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1 Class - X MATHEMATICS Time : 3 to 3½ hours Maximum Marks : 80 â Ø : 3 âð 3½ ƒæ ÅUð çï Ì : 80 General Instructions : 1. All questions are compulsory. Total No. of Pages : 13 é Ü ÂëcÆUæð è â Øæ : 13. The question paper consists of 34 questions divided into four sections A, B, C and D. Section - A comprises of 10 questions of 1 mark each, Section - B comprises of 8 questions of marks each, Section - C comprises of 10 questions of 3 marks each and Section - D comprises of 6 questions of 4 marks each. 3. Question numbers 1 to 10 in Section - A are multiple choice questions where you are to select one correct option out of the given four. 4. There is no overall choice. However, internal choice has been provided in 1 question of two marks, 3 questions of three marks each and questions of four marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculator is not permitted. 6. An additional 15 minutes time has been allotted to read this question paper only. âæ æ Ø çùîðüàæ Ñ 1. âöè ÂýàÙ çùßæøü ãñ Ð. â ÂýàÙ- æ ð 34 ÂýàÙ ãñ, Áæð æúu ¹ ÇUæð ð, Õ, â ß Î ð çßöæçáì ãñð ¹ ÇU - ð 10 ÂýàÙ ãñ æñúu ÂýˆØð ÂýàÙ 1 æ ãñ, ¹ ÇU - Õ ð 8 ÂýàÙ ãñ æñúu ÂýˆØð ÂýàÙ æð ð ãñ, ¹ ÇU - â ð 10 ÂýàÙ ãñ æñúu ÂýˆØð ÂýàÙ 3 æð æ ãñ, ¹ ÇU - Î ð 6 ÂýàÙ ãñ æñúu ÂýˆØð ÂýàÙ 4 æð æ ãñð 3. ÂýàÙ â Øæ 1 âð 10 Õãéçß ËÂèØ ÂýàÙ ãñ Ð çî» æúu çß ËÂæð ð âð âãè çß Ë éùð Ð 4. â ð æð ü Öè âßæðüâçúu çß Ë Ùãè ãñ, Üðç Ù æ ÌçÚU çß Ë 1 ÂýàÙ æð ð, 3 ÂýàÙ 3 æð ð æñúu ÂýàÙ 4 æð ð çî» ãñ Ð æâ çî» çß ËÂæð ð âð çß Ë æ ØÙ Úð UÐ 5. ñ Ü é ÜðÅUÚU æ ÂýØæð» ßçÁüÌ ãñð 6. â ÂýàÙ- æ æð ÂÉ Ùð ð çü 15 ç ÙÅU æ â Ø çîøæ»øæ ãñð â ßçÏ ð ÎæñÚUæÙ ÀUæ æ ð ßÜ ÂýàÙ- æ æð ÂÉ ð»ð æñúu ßð žæúu-âéçsì æ ÂÚU æð ü žæúu Ùãè çü¹ð»ðð 1 P.T.O.
2 SECTION - A Question numbers 1 to 10 carry one mark each. 1. Euclid s division lemma states that if a and b are any two 1ve integers, then there exists unique integers q and r such that (A) a5bq1r, 0 < r < b (B) a5bq1r, 0 d r d b (C) a5bq1r, 0 d r < b (D) a5bq1r, 0 < b < r. Which of the following is not defined? (A) cos 0 o (B) tan 45 o (C) sec 90 o (D) sin 90 o 3. The graph of y5p (x) given below. The number of zeroes of p(x) are : (A) 0 (B) (C) 4 (D) 3 4. If sinu5 1 3, then the value of cot u1 is : (A) 6 (B) 9 (C) 18 (D) 4 5. The mean and median of a data are 14 and 15 respectively. The value of mode is (A) 16 (B) 17 (C) 13 (D) In DLMN, L5608, M5508. If DLMN DPQR, then the value of R is (A) 408 (B) 308 (C) 708 (D) 1108
3 7. The value of tan45 sin30cos30 is : (A) 1 (B) 1 (C) 1 (D) 8. If 1 is zero of the polynomial p(x)5ax 3(a1) x1, then the value of a is (A) 1 (B) 1 (C) (D) 9. Which of the following is not an irrational number? (A) 5 3 (B) (C) 41 (D) (seca1tana) (1sinA) is equal to : (A) seca (B) sina (C) coseca (D) cosa. SECTION - B Question numbers 11 to 18 carry marks each. 11. In figure- AB& DE and BD& EF. Prove that DC 5CF3AC. 1. Find the zeroes of the quadratic polynomial 3 x 8x Write any two merits and demerits of mean. 3 P.T.O.
4 14. In DABC, AB5AC and D is a point on side AC such that BC 5AC. CD. Prove that BD5BC. 15. In figure-3, ABC is right triangle, D is mid point of BC. Show that tan M 1 tan > OR In DPQR right angled at Q, PR1QR55cm and PQ55cm. Find the value of sin P. 16. For which values of p does the pair of equations given below has unique solution. 4x1py1850; x1y Check whether 6 n can end with the digit 0 for any natural number n. 18. The mean of the following data is 7.5. Find the value of p. x i f i p 8 4 SECTION - C Question numbers 19 to 8 carry 3 marks each. 19. Prove that Prove that secm1 secm1 5 cosecu. secm1 secm1 OR 1 sina 5secA1tanA 1 sina 4
5 0. On dividing x 3 3x 1x1 by a polynomial g(x), the quotient and remainder were x and x14 respectively. Find g(x). 1. An army contingent of 616 members is to march behind an army band of 3 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? OR Find the HCF and LCM of 306 and 54. Verify that HCF3 LCM5Product of the two numbers.. Evaluate : sin68 cot15 cos 5tan75 3tan45 tan0 tan40 tan50 tan Solve for x and y 5 1 x 1 y 5 ; 6 3 x 1 y 51. OR For what values of a and b does the following pairs of linear equations have an infinite number of solutions. x13y57; a(x1y)b(xy)53a1b. 4. The perpendicular AD on the base BC of DABC intersects BC in D such that BD53CD. Prove that AB 5AC 1BC. 5. The given distribution shows the number of runs scored by some top batsmen of the world in one - day international cricket matches. Find the mode of the data. Runs Scored No. of batsmen Runs Scored No.of batsmen P.T.O.
6 6. In the figure-4 ABC is a right angled triangle, right angled at C. DE A AB. Prove that DABC DADE and hence find the lengths of AE and DE. 7. During the medical check up of 35 students of a class, their weights were recorded as follows. Draw a less than type ogive for the given data. Hence obtain Median weight from the graph. Weight (in kg) No. of students less than 38 0 less than 40 3 less than 4 5 less than 44 9 less than less than 48 8 less than 50 3 less than Prove that 3 5 is irrational. SECTION - D Question numbers 9 to 34 carry 4 marks each. 9. Solve the system of equations graphically. x1y55 ; x3y54. Also find the points where the lines meet the x - axis. tanm cot 30. Prove that : 1 sec.cosec 1 cot M 1tan M M M M 6
7 31. If the median of the distribution given below is 8.5, find the values of x and y. Class Intervals Total Frequency 5 x 0 15 y Prove that : coseca cota sina sina coseca cota OR 4 Prove that : sec sin Msin M u cos McosM 33. If the polynomial x 4 6x 3 116x 5x110 is divided by another polynomial x x1k, the remainder comes out to be x1a, find the values of k and a. 34. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. OR Prove that in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle. - o 0 o - 7 P.T.O.
8 ÂýàÙ â Øæ 1 âð 10 Ì ÂýˆØð ÂýàÙ 1 æ ãñð ¹ ÇU- 1. Øêç ÜÇU çßöæáù Âý ðçø æ âð ØçÎ a ÌÍæ b Îæð ÏÙæˆ Âê ææz ãñ, Ìæð q ÌÍæ r â Âý æúu Ü Âê ææz âð ÁéÇðU ãñ ç Ñ (A) a5bq1r, 0 < r < b (B) a5bq1r, 0 d r d b (C) a5bq1r, 0 d r < b (D) a5bq1r, 0 < b < r. çù Ù ð âð æñù âæ ÂçÚUÖæçáÌ Ùãè ãñ? (A) cos 0 o (B) tan 45 o (C) sec 90 o (D) sin 90 o 3. ÕãéÂÎ y5p (x) æð»ýæè mæúuæ æ ë çì (1) ð çî¹æøæ»øæ ãñð â ð àæê Øæ è â Øæ ãæð»è Ñ (A) 0 (B) (C) 4 (D) 3 4. ØçÎ sinu5 1 3 Ìæð cot u1 æ æù ãæð»æ Ñ (A) 6 (B) 9 (C) 18 (D) 4 5. é ÀU, æ Ç æð æ ŠØ æù ÌÍæ æçšø æ ý àæñ 14 ÌÍæ 15 ãñð â ð ÕãéÜ æ æù ãæð»æ Ñ (A) 16 (B) 17 (C) 13 (D) ç æöéá LMN ð L5608, M5508. ØçÎ DLMN DPQR, ÌÕ R æ æù ãæð»æ Ñ (A) 408 (B) 308 (C) 708 (D)
9 7. tan45 sin30cos30 æ æù ãñ Ñ (A) 1 (B) 1 (C) 1 (D) 8. ØçÎ p(x)5ax 3(a1) x1 æ àæê Øæ 1 ãæð Ìæð a æ æù ãæð»æ Ñ (A) 1 (B) 1 (C) (D) 9. çù Ù ð âð æñù âæ ÂçÚU ðø â Øæ Ùãè ãñ? (A) 5 3 (B) (C) 41 (D) (seca1tana) (1sinA) ÕÚUæÕÚU ãñ Ñ (A) seca (B) sina (C) coseca (D) cosa. ¹ ÇU-Õ ÂýàÙ â Øæ 11 âð 18 Ì ÂýˆØð ÂýàÙ æ ãñð 11. æ ë çì () ð AB& DE ÌÍæ BD& EF Ìæð çâh èçáøð ç Ñ DC 5CF3AC. 9 P.T.O.
10 1. 3 x 8x14 3 çmƒææì ÕãéÂÎ ð àæê Øæ ææì ÚUæðÐ 13. ŠØ æù ð æð ü Îæð ÂØéQ Ìæ ÌÍæ ÙéÂØéQ Ìæ çüç¹øðð 14. ç æöéá ABC ð, AB5AC ß D çõ Îé, ÖéÁæ AC ÂÚU â Âý æúu çsíì ãñ ç, BC 5AC. CD, Ìæð çâh Úð U ç BD5BC. 15. æ ë çì - 3 ð DABC â æð æ ç æöéá ãñ ÌÍæ D, BC æ ŠØ çõ Îé ãñð M tan > Ìæð Îàææü Øð tan 1 Øæ â æð æ DPQR æ Q â æð æ ãñð PR1QR55 âð. è. ÌÍæ PQ55 âð. è, Ìæð sin P æ æù ææì èçá Ð 16. p ð ç â æù ð çüøð çù Ù ÚñUç¹ â è ÚU æ Øé æ ãü Ü ãæð»æ? 4x1py1850; x1y Áæ èçá ç 6 n æ Ì àæê Ø ð ãæð â Ìæ ãñ, Áãæ n æð ü Âýæ ë Ì â Øæ ãæðð 18. çù Ù æ Ç æð æ æšø 7.5 ãñð p æ æù ææì èçá Ð x i ŒÍˆž Ÿ f i ¼ÍÁ <¼ÍÁ<³Í p
11 ÂýàÙ â Øæ 19 âð 8 Ì ÂýˆØð ÂýàÙ 3 æ ãñð ¹ ÇU-â 19. çâh ÚUæð çâh ÚUæð secm1 secm1 5 cosecu. secm1 secm1 1 sina 5secA1tanA 1 sina Øæ 0. x 3 3x 1x1 æð ÕãéÂÎ g(x) âð Öæ» ÎðÙð ÂÚU ý àæñ Öæ»È Ü ÌÍæ àæðáè Ü x ß x14 ãñ Ìæð ÕãéÂÎ g(x) ææì èçá Ð 1. ç âè ÂÚðUÇU ð 616 âîsøæð ßæÜè âðùæ ( æ èü) è ÅéU Ç è æð 3 âîsøæð ßæÜð æ èü Õñ ÇU ð ÂèÀðU æ ü ÚUÙæ ãñð ÎæðÙæð â êãæð æð â æù â Øæ ßæÜð SÌ Öæð ð æ ü ÚUÙæ ãñð Ù SÌ Öæð è çï žæ â Øæ Øæ ãñ çáâ ð ßð æ ü ÚU â Ìð ãñ Ð Øæ 306 ÌÍæ 54 æ HCF (.â.â) ÌÍæ LCM (Ü.â.Â) ææì èçá ÌÍæ âˆøæçâì èçá ç HCF3 LCM5ÎæðÙæð â Øæ æð æ»é æùè Ü. æù ææì èçá Ð sin68 cot15 cos 5tan75 3tan45 tan0 tan40 tan50 tan x ÌÍæ y ð çüøð ãü Úð U Ñ x y 5 ; 6 3 x 1 y 51. Øæ a ÌÍæ b ð ç â æù ð çüøð Ùè ð çü¹ð Øé ÚñUç¹ â è ÚU æ æ Ù Ì ãü ãñ? x13y57; a(x1y)b(xy)53a1b. 4. DABC ð æïæúu BC ÂÚU AD Ü Õ ãñ ÌÍæ çõ Îé D â Âý æúu çsíì ãñ ç BD53CD çâh èçá AB 5AC 1BC. 11 P.T.O.
12 5. ç ý ð ÅU ñ ð çù ÙçÜç¹Ì ÕæÚ UÕæÚUÌæ Õ ÅUÙ â âæúu ð Âýçâh ÕËÜðÕæÁæð mæúuæ ÚUÙæð è â Øæ ÎàææüÌè ãñð çù Ù æ Ç æð æ ÕãéÜ ææì ÚUæðÐ Á<ÀÍ ž ÏÉ h Í ¼²Ã ¼Í Í ž ÏÉ h Í Á<ÀÍ ž ÏÉ h Í ¼²Ã ¼Í Í ž ÏÉ h Í æ ë çì - 4 ð ABC â æð æ ç æöéá ãñ çáâ æ C â æð æ ãñð ØçÎ DEA AB Ìæð çâh èçá DABC DADE ÌÍæ ÖéÁæ AE ÌÍæ DE è Ü Õæ ü Öè ææì ÚUæðÐ 7. ÿææ ð 35 Õ ææð ð SßæS Ø Áæ ð â Ø Ù ð ÖæÚU æ çßßúu æ çù Ù Âý æúu çîøæ»øæ ãñð ÌæðÚU æ Ò Âý æúu æó ¹è ç Øð ÌÍæ ŠØ æù æ æù»ýæè mæúuæ æüê èçá Рȼ7É ßÈ¹Ò ÉÁNÈÉ ÞºÈÒ ÊÄ cºè 38 É ž ¾ 0 40 É ž ¾ 3 4 É ž ¾ 5 44 É ž ¾ 9 46 É ž ¾ É ž ¾ 8 50 É ž ¾ 3 5 É ž ¾ çâh èçáøð 3 5 ÂçÚU ðø â Øæ ãñð 1
13 ÂýàÙ â Øæ 9 âð 34 Ì ÂýˆØð ÂýàÙ 4 æ ãñð ¹ ÇU-Î 9. â è ÚU æ x1y55 ; ÌÍæ x3y54 æð»ýæè mæúuæ ãü ÚUæð, ÌÍæ ßã çõ Îé æüê ÚUæð Áãæ ÚðU¹æØð x - ÿæ âð ç ÜÌè ãñð tanm cotm secmcosecm 1cot M 1tanM. 30. çâh èçá ØçÎ çù Ù Õ ÅUÙ è æçšø æ 8.5 ãæð Ìæð x ÌÍæ y æ æù ææì èçá Ð Á Þ m ¼7Ⱦ Í È¼ 7 ȼ7 È 5 x 0 15 y çâh èçá coseca cota sina sina coseca cota Øæ çâh èçá sec u 4 sin Msin M 4 cos McosM ØçÎ ÕãéÂÎ x 4 6x 3 116x 5x110 æð ÕãéÂÎ x x1k âð çßöæçáì Úð U, Ìæð àæðáè Ü x1a æìæ ãñð k ÌÍæ a æ æù ææì ÚUæðÐ 34. çâh Úð U, Îæð â M  ç æöéáæð ð ÿæð æè Üæð æ ÙéÂæÌ Ù è â»ì ÖéÁæ æð ð ÙéÂæÌ ð ß»ü ð ÕÚUæÕÚU ãæðìæ ãñð Øæ ØçÎ ç âè ç æöéá ð ÖéÁæ æ ß»ü àæðá Ø Îæð ÖéÁæ æð ð ß»æðZ ð Øæð» ð ÕÚUæÕÚU ãæð Ìæð çâh ÚUæð ç ÕÇ è ÖéÁæ ð âæ Ùð æ æð æ â æð æ ãæð»æð - o 0 o - 13 P.T.O.
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