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SLC Model Question SLC _ Compulsory Math _ Area of Triangles and Parallelogram

[Short Answer Questions]

# Section I

a)Find the area of the triangle given below.

b) Find the area of an equilateral triangle whose side is 6cm.

c) The area of equilateral triangle is 4cm2, find the perimeter.

d) The perimeter of equilateral triangle is 24cm, find it's area.

e) In the given rhombus, PQ = 6cm and PT = 4cm, find the area of the rhombus PQRS.

f) In the adjoining figure, PQRS is a rhombus. If PR = 4cm and QS = 6cm, find the area of PQRS.

g) In the adjoining figure, ABCD is a rhombus. If the area of ABCD = 36cm2 and the diagonal AC = 8cm, find the

length of the diagonal BD.

h) Each side of rhombus measures 10cm, diagonal is 12cm long, find the length of other diagonal.

i) Find the perimeter of a rhombus ABCD whose diagonals AC and BD are 12cm and 16cm respectively.

j) Find the area of square ABCD in which diagonal BD = 3cm.

# Section II

a)Calculate the area of the given quadrilateral in which CX = 6cm, AY = 8cm and BD = 8cm.

b) PQRS is a quadrilateral in which PR = 10cm, perpendiculars from S and Q on PR are 3.4cm and 4.6 cm respectively.

Calculate the area of the quadrilateral.

c) Find the area of the given quadrilateral ABCD where AM and CN are perpendiculars to BD and 4AM = 2CN = BD =

12cm.

d) Find the area of the given quadrilateral PQRS where PA and RB are perpenducular to QS and 3RB = 2PA = QS =

6cm.

e) In a quadrilateral ABCD, diagonal AC = 17cm, perpendiculars from B and D on AC are 11cm and 9cm

respectively. Calculate the area of the quadrilateral.

f) The diagonals of rhombus are 10cm and 12 cm respectively. Find it's area.

g) If the area of rhombus is 90cm2 and its one diagonal is 30cm. Find its other diagonal.

# Section III

a) In the given figure, ABCD is a parallelogram, ADP = 7cm2 and BCP = 5cm2, find the area of parallelorgram ABCD.

b) M is the midpoint of the LN of KLN. If the area of KLN is 30cm2, what will be the area of KMN?

c) ABCD is a square and EBCF is a parallelogram. If AB = 4cm, calculate the area of the parallelogram EBCF.

d) If parm ABCD, BC = 2QC and P is any point on AD If area of parm ABCD is 48cm2, find the area of PQC.



# Section IV

a) In the given trapezium PQRS, PQTS is a rectangle. If PS = 8cm, QR = 11cm and SR = 5cm, find the area of trapezium

PQRS.

b) Calculate the area of the given trapezium ABCD.

c) In the given figure AB//FC, AB = 18cm, FC = 28 cm, EC = 23cm, AF = 13cm & AEFC. Find the area of the figure

ABCF.

d) In the given figure AB//DC, AB = 10cm, DC = 14 cm, AD = 5cm & AEDC. Find the area of trapezium ABCD.

e) The given figure is a trapezium ABCD in which AB//DC and BCD = 900, if CD = 7cm, AB = 17cm and AD = 26cm,

calculate the area of the trapezium ABCD.

f) The given figure is a trapezium PQRS in which PQ//SR and PQR = 900, If PQ = 10cm, SR = 18cm and PS = 17cm,

calculate the area of the trapezium PQRS.

# Section V

a) The areas of two parallelograms are equal and their altitudes are 6cm and 9cm. If the base of the first

parallelogram is 12cm, find the base of the second parallelogram.

b) The area of two parallelograms are equal. The altitude of one parallelogram is 4cm and the base of the other is

6cm. If the base of the first parallelogram is 9cm, find the height of the second parallelogram.

c) ABCD is a trapezium with area 60sq.cm. where AD//BC, AFBC. If BC = 10cm and AF = 8cm, find the value of AD.

d) The parallel sides of a trapezium are 4.3 cm and 5.7 cm If its area is 40sq.cm, find its height.

e) ABCD is a trapezium in which AB//CD, AD = BC = 13cm, AB = 18cm and DC = 28cm, Find the area.

f) Two adjacent sides of a parallelogram are 12 and 30 units and they include an angle of 1500. Find the area of the

parallelogram.

# Section VI

a) Find the area of a parallelogram whose base is (x + 2) and altitude is (2x -4)

b) If the sides of parallelogram are doubled, how does the area of the parallelogram The area of parallelogram is 2x2

+ 3x - 9 square units and the base is (2x - 3) units. Find the length of the altitude drawn to the base.

d) A parallelogram MNOP, M = 450, altitude PR = 5cm and diagonal PN = 13cm, Find the area of the parallelogram

e) Find the altitude of an equilateral triangle whose area is 36cm2

f) Find the area of an equilateral triangle whose altitude is 4units.

g) Find the area of an isosceles right angled triangle whose hypoteneous is 8 cm.

h) The sides of an equilateral triangle is 8cm, Find the length of other equilateral triangle, with twice the area.

# Section VII

a) In the given figure, parallelogram PQRD and EQR are on the same base and between the same parallel. If the

area of the EQR is 7 square cm, find the area of the parallelogram PQRD.

b) In the given figure, ABCD is a square and EBC a triangle. If AB = 6cm, calculate the area of the triangle EBC.

c) In given parallelogram ABCD, AMBC, ANCD BC = 15cm, AM = 8cm and AN = 10cm, find the perimeter of the

parallelogram.

d)In the parallelogram ABCD, BC = 12cm, DC = 10cm and area of parallelogram = 96 sq.cm. Calculate the lengths

of altitude AM and altitude AN.



# Section VIII

a) In the given figure, PORS is a parallelogram in which T is the middle point of PS. If the area of the PTO is 6cm2, what

will be the area of the quadrilateral OTSR.

b) In the given figure, BOY is triangle. in which M and N are the mid-points of OY and OM respectively if the area of

BOY = 30cm2. Find the area of MBN.

c) In the given figure, BD = DC = 2BE and ABE = 12cm2, find the area of ABC.

d) In the given figure, QR = RS and PM = RM. If MQR = 6.5 cm2, find the area of PQS.

e) In ABD, ED = 2AE, BC = CD and ABD = 64.8cm2 find the area of BEC.

• Find the area of PQR from the given figure.

# Section IX

a) In the given figure MN//PR, Prove that MOP = RON.

b) In the given figure, M is the midpoint of the diagonal BD of quadrilateral ABCD, then prove that area of

quadrilateral AMCD and quadrilateral AMCB are equal.

c) In ABC, P is any point on median AD then prove that APB = APC.

# Section X

a) The area of rhombus is equal to one half the product of its diagonal prove

b) Prove that area of a kite is equal to one half the product of diagonals.

c) Prove that area of trapezoid is half of the products of its altitude and sum of the bases.

d) Prove that the area of a quadrilateral is equal to one half the product of a diagonal and the sum of

perpendiculars on it's from the opposite vertices.

e) Prove that median of triangles divide the triangle into two triangles of equal area.

Answers

I. a) 126cm2 b) 9cm2 c) 12cm d) 16cm2 e) 24cm2 f) 12cm2

g) 9cm h) 16cm

i) 40cm j) 9cm2

II. a) 56cm2 b) 40cm2 c) 54cm2 d) 15cm2

e) 170cm2 f) 60cm2 g) 6cm

III. a) 24cm2 b) 15cm2 c) 16cm2 d) 12cm2

IV. a) 38cm2 b) 48cm2 c) 276cm2 d) 36cm2

e) 288cm2 f) 210cm2 g) 45cm2

V. a) 8cm b) 6cm c) 5cm d) 8cm

e) 276cm2 f) 180 sq. units

VI. a) (2x2-8)sq. units b) 3 times c) (x + 3)units d) 85cm2

e) 6cm f) 16sq.units g) 16sq.cm h) 8cm

VII. a) 14cm2 b) 18cm2 c) 54cm d) 9.6cm, 8cm

VIII. a) 18cm2 b) 7.5cm2 c) 48cm2 d) 26cm2

e) 21.6cm2 f) 120cm2



SLC Model Question SLC _ Compulsory Math _Theorems from Geometry

[Theorems]

# Section I

a) Prove that diagonals of a parallelogram divide it into two triangles of equal area.

b) Prove that diagonal PR of a given parallelogram PQRS divides two equal triangles PQR and PSR.

• Prove that, diagonal MO of given parallelogram MNOP divides two equal triangles MNO and MPO.

# Section II

a) Prove that, parallelograms on the same base between the same parallels are equal in area.

b) Parallelograms ABCD and EBCF standing on the same base BC between the same parallels BC and AF are

equal in area prove.

c) Parallelograms PQRS and MQRN standing on the same base QR between the same parallels QR and PN are

equal in area prove.

d) Prove that, rectagle MNPQ and parallelogram BNPA standing on the same base NP between the same parallels

NP and MA are equal in area.

e) Prove that, rectangle BNPA and parallelogram MNPQ are standing on the same base NP between the same

paralles NP and BQ equal in area..

f) In the given figure prove that area of rectangle MNPF = area of parallelogram ENPQ and area

of triangle MNE = area of triangle FPQ.

g) In the given figure, prove that area of rectangle ABED = area of parallelogram CBEF

and area of triangle ABC area of triangle DEF.

h) In the given figure prove that

i) DAF = CBE.

ii) ABCD = ABEF.

i) In the given figure prove that,

i) DXM = ZYN

ii) DXYZ = MXYN.



# Section III

a) Prove that the area of triangle is equals to half of the area of parallelogram standing on the same base and

between the same parallels.

b) Prove that the aera of triangle is equals to twice the area of triangle standing on the same base and between

the same parallels.

c) Prove that area of triangles MPQ is equal to one half of the area of parallelogram RPQN standing on the same

base PQ and between the same parallels PQ and MN.

d) Prove that area of paralegal RMNQ is equals to the twice the area of triangle PMN standing on the same base

MN and between the same paralles MN and PQ.

# Section IV

a) Prove that, trianlge standing on the same base between the same paralles are equal in area.

b) Prove that area of PEF and QEF standing on the same base EF and between the same parallels EF and PQ are

equal in area.

c) Prove that area of EPQ and FPQ standing on the same base PQ and between the same parallels PQ and EF are

equal in area.

# Section V

a) Prove that, central angle is double of inscribed angle if they stand on the same are in a circle.

b) Prove that inscribed angle is half of the central angle if they stand on the same are in circle.

c) P, Q and R are the three point lies on the circumference of circle with centre A prove that QAR is double of QPR

ofter joining PQ, PR AQ and AR

d) L, M and N are the three points lies on the circumference of circle with centre B prove that MBN is double of MLN

after joining LM, LN BM and BN.

# Section VI

a) Prove that inscribed angle standing on the same are in a circle area equal.

b) Prove that angles on the same segment of a circle area equal.

c) PQMN is a circle with centre R after joining PN, PM, MQ and NQ prove that PNQ is equal to PMQ.

d) WXYZ is a circle with centre A after WZ, WY, XY and XZ prove that WZX is equal to WYX.

# Section VII

a) Prove that opposite angles of cyclic quadrilateral are supplementary.

b) Prove that Sum of opposite angles of cyclic quadrilateral are 1800 or two right angles.

c) PQRS is a cyclic quadrilateral with centre M prove that P + R = 1800 and Q + S = 1800.

d) ABCD is a cyclic quadrilateral with centre N prove that A + C = 1800 and B + D = 1800.

e) W, X, Y & Z are 4 points lies on the circumference of a circle with centre P. After joining WX, XY, YZ & ZW, Prove

that WXY and WZY are supplementary.

f) W, X, Y & Z are 4 points lies on the circumference of a circle with centre Q. After joining WX, XY, YZ & ZW, Prove

that XWZ and XYZ are supplementary.

# SectionVIII

• Prove that angle on the semicircle being right angle.

# Section IX

• Prove that if one side of a cyclic quadrilateral is produced, prove that the exterior angle so formed is equal to

the opposite interior angle of quadrilateral.



SLC Model Question SLC _ Compulsory Math _ Geometric Deduction

[Short Questions]

# Section A

a) lbPsf] lrqdf ABCD df CD Pp7f ;r xf] / CD sf] s'g} ljGb' P 5 eg] APD + BCP = APB x'G5 egL k|dfl0ft ug{'xf];\ .

In the given figure, ABCD is a parallelogram and P is any point on CD. Prove that APD + BCP = APB.

b) ABCD Pp6f ;dfgfGt/ rt'e{'h xf] . AB sf] s'g} ljGb' P / AD sf] s'g} ljGb' Q 5 eg] QBC = APD + CPB x'G5 elg

k|dfl0ft ug{'xf];\ .

ABCD is a parallelogram. P is any point in AB and Q is any point on AD. Prove that. QBC = APD + CPB.

c) lbOPsf] lrqdf ABCD Pp6f ;=r= xf] h;df ljs0f{ BD df kg{] s'g} ljGb' X 5 . k|dfl0ft ug{'xf];\ .

AXB = rt'e{'h ABCX.

In the given figure, ABCD is a parallelogram is which X is any point on the diagonal BD. Prove that : AXB = quadrilateral ABCX.

d) ;Fu}sf] lrqdf PQRSPp6f ;dfgfGt/ rt'e{'h xf] . ljs0f{ PR sf] s'g} ljGb' M ;Fu Q / S hf]l8Psf 5g\ . l;4 ug{'xf];\ .

PQM sf] If]qkmn = PSM sf] If]qkmn

In the adjoining diagram, PQRS is a parallelogram, Q and S are joined to any point M on the diagonal PR of the

parallelogram Prove that are of the PQM = area of PSM

e) rt'e{'h ABCD df AO = Co eP 2ABD = ABCD x'G5 egL k|dfl0ft ug{'xf];\ .

In quadrilateral ABCD, AO = Co, prove that 2ABD = ABCD.

f) ABC sf] zLif{ljGb' A / BC sf] Pp6f ljGb' D hf]l8Psf] 5 / AD sf] dWoljGb' E 5 eg] ABC = 2EBC x'G5 elg k|dfl0ft ug{'xf];\ .

The vertex A of ABC is joined to a point D on the side BC. The midpoint of AD is E, Prove that ABC = 2EBC.

# Section B

a) ;Fu]sf] lrqdf ABCD Pp6f ;dfgfGt/ rt'e{'h xf] . CD df s'g} Pp6f ljGb' N 5 . AN n] BC nfO{ M ljGb'df 5'g] u/L a9fOPsf] 5 eg]

BNC sf] If]qkmn = DNM sf] If]qkmn x'G5 elg k|dfl0ft ug{'xf];\ .



In the adjoining figure, ABCD is a parallelogram, N is any point CD AN and BC are produced to meet the point M. Prove that are of BNC =

the area of the DNM.

b) lbOPsf] lrqdf ABCD Pp6f ;=r= xf] . olb DC = CQ eP PQC = BPQ x'G5 egL k|dfl0ft ug{'xf];\ .

In the given figure, ABCD is a parallelogram IF DC = CQ, prove that PQC = BPQ.

c) ;=r= ABCD sf] e'hf AD df s'g} ljGb' F 5 BF / CD nfO{ a9fP/ E ljGb'df ldnfOPsf] 5 . k|dfl0ft ug{'xf];\ . BCE = ABDE.

ABCD is a parallelogram F is any point on side AD.BF and CD are produced to meet at E. Prove that BCE = ABDE.

d) ABC sf] e'hf BC ;Fu ;dfgfGt/ /]vf XY 5 . BE//AC / CF//AB n] XY nfO{ E / F df e]6\5 eg] ABE = ACF x'G5 elg

k|dflf0ft ug{'xf];\ .

XY is a line parallel to side BC of ABC BE//AC and CF//AB meet XY in E and F respectively. Show that ABE = ACF

e) ABC df AB sf] dWoljGb' D / BC sf] s'g} ljGb' P 5 olb . CQ//DP eP 2BPQ = ABC x'G5 elg k|dfl0ft ug{'xf];\ .

In ABC, D is the mid-point of AB and P is any point on BC. IF CQ//DP, prove that: 2BPQ = ABC.

f) lbOPsf] lqe'h ABC df, BE / CD b'o{ dlWosfx¿ ljGb' O df k|lt5]lbt 5g\ . l;4 ug{'xf];\ . BOC sf] If]qkmn = rt'e{'h ADOE sf] If]qkmn

In the given triangle ABC, two medians BE and CD are intersect at O. Prove that area of BOC = area of quadrilateral ADOE.

g) ABC sf e'hfx¿ AB, BC, / CA sf dWoljGb'x¿ qmdzM P, Q, R 5g\ eg] k|dfl0ft ug{'xf];\ .

P, Q and R are the midpoints of sides AB, BC and CA respectively of ABC, prove that,

APR = BPQ = CQR = PQR = ABC.

h) lbOPsf] lrqdf AP, BP / CP sf dWoljGb'x¿ qmdzM L, M / N x'g\ eg] ABC = 4LMN x'G5 elg k|dfl0ft ug{'xf];\ .

In the given figure, L, M and N are the mid-points of AP, BP and CP respectively, prove that ABC = 4LMN.

# Section C

a) rt'e{'h ABCD sf] lzif{ljGb' C af6 ljs0f{ DB ;Fu ;dfgfGt/ x'g] u/L lvrLPsf] /]vfn] AB nfO{ nDafOPsf] /]vfsf] ljGb' E df

5f]Psf] 5 . l;4 ug{'xf];\ rt'e{'h ABCD sf] If]qkmn = DAE sf] If]qkmn

The line drawn through the vertex. C of the quadrilateral ABCD parallel to the diagonal DB meets AB product at E. Prove that the quad

ABCD = DAE in area.



b) lbOPsf] lrqdf PQRS ;dnDa rt'e{'h xf], h;df PQ//MN//SR 5g\ eg] l;4 ug{'xf];\ .

PSN sf] If]qkmn = QRM sf] If]qkmn

In the given figure, PQRS is a trapezium in which PQ//MN//SR. Prove that area of PSN = area QRM.

# Section D

a) rlqmo rt'e{'h ABCD df AP, BP, CR / DR qmdzM A, B, C / D sf cw{sx?åf/f ag]sf] rt'e{'h QPRS klg rlqmo rt'e{'h x'G5

elg k|dfl0ft ug{'xf];\ .

In a cyclic quadrilateral ABCD AP, BP, DR, CR are the bisectors of A, B, D, C respectively. Prove that PQRS is also a cyclic quadrilateral.

b) lbOPsf] lrqdf j[Qsf] s]Gb| ljGb' O, Jof; AB / DOAB eP AEC = ODA x'G5 elg l;4 ug{'xf];\ .

In the adjoining figure O is the center of a circle. AB is a diameter and DOAB. Prove AEC=ODA.

c) lbOPsf] lrqdf AD//BC 5 eg] AYC = BXD x'G5 egL k|dfl0ft ug{'xf];\ .

Given figure is a circle where AD//BC, Prove that AYC=BXD.

d) lbOPsf] lrqdf AC = BC / ABCD Pp6f rlqmo rt'e{'h xf] . eg] CD n] BDE nfO{ cfwf ub{5 egL k|dfl0ft ug{'xf];\ .

In the given figure, AC=BC and ABCD is a cyclic quadrilateral prove that DC bisects BDE.

e) b'O{ hLjfx? AB / CD k/:k/ nDa 5g\ AOD + BOC = 1800 x'G5 elg k|dfl0ft ug{'xf];\ .

Two chords AB and CD are perpendicular to each other. Prove AOD+BOC=1800.

f) lbOPsf] lrq Pp6f j[Qsf] xf] . h;df PMS = QNR eg] PQ//RS x'G5 egL k|dfl0ft ug{'xf];\ .

Given figure is a circle and PMS = QNR Prove PQ//RS.

g) lbOPsf] lrqdf s]Gb| ljGb' O ePsf] j[Qsf b'O{ Aof;x? AB / CD x'g\ . olb hLjf CE Aof; AB ;Fu ;dfgfGt/ / BOC clws

sf]0f x'g\ eg] rfk DBE sf] dWo ljGb' B x'G5 egL l;4 ug{'xf];\ .

In the figure, AB and CD are two diameters of a circle with center O. If the chord CE is parallel to AB and BOC is obtuse prove that B is the

mid point of arc DBE.



h) lbOPsf] lrqdf ABCD Pp6f rlqmo rt'e{'h xf] . olb BC = DE / CA n] BCD nfO{ ;dlåljefhg ub{5 eg] ∆ACE ;dlåjfx' lqe'h

x'G5 egL k|dfl0ft ug{'xf];\ .

In the given figure, ABCD is a cyclic quadrilateral. If BC=DE and CA bisects BCD, prove that ACE is an isosceles triangle.

i) lbOPsf] j[Qdf AB Aof; xf] . OC / OD Aof;fw{x? x'g\, hxfF rfk BC = rfk CD 5 eg] AD//OC x'G5 egL k|dfl0ft ug{'xf];\ .

In the adjoining figure, AB is the diameter. OC and OD are radii where Arc BC= Arc CD, prove that AD//OC.

# Section E

a) lbOPsf] j[Qdf AB Aof;, OC / OD Aof;fy{x? x'g\, hxfF BOC =COD x'g\ eg] AD//OC x'G5 egL l;4 ug{'xf];\ .

In the adjoining figure, AB is the diameter. OC and OD are radii where BOC=COD, prove that AD//OC.

b) lbOPsf] lrqdf BC, DBE sf] cw{s xf] . ABCD rlqmo rt'e{'h xf] eg] k|dfl0ft ug{'xf];\ AC = DC

In the given figure BC is a bisector of DBE. ABCD is a cyclic quadrilateral Prove that AC=DC.

c) O / P s]Gb|ljGb' ePsf b'O{ j[Qx? A / B df k|ltR5]lbt ePsf 5g\ . A / B tyf O / P hf]8\bf C df sf6LPsf 5g\ k|dfl0ft

ug{'xf];\ .

i) AC = BC ii) OCA = 900

Two circles having center O and P intersects at points A and B. AB meets OP at point C. Prove that (I) AC=BC (II) OCA=900.

d) O s]Gb|ljGb' ePsf] j[Q XPY / MQN nfO{ /]vf XY n] X / Y tyf M / N df sf6]sf] 5 k|dfl0ft ug{'xf];\ . XM = YN

O is the center of two circles XPY and MQN. XY cuts the circle XPY and MQN at X,Y and M,N respectively. Prove XM=YN.

e) ;Fu}sf] lrqdf ljGb' A af6 AB / AC :kz{ /]vfx? lvlrPsf 5g\ . :kz{ /]vf DE ljGb' F df :kz{ ePsf 5g\ eg]

k|dfl0ft ug{'xf];\ .

In the adjoining figure A is the external point of the circle. AB, AC and DE are tangents of the circle at B, C and F. prove that

AB+AC=AD+DE+AE.

f) lrqdf AB, BC, CD / DA :kz{ /]vfx? j[Qsf] 3]/fsf] qmdzM P,Q,R / S :kz{ ljGb'df :kz{ ePsf 5g\ k|dfl0ft ug{'xf];\ AB + CD =

BC + AD

In the given figure AB, BC, CD and DA are tangents of the circle at P, Q, R, S respectively. Prove that AB+CD=BC+AD.



g) lbOPsf] lrqdf b'O{ j[Qx? M / N df k|ltR5]lbt 5g\ . PQ / N df ljGb' M af6 kf/ ePsf 5g\ . R,P,S / Q qmdzM ljGb' N

;Fu hf]l8Psf 5g\ eg] l;4 ug{'xf];\ .

In the given figure two circles intersects at M and N respectively PQ and RS pass through M. R, P, S, Q are joined with N. Prove that

PNR=SNQ.

h) lbOPsf] lrqdf AEC = BFD 5 . l;4 ug{'xf];\ . i) ABC = BCD ii) AB//CD

In the given figure AEC=BFD. Prove that i) ABC = BCD ii) AB//CD

i) lbOPsf] lrqdf, PQ = PR 5g\ eg] QR//ST x'G5 egL b]vfpg'xf];\ .

In the given figure PQ=PR. Prove that QR//ST.

j) lbOPsf] lrqdf b'O{ hLjfx? PQ / RS ljGb' X df nDa x'g]u/L sfl6Psf 5g\ . k|dfl0ft ug{'xf];\ M

=

In the adjoining figure chords PQ and RS intersects at X. Prove that, =

k) tn lbOPsf] lrqdf ABCD Pp6f rlqmo rt'e{'h xf] . olb AB = AC eP BDE sf] cw{s AD xf] egL l;4 ug{'xf];\ .

In the given figure ABCD is a cyclic quadrilateral. If AB=AC. Prove that AD is the bisector of BDE.

l) lbOPsf] lrqdf ABC Pp6f lqe'h xf] . h;df AB = AC 5 . k|dfl0ft ug{'xf];\ . AD = AE

In the given figure ABC is a triangle in which AB=AC. Prove that AD=AE.

# Section F

a) lbOPsf] lrqdf PQ//RS eP PTR = QUS x'G5 egL k|dfl0ft ug{'xf];\ MIn the given figure PQ//RS prove that PTR=QUS.

b) ;Fu}sf] lrqdf ABCD Pp6f ;=r= xf] . a[Qn] AB nfO{ E / DC nfO{ F df sf6]sf] 5 eg], k|dfl0ft ug{'xf];\ EFD = ABC

In the adjoining figure ABCD is a parallelogram. The Inscribed circle cuts AB at E and CD at F Prove that EFD=ABC.



c) lbOPsf] lrqdf ABCD rlqmo rt'e{'h xf] / AB//DC 5 . l;4 ug{'xf];\ . (i) AD=BC. (ii) AC=BD.

In the adjoining figure ABCD is a cyclic quadrilateral AB//DC. Prove that (i) AD=BC. (ii) AC=BD.

d) lbOPsf] lrqdf, ∆ABC ;djfx' lqe'h 5 . BAC sf] cw{s AD eP ∆BCE ;dl4jfx' lqe"h xf] egL k|dfl0ft

ug{'xf];\ .

In the given figure, ABC is an inscribed equilateral triangle if AD is the bisector of BAC, prove that BCE is an isosceles triangle.

e) lbOPsf] lrqdf, b'O{ j[Qx? P / Q ljGb'x?df Ps cfk;df sflf6Psf 5g\ . P / Q af6 b'O{ /]vfx? APB / CQD uPsf 5g\

eg] k|dfl0ft ug{'xf];\ .

In the given figure, APB and CQD are the straight lines through the points of intersection of two circles. Prove that

(i) AC//BD (ii) CPD=AQB.

f) ;Fu}sf] lrqdf O s]Gb| ePsf] j[Qdf AMOXOZ eP egL k|dfl0ft ug{'xf];\ . OAZ=XYO.

In the adjoining figure, O is the center of circle. If AMOXOZ, then prove that OAZ=XYO.

g) lbOPsf] lrqdf DE Aof; 5 . olb BE = CE df AED = (ABC - ACB) x'G5 egL k|dfl0ft ug{'xf];\ .

In the given figure, DE is a diameter. If arc BE=arc CE, then prove that AED=½ (ABC- ACB)

h) lrqdf s]Gb|ljGb'x? X / Y ePsf j[Qx?sf] :kz{ljGb' P eP k|dfl0ft ug{'xf];\ . XQ//YR.

In the figure the point of contact of two circles having center X and Y is P. Prove that XQ//YR.

i) lbOPsf] j[Qdf O s]Gb|ljGb' xf] . AB / CD b'O{ :kz{ /]vfx? x'g\ . k|dfl0ft ug{'xf];\ BAC = 2OBC

In the given figure O is the centre of the circle AB and CD are two tangents Prove that BAC = 2OBC.



j) lrqdf rlqmo rt'e{'h ABCD sf] Ps e'hf BC nfO{ CP = AB x'g] u/L ljGb' P ;Dd nDafO{Psf] 5 . olb ABC sf] cw{s

BD 5 eg] ∆DBP ;dl4afx' lqe'h xf] egL l;4 ug{'xf];\ .

In the adjoining figure the side BC of a cyclic quadrilateral ABCD is produced to the point P making CP=AB. If BD bisects ABC, prove that

DBP is an isosceles triangle.

k) ;Fu}sf] lrqdf AB sf] dWoljGb' X 5 . SAT :kz{/f]vf xf] . ST//XY 5 . k|dfl0ft ug{'xf];\ . i) AYX = ABC

ii) BXYC \rlqmo rt'e{'h xf] .

In the adjoining figure X is the midpoint of AB, SAT is tangent & ST//XY. Prove that

(i) AYX=ABC (ii) BXYC is a cyclic quadrilateral.

l) ;Fu}sf] lrqdf PT Aof; xf] . rfk SR = rfk RT eP k|dfl0ft ug{'xf];\ . PS//OR.

In the adjoining figure PT is a diameter Arc SR = Arc RT. Prove that PS//OR.

m) lbOPsf] lrqdf j[Qsf hLjfx? MN / RS af\xo ljGb' X df sfl6Psf 5g\ . eg] l;4 ug{'xf];\ . MXR=(Arc MR - Arc NS).

In the given figure chords MN and RS intersect at external point X. Prove that MXR = (Arc MR - Arc NS).

# Section G

a) lrqdf P / Q qmdzM rfk AB / rfk AC sf dWoljGb'x? x'g\ eg] AX = AY x'G5 egL k|dfl0ft ug{'xf];\ .

In the figure P and Q are the mid-points of arc AB and arc AC respectively. Prove that: AX = AY.

Hints: - VII(a) 1. Join PA and QA. 2. PQA = PAB =a 3. CAQ = QPA = b

4. AXY = PAB + QPA = a + b 5. AYX = CAQ + AQP = b + a

6. AXY = AYX 7. AX = AY

b) lrqdf, hLjfx? PR / QS k/:k/ ;dsf]0f kf/Lsg ljGb' E df sfl6Psf 5g\ . QR sf] dWo ljGb' X / XE a9fpbf PS sf] Y df e]6\5 eg] EYPS x'G5

egL k|dfl0ft ug{'xf];\ .

In the figure chords PR and QS of a circle intersect at point E at right angles. X is the mid-point of QR and XE is produced

meets PS in Y Prove that EYPS.



c) lbOPsf] lrqdf PQ//AB eP AY = BY x'G5 egL k|dfl0ft ug{'xf];\ .

In the given figure, PQ//AB. Prove that AY = BY.

d) lbOPsf] lrqdf PB = QB eP MN//PQ x'G5 egL k|dfl0ft ug{'xf];\ .

In the given figure, PB = QB. Prove that MN//PQ.

e) olb lbOPsf] lrqdf CMAB / e'hf MN//DE eP AMNC Pp6f rlqmo rt'e{'h / CNAE x'G5 egL k|dfl0ft ug{'xf];\ .

In the given figure, CMAB and MN//DE, then prove that AMNC is a cyclic quadrilateral and CNAE.

Hints: VII(e) Join A & C (i) CAN = CMN (both are equals to CDE so, AMNC is a Cyclic quadrilateral (ii) AMC = CNA [since both are on the

same arc hence CNAE]

f) ABCD Pp6f rlqmo rt'e{'h xf], h;df e'hfx? AD / BC nDAofpFbf ljGb' L df / e'hfx? AB / DC nDAofpFbf ljGb' M df

e]6]sf 5g\ . ALB sf] cw{sn] DC nfO{ ljGb' P df / AB nfO{ R df / AMD sf] cw{sn] BC nfO{ ljGb' Q df / AD nfO{ S df e]6]sf 5g\ eg] k|dfl0ft

ug{'xf];\ .

i) PR / QS k/:k/ ;dsf]0f x'g]u/L sf6\5g\ .

ii) PQRS Ps ;djfx' rt'e{'h xf] .

ABCD is a cyclic quadrilateral in which the sides AD and BC when produced meet at L and the sides AB and DC, when produced meet

at M. AMD meets BC in Q and AD in S. Prove that i) PR and QS intersect at right angles ii) PQRS is a rhombus.

Hints VII(f) 1. In LAR & LPC i. ALR =PLC ii. LAR = LCP

2. LAR LPC [by A.A case]

3. ARL = DPR

4. MPR = MRP 5. MP = MR 6. PN = RN & MNPR

7. Similarly SN = NQ & LNSQ 8. PQRS is a rhombus

g) A / B df k|ltR5]lbt b'O{ j[Qx?df Pp6f j[Qdf s'g} ljGb' P 5 PA / PB nDAofpFbf csf{] j[Qdf Q / R df e]6\5 eg] P ljGb'df lvr]sf]

:kz{ /]vf QR ;Fu ;dfgfGt/ x'G5 egL k|dfl0ft ug{'xf];\ .

Two circles intersect at A and B. P is any point on one circle PA and PB produced & meet other circle at Q and R, prove that tangent drawn

on a point P is parallel to QR.

h) lbOPsf] lrqdf O j[Qsf] s]Gb|ljGb' CE :kz{/]vf / D :kz{ljGb' 5 eg] k|dfl0ft ug{'xf];\ .

In the given figure, O is the centre of the circle. CE is a tangent and D is the point of contact then prove that:

BOD = 2(AFD-ACD).

Hints : VII(h) Join BD & DA



1. Let AFD = ABD = a, BAD = PDC = b & BCD = c

2. BOD = 2BAD = 2b 3. a = b + c b = a - c 2b = 2a-2c

4. BOD = 2(a-c) or, BOD = 2(AFD-ACD)

# Section H

a) lrqdf A, B, C, D, P, Q, R, S kl/lwsf ljGb'x? eP P + Q + R + S = 5400 x'G5 egL k|dfl0ft ug{'xf];\ .

In the figure A, B, C, D, P, Q, R, S are the points at the circumference then prove that: P + Q + R + S = 5400

Hints: VIII(a) Draw BD then opposite angles of cyclic quadrilateral is 1800 so.

P + ADB = 1800, S + ABD = 1800, R + DBC = 1800 and Q + CDB = 1800

Adding all of them

(P + Q +S) + [(ADB + CBD) +(ABD + DBC)] = 7200

or, (P + Q + R + S) + (ADC + ABC) = 7200

or, (P + Q + R + S) + 1800 = 7200 P + Q + R + S = 5400

b) lrqdf ABC ;dafx' lqe'h xf] . D / E qmdzM rfk AB / rfk AC sf dWolaGb'x? eP k|dfl0ft ug{'xf];\ M 3XY = DE

In the adjoining figure, ABC is an equilateral triangle. D and E are the midpoints of arc AB and arc AC respectively. Prove that : 3XY = DE.

Hints: VIII(b)

(1) = = = =[Being ABC equilateral from given]

(2) DAX = ADX [Both are angles at the circumference and = ]

(3) DX = AX [From (ii)] (4) Similarly AY = YE[Same as above]

(5) AX = AY = XY [Sides of an equilateral triangle]

(6) DX = YE[From (iii), (iv) and (v)]

(7) DE = DX + XY + YE = XY + XY + XY = 3XY

c) ABC Pp6f ;dl4afx' lqe'h j[QcGtu{t 5 . B / C sf cw{sx? kl/lwsf laGb'x? X / Y df qmdzM e]6 ePsf 5g\ . ax'e'h BXAYC sf rf/cf]6f

e'hfx? a/fa/ x'G5g\ egL k|dfl0ft ug{'xf];\ .

ABC is an isosceles triangle inscribed in the circle, the bisectors of B and C meet the circumference at X and Y respectively. Shoe that the

polygon BXAYC must have four of its sides equal.

Hints: VIII(c) 1) ACY = BCX [Being CX is a bisector of ACB]

2) ABY = YBC [Being BY is a bisector of ABC]

3) arc AX = arc BX = arc AY = arc YX [Corresponding areas of (i) and (ii)]

4) AX = BX = AY = YC [Corresponding segment of (iii)]

d) lbOPsf] lrqdf BAC sf] cw{s AP 5 eg] EF / BC ;dfgfGt/ 5g\ egL k|dfl0ft ug{'xf];\ .

In the given figure, AP is bisector of BAC. Prove that EF and BC are parallel.

Hints:VIII(d) (1) Join AD then being ABP = EAF and BAP = EDF we get EAF = EDF and AEFD is a cyclic quad.

(2) EFA = EDA and EDA = ACB gives EFA = ACB and EF//BC

e) ABCD Pp6f ju{ / AEF Pp6f ;dafx' lqe'h Pp6} j[Qdf cGtu{t 5g\ EF//BD x'G5 elg k|dfl0ft ug{'xf];\ .



ABCD is a square and AEF is an equilateral triangle inscribed in the same circle. Prove that EF//DB.

Hints:VIII(g) (1) Being AD = AB:AD = AB (2) AE = AF:ADE = ABF

(3) DE = BF(subtracting (i) from (ii) hence BD/EF.)

f) ABC df, x, y z qmdzM BC, AC / AB sf dWoljGb'x? x'g\ olb AWBC eP W,X,Y, Z ljGb'x? rqmLo x'g egL k|dfl0ft ug{'xf];\ .

In ABC, X, Y and Z are the mid-points of BC, AC and AB respectively. If AWBC, prove that W, X, Y and Z are co cyclic.

Hints:VIII(f) Join WZ

(1) AZ = BZ = WZ [Being ABW rt. angled ]

(2) ZBXY is a parm. so B = ZYX and

(3) Being BZ = ZW, B = ZWB hence ZWB = XYZ and W, X, Y, Z are con-cyclic.

g) ;Fu}sf] lrqdf AB = BC, AC//XY, CX, AY, XD / YD x? ;Lwf/]vfx? eP k|dfl0ft ug{'xf];\ ACYX Pp6f rqmLo rt'e{'h xf] .

In the adjoining figure, AB = BC, AC//XY, CX, AY, XD and YD are straight lines. Prove that: ACYX is a cyclic quadrilateral

Hints: VIII(g) (1) AB = BC, BAC = BCA and

being AC//XY, BAC = BYX

(2) Combing all we get BCA = BYX ACYX is a cyclic quadrilateral

h) lrqdf ABC Pp6f cGtu{t lqe'h xf] . h;df PMAB,PNAC / PRBC 5 eg] MNR Pp6f ;Lwf/]vf x'G5 egL k|dfl0ft ug{'xf];\ .

ABC is an inscribed triangle, PMAB,PNAC and PRBC. Prove that MNR is a straight line.

Hints:VIII(h) Join PA and PC

1) Show ANPM is a cyclic quad

2) Show PNRC is a cyclic quad.

3) PCB = PAM[Being ABCP is a cyclic quad.]

4) PCR = PAM

5) PNM = PAM

6) PCR = PNM

7) PNR + PCR = 1800 [Being the opposite angles of a cyclic quad.]

8) PNR + PNM = 1800 [from 6 & 7]

9) RNM is a straight line [Form (8)]



SLC Model Question SLC _ Compulsory Math _ Measurement of Areas & Volumes of Solid Figures


# Type A Questions

1. Find the Volume of given solid figures.



2. Find the curve surface area & total surface area of the given solid figure.



# Type B Questions

3. a) Find the volume of the given solid figure.

Hints: III(a) r = 5cm, In FIE, EF = = 13cm

All cones are equal, so height (h) = 12 cm & slant height (l) = 13cm

Volume of one cone (V1) = cm3

Volume of cylinder (V2) = r2 × 28 cm3

Volume of hemisphere (V3) = cm3

V = 3V1 + V2 + V3 = r2 (12 + 28 + ) cm3

b) Find the curved surface & total surface area of the given solid figure.

Hints III (b)

r = 5cm, EI = 12cm, In FIE, EF = = 13cm

T.S.A. of one cone (A1) = r (r + 13), T.S.A. of cylinder (A2) = 2r(r+28)

T.S.A of hemisphere (A3) = 3r2, Area of one circle (A4) = r2

T.S.A of whole figure (A) = 3A1 + A2 + A3 - 6A4 = 270 + 430 + 75 - 150 = 625 = 1964.28cm2

C.S.A of whole figure = A or, 3s1 + s2 + s3 = 3rl + 2rh + 2pr2

c) A cone is placed on a cylinder so that the base of the cone covers exactly the base of the cylinder as shown in

the figure. The area of the base of the cylinder is 125 cm2 and the height of the cylinder is 3 cm. If the volume of

the whole solid is 625cm3, find the height of the solid.



• A cone is placed on a cylinder so that the base of the cone covers exactly the base of the cylinder as shown in

the figure. The area of the base of the cylinder is 125 cm2 and the height of the cylinder is 3 cm. If the volume of

the whole solid is 625cm3, find the volume of conical part only.

e) A tent is of the shape of right circular cylinder upto a height of 4 cm with radius 14 cm and then becomes a right

circular cone with height of 6 cm. Calculate the total surface area of tent.

f) A tent of height 33 m is in the form of a right circular cylinder of radius 60 m and height 22 m surmounted by a right

circular cone of the same radius. Find the total surface area of the tent.

g) In the given figure, cone, cylinder and hemisphere of same radius are joined. Find the ratio of volumes

of cone, cylinder and hemisphere.

h) In the given solid, the radii of cone and hemisphere are equal. Find the ratio of their volumes.

i) A circus tent is cylindrical to a height of 5 m and conical above it. If its diameter is 112 m and its slant height 73 m.

Calculate the total area of canvas required.



# Type C Questions

4. a) Three silver metallic sphere of radius 1cm, 6 cm and 8 cm respectively are melted and form a single new

metallic sphere. Find the diameter of that new sphere.

b) Three metallic sphere of radius 2cm, 12 cm and 16 cm respectively are melted and form a single new metallic

sphere. Find the diameter of that new metallic sphere.

c) The diameter of an orange is made one-third of its original size. What will be the difference in the volume?

d) If the radius of a sphere is doubled, what will be the difference in the volume.

e) A solid iron sphere of radius 3 cm is melted to form a solid right circular cylinder of diameter 6 cm. Find the

height of the cylinder.

f) A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base

diameter 8 cm. Find the height of the cone.

Answers

I. a) 11314.28 m3 b) 44869.52mm3 c) 4646.68cm3

d) 6146.64cm3 e) 553.13cm3 f) 2614.85cm3

g) 290400cm3 h) 11314.28cm3 i) 2258.66cm3

j) 6930cm3 k) 2310cm3 l) 1950.66cm3

m) 358.27cm3 n) 1961.13cm3

II. a) 1980cm2, 1980cm2 b) 314.28cm2, 364.57cm2

c)804.56cm2, 1005.71cm2 d) 4604.28m2, 4682.99m2

e) 9208.57mm2,9522.85mm2 f)1551.44cm2, 15551.44cm2

g) 19800cm2, 31114.28cm2 h) 2074.27cm2, 3331.43cm2

i) 985.6cm2, 985.6cm2 j) 858cm2, 858cm2

k) 282.85cm2, 282.85cm2 l) 829.71cm2, 829.71cm2

III. a) 3404.74cm3 b) 625cm2 or 1964.28cm2, 625cm2 or 1964.28cm2

c) 9cm d) 250cm3 e) 1020.8cm2 f) 19800m2 g) 1:3:1 h) 6:7

i) 14608m2 j) 9 cm k) 36 cm

IV. a) 9cm b) 18cm c) times of original volume d) 7 times of original volume e) 4cm f) 14 cm



SLC Model Question SLC _ Compulsory Math _ SETS


• If n(A) = 40, n(B) = 50 / n(AB) = 15, find the value of n(AB)

• n(U) = 80, n(P) = 50, n(Q) = 30 / n(PQ) = 10. Draw the Venn diagram to illustrate the above information and find

the value of n(PQ).

• If n(A) = 45, n(B) = 65 & n(AB) = 85 then,

• Find the value of n(AB)

• Find the value of n0 (B)

• Show it in venndiagram.

• A survey conducted shows that 75% like milk and 60% like curd. If they like atleast one of them then]

• Show it in a Venn diagram. • What percent were there, who like both ?

• What was the percent, who like curd only ?

• In survey of 170 girls it was found that 18 girls can neither dance nor can sing, 96 can dance and 69 can sing. By

representing these information in a Venn diagram, find the number of girls who can both dance and sing.

• Out of 100 students, 80 passed in science, 71 in Mathematics, 10 failed in both subjects and 7 didn't appear in an

examination. Find the number of students who passed in both subject using venn diagram.

• In a survey of 120 students, it was found that 17 drink neither tea nor coffee. 88 drink tea and 26 drink coffee. By

drawing Venn-diagram, find out the number of students who drink both tea and coffee.

• In a survey of 2400 tourists visited in Nepal, it was found that 1650 liked to visit Bhaktapur, 850 liked to visit Lalitpur

and 150 did not like to visit both places:

• Represent the above information in a Venn-diagram. • How many were there who liked to visit both places ?

• How many were there who liked to visit Bhaktapur only ?

• In a survey of 2000 tourists visited in Nepal, it was found that 1125 liked to visit pokhara, 750 liked to visit Janakpur

and 250 did not like to visit both places.

• Represent the above information in a venn-diagram • How many were there who like to visit both places ?

• How many were there who liked to visit janakpur only ?

• In a school of 80 students of class X were asked what they would like milk or tea 60 said they would like tea, 50

said they would like milk and 10 said they would like neither milk nor tea. By drawing Venn-diagram, find the

number of students. • (Who like both tea and milk ?)

• (Who like milk only ?)

• (Who like tea only ?)

• In a survey of group 100 students 55 like to read Muna magazine, 45 like Yuba Manch and 14 do not like to read

either of them, Draw a Venn diagram and give of answer to the following How many of them like to read

• Muna magazine only

• Yuba Manch only.

• Both of them.

• In a survey of 60 students, 30 drink milk, 25 drink curd and 10 drink milk as well as curd then

• Draw a venn -diagram of above information.

• Find the number of students who drink neither of them.

• Out of 100 students in a S.L.C. Examination, 80 passed in math, 60 passed in science and 50 passed in both

subjects.

• Draw a Venn diagram to illustrate the above information



• Find the number of students failed in both subjects.

• In a class of 25 students, 17 like Volleyball, 15 like Basketball and 10 like both games, Illustrate it with a Venn-

diagram and find the number of students who don't like any of the games

• In a survey of 200 students 125 liked to admit in science faculty, 100 in humanity faculty and 40 like to admit either

of faculties and the rest were found not to be admitted in both faculties.

• Show the above information in a venn diagram.

• Find how many were there who don't like to admit in both faculties.

• Out of 5000 Japanese who are on Nepal tour, 40% have already toured India 30% have also toured Pakistan. If

10% of them have toured both the countries by drawing Venn-diagram find the number of tourists those who

have not yet visited both the countries.

• Out of 1350 candidate 600 passed in Health, 700 in social, 350 in Nepali and 50 did not passed in all three

subjects. If 200 passed in Health and social, 150 in Health and Nepali, 100 in social and Nepali. • How many candidates passed in all subjects ?

• Illustrate the above information in a Venn-diagram.

• G, S & K represent the people who read Gorkhapatra, Samacharpatra & the Kantipur daily. If n(GSK) = 100%,

n(G) = 50%, n(S) = 55% n(K) = 35%, n(GS) = 15, n(SK) = 10%, n(GK) = 20%, then what percentage of the people read all these three daily papers ?

• Of the total candidates in an examination, 40% students passed in math 45% in science and 55% in Health. If 10%

passed in math and science, 20% in science & Health and 15% in Health and Maths.

• Draw a Venn-diagram to show the above information • Calculate the percentage. Who passed in all three subjects ?

• If n(A) = 65, n(B) = 50, n(C) = (AB) = 25, n(BC) = 20, n(CA) = 15, n(ABC) = 5 and n(U) = 100, find the value of

n( )

• In a survey of 100 people 65 read Kantipur, 45 read Gorkhapatra, 40 read Himalayan times, 25 read Kantipur as

well as Gorkhapatra, 20 read Kantipur as well as Himalayan times, 15 read Gorkhapatra as well as Himalayan

times and 5 read all three news papers.

• Draw the Venn-diagram to illustrate the above information.

• How many people don't read all three news papers.

• In a survey of a group of students it was found that 60 like to listen poems, 45 liked to listen stories, 35 liked to listen

both and 10 did not like to listen both. Then

• Present the above information in a Venn-diagram. • How many students were surveyed ?

• How many students were liked to listen the story only ?

• In a survey of a group showed that 60 liked tea, 45 liked coffee, 30 liked milk, 25 liked coffee as well as tea. 20

liked tea as well as milk, 15 liked Coffee as well as milk and 10 liked all three how many were asked this question.

Solve by drawing venn -diagram.

• Among the candidates appeared in an examination, 80% passed in English, 85% in Maths and 75% in both English

and Maths. Find the total number of candidates if 45 candidates were failed in both English and Maths.

• By filling the following information in a venn diagram, Find the cardinal number of n (U). n(L) = 14, n(M) = 13, n(N)

= 22, n(LMN) = 6, n(LM) = 7, n(MN) = 9, n(LN) = 11, n( ) = 4

• In a school 60 students passed in Mathematics, 45 passed in Science and 30 passed in both subjects. • How many students were there in the school ?

• How many students have passed in Mathematics only ?

• How many students had passed in Science only ?

• Represent the above information in a Venn-diagram

• In a college, 65 are studying in mathematics, 50 students are studying in science and 35 students are studying in

both subjects. • How many students are studying in the college ?

• How many students are studying in Mathematics only ?

• How many students are studying in Science only ?

• Represent the above information in a Venn - diagram.

• In an examination, it was found that 55% failed in Maths and 45% failed English. If there were 35% passed in both

subjects. • What percent failed in Maths only ?

• What percent failed in English only ?

• Represent the above information in a venn diagram.

• In a survey of a village, it was found that 85% of the people like Dashain Festival and 60% like Tihar Festival. If there

were 5% people who did not like both festival: • What percent liked Dashain Festival only ?



• What percent liked Tihar Festival only ?

• Represent the above information in a Venn-diagram.

• In a school, all students play either volleyball or football or both. 300 play football. 250 volleyball and 110 play

both game draw a Venn diagram to find.

• Number of students who play football only.

• Number of students who play volleyball only.

• The total number of students in the school.

• In a survey of a community of 240 people, it was found that 151 people could speak Newari language, 95 could

speak English language and 14 could speak neither of the languages. Represent the above information in a

Venn-diagram supposing x as the number of people who can speak both languages.

• Find the value of x.

• Find the number of people who can speak English only.

• 40% of the students of a school play football, 30% play volleyball and 20% play both. If 90 students play neither

football nor volleyball use Venn-diagram and find the number of students in the school. Also find the number of

students who play only football.

• Out of 20 staffs in an office, 15 can speak local Newari language other than Nepali language, 8 can speak

Bhojpuri, while 3 can not speak any local languages. By drawing venn diagram, find the number of staffs.

• Who can speak at least any one local language ?

• Who can speak any one local language only out at the two.

• 40 students were asked what they would like milk or curd. 30 said they would like curd, 25 said they would like milk

and 5 said they would neither like milk nor curd. By drawing Venn diagram, find the number of students.

• Who like milk only ?

• Who like curd only ?

• Who like either milk or curd, any one only ?

• Set P & Q are two subsets of a universal set U. Illustrate the given information in a venn -diagram & solve the

following problems.If n(U) = 40, n(P) = 25, n(Q) = 18, n(PQ) = 5 n(PQ) = ? n( ) = ?, n(P - Q) = ? n(Q-P) = ?

• If U = {X:X is a positive integer less than 16} A = {Y:Y is a prime number} B = {Z:Z is an odd number} then write down

the cardinal value by drawing venn diagram.

• n(AB)

• n( )

• n(A - B)

• n[U-(AB)]

• By asking question in a group, 110 answered TB spread through drugs, 75 said through smoking and 60 said

through food. Among them, 25 said from both drugs and smoking, 10 said from smoking & food & 10 said from

drug and food, while 5 said from all three. • How many pedestrians were involved ? Solve by drawing Venn-diagram.

• How many said TB spreads through drugs only ?

• In a group 30 students read Math, 24 read Economic, 22 read Statistics 14 read Math only,8 read Economics only,

6 read Math and Statistics only, 2 read Math and Economics only and 8 read neither of them • Find the number of students in their group ?

• How many students read Economics and Statistics only ?

• How many students read all three subjects ?

• Show all the information in Venn-diagram.

• In a class of 25 students, 12 have taken Mathematics, 8 have taken Mathematics but not Biology. Find the

number of students who have taken Mathematics and Biology and those who have taken Biology but not

Mathematics. Solve by making a Venn-diagram.

• In a interview of 50 people. 15 liked milk but not coffee, 5 liked coffee and milk and 5 did not like both. How many liked coffee only ? Represent all result in venn diagram also.

• Each of a group of 20 students study at least one of the three subjects Nepali, English and Maths. All those who

study English also study Nepali 3 study all three subjects. 4 students only study Nepali. 8 students study English. 14

students study Nepali. Draw Venn-diagram to illustrate above information. • How many students study only Nepali and Maths but not English ?

• How many students study only Maths ?

• Show all the information in Venn diagram ?

• Each group of 25 students studies at least one of the three subjects Nepali, English and Maths. All those who study

English also study Nepali, 4 students study all three subjects, 5 students study only Nepali. 7 students study English

and 15 students study Nepali.



• Draw the venn-diagram to illustrate above information. • How many students study Nepali and Math but not a English ?

• How many students study math only ?

• In a group of students 18 read Account, 19 read Maths, 16 read Science. 6 read Account only, 9 read Maths

only, 5 read Maths and Account only, 2 read Maths and Science only. By using venn-diagram, find the following • How many read all subjects ?

• How many read science only ?

• How many read account and science ?

• How many students are there altogether ?

• Out of 50 students in a class like Mathematics or Science or both. Among them 20 like both subjects and the ratio

of Maths and Science is 3:2

• Find the number of students who like Maths only.

• Find the number of students who like Science only.

• Illustrate the above information in a venn diagram.

• In a class of 75 students, 30 students liked Nepali but not Science and 25 students liked Science but not Neplai. If

10 students did not like both, how many students liked both subjects and represent the above information in

venn-diagram.

• In a class of 55 students, 15 students liked maths but not English and 18 students liked English but not Math. If 5

students did not like both, how many students liked both subjects and represent the above information in Venn-

diagram.

• In an election 1400 voters did not cast their votes and 400 votes are declared invalid. Two candidates A and B in

the election A defected B by 60 votes. It was found that A secured 2400 of total rolled votes. Find the number of

voters.

• In a survey among 100 people, 50 liked coffee, 30 liked milk, 40 liked tea, 20 like coffee only, 25 liked tea only, 10

liked tea and coffee and 5 liked tea, coffee and milk all. Using venn-diagram, find the number of people who

liked neither of these.

• An ice-cream shop sold three type of ice creams A, B & C out of the customers questioned, 470 liked A, 300 liked

B, 350 liked C, 130 liked A and B, 120 liked A and C, 70 liked B and C, 50 liked all three.

• Draw the venn-diagram to illustrate above information • How many liked exactly two types ?

• A survey was done about mobile and telephone facility 200 people had telephone facility, 25 had both facility 135 people did not have telephone facility and 300 did not have mobile facility ?

• How many participants in the survey ?

• How many had mobile facility only ?

• In a class of 65 students, 10 students liked Maths but not English and 20 students liked English but not Maths. If 5 students did not like both how many students liked both subjects ? Solve by Venn-diagram.

• In a result of first terminal examination of class ten 20% students are passed in all three subjects 46% failed in

Maths, 57% failed in English, 42% failed in dance, 35% failed in Maths and English, 25% failed in English and dance,

15% failed in dance and Maths.

• Draw venn diagram to show above information

• Find how many percent failed in all three subjects

• In a village of 140 house, 60 believe in Buddhism, 70 in Hindu Religion and 45 in other religion. Among then 17

house don't find any difference in Hindu and Buddhims, 18 house don't find any different in Hindu and other

religion while 16 houses don't find any difference between Buddhism and other religion. If 6 houses don't believe

in any religion, find how many houses do not find any difference in any religion. Show it in venn-diagram.

• In a group of players 40 play Volleyball out of which 25 play volleyball only. 10 play football only and 8 play

badminton only 27 play football, 30 play badminton out of which 15 play volleyball and badminton both. Using

venn-diagram find. • How many play all three games ?

• How many play two games only ?

• In an interview of 60 people space times, Rajdhani and Himalayan times are liked by 45, 30 and 15 people

respectively. If the total number of people who liked only two magazines in 22 and they liked at least one of the

magazines. Find the number of people who liked all the magazines.

• In an examination 48% failed in science, 39% in account and 33% in History, 12% in science and Account, 9% in

science and History and 13% in Account and History and 3% in all three subjects. Draw Venn-diagram to

represent the given information and find the.

• Percentage passed in all three subjects.

• Percentage failed exactly in two subjects.

• Percentage failed exactly in one subject.



Answers



SLC Model Question SLC _ Compulsory Math _ Time and work

[Short Questions]

# Important Questions for SLC

• Ram can do a work in 9 days and Bhawesh can do it in 12 days. In how many days, both together, can do the work ?

• A and B can do a work in 30 days and 40 days respectively. In how many days can A and B working together do it ?

• A and B can do a work in 12 days, which A alone can do in 30 days, In how many days B alone can do it ?

• A and B can do a piece of work in 12 days. B alone can do it in 30 days. Both started the work together but B leaves after 5 days. In how many days is the whole work completed ?

• Mohan and Sohan can do a piece of work in 12 days. Mohan alone can do it in 18 days. Both started the work together, but Mohan left the work after 6 days. How long will Sohan work alone to finish the remaining work ?

• A and B can finish a piece of work in 12 and 18 days respectively. After some days when both start to work

together A leaves and B completes the remaining work in6 days. How many days has A worked together before he left ?

• A, B and C working together can do a piece of work in 18 days. A could do it alone in 72 days. After working together for 6 days. A is fallen ill. How long will B and C together take to finish the remaining work ?

• A, B, and C can finish a piece of work in 30, 40 and 60 days respectively 10 days after they started to work

together B leaves. 4 days after B left, A leaves and C completes the remaining work. Find how many days C had

worked altogether.

• A and B can do a piece of work in 45 and 30 days respectively. They work at it together for some days then A leaves 5 days before completion the work. How many days would they work together ?

• A can do a piece of work in 20 days, B can do it in 30 days and C in 40 days. Three of them started the work

together. A left the work after 5 days and C left it in 10 days before its completion. Find in how many days the work

might have been finished.

• A, B and C can do a piece of work in 15 days, 20 days and 30 days respectively. Three of them started the work

together but A left after 5 days and B left 2 days before its completion. Find in how many days the work might

have been finished.

• A can do a piece of work in 10 days, B can do it in 20 days and C in 30 days Three of them started the work

together. A left the work after 4 days and C left it some days before its completion. If the work is completed in 9days, for how many days did C work ?

• X, Y and Z can finish a piece of work in 20, 30 and 40 days respectively. If x left the work after working for 5 days, in how many days can Y and Z together complete the remaining work ?

• If 12 labours can do a piece of work in 15 days working 8 hours a day, how many days will 10 labours take to finish half of it working 9 hours a day ?

• 20 men can do a piece of work in 24 days. After working for few days, 4 men are added and the work was finished 3 days earlier. After how many days were 4 men added ?

• A piece of work was to be completed in 40 days. Then 10 more men were employed and they finished of work in

20 days, then 10 more men added and work was completed in time, find the number of men employed the first.

• 20 men can finish a piece of work in 30 days. When should 5 men leave the work, so that if may be finished 35 days ?

• 40 men can do a piece of work in 24 days. After working for a few days 8 men are added and the whole work as completed 3 days earlier. After how many days were those 8 men added ?

• 6 labours can finish a work in 120 days working 7 hours daily. If the work is to be finished in 105 days by working 6 hours daily, how labours will be required ?

• A contractor undertakes to dig a canal 12 km. long in 350 days and employs 45 men. He finds that after 200 days

of working only 4 kilometers of canal have been completed. How many extra men must be employed to finish the work in time ?



• A certain number of men can complete a piece of work in 90 days. If, however, there are 15 men less, it will take 10 days more for the work to complete. How many men were there originally ?

• 12 men are engaged to do a piece of work in 15 days. After working for 6 days they found that only of the work is done. How many men should be added to finish the work in time ?

• 200 labours need 50 days to construct a road of 4 km. long working 8 hours a day. How many labours should be decreased to construct the same road in 80 days working 10 hours a day ?

• 55 men can finish a piece of work in 42 days. How many additional men must be added to complete the work 9 work 9 days earlier ?

• 16 men are engaged to do a piece of work in 20 days. After 8 days it is found that only of the work is done. How

many additional men should be taken to complete the work in time.

• A men undertakes to do a certain job in 32 days. He hires 12 men for the job At the end of 8 days, only one fifth of the work was completed. How many extra men must be employed in order to complete the work in time ?

• A contractor had to finish a work in 30 days and he employed some men to do the work. They finished half of the

work in 20 days. When 60 more men were added the work was finished on the specified time. How many men were employed in the beginning ?

• A piece of work was to be completed in 30 days. A number of men employed upon it did only half the work in 18

days. 12 more men were added and the work was completed in the specified time. How many men were employed at first ?

• A contractor had to finish a work in 60 days and he employed 60 laboures to do the work. They finished half of the work in 40 days. How many more labourers should be added to finish the work in specified time ?

• 20 men can do a piece of work in 24 days. After working for 6 days, an additional number of men are taken to

finish the work in 21 days from the beginning. Find the number of additional men.

• A piece of work has to be completed in 30 days. A number of men employed in it and did only half of the

work in 20 days. 20 more men were added and work was completed in the specific time. How many men were employed first ?

• A piece of work had to be completed in 60 days. A number of men employed upon it and they did only of the

work in 40 days. 20 more men were added and the work was completed in the specified days. How many men were employed at first ?

• A contractor undertook to do a work in 40 days and employed a certain number of men upon it who did only half of the work in 24 days. So, 16 more men added to do the work in time. How many men were employed initially ?

• A work was to be finished in 16 days and for this 30 persons started together But only work in finished after 10 days.

Find the number of additional men should betaken to do the work in given time.

• 20 men can do a work in 30 days; after how long time should 5 men leave so that the work may be completed in 35 days ?

• 4 men can do a work in 5 days. But due to Nepal-Band, no work is done on the third day. How many extra men should be added to complete the work in the fixed schedule ?

• A piece of work was to be completed in 30 days, A number of man employed on it, did only half of the work in 20

days, Again 20 more men were added and the work was completed in specified time. How many were employed at first ?

• If 10 pumps draw 430 liters of water is 10 minutes, how many liters will be drawn by 6 pumps in 20 minutes ?

• A tap can fill a cistern in 10 minutes while other can empty in 15 minutes. The cistern being empty and if both the taps are opened together, in how many minutes will the cistern be full ?

• Two taps A and B fill a tank in 8 hours and 10 hours respectively when they are opened separately. If both the taps are opened at the same time, how long will it take to fill the tank ?

• Three taps can fill a cistern in 36 minutes, 30 minutes and 20 minutes respectively. The cistern being empty all the

three taps are kept open and after 6 minutes the first tap is closed. In how many more minutes will the cistern be full ?

• Two pipes A and B can fill a tank in 24 min and 32 min. respectively. If both pipes are opened simultaneously, after how much time B should be closed so that the tank is full in 18 minutes ?

• A and B can fill a casern in 20 min. and 25 min. respectively. Both the pipes are kept open for 5 min. and then the second pipe is turned off. In how many minutes more is the tank completely filled ?

• A and B can fill a tank in 2 hrs. and 4 hrs respectively. Both were opened at once but after sometime the first was shut up and the tank was filled in 2 hrs more. When did the first pipe A shut up ?

• Two pipes A and B can fill a tank in 36 min. and 45 min. respectively. A waste pipe C cans empty the tank in 30 min. First A and B are opened. After 7 min, C is also opened. In how much time, the tank is full ?

• A cistern has a leak which would empty it in 8 hours. A tap is turned on which admits 6 liters a minute into the cistern and it is now emptied in 12 hours. How many liters does the cistern hold ?



• A and B can fill a cistern in 60 min, and 75 min. respectively. If all the pipes (including a waste -pipe C) are opened

together, the cistern is full in 50 minutes. Find the time required for C to empty the full cistern.

• Surav takes one day to do a work; Gaurav takes two days to do the same work. IF Gaurav takes 24 days to do 2/5 of the work, in how many days will both together finish the whole work ?

• Ram can do twice as much work as Shyam. If they together can complete a work in 24 days. In how many days will each take to complete the work ?

• A takes twice as much time as B and thrice as much time as C to finish a piece of work. If they work together they can finish the work in 4 days. Find the time each will take to finish the work ?

• A alone can do a certain work in the same time in which B and C together can do it. If A and B could together do it in 10 days and C alone in 50 days, in what time can B alone do it ?

• A is thrice as good a workman as B and is therefore able to finish a piece of work in 60 days less than B. Find the

time in which they can do it working together.

• A can do as much work in 2 days as B can do in 3 days and B can do as much work in 4 days as C in 5 days. It

takes 20 takes 20 days to complete the work if all of them work together. How long would B take to do all the work by himself ?

• A takes twice as much time as B and thrice as much time as C to finish a piece of work. If they work together they

can finish the whole work in 5 days. Find the time each will take to finish the work separately.

• Promod and Ramchandra can do a pieces can do a piece of work in 30 and 20 days respectively After they work

together for some days, Ramchandra left. If promod completes the remaining worked in 15 days how long had

Ramchandra worked together.

• A can do a work in 30 days and B in 40 days. They start working together but after 8 days B leaves. In how many days A can finish the remaining work ?

• Mohan and Sohan together can finish a piece of work in 30 days. After they worked for 20 days. Sohan left. If Mohan can finish the remaining work in 35 days, How long Sohan alone can finish the whole work ?

• Ram and Shyam can do a piece of work in 20 days and 25 days respectively. They work to together of 5 days and Shyam left the work. In how many days will Ram finish the remaining work ?

• A and B can do a piece of work in 24 days. A takes 40 days to complete the same work. After they work for 8 days, A leaves, If B completes the remaining work, in how many days was the whole work completed ?

• A can do a piece of work in 14 days. B can do the same work in 21 days. After they work for 7 days, A leaves. If B completes the remaining work; in how many days was the whole work completed ?

• Ram and Shyam together can finish a piece of work in 8 days. Ram alone can finish the work in 12 days. If he leaves after working for 4 days alone. In how many days will Shyam finish the remaining work alone ?

• A can do a piece of work in 18 days and B in 24 days. They both worked together and after some days A left. If B finished the remaining work in 3 days, how long did A work together ?

• A can do a piece of work in 36 days and B in 54 days. They both worked together and after some days B left. If A finished the remaining work in 11 days, how long did B work together ?

• C and D can finish a piece of work in 20 days. After they work for 12 days D left. If C alone can finish teh remaining work in 40 days, how long would D alone take to do whole work ?

• A and B can finish a piece of work in 40 days. After they worked for 25 days. B left. If A can finish the remaining work in 45 days. How long B alone can finish the whole work ?

• A and B can do a work in 12 days, B and C in 15 days C and A in 20 days. In how many days will they finish it together ?

• A can do a work in 20 days and B in 12 days. B worked for 9 days and left. In what time will A complete the remaining work ?

• A and B can do a work in 24 and 16 days respectively. With the help of C, they can finish it in 8days, Find in what time will A and C together do it ?

• Bhawesh and Abhiyan can do a work in 25 and 20 days respectively. Bhawesh started the work and was joined by Abhiyan after 10 days. How long was the work last for ?

• 12 men can dig a field in 8 days. Three days after they started to work together, 3 extra men joined them, Find the

time taken by all of them to complete the remaining work.

• A and B can together do a piece of work in 30 days. They worked for it for 20 days and then B left. The remaining

work was done by A alone in 20 more days. Find the time required for A to do the whole work alone.

• A and B can do a work in 45 and 40 days respectively. They started the work together, but A leaves after some days and B finished the remaining work in 23 days. After how long time did A leave ?

• A and B can do a work in 20 and 30 days respectively. They start to work together but A leaves 5 days before the completion. In how many days to do the whole work ?

• A and B can do a work in 72 days. If A works alone for the last 20 days, it is completed in 80 days. How many days will be required for B to do the whole work alone ?



• To do a work A would take twice as long as B and C together. B takes 3 times as along as C and A together. All of them together can do it in 12 days. Find in what time A, B and C alone can do it ?

• A can do as much work in 2 days as B in 3 days and B can do as much work in 4 days as C in 5 days. If all of them can do a work in 20 days, how long would each take to do the work separately ?

• While A does of a work. B does and while B does, C does .In how many hours will C finish a work which can be finished by A in 20 hours ?

• A, B and C can do a work in 20, 30 and 40 days respectively. All of them worked together but A left after 5 days and C left in 10 days before its completion. In how many days required doing the whole work ?

• A, B and C can do work in 10, 20 and 30 days, A left. C left it some days before completion. If total time taken for the whole work is days. For how many days did C work ?

• A, B and C can do a work 50, 70 and 90 days respectively. Only A started the work and B joined after 10 days, C

joined after 5 days from this and all of them continued till it was over. In how many total days taken for the whole

work.

• A, B and C can do a work in 60, 80 and 100 days respectively. A and B started the work. After 10 days A left but C joined. After 20 days from the join of C, B left but A rejoined. In how many total days to do the whole work ?

• A, B and C together can do a work in 6 days, which B alone can do in 16 days and B and C together in 10 days In how many days can A and B together do it ?

• A and B can do a work in 10 days, B and C in 15 days, A and C in 25 days. They all work at it together for 4 days, A

then leaves and B and C go on together for 5 days more and then B leaves, how many more days will C complete the work ?

• A can do work in 20 days, A and B together in days. A works alone for 8 days, A and C together for 6 days, and B finishes it in 3 days. Find in what time B and C together could do it ?

• A can do a work in 8 days, which B can destroy in 3 days, A has worked 6 days, during the last 2 of which B has been destroyed. How many days must A now work alone in order to complete his task ?

• A and B can do a work in 8 days and 7 days respectively. They work alternately for a day. If A starts the work, find in how many total days completed to do work ?

• 25 men are employed to do a work which they could finish it in 20 days but there is a drop off by 5 men at the end of every 10 days. In what time will the work be completed ?

• A cistern would be filled by a tap A in 3 hours, or emptied by a tap B in 3 hours. The cistern being half full A is turned on at 8:00 Am and B at 8:45 Am Find when the cistern will again be half full ?

• A is twice and B is just as good a workman as C. The three work together for two days, and then A works alone for

half a day and B for a day. How long would it have taken A and C together to complete as much as the three will

have thus performed?

slc _ compulsory math _ model question _ all

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