containing !

:h e right angle are 4x em and (2x - 1) cm. If ;: : . e area of the triangle is 30 cm-; calculate theiengths of its sides.= - 1he hypotenuse of a right-angled triangle is:6 em and the sum of other two sides is3.i em. Find the lengths of its sides.The sides of a right-angled triangle are.x - 1) em, 3x em and (3x + 1) ern, Find:ilthe value of x,ii) the lengths of its sides,m t its area.The hypotenuse of a right-angled triangle~ one side by 1 em and the other side by 18

Since breadth = x m :. Length = (x + 3)m(i) Given: Area of verandah = its perimeter

i.e. length x breadth = 2 (length + breadth)~ (x + 3) . x = 2(x + 3 + x)

. . - - _ o o; Numerically:

x2 + 3x = 4x + 6x- -.. t - , _ . II ~r"x2-x-6 = 0 r-:.----_.-(x - 3) (x + 2) = 0 . On factorising :.- - - .. , - - ......

.\U:,.and_ . ._ - _ . .. . . . . . . . ., _ . _ . . . . _ . . _ - . . . _ . . . . . , , _ -. . . . . . . . . . . . .. . ._ . ._ . . . .I I . -

(ii)

em; find the lengths of the sides of the triangle.5. The diagonal of a rectangle is 60 m more thanits shorter side and the larger side is 30 mmore than the shorter side. Find the sides ofthe rectangle.

6. The perimeter of a rectangle is 104 m and itsarea is 640 m2 Find its length and breadth.

7. A footpath of uniform width runs round theinside of a rectangular field 32 m long and24 m wide. If the path occupies 208 m2, findthe width of the footpath.

8. Two squares have sides x em and (x + 4) em.The sum of their areas is 656 sq. em, Expressthis as an algebraic equation in x and solve theequation to find the sides of the squares.

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9. The dimensions of a rectangular field are50 m by 40 m. A flower bed is prepared insidethis field leaving a gravel path of uniformwidth all around the flower bed. The total costof laying the flower bed and gravelling thepath at ~ 30 and ~ 20 per square metre,respectively, is ~ 52,000. Find the width of thegravel path.

10. An area is paved with square tiles of a certainsize and the number required is 128. If thetiles had been 2 em smaller each way, 200 tileswould have been needed to pave the samearea. Find the size of the larger tiles.

11. A farmer has 70 m of fencing, with which heencloses three sides of a rectangular sheep pen;the fourth side being a wall. If the area of thepen is 600 sq. m, find the length of its shorterside.

12. A square lawn is bounded on three sides by apath 4 m wide. If the area of the path is i - thatof the lawn, find the dimensions of the lawn.13. The area of a big rectangular room is 300 m2.If the length were decreased by 5 m and thebreadth increased by 5 m; the area would beunaltered. Find the length of the room.

8.SG By increasing the speed of a car by 10 kmIhr , the time of journey for a distance of72 km is reduced by 36 minutes. Find the original speed of the car. [2005]

Let the original speed of the car = x kmIhr.. Time taken by it to cover 72 km = 72 hrs [Time = = Distance]x Speed

New speed of the car = (x + 10) kmIhr:. New time taken by the car to cover 72 km 72= x+lO hrs

Given : Time is reduced by 36 minutes :72 72 36~----=x x+lO 60

On solving, we get : x = - 40 or x = 30Since speed cannot be negative hence the value of x = 30i.e. =o Car A travels x km for every litre of petrol, while car B travels (x + 5) km for. every litre of petrol.

(i) Write down the number of litres of petrol used by car A and car B in coveringa distance of 400 km.

(ii) If car A uses 4 litres of petrol more than car B in covering the 400 km, write'down an equation in x and solve it to determine the number of litres of petrol .used by car B for the journey. [1997]

Illi I(i) = -II~,., I\~u.,':

: '= , __'.108


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ich hep pen;of theshorters by a78 " that

m2nd theuld be

400 400 400x + 2000 - 400 xGiven: ---- = 4 i.e. = 4x x + 5 x(x + 5)4 (x2 + 5x) = 2000 i.e. x2 + 5x - 500 = 0x = -25, or x = 20 On solving

400I II = (x + 5) litres400--- litres = ii i Ar20+5

n md ..

(ii)-= >

= >= > x = 20 .; Distance cannot be negative

C D By selling an article for ~ 24, a trader loses as much per cent as the cost priceof the article. Calculate the cost price. - - - - - - - - - - - - - - ~ - - - - - - - - - - - - - - ~r '. "Let C.P. of the article be ~ x.

Loss = x% of C.P.2x x

=-x~x=~-100 100x - = 24

= >100

lOO x - x 2 = 2400 i.e. x 2 - lOOx + 2400 = 0c.P. - Loss = S.P.

On solving, we get : x = 60 and x = 40

(ii) If the second train takes 3 hours more thanthe first train, find the speed of each train.

5. A girl goes to her friend's house, which isat a distance of 12 km. She covers half ofthe distance at a speed of x km/hr. andthe remaining distance at a speed of(x + 2) kmIhr. If she takes 2 hrs 30 minutesto cover the whole distance, find x' .

6. A car made a run of 390 km in 'x' hours. Ifthe speed had been 4 kmlhour more, it wouldhave taken 2 hours less for the journey.Find 'x'.

EXERCISE S(C)1. The speed of an ordinary train is x km per hrand that of an express train is (x + 25) km perhr.(i) Find the time taken by each train to cover

300 km.(ii) If the ordinary train takes 2 hrs more than

the express train; calculate speed of theexpress train.

2. If the speed of a car is increased by 10 kmper hr, it takes 18 minutes less to cover adistance of 36 km. Find the speed of the car.

3. If the speed of an aeroplane is reduced by40 km per hr, it takes 20 minutes more to cover1200 km. Find the speed of the aeroplane.4. A train covers a distance of 300 km between two stations at a speed of 'x' kmIh. Another'train covers the same distance at a speed of ;(x - 5) kmIh.(i) Find the time which each train takes to

cover the distance between the stations.

7. A goods train leaves a station at 6 p.m.,followed by an express train which leaves at8 p.m. and travels 20 kmlhour faster than thegoods train. The express train arrives at astation, 1040 km away, 36 minutes before thegoods train. Assuming that the speeds of boththe trains remain constant between the twostations; calculate their speeds.

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8. A man bought an article for ~ x and sold it Ifor ~ 16. If his loss was x per cent, find the :cost price of the article. .

9. A tr~der bought an article. for ~ x an~sold It for ~ 52, thereby making a profit of. . - - . - "" _ . . _ - - _ " - . . . - -(x - 10) per cent on his outlay. Calculate thecost price.

10. By selling a chair for ~ 75, Mohan gained asmuch per cent as its cost. Calculate the costof the chair .. .--- . . . .- - - - - ~ . . - . . . . . ..- . ..". . . . . . . . . _ . _ . . . . .

.. . . . " -.- .._-.G > The sum S of first n natural numbers is given by the relation : S = n (n + 1).Find n, if the sum is 276. .__ . _ ..._ -_ ..... I J II I I( ( )11 Given: S = 276

1 n2 + n - 552'2n (n + 1) -276 i.e. = 0- ... . __ .~ (n + 24) (n - 23) = 0 On factorising :- . . ._ _ . _ . . . . . _ - - _ . -~ n = -24, or n = 23 . Zero product rule :Since n is a natural number, reject n = -24.

'! - ~ I12. When 36 IS l

...._____"j

Let the required two digit number be 1Ox + yGiven: .xy = 12 and lOx + y + 36 = lOy + x

lOx + y + 36 = lOy + x~ 9y = 9x + 36 i.e. y = x + 4Now, .xy = 12 ~ x(x + 4) = 12

~ r + 4x - 12 = 0~ x = - 6, or x = 2Taking x = 2, we get : y = x + 4 = 2 + 4 = 6.,,'_.:.....:',' t r ..111'\';' = lOx + y

On solving

= 10 x 2 + 6 = u. \: . .:.o Five years ago, a woman's age was the square of her son's age. Ten years hence'her age will be twice that of her son's age. Find:(i) the age of the son five years ago.(ii) the present age of the woman. [2007]___ , _00' .. _ _ .. ___

Let the age of the son 5 years ago = x years... The woman's age 5 years ago = r years.~ The present age of the woman = (x2 + 5) years

and, the present age of her son = (x + 5) years110


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--_-_-_- --- -~- ~~ - --=-_

On solving, it gives x = 24 and x = - 16But the number of students cannot be negative, :. x = 24

. r - , I :l!1,...II

Let x students went for picnic480= > Each student paid = ~x

_. !o. pk,!,,_ = X - 8 = 24 - 8 = 1' _

Clearly, no. of students who planned for picnic = x + 8= > Each was to pay = ~ 480x+8Given: x x+8 - 10 i.e. x(x+8) = 10= > 3840 = 10(x2 + 8x) and x2 + 8x - 384 = 0On solving it gives x = - 24 and x = 16Since, the number of students cannot be negative, :. x = 16

480 480 480x +3840 - 480x

EXERCISE SeD) ,..1. The sum S of n successive odd numbersstarting from 3 is given by the relation :S = n(n + 2). Determine n, if the sum is 168.

2. A stone is thrown vertically downwards andthe formula d = 16t2 + 4t gives the distance,d metres, that it falls in t seconds. How longdoes it take to fall 420 metres?

3. The product of the digits of a two digitnumber is 24. If its unit's digit exceeds twiceits ten's digit by 2; find the number.

4. The ages of two sisters are 11 years and 14years. In how many years time will theproduct of their ages be 304 ?

5. One year ago, a man was 8 times as old ashis son. Now his age is equal to the squareof his son's age. Find their present ages.

6. The age of a father is twice the square of theage of his son. Eight years hence, the age of thefather will be 4 years more than three times theage of the son. Find their present ages.

7. The speed of a boat in still water is15 kmlhr. It can go 30 km upstream andreturn downstream to the original point in 4hours 30 minutes. Find the speed of thestream.

8. Mr. Mehra sends his servant to the market tobuy oranges worth ~. 15. The servant havingeaten three oranges on the way, Mr. Mehra

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pays 25 paise per orange more than the marketprice.Taking x to be the number of oranges whichMr. Mehra receives, form a quadratic equationin x. Hence, find the value of x.

9. ~ 250 is divided equally among a certainnumber of children. If there were 25 childrenmore, each would have received 50 paise less.Find the number of children.

10. An employer finds that if he increases the weeklywages of each worker by ~ 5 and employs fiveworkers less, he increases his weekly wage billfrom ~ 3,150 to ~ 3,250. Taking the originalweekly wage of each worker as ~ x; obtain anequation in x and then solve it to find the weeklywages of each worker.

11. A trader bought a number of articles for~ 1,200. Ten were damaged and he sold eachof the remaining articles at ~ 2 more thanwhat he paid for it, thus getting a profit of~ 60 on the whole transaction.Taking the number of articles he bought as x,form an equation in x and solve it.

12. The total cost price of a certain number ofidentical articles is ~ 4,800. By selling thearticles at ~ 100 each, a profit equal to thecost price of 15 articles is made. Find thenumber of articles bought.

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EXERCISE 8(E)1. The distance by road between two towns Aand B is 216 km, and by rail it is 208 km. Acar travels at a speed of x kmIhr and the traintravels at a speed which is 16 kmlhr fasterthan the car. Calculate :(i) the time taken by the car to reach town

B from A, in terms of x;(ii) the time taken by the train, to reach town

B from A, in terms of x.(iii) If the train takes 2 hours less than the car,

to reach town B, obtain an equation in x,and solve it.

(iv) Hence, find the speed of the train. i. "1.2. A trader buys x articles for a total cost of~ 600.(i) Write down the cost of one article in

terms of x.If the cost per article were ~ 5 more, the Inumber of articles that can be bought for~ 600 would be four less.

(ii) Write down the equation in x for theabove situation and solve it for x. 1 ';'N i

3. A hotel bill for a number of people forovernight stay is ~ 4,800. If there were4 people more, the bill each person had topay, would have reduced by ~ 200. Find thenumber of people staying overnight. p' I)

-t An aeroplane travelled a distance of 400 Ianat an average speed of x kmIhr. On the returnjourney, the speed was increased by 40 kmIhr.Write down an expression for the time takenfor:(i) the onward journey;[ii) the return journey.If the return journey took 30 minutes less thanthe onward journey, write down an equation in..(and find its value. I ~f\1 ~

~. ~ 6,500 was divided equally among a certainnumber of persons. Had there been 15 personsmore, each would have got ~ 30 less. Find theoriginal number of persons...\.plane left 30 minutes later than the scheduletime and in order to reach its destination1500 km away in time, it has to increase itsspeed by 250 kmIhr from its usual speed. Findi1 s usual speed.

7. Two trains leave a railway station at the sametime. The first train travels due west and thesecond train due north. The first train travels5 km/hr faster than the second train. If after2 hours, they are 50 km apart, find the speedof each train.

'8. The sum S of first n even natural numbers isgiven by the relation S = n(n + 1). Find n, ifthe sum is 420.

9. The sum of the ages of a father and his sonis 45 years. Five years ago, the product oftheir ages (in years) was 124. Determine theirpresent ages.

10. In an auditorium, seats were arranged in rowsand colunms. The number of rows was equalto the number of seats in each row. When thenumber of rows was doubled and the numberof seats in each row was reduced by 10, thetotal number of seats increased by 300. Find:(i) the number of rows in the original

arrangement.(ii) the number of seats in the auditorium

after re-arrangement. ( 2 1 1 11\.11. Mohan takes 16 days less than Manoj to do

a piece of work. If both working together cando it in 15 days, in how many days willMohan alone complete the work ?

12. Two years ago, a man's age was three timesthe square of his son's age. In three yearstime, his age will be four times his son's age.Find their present ages.

13. In a certain positive fraction, the denominatoris greater than the numerator by 3. If 1 issubtracted from the numerator and thedenominator both, the fraction reduces by 1~'Find the fraction.

14. In a two digit number, the ten's digit is bigger.The product of the digits is 27 and thedifference between two digits is 6. Find thenumber.

15. Some school children went on an excursionby a bus to a picnic spot at a distance of300 km. While returning, it was raining andthe bus had to reduce its speed by 5 km/hrand it took two hours longer for returning.Find the time taken to return.- - - - - . - - - - - , - - - - . . . . . . . - . . . . . . . .~-".--.----~-~----.

1 1 5


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Exercise 7 (A)1 5 1 11. 12, - 2 2. 4, - 4 3. 0'"4 4. 8, - 3 5'"2' 2 6. 2, - 3 7. 1, -"4 8. 2'"2

9. 5, - 2 10. 0, % 11 . J 1 3 = 361, - . J 1 3 = - 361 12. a, b 13. 2, - 4 14. i,~2 715. 1, 103 16.20, .; 25 17. 9, - 9 18. 2, - 2 " 19. 12, - 2 20. 5, - 5

21. (i) x2 - 8x + 15 = 0 (ii) x2 - x - 6 = 0 (iii) x2 - x - 20 = 0 (iv) 5x2+ 17x + 6 = 02 5 1 19 1

22. 3, - 2 23. 3' - "4 24. - " 2 ' - If 25. Yes 26. No 27. -"6 28. m = 3 and n = - 629. m = 4 and other root = 2

1 .

6.10.U.

1 .Exercise 7 (8)

1. (i) 9, - 3 (ii) 3, 7 (iii) - 3 .J19 = 136 or - 736 (iv) - 1..fi (v) - 1,t(vi) - 1, - 2t (vii) - 2 . J 2 = - 059 or - 341 (viii) 5, 5 (ix) 1 6 = 245,-1= - 082(x) 3.J2 = 423, -.J2 = -141 (xi) - 2, 1 (xii) 1 6 = 245 (xiii) 6, 3 / 3 (xiv) 5, 25

2. (i) rational and unequal (ii) irrational and unequal (iii) rational (real) and equal (iv) irrationaland unequal (since, b = 2.fj is irrational) (v) irrational and unequal (vi) imaginary roots

3. p = 6 or -2 4. n = 17 5. m = 51 or 3

4 .7 .

1 .6.10.

Exercise 7 (C)1. (i) 73 and 07 (ii) 03 and -23 (iii) 38 and -032. (i) 444 and 056 (ii) -056 and -269 (iii) 485 and -1853. (i) 4082 and -0082 (n) 15616 and 0384 (iii) -0392 and -5108

5 f 2 94. (i) J3 (ii) 3; 1 (iii) 4; 3 5. (i) 6, - 1, (ii) No real solution2(iii) 6, - 3, 2, 1 6. (i) 4, - 1 (ii) - i , 1j (ii) 1, - t (iv) 45, % 7. 364 or - 014

Exercise 7 (D)1. -1 2.3, -6 3.;. ~ 4. % . % 5. -2 6.1, -1, /{ =122, - /{ =-1227. /3 = 173, -/3 = - 173 (x2 = -1 gives no real solution) 9. -2, -~, 3 ~ J 5 = 262or 038

13 5J85 616. (i) 12

3+.J13 110. 1, - 1, - 2 = 330 or - 030 11.2, - " 2 ' 1= 237 or - 070 14. (i) 653 or - 153 (ii) 170or - 137

12. - 6, 1, 215. -3 or -23"

L2 .

4 .6.9 .12.

1.

4 .

(ii) 6 or - 2 (iii) 25 or 2 3 -217. a + b or a + b 18. - p or - q 9 .5 119. (i) - 6 or 3 (ii) 0 or - " 2 20. m = 7 and solution = " 2 (ii) m = 5 and solution = -2

4(iii) m = 4. For m = 4, solution = 3 and for m = -4, solution = -1 21'"3Exercise 8 (A)

1 .1. 7 and 8 or -8 and -7 2. 4 and 5 3. 4 and 9 4.4 or"4 5. 2 and 5 6. 5 and 1027. 8 and 12 8. 4 and 6 9. 5 and 7 10. " 5 11. 12, 8 and 6 12. 3 and 17 13. x2 - 4x - 60 = O.

Numbers are 9, 10 and 11. 14. 5 15.24 16. 12hrs and 15hrs 17.6

17

18


12/13

Exercise 8 (8)1. 12 ern, 5 em and 13 em 2. 10 em; 24 ern and 26 em 3. (i) 8 (ii) 7 em, 24 emand 25 em (iii) 84 sq. em 4. 25 em, 24 em and 7 ern 5. 90 m and 120 m

6. 32 m and 20 m 7. 2 m 8. x2 + 4x - 320 = 0; 16 cm and 20 em 9. 5 m10. 10 cm 11. 15 m when longer side is 40 m and 20 m when longer side is 30 m12. Each side is 16 m 13. 20 m.

0Exercise 8 (C)1. (i) 300 hrs and 30205hrs (ii) 75 km per hour 2. 30 km/hr 3. 400 km/hrx x+

4. (i) 300 hrs and 3005 hrs (H) 25 km/hr and 20 km/hrx x-7. 80 km/hr and 100 km/hr 8. ~ 20 or ~ 80 9. ~ 40

5.4 6. 1510. ~ 50

25Exercise 8 (0)

1. n = 12 2.5 seconds 3.38 4.5 years 5. Age of man = 49 years and age of son = 7 years6. Father = 32 years and son = 4 years 7. 5 km/hr 8. x l + 3x - 180 = 0; x = 12 9. 10010. x l + 25x - 3150 = 0; ~ 45 n..l - 40x - 6000 = 0; 100 12.60

Exercise 8 (E)1 (.)216 1X hrs2. (i) ~ 6~0

208(ii) x + 16 hrs (iv) 52 kmlhriii) 216 - ~= 2' x = 36x x+16 '(ii) 6004 - 600 = 5; x = 24 3. 8x- x

4. (i) 400 hrs; (ii) 400 hrs; 400 - 400 = 1 .. and x = 160 5. 50x x+40 x x+40 26. 750 km/hr 7. 1st train = 20 km/hr; 2nd train = 15 km/hr 8. 209. Father = 36 years and son:: 9 years 10. (i) 30 (H) 1200 11. 24 days12. Father = 29 years and son = 5 years 413. " 1 14.93 15. 12 hrs.

22262

Exercise 9 (A)1. (a) Reflection in origin (b) (4, -2) (c) (0, 6) (d) reflection in origin

(e) Reflection in y-axis 2. The point P lies in the line l. 3. (i) (3, -2) (ii) (-5, -4) (iii) (0, 0)4. (i) (-6, -3) (ii) (1, 0) (iii) (8, -2) 5. (i) (2, 4) (H) (2, -7) (iii) (0, 0) 6. (i) (6, 4)

(ii) (0, 5) (iii) (-3, -4) 7. (i) (-3, 0) (ii) (8, 5) (iii) (-1, 3) 8. (i) (-4, -5) (ii) (4, -5)9. (i) (2, -7) (ii) (2, 7) 10. a = 4 and b = -6 11. x = 8 and y = 5 12. (i) (3, 2)

(ii) Reflection in y-axis 13. (i) (4, -6) (ii) Reflection in x-axis 14. (i) (2, -6); (-3, -5)and (4, -7) (ii) Reflection in x-axis 15. P' = (-2 , -3) and Q ' = (-5 , 4)17. (i) (-4, 6); reflection in origin (ii) (-4, 6); reflection in origin (Hi) (-4, -6); reflection iny-axis (iv) (-4, -6); reflection in y-axis (v) (4, 6); reflection in x-axis

(vi) (4, 6); reflection in x-axis (a) True (b) True18. (-4, -1) and (-2, -5) 19. {a) y-axis (b) (-5, -8)

-2

10= O .

483


13/13

radius Exercise 18 (C)1. (i) Both are equal (ii) Both are equal (iii) Chord A D is greater than ehord AC (iv) 254. (i) 20 (ii) 130 8. (i) 150 (ii) 50 (iii) 130 9. (i) 60 (ii) 30 (iii) (22i J = 22 30'10. I: 11. (i) 24 (ii) 72 (iii) 48 12. (i) 27 (ii) 126 13. LABC = 30, LACB = 36 and

LBAC = 114 14. (i) 54 (ii) 60 (iii) 54 (iv) 1260

Exercise 18 (O)1. 60 2. 7 em 3. 4 j6 em = 9.80 em 6. 25 11. 108 12. (i) 33 (ii) 40 (iii) 12313. (i) x (ii) xiS. 116 16. LA = 60 amd LB = 30 19. LA = 36, LB = 84,

LC = 144 and LD = 96 20. 24 em 21. (a) LBAC = 30 (b) LABC = 6022. (i) 80 (ii) 50 (iii) 100 (iv) 80 23. a = 105, b = 130 and c = 62"_24. (i) 5SO (ii) 55 (iii) 100 27. (i) 80 (ii) 50 (iii) 100 (iv) 50 28. 21 em

eentreeentreeentreto the

given

Exercise 19 (A)1. 6 em 4. 8 em 5. 35 em, 25 em and 55 em 10. (i) 99 em (ii) 27 em. 15. 2 em16. (i) 120 (ii) 60 17. (i) 120 (ii) 30 (iii) 60 18. 58 19. 14 em20. (ii) x + 2y = 90 21. (i) 90 (ii) 30 (iii) 10 22. LBAC = 60.

BC. Exercise 19 (8)1. (i) 6 em (ii) 20 em (iii) 5625 em 2. (i) 30 (ii) 60 (iii) 30 3. (i) 60 (ii) 609. (i) 54 (ii) 108 13. (i) D E = 12 em 15. 40

when

-.t

Exercise 19 (C)2. (i) 50 - J 3 em2 = 8660 qn2 (ii) 8 em 3.4 j6 em = 980 em 6.8 em 7.60; 10 8.409. LA = 135; LB = 30; LC = L45; LD = 150 11. 60 16. (i) 40 (ii) 50 (iii) 3020. (i) 30 (ii) 60 21. 110 23. LPAB = 130; LAQB = 25 24. (i) 40 (ii) 50 (iii) no-

(iv) 30 30.7 em 31.8 em 32.14 em 33. LBAY = 30 and LAPY = 1034. (i) 36 (ii) 78 (iii) 78 (iv) 114 35. (i) 56 (ii) 22 36. (i) 110 (ii) 7037. (i) 112 (ii) 68 39. 15 em 40.60 41. 15 em

Exercise 201. 4 em 2. 8 em 5. 26 em 6. (ii) 18 em 7. 1-4 em 9. 2 em10. (i) Cireumeentre (ii) OA = OB = OC (iii) yes11. (i) Ineentre (ii) OR = O Q (iii) LACO = LBCO 12. (iii) Approx. 19 em

Exercise 21 (A)1. (i) 70 m (ii) 15,400 sq. m (iii) ~. 2,69,500 2. ~. 2,112 3. 176 m

613. 326'7 em = 32686 em 5. (i) 3: 5 (ii) 3: 5 6.16: 81 7.26565 sq. m 8. ~ 7,651889. 600 (ii) 792 kmIhr . 10. 40 11. 48 kmIhr . 12. (i) 3850 sq. em (ii) 132 em and8778 sq. em 13. 35 m 14. (i) 28 em (ii) 54635 sq. em 15. 482944 em? = 4829 em-

4 216. 121 : 81 17.2464 em? 18.500 19.628'7 em? and 314'7 em? 20. 616 cnr'21. (i) 300 (ii) 200 (iii) 120 (iv) 48

5560

H o

3 0=

495

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