Uttarakhand Board Class 10 Sample Paper 2026 Maths

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UBSE Sample Question Papers 2026
Roll No. (vuqØeakd) -

M.M. - 80 SAMPLE PAPER ¼izfrn”kZ iz”u i=½ 2026 TIME – 3.00 H
iw.kkZad&80 le;& 3-00 ?ka0
MATHEMATICS (xf.kr ) CLASS - 10

Instructions – funsZ”k

1 - There are in all 23 questions in this paper . All questions are compulsory.

bl iz”u i= esa dqy 23 iz”u gSA lHkh iz”u vfuok;Z gSA
2- Question no. 1 in multiple choice question-four option are given in answer of each part of the question. Write
correct option in your answer book..

ç'u la 1 cgqfodYih; ç'u gSA bl ç'u ds çR;sd [k.M ds mÙkj esa pkj fodYi fn;s x, gSaA lgh fodYi
viuh mÙkj iqfLrdk esa fyf[k,A
3- Question no. 6 is related to Assertion/Reason question. Four option are given in answer of each part of the question.
Write correct option in your answer book

ç'u l-6 [email protected] ls lEcfUèkr ç'u gS ftlesa çR;sd [k.M ds fy, pkj fodYi fn;s x, gSaA lgh
fodYi viuh mÙkj&iqfLrdk esa fyf[k,A

4 - Questions no. 18 is case study based questions carrying 4 marks each.
ç'u la[;k 18 çdj.k vè;;ua vkèkkfjr ¼case study based½ 4 vadksa dk ç'u gSA
5 - Marks for all questions are indicate against them .

lHkh iz”ukas ds vad muds lEeq[k vafdr fd;s x;s gSA
6 – There is no overall choice in the questions . However , an internal choice has been provided in different questions .
iz”ukas esa dksbZ lexz fodYi ugh gSA gkykafd fofHkfUu iz”uks esa vkarfjd fodYi iznku fd;s x;s gSA

7 – Draw neat diagram wherever required , Take π = wherever required , if not stated .

tgkW vko”;d gks] LoPN vkjs[k cuk;s tgkW vko”;d gks π = ysa] ;fn ugh fn;k x;k gSA

1. (a) If x= -3 is a solution of the quadratic equation x2 + (k-2) x + 9 = 0 , then the value of k is ; 1
;fn x= -3 f}?kkr lehdj.k x2 + (k-2) x + 9 = 0 dk ,d gy gS] rks k dk eku gSA
(i) -8 (ii) 8 (iii) 4 (iv) -4

(b) Two positive integers x and y are expressed as x= m 3n4 cand y= m5n2 , where m and n are 1
prime numbers. The LCM of x and y is

nks /kukRed iw.kkZd x vkSj y dks x= m3n4 vkSj y= m5n2 ds :i esa O;Dr fd;k x;k gS] tgkW m vkSj n vHkkT;

la[;k,a gSaA x o y dk y?kqRre lekioR;Z gSA
(i) m2n2 (ii) m3n4 + m5n2 (iii) m8n6 (iv) m5n4

(c) Coordinate of the mid point of a line segment join the points (3, -2) and (-7,-6) will be ; 1
fcUnqvks (3, -2) o (-7,-6) dks feykus okys js[kk[k.M ds e/; fcUnq ds funsZ”kkWd gksxa&
s
(i) (-2,-4) (ii) (-5,-4) (iii) (2,4) (iv) (3,-7)

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(d) The perimeter of a sector of a circle with radius 28 cm and central angle 90 0 is ; 1
28 lseh0 f=T;k ds vkSj 900 ds dsUnzh; dks.k okys o`Rr ds ,d f=T;[k.M dk ifjeki gSA

(i) 661 cm2 (ii) 616 cm2 (iii) 660 cm2 (iv) 540 cm2

(e) If the mean of 10 numbers is 20, what is the sum of these numbers ? 1
;fn 10 la[;kvks dk ek/; 20 gS] rks bu la[;kvks dk ;ksx D;k gksxkA
(i) 100 (ii) 200 (iii) 300 (iv) 400

(f) A tangent to a circle is __________ to the radius through the point of contact. 1
fdlh o`Rr dh Li”kZ js[kk] Li”kZ fcUnq ls xqtjus okyh f=T;k ij __________ gksrh gSA
(i) parallel (ii) perpendicular (iii) perpendicular bisector (iv) bisector
(i) lekukUrj (ii) yEcor (iii) yEc lef}Hkktd (iv) lef}Hkktd

(g) the discriminant of the quadratic equation 2x2−4x+3=0 is ; 1
f}?kkr lehdj.k 2x2−4x+3=0 dk fofoDrdj gSA
(i) -4 (ii) 4 (iii) 8 (iv) -8

(h) The distance of the point (9 , -3) from the origin is ; 1
ewy fcUnq ls fcUnq (9 , -3) dh nwjh gSA
(i) 3√10. (ii) √8 (iii) -3 (iv) √12

(i) A cylindrical drum has a diameter is 1.4 m and a height of 2.1 m. How much water can it hold ; 1
,d csyukdkj Mªe dk O;kl 1-4 ehVj vkSj ÅapkbZ 2-1 ehVj gSA blesa fdruk ikuh vk ldrk gSA
(i) 4.231 m3 (ii) 5.42 m3 (iii) 2.342 m3 (iv) 3.234 m3

(j) If the common difference of an A.P. is 5, then what is 𝑎18 − 𝑎13 ; 1

;fn ,d lekUrj Js.kh ¼ A.P. ½ dk lkoZ vUrj 5 gS] rks 𝑎18 − 𝑎13 dk eku gksxkA
(i) 5 (ii) 25 (iii) 20 (iv) 10

2. Find the 5th term from the last term (towards the first term) of an AP = 11, 8, 5,…, – 55 . 1
,d lekUrj Js.kh AP = 11, 8, 5,…, – 55 ds ¼vfUre in ls izFke in dh vksj½ 5 okW in Kkr dhft,A

3. In the given figure, PQ and PR are tangents to the circle such that 1
PQ = 7 cm and RPQ = 60° Find the length of chord QR .

fn;s x;s fp= esa] PQ vkSj PR ,d o`Rr dh Li”kZ js[kk;s gSA tgkW PQ = 7 lseh0 vkSj
RPQ = 60° gSA thok QR dh yEckbZ Kkr dhft,A

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4 . Find the HCF of integers 26 and 91 by using the prime factorisation method. 1
vHkkT; xq.ku[k.M fof/k dk mi;ksx djds iw.kkZd 26 o 91 dk HCF Kkr dhft,A
5. In the given figure, ∠CAB = 90° and AD ⊥ BC. If AC = 25 cm, AB = 1 m 1
and BD = 96.08 cm, then find the value of AD.

nh x;h vkd`fr esa] CAB = 90° vkSj AD ⊥ BC. gSA ;fn AC = 25 lseh0]

AB = 1 ehVj vkSj BD = 96.08 lseh0 gS rks AD dk eku Kkr dhft;sA

6. Questions number 6 (a) and (b) are Assertion and Reason based questions. Two
statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct
answer to these questions from the codes (i), (ii), (iii) and (iv) as given below.
(i) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(ii) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(iii) Assertion (A) is true, but Reason (R) is false.
(iv) Assertion (A) is false, but Reason (R) is true.

iz”u la[;k 6 (a) vkSj (b) vfHkdFku vkSj dkj.k vk/kkfjr iz”u gSA nks dFku fn;s x;s gS] ,d dks vfHkdFku
(A) vkSj nwljs dks dkj.k (R) ds :i esa fy[kk x;k gSA bu iz”uks ds fy;s uhps fn;s x;s dksM (i), (ii), (iii) vkSj
(iv) esa ls lgh mRrj dk p;u djasA
(i) vfHkdFku (A) vkSj dkj.k (R) nksuks lR; gS vkSj dkj.k] (R) vfHkdFku (A) dk lgh Li’Vhdj.k gSA
(ii) vfHkdFku (A) vkSj dkj.k (R) nksuks lR; gS vkSj dkj.k] (R) vfHkdFku (A) dk lgh Li’Vhdj.k ugh
gSA
(iii) vfHkdFku (A) lR; gS] ysfdu dkj.k (R) vlR; gSA
(iv) vfHkdFku (A) vlR; gS] ysfdu dkj.k (R) lR; gSA

(a). Assertion (A) : If we join two hemispheres of same radius along their bases, then we get a sphere. 1
Reason (R) : Total Surface Area of a sphere of radius r is 3πr .
2

vfHkdFku (A) % ;fn ge ,d gh f=T;k ds nks v/kZ xksyks dks muds vk/kkjkas ls tksM+rs gSa] rks ges ,d xksyk
feyrk gSA
dkj.k (R) % r f=T;k okys ,d xksys dk dqy i`’Bh; {ks=Qy 3πr2 gksrk gSA

1

(b). Assertion (A) : Two tangents drawn from an external point to a circle are equal in length.
Reason (R) : The angle between these tangents is equal to the angle subtended by the line
joining the center and the external point.

vfHkdFku (A) % fdlh ckgjh fcUnq ls ,d o`Rr ij [khph x;h nks Li”kZ js[kkvksa dh yEckbZ;ka cjkcj gksrh gSA
dkj.k (R) : bu Li”kZ js[kkvksa ds chp dk dks.k] dsUnz o ckgjh fcUnq dks feykus okyh js[kk }kjk cuk;s x;s
dks.k ds cjkcj gksrk gSA

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7. A card is drawn from a well shuffled deck of 52 cards. Find the probability that the card drawn is 2
either a red card or face card.
52 rk”k ds iRrkssa dh vPNh rjg ls QsaVs x;s xM~Mh esa ls ,d iRrk fudkyk tkrk gSA izkf;drk Kkr dhft,s fd
fudkyk x;k iRrk ;k rks yky jax dk gS ;k Qsl dkMZ gSA

8. If cos (A+B) = and tan (A-B) =
√
where 0 ≤ A+B ≤ 900 , then find the value of sec (2A – 3B) . 2
;fn cos (A+B) = vkSj tan (A-B) = √ gS] rks sec (2A – 3B) dk eku Kkr djksA ;fn 0 ≤ A+B ≤ 900

OR (vFkok)

A square field ABCD is divided into two right- angled isosceles triangles by drawing a diagonal AC. In such a
(sin A cos A) (1 tan A)
triangle, one of the acute angles is 450. Show that = 2
(sin A cos A) (1 tan A)
,d oxkZdkj eSnku ABCD dks ,d fod.kZ AC [khpdj nks ledks.k lef}ckgq f=Hkqtksa esa foHkkftr fd;k tkrk
(sin A cos A) (1 tan A)
gSA ,sls f=Hkqt esa ,d U;wudks.k 450 gSAfl) fdft;s fd =
(sin A cos A) (1 tan A)

9. In an A.P., if the common difference (𝑑) = - 4 and the 7th term (𝑎7) is 4, then find the 2
first term (𝑎).
,d lekUrj Js.kh (A.P. ) esa ;fn lkoZ vUrj (𝑑) = - 4 vkSj 7 okW in (𝑎7) ] 4 gS] rks igyk in Kkr dhft;sA

10. The following data represents the number of hours studied by 12 students: 2
fuEufyf[kr caVu 12 Nk=ks }kjk v/;;u fd;s x;s ?k.Vks dh la[;k dks n”kkZrk gSA
{4, 6, 8, 10, 10, 12, 14, 14, 16, 18, 20, 22}
Calculate ¼x.kuk dhft,½ a) Mean ¼ek/;½ b) Median ¼ek/;d½
11. The revenue (in ₹) generated by a small business is modeled by the polynomial P(x) = x² - 5x - 6. If x 2
represents the number of units sold, find the number of units at which the revenue becomes zero.

,d y?kq O;olk; dh gksus okyh vk; (:0 esa ) dks cgqin }kjk n”kkZ;k x;k gSA ;fn csph x;h oLrqvks dh
la[;k dks x }kjk n”kkZ;k x;k gS] rks Kkr dhft, dh fdruh oLrq,a cspus ij vk; “kwU; gks tkrh gSA
12. If p and q are zeroes of the polynomial p(y) = 21y2- y- 2, then find the value of p ÷ q and p×q . 2
;fn p vkSj q cgqin p(y) = 21y2- y- 2 ds “kwU;d gS rks p ÷ q o p×q dk eku Kkr dhft,A
13. A sector of a circle has a central angle of 90° and radius 14 cm. Find the area of the corresponding segment 2
,d o`Rr ds f=T;[k.M dk dsUnzh; dks.k 90° vkSj f=T;k 14 lseh0 gSA laxr o`Rr [k.M dk {ks=Qy Kkr
dhft,A
14. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the 4
dimensions of the garden.
,d vk;rkdkj cxhps dk v/kZ ifjeki] ftldh yEckbZ mldh pkSM+kbZ ls 4 ehVj vf/kd gS] 36 ehVj gSA
cxhps dh foek,a Kkr dhft,A
15. Prove that 5√3 + 2 is an irrational number given that √3 is an irrational number. 4
fl) djsa fd 5√3 + 2 ,d vifjes; la[;k gS] ;g ns[krs gq, fd √3 ,d vifjes; la[;k gSA
16. (a) By using trigonometric identities Prove that sec A (1 – sin A)(sec A + tan A) = 1. 4
f=dks.kferh; loZlehdkvksa dk mi;ksx djds fl) dhft;s fd sec A (1 – sin A)(sec A + tan A) = 1

OR ¼vFkok½
(b) If cosec A = x + , prove that cosec A + cot A = 2x or .
;fn cosec A = x + gS] rks fl) dhft, fd cosec A + cot A = 2x ;k gSA

17. Find the coordinate of house of a student which is on the x-axis and is equidistant from his 4
college A (2, –5) and barber shop B (–2, 9).
,d Nk= ds ?kj dk funsZ”kkad Kkr dhft, tks x v{k ij gS vkSj mlds dkyst A (2, –5) vkSj ukbZ
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dh nqdku B (–2, 9) ls lenwjLFk gSA

18. Case Study
Vijay is trying to find the height of a tower near his house. He uses the properties of similar triangles 1+1+2
to solve this. The height of Vijay's house is 20m, and it casts a shadow 10m long on the ground. At the
same time, the tower casts a shadow 50m long. Assume both the house and the tower are perpendicular
to the ground.
Based on the information above, answer the following questions:
(a) What is the height of the tower ?
(b) If Vijay's house casts a shadow of 12m at a different time, what would be the length of the tower's shadow at
that same time ?
(c) If the tower's shadow becomes 40m long, how long would Vijay's house's shadow be at that moment ?

fot; vius ?kj ds ikl ,d Vkoj dh ÅapkbZ Kkr djus dh dksf”k”k dj jgk gSA bls gy djs ds fy;s og
le:i f=Hkqtksa ds xq.kksa dk mi;ksx djrk gSA fot; ds ?kj dh Åapkb 20 ehVj gS] vkSj ;g tehu ij
10 ehVj yEch Nk;k Mkyrk gSA mlh le; Vkoj 50 ehVj yEch Nk;k Mkyrk gSA eku yas fd ?kj o Vkoj
nksuks tehu ij yEcor gSaA Åij nh x;h tkudkjh ds vk/kkj ij fuEufyf[kr iz”uksa ds mRrj nhft,A
(a) Vkoj dh ÅapkbZ D;k gS?
(b) ;fn fot; dk ?kj ,d vyx le; ij 12 ehVj dh Nk;k Mkyrk gS] rks mlh le; Vkoj dh Nk;k
dh yEckbZ D;k gksxh?
(c) ;fn Vkoj dh Nk;k dh yEckbZ 40 ehVj gks tkrh gS] rks ml {k.k fot; ds ?kj dh Nk;k fdruh yEch
gksxh?

19. (a) A 2-digit number is seven times the sum of its digits and two more than 5 times the product of its 6
digits. Find the number.

,d 2 vadh; la[;k mlds vadks ds ;ksx dk 7 xquk gS vkSj mlds vadks ds xq.kuQy ds 5 xqus ls 2 vf/kd
gSA og la[;k Kkr dhft,A
OR ¼vFkok½

(b) Find the value (s) of p for which the quadratic equation given as (p+4) x² - (p + 1) x + 1 = 0 has real and
equal roots. Also, find the roots of the equation(s) so obtained.
p dk@ds eku Kkr djsa ftlds fy;s f}?kkr lehdj.k (p+4) x² - (p + 1) x + 1 = 0 ds
ewy okLrfod vkSj cjkcj gSA lkFk gh bl izdkj izkIr lehdj.k dk@ds ewy Hkh Kkr dhft,A

20. The runs scored by 60 players in a cricket tournament are given below: 6
,d fØdsV VwukZe.s V esa 60 f[kykfM+;ks }kjk cuk;s x;s ju uhps fn;s x;s gSaA
Salary ¼osru½ 10- 20-30 30-40 40-50 50-60 60-70 70-80 80-90
20
No. of Employees 2 8 12 20 30 18 7 3
¼deZpkfj;ksa dh la[;k½
Determine the modal class and mode of the given data
fn;s x;s caVu dk cgqyd oxZ (Modal Class) vkSj cgqyd (Mode) Kkr dhft,A

21. Shanta runs an industry in a shed which is in the shape of a cuboid surmounted by a half cylinder . 6
If the base of the shed is of dimension 7 m × 15 m, and the height of the cuboidal portion is 8 m, find the
Volume of air that the shed can hold. Further, suppose the machinery in the shed occupies a total space

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of 300 m3, and there are 20 workers, each of whom occupy about 0.08 m3 space on an average. Then, how
much air is in the shed? (Take π = ).
“kkUrk ,d “ksM esa ,d m|ksx pykrh gS] tks ,d ?kukHk ds vkdkj dk gSA ftlds Åij ,d v/kZ csyu gSA
;fn “ksM dk vk/kkj 7 ehVj × 15 ehVj dk gS] vkSj ?kukHk okys fgLls dh ÅapkbZ 8 ehVj gSA rks “ksM esa lek
ldus okyh gok dk vk;ru Kkr dhft,A lkFk gh eku yhft, fd “ksM esa e”khujh dqy 300 ehVj 3 txg ?ksjrh
gS vkSj 20 Jfed gS ftuesa ls izR;sd vkSlru yxHkx 0-08 ehVj 3 txg ?ksjrk gS rks “ksM esa gok dk vk;ru Kkr
dhft,A ( π = yas )
22. State the converse of Basic Proportionality Theorem . Also find 6
in the following figure , given that AB ǀǀ DC ǀǀ EF and
= . Also find the length of EF if AB = 10 cm and DC = 15 cm.

vk/kkjHkwr vkuqikfrdrk izes; ds foykse dks fyf[k;sA lkFk gh fn;s x;s fp= esa Kkr dhft,] ;g ns[krs gq,
fd AB ǀǀ DC ǀǀ EF vkSj = gSA lkFk gh ;fn AB = 10 lseh0 vkSj DC = 15 lseh0 gS rks EF dh yEckbZ Kkr
dhft,A

23. (a) Two observers A and B are standing on opposite sides of a tower. The angles of elevation of 6
the top of the tower from A and B are 30° and 45° respectively. If the distance between A and
B is 100 meters, find the height of the tower.
nks isz{kd A vkSj B ,d Vkoj ds foijhr fdukjksa ij [kM+s gSaA A vkSj B ls Vkoj ds “kh’kZ ds mUu;u
dks.k Øe”k% 30° vkSj 45° gSA ;fn A vkSj B ds chp dh nwjh 100 ehVj gS] rks Vkoj dh ÅapkbZ Kkr dhft,A
OR ¼vFkok½
(b) A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a 6
car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed.
Six seconds later, the angle of depression of the car is foundto be 60°. Find the time taken by the car to
reach the foot of the tower from this point.

,d lh/kk jktekxZ ,d ehukj ds ikn dh vksj tkrk gSA ehukj ds “kh’kZ ij [kM+k ,d O;fDr ,d dkj dks 30°
ds voueu dks.k ij ns[krk gSA tks ,dleku xfr ls ehukj ds ikn dh vksj vk jgh gSA 6 lsds.M i”pkr
dkj dk voueu dks.k 60° ik;k x;kA bl fcUnq ls ehukj ds ikn rd igqpus esa dkj }kjk fy;k x;k le;
Kkr dhft,A

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jksy ua- eqfnzr i`"Bksa dh la[;k % 8

031 2026
izfrn'kZ iz'u i=&2026
Model Question Paper
xf.kr (MATHEMATICS)
le; % 3 ?k.Vs] [iw.kkZd
a % 80
Time : 3 Hours] [Max. Marks : 80

funsZ'k% (i) bl iz”u i= esa dqy 23 iz”u gSaA lHkh iz”u vfuok;Z gSaA
Directions:% There are in all 23 Questions in this question paper. All questions are
compulsory.
(ii) iz”uksa gsrq fu/kkZfjr vad muds lEeq[k vafdr gSaA
Marks allotted to the questions are mentioned against them.
(iii) izR;sd iz”u dks /;kuiwoZd if<+;s vkSj leqfpr mÙkj nhft,A
Read each question carefully and answer to the point.
(iv) iz”u la- 1 cgqfodYih; iz”u gSA bl iz”u ds izR;sd [k.M ds mÙkj esa pkj
fodYi fn;s x, gSaA lgh fodYi viuh mÙkj&iqfLrdk esa fyf[k,A
Question no.1 is multiple choice question four option are given in answer
of each part of the question. Write correct option in your answer book.
(v) iz”u la- 2 [email protected] ls lEcfU/kr iz”u gS ftlesa izR;sd [k.M ds fy, pkj
fodYi fn;s x, gSaA lgh fodYi viuh mÙkj&iqfLrdk esa fyf[k,A
Question no. 2 is related to Assertion/Reason question. Four option are
given in answer of each part of the question. Write correct option in your
answer book.
(vi) iz”u la- 1 o iz”u la- 2 dk izR;sd [k.M ,d ¼1½ vad dk gSA iz”u la- 3 ls 6
rd ,d ¼1½ vad ds iz”u gSaA iz”u la- 7 ls 13 rd nks ¼2½ vad ds iz”u gSaA
iz”u la- 14 ls 17 rd pkj ¼4½ vad ds iz”u gSAa iz”u la- 18 ls 22 rd N%
¼6½ vad ds iz”u gSA iz”u la- 23 dsl LVMh ls lEcfU/kr iz”u pkj vad
¼1+1+2½ dk gSA
Each parts of question no.1 and question no. 2 carries one (1) mark.
Question no. 3 to 6 are of one (1) mark each. Question no. 7 to 13 are of
two (2) mark each. Question no. 14 to 17 are four (4) mark each.
Question no 18 to 22 are of Six (6) mark each. Question no. 23 related to
case study question carries 4 marks (1+1+2).
(vii) bl iz”u i= esa lexz ij dksbZ fodYi ugha gSA rFkkfi dfri; iz”uksa esa
vkUrfjd fodYi iznku fd;k x;k gSA ,sls iz”uksa esa dsoy ,d fodYi dk gh

mÙkj nhft,A
There are no overall choice in this question paper, however, an internal
choice has been provided in few questions. Attempt only one of the given
choices in such questions.
[1] [P.T.O.

1- ¼d½ fuEu esa dkSu lh la[;k vifjes; la[;k gS\ 1
Which of the following numbers is an irrational number?

(i) √25 (ii) (iii) 0 (iv) √81
√

¼[k½ f}?kkr cgqin − 25 ds “kwU;dksa dk ;ksx gksxkA 1
The sum of the zeroes of the quadratic polynomial − 25 will be:

(i) 0 (ii) 1 (iii) 25 (iv) -25

¼x½ f}?kkr lehdj.k + + = 0 ds ewy okLrfod o vleku gksaxsA ;fn 1
The roots of quadratic equation + + = 0 will be real and unequal if:

(i) −4 <0 (ii) −4 >0
(iii) −4 =0 (iv) −4 >0
¼?k½ A.P; 10]7]4] -----------dk 30 ok¡ in gSA 1
The 30th term of an A.P; 10, 7, 4, …… is
(i) 97 (ii) 77 (iii) -77 (iv) - 87
¼M+½ fuEu esa dkSu lh f=Hkqtksa dh le:irk dh dlkSVh ugha gS\ 1
Which of the following is not a criterion of similarity of triangles?
(i) AAA (ii) SSS (iii) SAS (iv) ASA
¼p½ fuEu vkd`fr esa ;fn ∥ rks dk eku D;k gksxk\ 1
In the given figure if ∥ . Then what will be the value of AE.
;fn (If) AB = 6cm, AC = 9cm, AD = 2cm

(i) 1.5 cm (ii) 2.5 cm (iii) 3 cm (iv) 4 cm
¼N½ fcUnq ¼5]&3½ fdl prqFkkZa”k eas fLFkr gSa\ 1

In which quadrant point (5, -3) lies?
(i) I prqFkkZ”a k (ii) II prqFkkZ”a k (iii) III prqFkkZa”k (iv) IV prqFkkZ”a k
(i) I quadrant (ii) II quadrant (iii) III quadrant (iv) IV quadrant
[2] [P.T.O.

¼t½ R f=T;k okys ml f=T;[k.M dk {ks=Qy D;k gksxk] ftldk dsUnz dks.k PO gS\ 1
Area of a sector of angle P (in degrees) of a circle with radius R is

(i) ×2 (ii) × (iii) ×2 (iv) ×2

¼>½ fuEu esa dkSu fdlh ?kVuk dh izkf;drk ugha gks ldrh gS\ 1
Which of the following can not be the probability of an event?

(i) (ii) 15% (iii) 150% (iv) 0.5

¼´½ fuEu vkd`fr esa TP o TQ dsUnz O okys fdlh o`Rr ij nks Li'kZ js[kk,¡ bl izdkj gSa fd
∠ = 110 rks ∠ cjkcj gSAa 1
In this figure, if TP and TQ are the two tangents to a circle with centre O so that
∠ = 110 then ∠ is equal to

(i) 60O (ii) 70O (iii) 80O (iv) 90O
2. bl iz'u ds nksuksa [k.M [email protected] ls lEcfU/kr gSAa izR;sd [k.M ds fy, fuEu esa ls lgh
fodYi dk p;u dhft,A
Both parts of this section related to assertion/reason. Choose the correct option for both
part from the following options.
(i) A o R nksuksa lgh gSa vkSj R, A dh lgh O;k[;k djrk gSA
A and R are both true and R is the correct explanation of A.
(ii) A o R nksuksa lgh gS]a ysfdu R, A dh lgh O;k[;k ugha djrk gSA
A and R are both true and R is not the correct explanation of A.
(iii) A lgh gS] ysfdu R xyr gSA
A is true, but R is false.
(iv) A xyr gS] ysfdu R lgh gSA

A is false, but R is true.

[3] [P.T.O.

¼d½ dFku ¼A½ % ,d ifjes; la[;k vkSj ,d vifjes; la[;k dk ;ksx lnSo vifjes; gksrk gSA 1
Assertion (A) : The sum of a rational number and an irrational number is always an
irrational number.

dkj.k (R) % 7 + √3 ,d vifjes; la[;k gSA
Reason (R) : 7 + √3 is an irrational number.
¼[k½ dFku ¼A½ % lHkh oxZ le:Ik gksrs gSaA 1
Assertion (A) : All squares are similar.
dkj.k (R) % lHkh le:Ik vkd`fr;ka lokZaxle gksrh gSAa
Reason (R) : All similar figures are congruent.

3- og f}?kkr cgqin Kkr dhft, ftlds 'kwU;dksa dk ;ksx o xq.kuQy Øe'k%− o gSA 1
Find the quadratic polynomial whose sum and product of zeroes is respectively − and

4- fcUnq ¼3] 4½ dh y&v{k ls nwjh D;k gS\ 1
What is the distance of point (3, 4) from y-axis?
5- o`Ùk dks nks fcUnqvksa ij izfrPNsn djus okyh js[kk dks D;k dgrs gS\a 1
What is the line that intersects a circle at two points called?
6- f=T;k 3-5 lseh- okys v)Zxksys dk oØ i`"Bh; {ks=Qy Kkr dhft,A 1
Find the curved surface area of a hemisphere with radius 3.5cm.
7- ;fn f}?kkr cgqin 7 − 50 + ds 'kwU;d ,d&nwljs ds O;qRØe gSa rks k dk eku Kkr
dhft,A 2
If zeroes of the quadratic polynomial 7 − 50 + are reciprocal to each other, then
find the value of k.
8- nks jsy iVfj;k¡ lehdj.kksa + 2 − 4 = 0 vkSj 2 + 4 − 12 = 0 }kjk fu:fir dh xbZ gaAS
D;k jsy iVfj;k¡ ,d&nwljs dks dkVsaxh\ 2
Two rails are represented by the equations + 2 − 4 = 0 and 2 + 4 − 12 = 0. Will
the rails cross each other?
9- f}?kkr lehdj.k 2 − 4 + 3 = 0 dk fofoDrdj Kkr dhft, vkSj ewyksa dh izd`fr Kkr
dhft,A 2
Find the discriminant of the quadratic equation 2 − 4 + 3 = 0 and then find the
nature of roots.

10- ;fn = rks o ds eku Kkr dhft,A 2
If = then find the value of and .

[4] [P.T.O.

11- ,d ?kM+h dh feuV dh lqbZ dh yEckbZ 14 lseh- gSA bl lqbZ }kjk 20 feuV esa jfpr {ks=Qy
Kkr dhft,A 2
The length of a minute hand of a clock is 14 Cm. Find the area swept by the minute
hand in 20 minutes.
12- fuEu ckjEckjrk lkj.kh esa vKkr jkf'k;ksa x, y, z vkSj t ds eku Kkr dhft,A 2
Find the unknown quantities x, y, z and t in the following frequency table.

oxZ (Class) 0-10 10-20 20-30 30-40 40-50 50-60

ckjEckjrk (f) 6 5 y 4 z 8

lap;h ckjEckjrk 6 x 15 19 22 t
(Cumulative frequency)

13- ,d isVh esa 30 fMLd gSaA ftuesa 1 ls 30 rd dh la[;k,sa vafdr gSaA ;fn bl isVh esa ls
;kn`PN;k ,d fMLd fudkyh tkrh gS] rks bldh D;k izkf;drk gS fd fMLd ij vafdr gksxh&
(i) nks vadksa dh ,d la[;k (ii) ,d iw.kZ oxZ la[;k 2
A box contains 30 discs which are numbered from 1 to 30. If one disc is drawn at
random from the box, then find the probability that it bears-
(i) A two-digit number (ii) A perfect square number
vFkok (OR)
gjizhr nks fHkUu&fHkUu flDdksa dks ,d lkFk mNkyrh gSA bldh D;k izkf;drk gS fd og de
ls de 1 fpÙk izkIr djsxh\
Harpreet tosses two different coins simultaneously. What is the probability that she gets
at least one head?

14- fl) dhft, 6 + √3 ,d vifjes; la[;k gS\ 4
Prove that 6 + √3 is irrational number?
vFkok (OR)
O;k[;k dhft, fd 7 × 11 × 13 + 13 vkSj 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 HkkT; la[;k,¡
D;ksa gSa\
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite
numbers?
15- fl) dhft,& 4
Prove that -

=
vFkok (OR)
[5] [P.T.O.

1+
= +
1−

16- fuEu vkd`fr esa = gS rFkk ∠ =∠ gSA fl) dhft, fd ∆ ,d lef}ckgq
f=Hkqt gSA 4

In this figure = and ∠ =∠ . Prove that ∆ is an isosceles triangle.

17- ;fn fcUnq ¼1] 2½] ¼4] y½ ¼ x] 6½ rFkk ¼3] 5½ blh Øe esa ysus ij ,d lekUrj prqHkqZt ds 'kh"kZ
gksa rks x vkSj y ds eku Kkr dhft,A 4
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x
and y.
18- nks O;fDr;ksa dh vk; dk vuqikr 9 % 7 gS vkSj muds [kpksZa dk vuqikr 4 % 3 gSA ;fn izR;sd
O;fDr izfr ekg 2000 #i;s cpkrk gS rks nksuksa O;fDr;ksa dh ekfld vk; Kkr dhft,A 6
The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If
each of them manages to save Rs. 2000. Find their monthly incomes.
vFkok (OR)
13 ehVj O;kl okys ,d o`Rrkdkj ikdZ dh ifjlhek ds ,d fcUnq ij ,d [kEHkk bl izdkj
xkM+uk gS fd bl ikdZ ds ,d O;kl ds nksuksa vUr fcUnqvksa ij cus QkVdksa A vkSj B ls [kEHks
dh nwfj;ksa dk vUrj 7 ehVj gksA D;k ,slk djuk lEHko gS\ ;fn gS rks nksuksa QkVdksa ls
fdruh nwfj;ksa ij [kEHkk xkM+uk gS\
A pole has to be erected at a point on the boundary of a circular park of diameter 13
metres in such a way that the differences of its distances form two diametrically
opposite fixed gates A and B on the boundary is 7 meters. Is it possible to do so? If yes,
at what distance from the two gates should the pole be erected?
19- ,d unh ds iqy ds ,d fcUnq ls unh ds lEeq[k fdukjkas ds voueu dks.k Øe'k% 30O o 45O

gSA ;fn iqy fdukjksa ls 5 ehVj dh Å¡pkbZ ij gks rks unh dh pkSM+kbZ Kkr dhft,A 6

[6] [P.T.O.

From a point on a bridge across a river, the angles of depression of the banks on
opposite sides of the river are 30O and 45O respectively. If the bridge is at a height of 5
metre form the banks. Find the width of the river.
vFkok (OR)
,d lh/kk jktekxZ ,d ehukj ds ikn rd tkrk gSA ehukj ds f'k[kj ij [kM+k ,d O;fDr ,d
dkj dks 30O ds voueu dks.k ij ns[krk gS] tks ehukj ds ikn dh vksj ,d leku pky ls
tkrh gSA 8 lsds.M ckn dkj dk voueu dks.k 60O gks x;kA bl fcUnq ls ehukj ds ikn rd
igaqpus esa fy;k x;k le; Kkr dhft,A
A straight highway leads to the foot of a tower, A man standing at the top of the tower
observes a car at an angle of depression of 30O, which is approaching the foot of the
tower with a uniform speed. 8 second later, the angle of depression of the car is found to
be 60O. Find the time taken by the car to reach the foot of the tower from this point.
20- Å¡pkbZ 2-4 lseh- vkSj O;kl 1-4 lseh- okys ,d Bksl csyu esa ls blh Å¡pkbZ vkSj O;kl okyk
,d 'kaDokdkj [kksy dkV fy;k tkrk gSA 'ks"k cps Bksl dk i`"Bh; {ks=Qy Kkr dhft,A 6
From a solid cylinder whose height is 2.4cm and diameter 1.4cm, a conical cavity of the
same height and same diameter is hollowed out. Find the total surface area of the
remaining solid.
21- fuEu caVu ,d eksgYys ds cPpksa ds nSfud tsc [kpZ dks n'kkZrk gSA ek/; tsc [kpZ 18 #i;s
gSA yqIr ckjEckjrk f Kkr dhft,A 6
The following distribution shows the daily pocket allowance of children of a locality.
The mean pocket allowance is Rs. 18. Find the missing frequency f.

nSfud tsc HkRrk ¼#-esa½
Daily pocket 11-13 13-15 15-17 17-19 19-21 21-23 23-25
allowance (In Rs.)

cPpksa dh la[;k 7 6 9 13 f 5 4
Number of children

22- nh xbZ vkd`fr esa 5 lseh- f=T;k ds ,d o`Rr dh 8 lseh- yEch ,d thok PQ gSA P o Q ij
Li'kZ js[kk,sa ijLij ,d fcUnq T ij izfrPNsn 6
djrh gSaA TP dh yEckbZ Kkr dhft,A
In the figure PQ is a chord of length 8 Cm
of a circle of radius 5 Cm. The tangents at P
and Q intersects at a point T. Find the length
TP.

[7] [P.T.O.

vFkok (OR)
f=Hkqtksa dh le#irk dh fdruh dlkSfV;k¡ gSa\ uke fyf[k,A
,d f=Hkqt ABC dh Hkqtk BC ij ,d fcUnq D bl izdkj fLFkr gS fd ∠ = ∠ rks
n'kkZb;s fd = .
How many criterion of similarity of triangles? Name them.
D is the point on the side BC of a triangle ABC such that∠ = ∠ . Show that
= .

dsl LVMh (Case Study)
23- jfo vius fy, lkbfdy [kjhnuk pkgrk gSA mlds firk mls gj eghus dqN cpr djus dh
lykg nsrs gSaA jfo igys eghus esa #- 500 cpr djds 'kq#vkr djrk gS vkSj mlds ckn izfr
ekg viuh cpr esa #- 100 dh o`f) djrk gSA

Ravi wants to buy a bicycle. His father suggests saving money every month. Ravi
decides to start saving with Rs. 500 in the first month and increase his saving by Rs.
100 each month.
iz'u 1- jfo dh cpr ds vuqØe esa lkoZUrj D;k gS\ 1
What is the common difference (d) of Ravi's saving pattern?
iz'u 2- ikapos eghus esa jfo fdruh cpr djsxk\ 1
How much will Ravi save in the 5th month?
iz'u 3- fdl eghus esa jfo dh cpr #- 1500 gksxh\ 2
In which month Ravi's saving will be Rs. 1500?

**********

[8] [P.T.O.

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