Rajasthan Board Praveshika Question Paper 2019 Mathematics

Zm_m§H$ Roll No.

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Sl.No. :

No. of Questions – 30 P–09–Mathematics
No. of Printed Pages – 11

àdo{eH$m narjm, 2019
PRAVESHIKA EXAMINATION, 2019

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àíZ nÌ H$mo ImobZo Ho$ {bE `hm± \$m‹S>|
MATHEMATICS
g_` : 3¼ KÊQ>o
nyUmªH$ : 80

narjm{W©`m| Ho$ {bE gm_mÝ` {ZX} e …
GENERAL INSTRUCTIONS TO THE EXAMINEES :
1) narjmWu gd©àW_ AnZo àíZ nÌ na Zm_m§H$ A{Zdm`©V… {bI| &
Candidates must write first his / her Roll No. on the question paper
compulsorily.
2) g^r àíZ H$aZo A{Zdm`© h¢ &
All the questions are compulsory.
3) àË`oH$ àíZ H$m CÎma Xr JB© CÎma-nwpñVH$m _| hr {bI| &
Write the answer to each question in the given answer-book only.
4) {OZ àíZmo§ ‘| AmÝV[aH$ IÊS> h¡§, CZ g^r Ho$ CÎma EH$ gmW hr {bI|&
For questions having more than one part, the answers to those parts
`hm± go H$m{Q>E

are to be written together in continuity.

P–09–Mathematics 5021 [ Turn Over

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5) àíZnÌ Ho$ {hÝXr d A§J«oOr ê$nmÝVa ‘| {H$gr àH$ma H$s Ìw{Q> / AÝVa / {damoYm^mg hmoZo na {hÝXr ^mfm Ho$
àíZ H$mo hr ghr ‘mZ|&
If there is any error / difference / contradiction in Hindi & English versions of
the question paper, the question of Hindi version should be treated valid.

6) ^mJ> àíZ g§»¶m A§H$ à˶oH$ àíZ
A 1 - 10 1

~ 11 - 15 2

g 16 - 25 3

X 26 - 30 6

Part Q. Nos. Marks per question

A 1 - 10 1

B 11 - 15 2

C 16 - 25 3

D 26 - 30 6

7) àíZ H«$‘m§H$ 27 d 29 ‘| AmÝV[aH$ {dH$ën h¢&
There are internal choices in Question Nos. 27 and 29.

8) AnZr CÎma-nwpñVH$m Ho$ n¥ð>m| Ho$ XmoZm| Amoa {b{IE& ¶{X H$moB© a’$ H$m¶© H$aZm hmo, Vmo CÎma-nwpñVH$m Ho$
A§{V‘ n¥ð>m| na H$a| Am¡a BÝh| {VaN>r bmBZm| go H$mQ>H$a CZ na "a’$ H$m¶©' {bI X|&
Write on both sides of the pages of your answer-book. If any rough work is
to be done, do it on last pages of the answer-book and cross with slant lines
and write ‘Rough Work’ on them.

9) àíZ H«$‘m§H$ 26 H$m boIm{MÌ J«m’$ nona na ~ZmBE&
Draw the graph of Question No.26 on graph paper.

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^mJ> - A
PART - A

1) ÛÝÛ ¶moJ {d{Y go 62 H$m dJ© kmV H$s{OE& [1]

Find the square of 62 by Dwandwa yoga method.

2) hb H$s{OE : (x+1) (x+2) = (x–5) (x–6) [1]

Solve : (x+1) (x+2) = (x–5) (x–6)

3) 68 VWm 119 H$m ‘hÎm‘ g‘mndV©H$ kmV H$s{OE& [1]

Find the HCF of 68 and 119.

4) tan260° + 3cos2 30° H$m ‘mZ kmV H$s{OE& [1]

Find the value of tan260° + 3cos2 30°

5) ¶{X sin2A = cos (A–18°) hmo, Vmo A H$m ‘mZ kmV H$s{OE& [1]

If sin2A = cos (A–18°), then find the value of A.

6) g‘Vb ‘| bwT>H$Zo dmbo d¥Îm Ho$ Ho$ÝÐ H$m {~ÝXþ nW {b{IE& [1]

Write the locus of the centre of rolling circle in a plane.

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7) ¶{X ABC  DEF h¡ {OZ‘| AB = 1.6 go‘r Am¡a DE = 2.4 go‘r hmo Vmo ABC Am¡a DEF Ho$
joÌ’$bm| H$m AZwnmV kmV H$s{OEo& [1]

If ABC  DEF in which AB = 1.6cm and DE = 2.4cm. Find the ratio of the
areas of ABC and DEF.

8) Xmo {IbmS>r A Am¡a B eVa§O H$m EH$ ‘¡M IobVo h¡ ¶h kmV h¢ {H$ A Ûmam ‘¡M {OVZo H$s àm{¶H$Vm 56 h¡& B
Ho$ OrVZo H$s àm{¶H$Vm kmV H$s{OE& [1]

Two players A and B plays a chess match. It is given that probability of winning
the match by A is 5
6 . Find the probability of winning the match by B.

9) ¶{X EH$ H$ma H$m {H$am¶m àW‘ {H$bmo‘rQ>a Ho$ {bE 20 ê$. VWm BgHo$ ~mX à{V {H$‘r Ho$ {b¶o 11 ê$. h¡ Vmo
15 {H$‘r MbZo Ho$ {b¶o Hw$b {H$am¶m kmV H$s{OE& [1]

If fare of a car for first kilometer is 20 and for after 1 kilometer is 11, then find
the total fare for 15 kilometers.

10) EH$ nV§J ^y{‘ go 75 ‘rQ>a H$s C±MmB© na CµS> ahr h¡& {OgHo$ YmJo H$m ^y{‘ Ho$ gmW PwH$md 60º h¡ Vmo YmJo H$s
bå~mB© kmV H$s{O¶o& [1]

A kite is flying at a height of 75 metres from the level of ground attached to a string
inclined at 60º to the horizontal. Find the length of the string.

P–09–Mathematics 5021

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^mJ> - ~

PART - B

11) CngyÌ AmZwê$߶oU Ûmam 42 H$m KZ’$b kmV H$s{OE& [2]

Find the cube of 42 by using Upsutra Anurupyena.

12) {gÕ H$s{OE {H$ 7 5 EH$ An[a‘o¶ g§»¶m h¡& [2]

Prove that 7 5 is an irrational number..

13) EH$ d¥Îm H$s {ÌÁ¶m 9 go‘r Am¡a {ÌÁ¶ IÊS> H$m H$moU 70º h¡ d¥Îm Ho$ bKw {ÌÁ¶ IÊS> H$m joÌ’$b kmV H$s{OE&
 22 
   . [2]
 7 

Radius of a circle is 9cm and the angle of the sector is 70º. Find the area of the
 22 
minor sector of the circle.    
 7 

14) EH$ ~obZ H$s C±MmB© 21 go‘r VWm CgH$m dH«$ n¥îR>r¶ joÌ’$b 924 dJ© go‘r h¡& ~obZ H$s {ÌÁ¶m kmV H$s{OE&
[2]

The height of a cylinder is 21cm and its curved surface area is 924cm2. Find the
radius of cylinder.

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15) A d B Ho$ ‘ܶ H$s Xÿar 125 {H$‘r. h¡ VWm BgHo$ ‘ܶ 8 ¶mVm¶mV {g½Zb {‘bVo h¢& ¶{X EH$ H$ma 50 {H$‘r
à{V K§Q>m H$s g‘mZ J{V go g^r hao {g½Zbm| H$mo nma H$aVo hþE dh B {~ÝXþ na 2 KÊQ>o 30 {‘ZQ> na nh±ÿM OmVr
h¡& bo{H$Z Aݶ {XZ ^mar ¶mVm¶mV Ho$ H$maU {ZåZmZwgma ê$H$Zm nµS>Vm h¡& àW‘ ¶mVm¶mV {g½Zb 1 {‘ZQ>
{ÛVr¶ ¶mVm¶mV {g½Zb 2 {‘ZQ> ......... 8 do {g½Zb VH$ ...... 8 {‘ZQ> Cgr H$ma Ûmam {b¶ J¶o Hw$b g‘¶
H$s JUZm H$s{OE& ¶{X dh g^r ¶mVm¶mV {g½Zbm| H$s AZwnmbZm H$aVr h¡& [2]
The distance between A and B is 125 km and there are Eight(8) traffic signals in
between A and B. If a car by 50 km/h speed reaches point B crossing all green
signals in 2 hours and 30 minutes but on other days due to heavy traffic it happens
to stop as follows. First Traffic Signal - 1 minute, Second Traffic Signal - 2 minutes
and upto Eight(8th) signal - 8 minutes. Calculate the total time taken by that car if it
follows the total traffic signals.

^mJ> - g
PART - C

1 2 6
16) JwUZ IÊS> {d{Y go g‘rH$aU   Ohm± x  1, 2 H$mo hb H$s{OE& [3]
x  2 x 1 x

1 2 6
Solve the equation   where x  1, 2 by factorisation method.
x  2 x 1 x

17) 2 Am¡a 101 Ho$ ‘ܶ 5 go {d^m{OV (^mÁ¶) hmoZo dmbr g^r àmH¥$V g§»¶mAm| H$m ¶moJ ’$b kmV H$s{OE& [3]

Find the sum of all the natural numbers divisible by 5 between 2 and 101.

18) {H$gr AnyU© ‘rZma Ho$ AmYma go 120 ‘rQ>a Xÿa {H$gr {~ÝXþ go ‘rZma Ho$ {eIa H$m CÝZ¶Z H$moU 30º h¡& kmV
H$s{OE {H$ ‘rZma H$mo {H$VZm D±$Mm ~Zm¶m Om¶o {Oggo Cgr ñWmZ na CgH$m CÝZ¶Z H$moU 60º hmo Om¶o? [3]

From a point on the ground which is 120m away from the foot of the unfinished
tower, the angle of elevation of the top of the tower is found to be 30º. Find how
much height of tower have to increased so that its angle of elevation at same point
become 60º?

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19) [gÕ H$s{OE {H$ ¶{X Xmo g‘ê$n {Ì^wOm| Ho$ joÌ’$b g‘mZ hmo, Vmo XmoZmo§ {Ì^wO gdm©Jg‘ hmoVo h¢& [3]

Prove that if area of two similar triangles are equal, then they are congruent.

20) AmH¥${V ‘| AOB d¥Îm H$m ì¶mg h¡ VWm C,D Am¡a E AY©d¥Îm na pñWV H$moB© VrZ {~ÝXþ h¢ ACD + BED
H$m ‘mZ kmV H$s{OE? [3]

In fig. AOB is the diameter of a circle and C,D and E are three points situated on
semi circle. Find the value of ACD + BED.

21) EH$ O Ho$ÝÐ dmbm d¥Îm MVw^©wO ABCD H$s Mmamo ^wOmAm| H$mo AÝV: ñne© H$aVm h¡ ¶{X AB H$mo ñne© {~ÝXþ
3:1 ^mJm| ‘| {d^m{OV H$ao VWm AB = 12 go‘r h¡ Vmo d¥Îm H$s {ÌÁ¶m kmV H$s{OE O~ {H$ OA = 15 go‘r h¡&
[3]

A circle with centre ‘O’ touches all the four sides of a quadrilateral. ABCD internally
in such a way that it divides AB in 3:1 and AB = 12cm then find the radius of circle
where OA = 15 cm.

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22) 5 go‘r ^wOm dmbo g‘~mhþ {Ì^wO Ho$ AÝVJ©V d¥Îm H$s aMZm H$s{OE& [3]

Construct an incircle of an equilateral triangle with side 5 cm.

23) {ÌÁ¶m 10cm dmbo EH$ d¥Îm Ho$ AÝVJ©V ItMo Om gH$Zo dmbo dJ© H$m joÌ’$b kmV H$s{OE& [3]

Find the area of the square inscribed in a circle of radius 10cm.

24) 6 go‘r ì¶mg H$m EH$ Jmobm 12 go‘r ì¶mg Ho$ ~obZmH$ma ~V©Z ‘| {Og‘| nmZr h¡ S>mbm OmVm h¡ ~V©Z ‘| nmZr
{H$VZm D$na MT> Om¶oJm? [3]

A sphere of 6cm diameter is dropped into cylindrical vessel of diameter 12cm.
Find the rise in water in the vessel.

25) EH$ W¡bo ‘| 15 H$mS>© h¡ {OZ na g§»¶mE± 1, 2, 3, 4, ........, 15 A§{H$V h¡& W¡bo ‘| go ¶mXpÀN>H$ EH$ H$mS>©
{ZH$mbm OmVm h¡& àm{¶H$Vm kmV H$s{OE {H$ Bg na [3]

i) EH$ A^mÁ¶ g§»¶m h¡&

ii) 2 go {d^m{OV hmoZo dmbr g§»¶m h¡&

A bag contains 15 cards. The numbers 1,2,3,4,........,15 are printed on them. A
card is at random drawn from the bag. Find the probability that the number on the
card is

i) a prime number

ii) a number is divisible by 2

P–09–Mathematics 5021

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^mJ> - X
PART - D

26) {ZåZ a¡{IH$ g‘rH$aU ¶w½‘ H$mo AmboIr¶ {d{Y Ûmam hb H$s{OE& [6]

3x – 5y = –1

2x – y = –3

AV: BgH$s ghm¶Vm go g§~§Y (x + y)2 = A ‘| A H$m ‘mZ kmV H$s{OE&

Solve the following pair of linear equation by graphical method.

3x – 5y = –1

2x – y = –3

Thus find the value of A in the relation (x + y)2 = A.

27) {gÓ H$s{OE& [6]

1  secA sin 2 A
i) 
secA 1  cosA

cos 2 θ sin 3 θ
ii)   1  sin θ cos θ .
1  tan θ sin θ  cosθ

AWdm

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{gÕ H$s{OE& [6]

sin θ 1  cosθ
i)   2cosecθ
(1  cos θ) sin θ

sin θ  2sin 3 θ
ii)  tan θ
2cos 3 θ  cosθ

Prove that :

1  secA sin 2 A
i) 
secA 1  cosA

cos 2 θ sin 3 θ
ii)   1  sin θ cos θ
1  tan θ sin θ  cosθ

OR

Prove that :

sin θ 1  cos θ
i)   2cosecθ
(1  cos θ) sin θ

sin θ  2sin 3 θ
ii)  tan θ
2cos3 θ  cosθ

28) {H$gr g‘Vb ‘| Mma {~ÝXþ P(2, –1), Q(3, 4), R(–2, 3) Am¡a S(–3, –2) h¡ Vmo {gÕ H$s{OE {H$
PQRS dJ© Zhr EH$ g‘MVw^©wO h¡& [6]

If there are four points P(2, –1), Q(3, 4), R(–2, 3) and S(–3, –2) in a plane, then
prove that PQRS is not a square but a rhombus.

P–09–Mathematics 5021

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29) {gÕ H$s{OE {H$ MH«$s¶ MVw^©wO Ho$ gå‘wI H$moU ¶w½‘ gånyaH$ ¶m CZH$m ¶moJ 180º hmoVm h¡& [6]

AWdm

{gÓ H$s{OE {H$ ¶{X EH$ d¥Îm H$s ñne© aoIm go EH$ Ordm ItMr OmE Vmo Bg Ordm Ûmam Xr JB© ñne© aoIm go
~ZmE JE H$moU H«$‘e: Cgr Ordm Ûmam EH$mÎma d¥ÎmIÊS>m| ‘| ~Zo H$moUmo§ Ho$ ~am~a hmoVo h¢& [6]

Prove that the opposite angles of cyclic quadrilateral are supplementary or sum is
180º.

OR

Prove that if a chord is drawn from a point of contact of the tangent of the circle
then angle made by this chord with the tangent are equal to the respective alternate
angles made by segments with this chord.

30) {ZåZ ~ma§~maVm ~§Q>Z Ho$ ‘mܶH$ d ~hþbH$ kmV H$s{OE& [6]

dJ© 10-25 25-40 40-55 55-70 70-85 85-100

fi 6 20 44 26 3 1

Find the median and mode of the following frequency distribution.

Class 10-25 25-40 40-55 55-70 70-85 85-100

fi 6 20 44 26 3 1



P–09–Mathematics 5021

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