बीजगणित का प्रारंभ | 03. गुणा

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बीजगणित का प्रारंभ 03

Multiplication (गुणा)

xq.kk djrs oDr fuEu ckrksa dks /;ku esa j[krs gSaA 

  1. dks”Bd dk vFkZ xq.kk gksrk gSA 

  2. ftl in ij dksbZ fpUg ugha gksrk ml ij tek ;k Iyl dk fu’kku ;k fpUg ekuk tkrk gSA 

  3. bl in dh ?kkr] gj rFkk xq.kkad lnSo ,d ekuk fy;k tkrk gSA 

  4. lcls igys fpUgksa dh xq.kk djrs gSaA blds fy, fu;e ua0 02 nsa[ksaA 

  5. blds mijkar vadksa dh xq.kk djrs gSaA 

  6. blds mijkar leku inksa dh xq.kk djrs gSaA leku inksa dh xq.kk djrs oDr leku inksa dh ?kkrkas dks tksM nsrs gSaA 

  7. ifjes; ;k fHkUu la[;kvsa ds val dh val esa rFkk gj dh gj ls xq.kk djrs gSaA 

  8. नौ


vkbZ;s xq.kk djds fuEu lkjf.k;ksa dks iwjk djsaA 

गुणा करते वक्त निम्न बातों को ध्यान में रखते हैं। 

1. कोष्ठक का अर्थ गुणा होता है। 

2. जिस पद (अथवा चर या अचर) पर कोई चिन्ह नहीं होता तो उस पद (अथवा चर या अचर) पर जमा या प्लस का निसान या चिन्ह माना जाता है। 

3. साथ ही इस पद की घात, हर तथा गुणांक को सदैव एक (1) माना लिया जाता है। 

4. सबसे पहले चिन्हों की गुणा करते हैं। इसके लिए नियम नं0 02 देंखें। 

5. इसके उपरांत अंकों की गुणा करते हैं। 

6. इसके उपरांत समान पदों की गुणा करते हैं। समान पदों की गुणा करते वक्त समान पदों की घातों को जोड देते हैं। 

7. परिमेय या भिन्न संख्याअें के अंस की अंस में तथा हर की हर से गुणा करते हैं। 

8.

आईये गुणा करके निम्न सारणियों को पूरा करें। 


1. Find the product of the following pairs of monomials.

fuEu ,d inh; O;atdksa ds xq.kuQy Kkr djksA 

  1. 4 ×7p

  2. – 4p × 7p 

  3.  – 4p × 7pq

  4.  4p3 × – 3p

  5. 4p × 0

  6.  (p × q); 

  7. (10m × 5n);

  8.  (20x2 × 5y2); 

  9. (4x × 3x2); 

  10. (3mn× 4np)

  11.  xy × yz × zx 

  12. a× – a2 × a3

  13.  2× 4y× 8y2× 16y3

  14.  a × 2b × 3c × 6abc

  15.  m × – mn × mnp

  16. 5a × 3a2 × 7a4

  17. 2p × 4q × 8r

  18. xy × 2x2y × 2xy2

  19. xy × 2xy × 2xy

  20.  a × 2b × 3c


Find the product 

  1. 2x (3x + 5xy) 

  2. a2 (2ab – 5c)

  3. 4p, q + r 

  4. ab, a – b 

  5.  a + b, 7a2b2 

  6.  a2 – 9, 4a

  7.  pq + qr + rp, 0

  8. (a2) × (2a22) × (4a26) 

  9.  x × x2 × x3 × x4


Multiply the binomials.

  1. (2x + 5) and (4x – 3)

  2. (y – 8) and (3y – 4)

  3. (2.5l – 0.5m) and (2.5l + 0.5m) 

  4. (a + 3b) and (x + 5)

  5. (2pq + 3q2) and (3pq – 2q2)


Find the product.

  1. (5 – 2x) (3 + x) 

  2. (x + 7y) (7x – y)

  3. (a2 + b) (a + b2) 

  4. (p2 – q2) (2p + q)

  5. (x + 3) (x + 3) 

  6. (2y + 5) (2y + 5) 

  7.  (2a – 7) (2a – 7)

  8. (2a – 3b) (2a – 4b)

  9. (2a – 5b) (2a – 7b)

  10.  (a2 + b2) (– a2 + b2) 

  11. (6x – 7) (6x + 7) 

  12. (– a + c) (– a + c)

  13. (x + 3) (x + 7)

  14. (4x + 5) (4x + 1)

  15. (4x – 5) (4x – 1) 

  16. (4x + 5) (4x – 1)

  17. (2x + 5y) (2x + 3y) 

  18. (2a2 + 9) (2a2 + 5)

  19. (xyz – 4) (xyz – 2)


Find the product.

  1. (5 – 2x) (3 + x + y)

  2.  (x + 7y) (7x – y+3)

  3. (a2 + b) (a + b – 5) 

  4. (p2 – q2) (2p + q+7)

  5.  (x + 3) (x2 + x + 3) 

  6. (2y + 5) (2y 2 +  y + 5) 

  7. (2a – 7) (2a2 + a – 7)

  8. (2a – 3b) (2a – 4b+6)

  9. (2a – 5b) (2a – 7b+1)

  10. (a2 + b2) (5– a2 + b2) 

  11. (6x – 7) (6x 2 + x + 7) 

  12. (– a + c) (8 – a + c)

  13. (x + 3) (x 2 + x + 7)

  14. (4x + 5) (4x2 +x + 1) 

  15. (a + b + c)(a + b – c+ 8) 

  16. (a + b + c)(a + b – c)


Divide the followings: 

  1. (3x + 5xy) ÷ 2x

  2. (2ab – 5c) ÷ a2 

  3. (q + r

  4. ab, a – b 

  5. a + b, 7a2b2 

  6. a2 – 9, 4a

  7.  pq + qr + rp, 0

  8. (a2) × (2a22) × (4a26) 

  9.  x × x2 × x3 × x4


fuEufyf[kr foHkktu dhft, %

(i) 28x4 ÷  56x 

(ii) –36y3 ÷  9y2 

(iii) 66pq2r3 ÷  11qr2

(iv) 34x3y3z3 ÷  51xy2z3 

(v) 12a8b8 ÷  (– 6a6b4)

(v) –20x ÷  10x2 

(ii) 7x2y2z2 ÷  14xyz



2. fn, gq, cgqin dks fn, gq, ,dinh ls Hkkx nhft, %

  1. (5x2 – 6x) ÷ 3x 

  2. (3y8 – 4y6 + 5y4) ÷ y4

  3. 8(x3y2z2 + x2y3z2 + x2y2z3) ÷ 4x2y2z2 

  4. (x3 + 2x2 + 3x) ÷ 2x

  5. (p3q6 – p6q3) ÷ p3q3

  6. Divide (4y3 + 5y2 + 6y) by  2y ls foHkktu

  8. 24 (x2yz + xy2z + xyz2) ÷ 8xyz

  9. Divide  from

  10. (7x2 + 14x) ÷ (x + 2)

  11. Divide, by. 

  12. (y2 + 7y + 10) ÷ (y + 5) 

  13. (m2 – 14m – 32) ÷ (m + 2)

  14. (5p2 – 25p + 20) ÷ (p – 1) 

  15. Divide, from.

  16.  Dividefrom. 


(07.)  gy %& 

 


(08.)  gy %& 

(09.)  gy %& 




Divide the following polynomials with x + 1.

fuEu cgqinksa dks x + 1 ls Hkkx djksA   

  1. x2 + 3x + 2

  2. x2 – 1

  3. x2 – 2x –3

  4. x3 + x2  + x + 2

  5. x4 + x3 + x2 + x + 1

  6. x4 + 3x3 + 3x2 + x + 1


3. fuEufyf[kr foHkktu dhft, %

  1. (10x – 25) ÷ 5 

  2. (10x – 25) ÷ (2x – 5)

  3. 10y(6y + 21) ÷ 5(2y + 7) 

  4. 9x2y2(3z – 24) ÷ 27xy(z – 8)

  5. 96abc (3a – 12) (5b – 30) ÷ 144(a – 4) (b – 6)


4. funsZ'kkuqlkj Hkkx nhft, %

  1. 5(2x + 1) (3x + 5) ÷ (2x + 1) 

  2. 26xy(x + 5) (y – 4) ÷ 13x(y – 4)

  3. 52pqr (p + q) (q + r) (r + p) ÷ 104pq(q + r) (r + p)

  4. 20(y + 4) (y2 + 5y + 3) ÷ 5(y + 4) 

  5. x(x + 1) (x + 2) (x + 3) ÷ x(x + 1)

  6. 4yz(z2 + 6z – 16) ÷ 2y(z + 8)

  7. 5pq(p2 – q2) ÷ 2p(p + q)

  8. 12xy(9x2 – 16y2) ÷ 4xy(3x + 4y) 

  9. 39y3(50y2 – 98) ÷ 26y2(5y + 7) 

  10. 44(x4 – 5x3 – 24x2) dks 11x (x – 8) ls Hkkx nhft,A

  11. z(5z2 – 80) dks 5z(z + 4) ls Hkkx nhft,A


Equation 

In an equation there is always an equality sign. The equality sign shows that the value of the expression to the left of the sign (the left hand side or L.H.S.) is equal to the value of the expression to the right of the sign (the right hand side or R.H.S.).

,d lehdj.k esa] lerk ;k lfedk (equality) ;k cjkcj dk fpUg lnSo gksrk gSA lerk dk fpUg ;g n'kkZrk gS fd bl fpUg ds ckb± vksj ds O;atd ^ck;k¡ i{k* ;k (LHS) dk eku fpUg ds nkb± vksj ds O;atd ^nk;k¡ i{k* ;k (RHS) ds eku ds cjkcj gSA


Equation:- An equation is a condition on a variable such that two expressions in the variable should have equal value.

OR

In short, an equation is a condition on a variable. The condition is that two expressions should have equal value. Note that at least one of the two expressions must contain the variable.


lehdj.k %& ,d lehdj.k] ,d pj ij ,slk izfrca/k gksrk gS ftlesa nksuksa i{kksa esa O;atdksa dk eku cjkcj gksuk pkfg, A

vFkok

laf{kIr :i esa] ,d lehdj.k pj ij ,d izfrca/k gksrk gSA izfrca/k ;g gS fd nksuksa O;atdksa ds eku cjkcj gksus pkfg,A è;ku jgs fd bu nksuksa O;atdksa esa ls de ls de ,d esa pj vo'; gksuk pkfg,A


Roll No. 04 : - An equation remains the same, when the expression on the left and on the right are interchanged. This property is often useful in solving equations.

lehdj.k 04 : - fdlh lehdj.k ds ck,¡ vkSj nk,¡ i{kksa ds O;atdksa dks vkil esa cnyus ij] lehdj.k ogh jgrh gSA ;g xq.k cgq/kk lehdj.kksa dks gy djus esa mi;ksxh jgrk gSA


An equation remains the same if the L.H.S. and the R.H.S. are interchanged.

fdlh lehdj.k ds ck,¡ vkSj nk,¡ i{kksa dks ijLij cnyus ij] lehdj.k ugha cnyrkA


01. Write the following statements in the form of equations:

  1. The sum of three times x and 11 is 32.

  2. If you subtract 5 from 6 times a number, you get 7.

  3. One fourth of m is 3 more than 7.

  4. One third of a number plus 5 is 8.

01. fuEufyf[kr dFkuksa dks lehdj.kksa ds :i esa fyf[k, %

  1. x ds  frxqus vkSj 11 dk ;ksx 32 gSA

  2. ;fn fdlh la[;k ds 6 xqus esa ls vki 5 ?kVk,¡] rks 7 izkIr gksrk gSA

  3. m dk ,d pkSFkkbZ 7 ls 3 vf/kd gSA

  4. fdlh la[;k ds ,d frgkbZ esa 5 tksM+us ij 8 izkIr gksrk gS A


02. Write equations for the following statements:

  1. The sum of numbers x and 4 is 9.

  2. The difference between y and 2 is 8.

  3. Ten times a is 70

  4. The number b divided by 5 gives 6.

  5. Three fourth of t is 15. (vi) Seven times m plus 7 gets you 77.

  6. One fourth of a number minus 4 gives 4.

  7. If you take away 6 from 6 times y, you get 60.

  8. If you add 3 to one third of z, you get 30.

02. fuEufyf[kr dFkuksa ds fy, lehdj.k nhft, %

  1. la[;kvksa x vkSj 4 dk ;ksx 9 gSA

  2.  y esa ls 2 ?kVkus ij 8 izkIr gksrs gSaA

  3. a dk 10 xquk 70 gSA (iv) la[;k b dks 5 ls Hkkx nsus ij 6 izkIr gksrk gSA

  4. t dk rhu&pkSFkkbZ 15 gSA

  5. m dk 7 xquk vkSj 7 dk ;ksxiQy vkidks 77 nsrk gSA

  6. ,d la[;k x dh pkSFkkbZ esa ls 4 ?kVkus ij vkidks 4 nsrk gSA

  7. ;fn vki y ds 6 xqus esa ls 6 ?kVk,¡] rks vkidks 60 izkIr gksrk gSA

  8. ;fn vki z ds ,d&frgkbZ esa 3 tksM+sa] rks vkidks 30 izkIr gksrk gSA


03. Convert the following equations in statement form: 

03. fuEufyf[kr lehdj.kksa dks lkekU; dFkuksa ds :i esa cnfy, %

  1. x – 5 = 9

  2. 5p = 20 

  3. 3n + 7 = 1 

04. Convert the following equations in statement form: 

04. fuEufyf[kr lehdj.kksa dks lkekU; dFkuksa ds :i esa fyf[k, %

  1. p + 4 = 15 

  2.  m – 7 = 3 

  3. 2m = 7 

  4.   

  5.  

  6. 3p + 4 = 25 

  7. 4p – 2 = 18 


05.  Father’s age is 5 years more than three times Raju’s age. Raju’s father is 44 years old. Set up an equation to find Raju’s age.

05.  jktw ds firk dh vk;q jktw dh vk;q ds rhu xqus ls 5 o"kZ vf/kd gSA jktw ds firk dh vk;q 44 o"kZ gSA jktw dh vk;q Kkr djus ds fy,] ,d lehdj.k cukb, A 


06. Set up an equation in the following cases:

  1. Irfan says that he has 7 marbles more than five times the marbles Parmit has. Irfan has 37 marbles. (Take m to be the number of Parmit’s marbles.)

  2. Laxmi’s father is 49 years old. He is 4 years older than three times Laxmi’s age. (Take Laxmi’s age to be y years.)

06.  fuEufyf[kr fLFkfr;ksa eas lehdj.k cukb, %

  1. bjQku dgrk gS fd mlds ikl] ijehr ds ikl ftrus daps gSa muds ik¡p xqus ls 7 vf/kd daps gSaA bjQku ds ikl 37 daps gSaA  ¼ijehr ds d¡pksa dh la[;k dks m yhft,A½

  2. y{eh ds firk dh vk;q 49 o"kZ gSA mudh vk;q] yM+dh dh vk;q ds rhu xqus ls 4 o"kZ vf/kd gSA ¼y{eh dh vk;q dks y o"kZ yhft,A½


07. Set up an equation in the following cases:

  1. The teacher tells the class that the highest marks obtained by a student in her class is twice the lowest marks plus 7. The highest score is 87. (Take the lowest score to be l.)

  2. In an isosceles triangle, the vertex angle is twice either base angle. (Let the base angle be b in degrees. Remember that the sum of angles of a triangle is 180 degrees).


07.  fuEufyf[kr fLFkfr;ksa eas lehdj.k cukb, %

  1. vè;kfidk crkrh gSa fd mudh d{kk eas ,d fo|kFkhZ }kjk izkIr fd, x, vf/kdre vad] izkIr fd, U;wure vad dk nqxquk /ku 7 gSaA izkIr fd, x, vf/kdre vad 87 gSaA ¼U;wure izkIr fd, x, vadksa dks l yhft,A½

  2. ,d lef}ckgq f=Hkqt esa 'kh"kZ dks.k izR;sd vk/kkj dks.k dk nqxquk gSA ¼eku yhft, izR;sd vk/kj dks.k b fMxzh gSA ;kn jf[k, fd f=Hkqt ds rhuksa dks.kksa dk ;ksx 180 fMxzh gksrk gSA½


08.  A shopkeeper sells mangoes in two types of boxes, one small and one large. A large box contains as many as 8 small boxes plus 4 loose mangoes. Set up an equation which gives the number of mangoes in each small box. The number of mangoes in a large box is given to be 100.

08.  ,d nqdkunkj nks izdkj dh isfV;ksa esa vke csprk gSA ;s isfV;k¡ NksVh vkSj cM+h gSaA ,d cM+h isVh esa 8 NksVh isfV;ksa ds cjkcj vke vkSj 4 [kqys vke vkrs gSaA izR;sd NksVh isVh esa vkeksa dh la[;k crkus okyk ,d lehdj.k cukb,A fn;k gqvk gS fd ,d cM+h isVh esa vkeksa dh la[;k 100 gSA


Rule No. 13: - Thus if we add or subtract the same number to both sides of a balance equation, the balance is undisturbed.

fu;e 13 :-  

;fn ge ,d lfedk ds nksuksa i{kksa esa ,d gh la[;k tksM+sa ;k ?kVk,sa rks Hkh og lfedk lR; gksrh gSA


Rule No. 14: - If we multiply or divide both sides of the equation by the same number, the balance is undisturbed.

fu;e 14 :-  

;fn ge ,d lfedk ds nksuksa i{kksa dks ,d gh 'kwU;srj (non-zero) la[;k ls xq.kk djsa ;k Hkkx nsa] rks Hkh og lfedk lR; gksrh gSA


Note :- In case of the balanced equation, if we

  1. add the same number to both the sides, or

  2. subtract the same number from both the sides, or 

  3. multiply both sides by the same number, or 

  4. divide both sides by the same number,


The balance remains undisturbed, i.e., the value of the L.H.S. remains equal to the value of the R.H.S.

uksV%&  ,d larqfyr lehdj.k dh fLFkfr esa ;fn ge

  1. nksuksa i{kksa esa ,d gh la[;k tksM+sa ;k 

  2. nksuksa i{kksa esa ls ,d gh la[;k ?kVk,¡ ;k

  3.  nksuksa i{kksa dks ,d gh la[;k ls xq.kk djsa ;k 


nksuksa i{kksa dks ,d gh la[;k ls Hkkx nsa rks larqyu esa dksbZ ifjorZu ugha gksrk gS vFkkZr~ LHS vkSj RHS ds eku leku jgrs gSa A


Transposing : - Changing side is called transposing. While transposing a number, we change its sign. See rule no.  03. 

LFkkukiUu %& i{k cnyus dks LFkkukiUu (transposing) djuk dgrs gSaA LFkkukiUu djus esa] la[;k dk fpUg cny tkrk gSA blds fy, fu;e uEcj 03 nsas[ksaA 


Note: - Transposing means moving to the other side. Transposition of a number has the same effect as adding same number to (or subtracting the same number from) both sides of the equation. When you transpose a number from one side of the equation to the other side, you change its sign. For example, transposing +3 from the L.H.S. to the R.H.S. in equation x + 3 = 8 gives x = 8 – 3 (= 5). We can carry out the transposition of an expression in the same way as the transposition of a number.

uksV%&  LFkkukiUu dk vFkZ gS ,d i{k ls nwljs i{k esa tkukA fdlh la[;k dks LFkkukiUu djuk] la[;k dks nksuksa i{kkas esa tksM+us ;k nksukas i{kksa esa ls ?kVkus ds leku gh gSA tc vki ,d la[;k dks ,d i{k ls nwljs i{k esa LFkkukiUu djrs gSa rks vki mlds fpfUgr dks cny nsrs gSa A mnkgj.kkFkZ] lehdj.k x + 3 = 8 esa + 3 dk LFkkukiUu LHS ls RHS djus ij x = 8 – 3 = 5 izkIr gksrk gS A ge O;atdksa dk Hkh LFkkukiUu mlh fof/k ls djrs gSa tSls ,d la[;k dk LFkkukiUu djrs gSa A


10. Solve the following:

10. fuEu dks ljy djks %


  1. 6x = 18

  2. 7y = 14

  3. 2x = 6

  4. 4m = 16

  6.  

  8.     x + 4 = 13

  9. x + 5 =  18

  10. y – 5 = 7

  11. x + 10 = 12

  12. x – 10 = 14

  13. 7y – 10 = 0

  14. 4y – 16 = 0

  15. 5m – 25 = 0

  16. 5m + 25 = 0

  17. 3x + 9 = 0

  18. 2x – 6 = 0


The value of the variable for which the equation is satisfied is called the solution of the equation.

pj dk og eku ftlds fy, lehdj.k larq"V gksrk gS] lehdj.k dk gy dgykrk gSA


11. Solve the following:

11. fuEu dks ljy djks %

  1. 2m + 4 = 8

  2. 4m – 5 = 11

  3. 11 + 2a = 25

  4. 12x + 4 = 3x

  5. 75 – 3a =35

  6. 2x – 3 = 5

  7. 7 – 4 y = 3

  8. 20x = x + 95


The above property gives a systematic method of solving an equation. We carry out a series of identical mathematical operations on the two sides of the equation in such a way that on one of the sides we get just the variable. The last step is the solution of the equation.

mijksDr xq.kksa }kjk lehdj.k dks pj.k fof/k ls gy fd;k tk ldrk gSA gesa nksuksa i{kksa esa ,d ls vf/d xf.krh; lafØ;k,¡ djuh iM+rh gSa] ftlls fd nksuksa esa ls ,d i{k esa gesa dsoy pj izkIr gksA vafre pj.k lehdj.k dk gy gSA


12. Give first the step you will use to separate the variable and then solve the equation:

12. igys pj dks i`Fkd djus okyk pj.k crkb, vkSj fQj lehdj.k dks gy dhft, %

(a) x – 1 = 0 (b) x + 1 = 0

(c) x – 1 = 5 (d) x + 6 = 2 

(e) y – 4 = – 7 (f) y – 4 = 4

(g) y + 4 = 4 (h) y + 4 = – 4


13. Give first the step you will use to separate the variable and then solve the equation:

13. igys pj dks i`Fkd djus ds fy, iz;ksx fd, tkus okys pj.k dks crkb, vkSj fQj

lehdj.k dks gy dhft, %

(a) 3l = 42 (b)

 (c)(d) 4x = 25

(e) 8y = 36 (f) 

(g)(h) 20t = – 10


14. Give the steps you will use to separate the variable and then solve the equation:

14. pj dks i`Fkd djus ds fy,] tks vki pj.k iz;ksx djsaxs] mls crkb, vkSj fQj lehdj.k dks gy dhft, %

(a) 3n – 2 = 46 (b) 5m + 7 = 17

(c)(d)


15. Solve the following equations:

15. fuEufyf[kr lehdj.kksa dks gy dhft, %

(a) 10p = 100 (b) 10p + 10 = 100

(c)(d)

(e)(f) 3s = –9

(g) 3s + 12 = 0 (h) 3s = 0

(i) 2q = 6 (j) 2q – 6 = 0

(k) 2q + 6 = 0 (l) 2q + 6 = 12


16. Solve the following equations.

16. fuEufyf[kr lehdj.kksa dks gy dhft, %


(a)(b) 5t + 28 = 10

(c)2 (d)

(e)(f) 

(g)(h) 6z + 10 = –2

(i)(j)


17. Solve the following equations.

17. fuEufyf[kr lehdj.kksa dks gy dhft, %


  1. 2(x + 4) = 12 

  2. 3(n – 5) = 21

  3. 3(n – 5) = – 21

  4. 3 – 2(2 – y ) = 7 

  5.  – 4(2 – x) = 9 

  6. 4(2 – x) = 9

  7. 4 + 5 (p – 1) = 34 

  8. 34 – 5(p – 1) = 4


18. Solve the following equations.

18. fuEufyf[kr lehdj.kksa dks gy dhft, %

  1. 4 = 5(p – 2) 

  2.  – 4 = 5(p – 2) 

  3. –16 = –5 (2 – p)

  4. 10 = 4 + 3(t + 2) 

  5. 28 = 4 + 3(t + 5) 

  6. 0 = 16 + 4(m – 6)

  7. 5 (3 – x) = 10

  8. 3 (2 + x) = 12

  9. 3 (x + 2) = 2

  10. 3y + 7 = 5 – y

  11.   8y – 3 = – 2y – 3 

  12. 6y – 7 = 2 (y + 3)

  13. 10 (x + 2) = 5 (x – 3)

  14. 3 (x + 1) = x + 5


19. Solve the following equations.

19. fuEufyf[kr lehdj.kksa dks gy dhft, %




SIMPLE     MATH’S

12. The formulae of following things are given then find the other terms of the formulae: 

12. fuEu lw=ksa ds mi;ksx djds mlds vU; vO;oksa dks Kkr djksA 


  1. Area = Length 🞨 Breadth 

  3. A =  π r2

  4. V= π r2h

  5.                                           

  8. ∠A + ∠B + ∠C = 1800

  10. Interest = Amount + Principal

  11. 1st No. 🞨 2nd No. = LCM 🞨 HCF


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