COMPULSORY MATHS
Model Set 1
Time: 3.00 hours
Full Marks: 100
Pass Marks: 32
;d'x …sÚ (Group ‘A’)
1.
a.
[9×(2+2)=36]
dfg lgsfNg'xf];\ (Evaluate):
45a 2  80a 2  6a 5
5 5a 2
2.
a 2 p  a p 1  a p
a3 p  a p 1
b.
;/n ug{'xf];\ (Simplify):
a.
xn ug{'xf];\ (Solve) : 3 2 x  7  3  0
b.
olb Pp6f wgfTds ;ª\Vofsf] ju{af6 7 36fpFbf kl/0ffd 9 x'G5 eg] pQm
;ª\Vof kQf nufpg'xf];\ .
If 7 is subtracted from the square of a positive number, the result is 9. Find the
number.
3.
olb  fx = 400+20a,
dfg lgsfNg'xf]; .
a.
 f =18+2a / lbOPsf] >]0fLsf] dWos @) eP a sf]
If  fx = 400+20a ,  f =18+2a, and the mean of the given series is 20, find the
8
value of a.
7
;Fu} lbOPsf ;l~rt jf/Djf/tf js|jf6 dlWosf /
dlWsfsf] ju{
b.
6
5
4
3
kQf nufpg'xf];\ .
Find the median class and value of median from the
adjoining cumulative frequency curve.
4.
a.
2
1
0
10 20
30 40 50 60 70
->]0fLcG
t /_Class interval
/fd|/L lkml6Psf] 52 kQL ePsf] tf;sf] u8\8Laf6 Pp6f kQL
lgsflnPsf] 5 eg] pQm klQ /fgL cyjf sfnf] /ªsf] PSsf kg{] ;DefJotf
kQf nufpg'xf];\ .
80 90
A card is drawn from a well-shuffled deck of 52 cards. Find the probability of
getting such card is queen or a black ace.
b.
Pp6f l;SsfnfO{ b'O{k6s ;Dd pkmfbf{ aGg] ;DefJotfnfO{ j[Iflrqdf
k|:t't ug{'xf];\ .
A coin is tossed two times and represents the probabilities in a tree diagram.
a.
5.
lqe'h ABC df AB=8 ;]=ld= BC= 12 ;]=ld= /ABC=60° eP ;f] lqe'hsf]
If]qkmn kQf nufpg'xf];\ .
In ∆ABC, AB=8cm, BC= 12 cm and ABC=60°.
Find the area of the triangle.
b.
lbOPsf] lrq lqe'hfsf/ cfwf/ ePsf] Pp6f 7f]; lk|Hd xf] . olb
AB= 3;]=ld= BC=5;]=ld= / CF= 12;]=ld= eP pQm lk|Hdsf]
cfotg kQf nufpg'xf];\ .
A
C
B
D
The adjoining diagram is a triangular based solid prism. If AB=
3cm, BC=5cm and CF= 12cm find the volume of the prism.
F
E
Pp6f a]ngfsf] cw{Jof; / prfOsf] of]ukmn 12 ;]=ld / cfwf/sf] kl/lw 416
;]=ld= eP pQm
6. a .
a]ngfsf] k"/f ;txsf] If]qkmn lgsfNg'xf];\ .
If the sum of radius and height of cylinder is 12 cm and circumference of base is
416 cm find the total surface area of that cylinder.
b.
;Fu} lbOPsf] lrq Pp6f 7f]; uf]nfsf] xf] . olb cfwf/sf]
Jof; - AB)= 28 ;]=ld= eP pQm uf]nfsf] cfotg
lgsfNg'xf];\ .
B
A
28cm
The diagram given alongside is of a solid sphere. If AB= 28 cm, Find the volume
of the sphere.
7.
a.
Pp6f j:t'sf] jf:tljs d"Nodf15% a9fO{ clª\st d"No ?= 2760 sfod ul/of] .
;f] j:t'sf] jf:tljs d"No kQf nufpg'xf];\ .
The marked price of an article was fixed to Rs. 2760 by increasing 15% in its
actual price. Find its actual price.
b.
?= 5000 sf] k|ltjif{ 10% jflif{s rqmLo Aofhb/n] 2 jif{df x'g] rlqmo
ld>wg kQf nufpg'xf];\ .
Find the compound amount on Rs. 5000 in 2 years at 10% per annum.
8.
a.
lbOPsf] lrqdf, WXYZ Pp6f ;dfgfGt/ rt'e{'h 5 . olb WX
V ;Dd
a9fpFbf ag]sf] ∆ZYV sf] If]qkmn 15cm2 eP ∆WXT sf]
If]qkmn kQf nufpg'xf];\ .
nfO laGb'
T
Z
Y
If the given figure, WXYZ is a parallelogram . WX is
W
extended
15cm2, find
b.
V
X
up to U and ∆ZYU is formed. If Area of ∆ZYU is
the area of ∆WXT.
P
lbOPsf] lrqdf PRQS 5 . olb PQS=46° eP QSR kQf
nufpg'xf];\ . In the given figure, PRQS. If PQS=46°,
46°
S
Q
R
find QSR.
9.
a.
D
lbOPsf] lrqdf laGb'x? A, D, C / B cw{j[Qdf 5g\ . olb
DAC=30° / ABC= 70° eP ACD kQf nufpg'xf];\ .
In the given figure, points A, D, C and B are on semicircle. If DAC=30° and ABC= 70°, find ACD.
C
30 °
70°
A
b. lbOPsf] lrqdf PT n] laGb' A df j[QnfO{ :kz{ u/]sf] 5 .
olbBAT=60° / BAC=50° eP ABC kQf nufpg'xf];\ .
In the given figure, PT touches a circle at a point A. If BAT=60° and
BAC=50°, find ABC.
B
O
B
C
50° 60°
A
P
;d'x …vÚ (Group ‘B’)
10
[16×4=36]
Pp6f ljBfyL{x¿sf] ;d"xdf ul/Psf] ;j{]If0fdf 70% ljBfyL{x¿n] j}1flgsx¿sf
;DaGwdf , 65% n] v]nf8Lx?sf ;DaGwdf / 430 hgfn] b'j}sf af/]df cWoog
u/]sf] kfOof] . olb 8% n] s'g} sf] af/]df klg cWoog u/]sf] kfOPg eg] M
T
In a survey of the group of students, it was found that 70% of students studied about
scientists, 65% about players and 430 studied about both scientists and players. If 8% did
not study about scientists and players, then,
i)
dflysf] tYofªsnfO{ e]glrqdf k|:t't ug{'xf];\ .
Represent the above information in a Venn-diagram.
ii)
;j{]If0fdf efu lnPs]f hDdf ljBfyL{;ª\Vof kQf nufpg'xf];\ .
Find the total number of students who took part in the survey.
11.
dxQd ;dfkjt{s lgsfNg'xf];\ . (Find the H.C.F of):
6 x 2  2 x,6 x 3  10 x 2  4 x and 23 x  1
2
12.
xn ug{'xf];\ . (Solve) : 5x-1 + 5-x = 1
13.
;/n ug{'xf];\ . (Simplify):-
1
5
2
1
3y
xy

 2
 3
2
x y x y y x
x  y3
14. b'O{ cÍsf] s'g} Pp6f ;ª\Vof, Tof] ;ª\Vofsf] cª\sx¿sf] of]usf] rf/ u'0ff 5 . olb
Tof] ;ª\Vofsf] cª\sx¿sf] :yfg abn]/ aGg] ;ª\Vof / 9 sf] of]ukmn] pQm
;ª\Vofsf] b'O{ u'0ff x'G5 eg], ;f] ;ª\Vof kQf nufpg'xf];\ .
A two digit number is four times the sum of its digits. If the sum of the number formed
by reversing its digits and 9 is two times the original number, find the original number.
15.
olb tn lbOPs]f cfFs8fsf] dWos 68 eP x sf] dfg kQf nufpg'xf];\ .
If the mean of the following data is 68, find the value of x.
Marks obtained 40-50
50-60
60-70 70-80
80-90
90-100
x
12
-k|fKtfÍ_
No.of students
17
22
28
26
-ljBfyL{sf]
;+Vof_
16.
Pp6f gbLsf] lsgf/df ePsf] 40 ld6/ cUnf] ¿vsf] 6'Kkf]df pQm gbLsf] csf{]
lsgf/af6 cjnf]sg ubf{ pGgtf+z sf]0f 30° kfOof] eg] ;f] gbLsf] rf}8fO kQf
nufpg'xf];\ .
The angle of elevation of the top of a tree, 40m high situated at the bank of a river when
observed from the opposite bank of the river is found to be 30°, find the breadth of the
river.
17.
lbOPsf] 7f]; a:t' j]ngf / cw{uf]nf ldn]/ ag]sf] 5 .
h;sf] cfwf/sf] Jof; 14;]=ld= / k"/f nDafO 21;]=ld= 5
eg] ;f] 7f]; a:t'sf] k"/f ;txsf] If]qkmn kQf nufpg'xf];\
.
14cm
21cm
The given solid object is made up of a cylinder and a hemisphere whose diameter of the
base is 14cm and total length is 21 cm. Find the total surface area of the solid object.
O
18.
tn lbOPsf ju{ cfwf/ ePsf] lk/fld8sf] k"/f ;txsf]
If]qkmn lgsfNg'xf];\ .
C
D
Find the total surface area of the following
E
A
square based pyramid.
19.
F
7cm
B
A / B n] Pp6f sfd s|dz M 12 / 16 lbgdf ug{ ;S5g\ . A / B b'j}hgf ldn]/ 4 lbg
sfd u/]kl5 B n] ;f] sfd 5f]8\5 eg] afFsL sfd ug{ A nfO{ slt lbg nfUnf < kQf
nufpg'xf];\ .
A and B can do a piece of work in 12 and 16 days respectively. A and B work together
for 4 days and B leaves the work, find how long will A take to complete the remaining
work.
20.
Pp6f Sofd/f 25% 5'6 lbP/ 10% d"No clej[l4 s/ nufO{ a]lrof] . olb 5'6 /sd
?= 750 eP pQm Sofd/fsf] d"Nodf d"No clej[l4 s/ /sd slt lyof] kQf
nufpg'xf];\ .
A camera was sold after allowing 25% discount on the marked price and then levying
10% value added tax (VAT). If the discounted amount was Rs.750, how much value
added tax (VAT) was levied on the price of the camera?
21.
Pp6f ufpFsf] hg;ª\Vof k|To]s jif{ 5 k|ltztn] a9\b} hfG5 . olb b'O{ jif{sf]
cGTodf 1025 hgf a;fOF ;/]/ cGoq hfFbf ;f] ufpFsf] hg;ª\Vof 10,000 eof] eg]
;'?df ;f] ufpFsf] hg;ª\Vof slt lyof] <
The population of a village increases every year by 5%. If 1025 people leave the village
at the end of two years and the population of the village is 10,000, find the population of
the village in the beginning.
22.
Pp6} cfwf/ QR / pxL ;dfgfGt/ /]vfx? QR / PS sf] jLrdf ag]sf lqe'hx¿ PQR /
SQR sf] If]qkmn a/fa/ x'G5g egL k|dfl0ft ug'{xf];\ .
Prove that the triangles PQR and SQR standing on same base QR and between same
parallel lines QR and PS are equal in area.
23.
;Fu}sf] lrqdf PT j[Qsf] Jof; xf] / O s]Gb|laGb' xf] olb
rfk SR= rfk RT eP PS//OR.x'G5 egL l;4 ug'{xf];\ .
In alongside diagram, PT is the diameter and O is centre of
circle. If arc SR=arc RT, prove that PS//OR.
24.
S
P
R
O
Pp6f j[Qsf] pxL rfkdf cfwfl/t kl/lwsf]0fx¿ a/fa/ x'G5g\ egL k|of]u4f/f l;4
ug'{xf];\ . -slDtdf 3;]=ld= cw{Jof; ePsf b'O{ cf]6f j[Qx¿ cfjZos 5g\._ Verify
experimentally that the angles at the circumference standing on the same arc of a circle
are equal.(Two circles of radii at least 3 cm are necessary.)
25.
T
rt'e'{h ABCD sf] /rgf ug'{xf];\ h;df AB=5.8 ;]=ld, BC=6.2;]=ld , CD=5.1;]=ld ,
DA=4.8;]=ld / BAD=60° 5g\ .;fy} pQm rt'e'{h;Fu If]qkmn a/fa/ x'g]
lqe'hsf] /rgf ug'{xf];\ .
Construct a quadrilateral ABCD in which AB=5.8cm, BC=6.2cm , CD=5.1cm ,
DA=4.8cm and BAD=60°.also construct a triangle equal in area to the quadrilateral
ABCD.
Marking Scheme
Model set 1
1. a.
9a 5  4a 5
………………………………….. (1)
5a 5
1 ……………………………………… …… (1)
i)
ii)
a2 p
 1)
a
b. i)
a p (a 2 p  a)
ap(
ii)

3
2. a. i)
……………… …………………. (1)
1
…………………………………………………. (1)
a

3
2 x  7  (3)3
 2 x +7=27 ………………………………………(1)
ii) 2 x =20 x =10………………………………..........(1)
b. Let the positive number be x .
2
i) x  7  9 ……………………………………………(1)
ii) x  4 …………………………………………………..(1)
3.
a. (i)
20=
400  20a
……………………………..……. (1)
18  2a
(ii)
a=2………………………………………………. (1)
b . (i)
Median class = 30-40 …………………………. (1)
Median = 4…………….. ……………………………..…(1)
4. a
i)
ii)
4
2
, P(B.ace) =
……….……………….1
52
52
3
P(Queen or B.ace) =
…………….…………...……….1
26
P(Queen) =
b.
First outcome………...1
Second outcome …..1
(i)
(ii)
5. a. (i)
Area of triangle ABC =
1
 8cm 12cm  sin 60 ………….(1)
2
(ii)
Area of triangle ABC=55.42cm2 …………………………(1)
b. (i)
AC = 4 cm and area of triangle (A)=6 cm2……..…………....(1)
(ii)
Volume of prism (V) = 72 cm3……………..………………..(1)
6. a.
i)
ii)
T.S.A. of the cylinder= 2r(r+h)………………….(1)
T.S.A. of the cylinder=416×12=4992cm2……..….(1)
b.
3
4 22  28 
i) V=     ……….(1)
3 7  2 
ii) V= 11498.67 cm3……….(1)
7. a.
i) x +15% of x = Rs. 2760………………….(1)
ii) Actual price( x ) = Rs. 2400 ………………(1)
b.
2
10 

i) CA = Rs. 5000 1 
 ………………….(1)
 100 
ii) CA = Rs. 6050 …………….………………(1)
8. a.
i)
ii)
iii)
Finding area of WXYZ = 30 cm2………… .. 1
Finding area of ∆WXT=15cm2……………….1
b. (i) Finding QPR=44°………………… ………....1
(ii)
Finding QSR= 44°………………………1
9 a. i) Getting ADC=110° with correct reason…………1
ii) Getting ACD=40° with correct reason …………….1
a. i) BAC= 60, with correct reason……………………1
ii) ABC= 70°, with correct reason……………………1
10.
Let total number of students be x.
i)
Representation of the information in Venndiagram………1+1
ii)
iii)
70 x
65 x
8x
 430 
 430  430 
 x …………..…1
100
100
100
𝑥 =1000 [i.e. Total number of students =1000]……….1
Alternatively
Let n(sp) be a% of total students.
i)
Representation of information in Venndiagram…………....(1+1)
ii)
70-a+a+65-a=100-8 or
a=43…………………….…….(1)
43% of total students =430 or, total students
=1000….(1)
iii)
11.
i.
ii.
iii.
iv.
2x(3x+1) …………………………………………… ....(1)
2x(x-2) (3x+1) ………………………………………. (1)
2(3x+1) (3x+1) ……………………………………..… (1)
H.C.F = 2(3x+1) …………………………………...……(1)
12.
i. Let , 5x = a  a2-6a+5=0 …………………………….(1)
ii. (a-1) (a-5) = 0 …………………………..……………(1)
iii. a-1=0  x=0 …………..……………….…………(1)
a-5=0  x-1
iv. x=0, 1 ……………….………………………………..(1)
2x  2 y  x  y
3y
xy
i.
………………….(1)
 2
 3
2
2
2
x y
x y
x  y3
13.
x

xy
………..……….(1)
x  y  x  xy  y 2


ii.
x  y x  y 
iii.
x 3  x 2 y  xy2  x 2 y  xy2
…………..………………(1)
x  y x  y  x 2  xy  y 2

2

3
iv.
14.
x
……………….……………………..(1)
x  y  x 3  y 3


Let a two digit number be 10x+y where x and y are the digits of ten's
place and unit place respectively.
1. 10 +y=4( +y)
6
=3y
y=2
2. 10y+
 19
3.
4.
15
…………………………………………..(1)
+9=2(10
+y)
=8y+9 ………………………………..………(1)
=3 and y = 6……………….………..………………(1)
The original number = 10×3+6=36……………………(1)
Mark obtain
40-50
50-60
60-70
70-80
80-90
90-100
No. of students
17
22
28
26
x
12
Mid value
45
55
65
75
85
95
F(m)
765
1210
1820
1950
85x
1140
 f  105  x
 f .m  6885  85x ………………..….1+1
6885  85 x
…………………………….1
105  x
ii. x=15 ………………………………………..1
i. 68=
16.
(i)
(ii)
Appropriate figure with description ………….. (1)
40
In right angled triangle CAB, Tan30°=
… (1)
AB
(iii)
AB= 40 3 meter.….………..……………….. (1)
(iv)
The breadth of the river (AB) = 69.28…….... (1)
….meter………….…..(1)
17.
18.
i) Radius of the cylinder (r)= Radius of the hemisphere
(r)=7cm………………………………………………..…...1
ii) T.S.A. of cylinder without one circular part(A1)=r2+2rh=
22 2
22
 7  2   7  21 = 1078 cm2…………………………1
7
7
22
iii) Curved surface area of the hemisphere (A2)= 2r2= 2   7 2
7
=308cm2………………………………………..1
iv) T.S.A. of the solid(A)= A1=A2
1078+308=1386cm2 ……………………………..……..…1
i)
ii)
iii)
iv)
Slant height of pyramid =25 ……………..………………..(1)
Area of base of pyramid (A1)= 196 cm2 ………….………(1)
Area of 4 triangles(A2) = 700 cm2 ………..………………(1)
Total surface area of pyramid (A)=896 cm2………….…..(1)
19.
7
of work in 1day …………..………..(1)
48
5
ii) Remaining work after4 days=
works ………….………(1)
12
5
iii) A do
works in 5 days ………..………….……………(1)
12
iv) A can do remaining in 5 days………………………….…..(1)
i) A and B can do
20.
i) PT = 10000+1025= 11025 …………………………..….(1)
2
5

ii) 11025 = P 1   ………...………….…………...…(1)
 10 
iii) 11025 = P × 1.1025 …………….………….…...…...…(1)
iv) P = Rs. 10,000 …………….………………...…..........…(1)
21.
i) Let the MP be Rs x
Discount amount = 25% of x = Rs
x
……………….(1)
4
x
, o r x=Rs3000 …………….…………...…(1)
4
x  3x

iii) VAT amount (VAT) = 10% of  x   = …...…...…(1)
40
4

iv) VAT amount = Rs 225 ……………………...…..........…(1)
ii) Rs. 750=
22.
i)
ii)
iii)
iv)
Correct figure with given to prove and construction if
necessary…………………………1
Ar(∆PQR) = ½ ar( PQRT)……..1
Ar(∆SQR) = ½ ar( PQRT)……..1
Ar(∆PQR) = Ar(∆SQR) …………1
Full marks will be given for correct and appropriate proof:
Note: If the first step is wrong zero score for this answer.
23
Following statements with correct reason.
i)
ii)
iii)
iv)
SR=RT ……………………..………….1
SOR=ROT……………….…….1
SPT= ½ SOT= ROT…………1
PS//OR……………………………...1
NB:Full marks will be given for alternative correct
and appropriate proof.
24.
i)
ii)
iii)
25.
For the correct figures…………………………..….…..1
For the correct measurements with table……… .. …1+1
For the correct conclusion…………………………..…..1
(i)
Construction of BAD=60° on the line AB= 5.8 cm…...1
(ii)
Construction of quadrilateral ABCD. …………………..1
(iii)
Draw a line parallel to BD through the point C…….……1
(iv)
Construction of required ∆ADE…………………………1