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10th maths unsolved_sample_papers_-_3-min (1)

  1. 130/1 P.T.O. ·¤æðÇU Ù´. Code No. 30/1 SET-1Series JSR ÚUæðÜ Ù´. Roll No. ÂÚUèÿææÍèü ·¤æðÇU ·¤æð ©žæÚU-ÂéçSÌ·¤æ ·ð¤ ×é¹-ÂëcÆU ÂÚU ¥ßàØ çܹð´Ð Candidates must write the Code on the title page of the answer-book. · ·ë¤ÂØæ Áæ¡¿ ·¤ÚU Üð´ ç·¤ §â ÂýàÙ-˜æ ×ð´ ×éçÎýÌ ÂëcÆU 15 ãñ´Ð · ÂýàÙ-˜æ ×ð´ ÎæçãÙð ãæÍ ·¤è ¥æðÚU çΰ »° ·¤æðÇU ِÕÚU ·¤æð ÀUæ˜æ ©žæÚU-ÂéçSÌ·¤æ ·ð¤ ×é¹-ÂëcÆU ÂÚU çܹð´Ð · ·ë¤ÂØæ Áæ¡¿ ·¤ÚU Üð´ ç·¤ §â ÂýàÙ-˜æ ×ð´ 31 ÂýàÙ ãñ´Ð · ·ë¤ÂØæ ÂýàÙ ·¤æ ©žæÚU çܹÙæ àæéM¤ ·¤ÚUÙð âð ÂãÜð, ÂýàÙ ·¤æ ·ý¤×æ´·¤ ¥ßàØ çܹð´Ð · §â ÂýàÙ-Â˜æ ·¤æð ÂɸÙð ·ð¤ çÜ° 15 ç×ÙÅU ·¤æ â×Ø çÎØæ »Øæ ãñÐ ÂýàÙ-Â˜æ ·¤æ çßÌÚU‡æ Âêßæüq ×ð´ 10.15 ÕÁð ç·¤Øæ Áæ°»æÐ 10.15 ÕÁð âð 10.30 ÕÁð Ì·¤ ÀUæ˜æ ·ð¤ßÜ ÂýàÙ-Â˜æ ·¤æð Âɸð´»ð ¥æñÚU §â ¥ßçÏ ·ð¤ ÎæñÚUæÙ ßð ©žæÚ-ÂéçSÌ·¤æ ÂÚU ·¤æð§ü ©žæÚU Ùãè´ çܹð´»ðÐ · Please check that this question paper contains 15 printed pages. · Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. · Please check that this question paper contains 31 questions. · Please write down the Serial Number of the question before attempting it. · 15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. â´·¤çÜÌ ÂÚUèÿææ - II SUMMATIVE ASSESSMENT - II »ç‡æÌ MATHEMATICS çÙÏæüçÚUÌ â×Ø Ñ 3 ƒæ‡ÅðU ¥çÏ·¤Ì× ¥´·¤ Ñ 90 Time allowed : 3 hours Maximum Marks : 90
  2. 230/1 âæ×æ‹Ø çÙÎðüàæ Ñ (i) âÖè ÂýàÙ ¥çÙßæØü ãñ´Ð (ii) §â ÂýàÙ-˜æ ×ð´ 31 ÂýàÙ ãñ´ Áæð ¿æÚU ¹‡ÇUæð´ - ¥, Õ, â ¥æñÚU Î ×ð´ çßÖæçÁÌ ãñ´Ð (iii) ¹‡ÇU ¥ ×ð´ °·¤-°·¤ ¥´·¤ ßæÜð 4 ÂýàÙ ãñ´Ð ¹‡ÇU Õ ×ð´ 6 ÂýàÙ ãñ´ çÁâ×ð´ âð ÂýˆØð·¤ 2 ¥´·¤ ·¤æ ãñÐ ¹‡ÇU â ×ð´ 10 ÂýàÙ ÌèÙ-ÌèÙ ¥´·¤æð´ ·ð¤ ãñ´Ð ¹‡ÇU Î ×ð´ 11 ÂýàÙ ãñ´ çÁÙ×ð´ âð ÂýˆØð·¤ 4 ¥´·¤ ·¤æ ãñÐ (iv) ·ñ¤Ü·é¤ÜðÅUÚU ·¤æ ÂýØæð» ßçÁüÌ ãñÐ General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections – A, B, C and D. (iii) Section A contains 4 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each. (iv) Use of calculators is not permitted.
  3. 330/1 P.T.O. ¹‡ÇU - ¥ SECTION - A Âýà٠ⴁØæ 1 âð 4 Ì·¤ ÂýˆØð·¤ ÂýàÙ 1 ¥´·¤ ·¤æ ãñÐ Question numbers 1 to 4 carry 1 mark each. 1. ¥æ·ë¤çÌ 1 ×ð´ O ·ð¤‹Îý ßæÜð ßëžæ ·ð¤ çÕ´Îé C ÂÚU PQ °·¤ SÂàæü ÚðU¹æ ãñÐ ØçÎ AB °·¤ ÃØæâ ãñ ÌÍæ ÐCAB5308 ãñ, Ìæð ÐPCA ™ææÌ ·¤èçÁ°Ð ¥æ·ë¤çÌ 1 In fig.1, PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and ÐCAB5308, find ÐPCA. Figure 1 2. k ·ð¤ ç·¤â ×æÙ ·ð¤ çÜ° k19, 2k21 ÌÍæ 2k17 °·¤ â×æ´ÌÚU ŸæðÉ¸è ·ð¤ ·ý¤×æ»Ì ÂÎ ãñ´? For what value of k will k19, 2k21 and 2k17 are the consecutive terms of an A.P. ? 3. °·¤ ÎèßæÚU ·ð¤ âæÍ Ü»è âèɸè ÿæñçÌÁ ·ð¤ âæÍ 608 ·¤æ ·¤æð‡æ ÕÙæÌè ãñÐ ØçÎ âèÉ¸è ·¤æ ÂæÎ ÎèßæÚU âð 2.5 ×è. ·¤è ÎêÚUè ÂÚU ãñ, Ìæð âèÉ¸è ·¤è ܐÕæ§ü ™ææÌ ·¤èçÁ°Ð A ladder, leaning against a wall, makes an angle of 608 with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.
  4. 430/1 4. 52 žææð´ ·¤è ¥‘ÀUè Âý·¤æÚU Èð´¤ÅUè »§ü Ìæàæ ·¤è »aè ×ð´ âð ØæÎë‘ÀUØæ °·¤ žææ çÙ·¤æÜæ »ØæÐ ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° ç·¤ çÙ·¤æÜæ »Øæ žææ Ù Ìæð ÜæÜ Ú´U» ·¤æ ãñ ¥æñÚU Ù ãè °·¤ Õð$»× ãñÐ A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting neither a red card nor a queen. ¹‡ÇU - Õ SECTION - B Âýà٠ⴁØæ 5 âð 10 Ì·¤ ÂýˆØð·¤ ÂýàÙ 2 ¥´·¤ ·¤æ ãñÐ Question numbers 5 to 10 carry 2 marks each. 5. ØçÎ çmƒææÌè â×è·¤ÚU‡æ 2x21px21550 ·¤æ °·¤ ×êÜ 25 ãñ ÌÍæ çmƒææÌè â×è·¤ÚU‡æ p(x21x)1k50 ·ð¤ ×êÜ â×æÙ ãñ´, Ìæð k ·¤æ ×æÙ ™ææÌ ·¤èçÁ°Ð If 25 is a root of the quadratic equation 2x21px21550 and the quadratic equation p(x21x)1k50 has equal roots, find the value of k. 6. ×æÙæ P ÌÍæ Q, A(2, 22) ÌÍæ B(27, 4) ·¤æð ç×ÜæÙð ßæÜð ÚðU¹æ¹´ÇU ·¤æð §â Âý·¤æÚU â×ç˜æÖæçÁÌ ·¤ÚUÌð ãñ´ ç·¤ P, çÕ´Îé A ·ð¤ Âæâ ãñÐ P ÌÍæ Q ·ð¤ çÙÎðüàææ´·¤ ™ææÌ ·¤èçÁ°Ð Let P and Q be the points of trisection of the line segment joining the points A(2, 22) and B(27, 4) such that P is nearer to A. Find the coordinates of P and Q. 7. ¥æ·ë¤çÌ 2 ×ð´ °·¤ ¿ÌéÖüéÁ ABCD, O ·ð´¤Îý ßæÜð ßëžæ ·ð¤ ÂçÚU»Ì §â Âý·¤æÚU ÕÙæ§ü »§ü ãñ ç·¤ ÖéÁæ°¡ AB, BC, CD ÌÍæ DA ßëžæ ·¤æð ·ý¤×àæÑ çÕ´Îé¥æð´ P, Q, R ÌÍæ S ÂÚU SÂàæü ·¤ÚUÌè ãñ´Ð çâh ·¤èçÁ° ç·¤ AB1CD5BC1DA Ð ¥æ·ë¤çÌ 2
  5. 530/1 P.T.O. In Fig.2, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that. AB1CD5BC1DA. Figure 2 8. çâh ·¤èçÁ° ç·¤ çÕ´Îé (3, 0), (6, 4) ÌÍæ (21, 3) °·¤ â×çmÕæãé â×·¤æð‡æ ç˜æÖéÁ ·ð¤ àæèáü ãñ´Ð Prove that the points (3, 0), (6, 4) and (21, 3) are the vertices of a right angled isosceles triangle. 9. °·¤ â×æ´ÌÚU ŸæðÉ¸è ·¤æ ¿æñÍæ ÂÎ àæê‹Ø ãñÐ çâh ·¤èçÁ° ç·¤ §â·¤æ 25 ßæ´ ÂÎ, ©â·ð¤ 11 ßð´ ÂÎ ·¤æ ÌèÙ »éÙæ ãñÐ The 4th term of an A.P. is zero. Prove that the 25th term of the A.P. is three times its 11th term. 10. ¥æ·ë¤çÌ 3 ×ð´ °·¤ Õæs çÕ´Îé P âð, O ·ð¤‹Îý ÌÍæ r ç˜æ’Øæ ßæÜð ßëžæ ÂÚU Îæð SÂàæü ÚðU¹æ°¡ PT ÌÍæ PS ¹è´¿è »§ü ãñ´Ð ØçÎ OP52r ãñ, Ìæð Îàææü§° ç·¤ ÐOTS5ÐOST5308Ð ¥æ·ë¤çÌ 3
  6. 630/1 In Fig. 3, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP52r, show that ÐOTS5ÐOST5308. Figure 3 ¹‡ÇU - â SECTION - C Âýà٠ⴁØæ 11 âð 20 Ì·¤ ÂýˆØð·¤ ÂýàÙ ·ð¤ 3 ¥´·¤ ãñ´Ð Question numbers 11 to 20 carry 3 marks each. 11. ¥æ·ë¤çÌ 4 ×ð´ O ·ð¤‹Îý ßæÜð ßëžæ ·¤æ ÃØæâ AB513 âð×è ãñ ÌÍæ AC512 âð×è ãñÐ BC ·¤æð ç×ÜæØæ »Øæ ãñÐ ÀUæØæ´ç·¤Ì ÿæð˜æ ·¤æ ÿæð˜æÈ¤Ü ™ææÌ ·¤èçÁ°Ð ( p53.14 ÜèçÁ° ) ¥æ·ë¤çÌ 4 In fig.4, O is the centre of a circle such that diameter AB513 cm and AC512 cm. BC is joined. Find the area of the shaded region. (Take p53.14) Figure 4
  7. 730/1 P.T.O. 12. ¥æ·ë¤çÌ 5 ×ð´ °·¤ Åñ´UÅU ÕðÜÙ ·ð¤ ª¤ÂÚU Ü»ð ©âè ÃØæâ ßæÜð àæ´·é¤ ·ð¤ ¥æ·¤æÚU ·¤æ ãñÐ ÕðÜÙæ·¤æÚU Öæ» ·¤è ª¡¤¿æ§ü ÌÍæ ÃØæâ ·ý¤×àæÑ 2.1 ×è. ÌÍæ 3 ×è. ãñ´ ÌÍæ àæ´€Ãææ·¤æÚU Öæ» ·¤è çÌÚUÀUè ª¡¤¿æ§ü 2.8 ×è. ãñÐ Åñ´UÅU ·¤æð ÕÙæÙð ×ð´ Ü»ð ·ñ¤Ùßæâ ·¤æ ×êËØ ™ææÌ ·¤èçÁ°, ØçÎ ·ñ¤Ùßæâ ·¤æ Öæß ` 500 ÂýçÌ ß»ü ×è ãñÐ (p5 22 7 ÜèçÁ° ) ¥æ·ë¤çÌ 5 In fig. 5, a tent is in the shape of a cylinder surmounted by a conical top of same diameter. If the height and diameter of cylindrical part are 2.1 m and 3 m respectively and the slant height of conical part is 2.8 m, find the cost of canvas needed to make the tent if the canvas is available at the rate of ` 500/sq.metre. (Use p5 22 7 ) Figure 5
  8. 830/1 13. ØçÎ çÕ‹Îé P(x, y) çÕ´Îé¥æð´ A(a1b, b2a) ÌÍæ B(a2b, a1b) âð â×ÎêÚUSÍ ãñ, Ìæð çâh ·¤èçÁ° ç·¤ bx5ay. If the point P(x, y) is equidistant from the points A(a1b, b2a) and B(a2b, a1b). Prove that bx5ay. 14. ¥æ·ë¤çÌ 6 ×ð´, Îæð â·ð¤‹ÎýèØ ßëžææð´, çÁâ·¤è ç˜æ’Øæ°¡ 7 âð×è ÌÍæ 14 âð×è ãñ´, ·ð¤ Õè¿ çƒæÚðU ÀUæØæ´ç·¤Ì ÿæð˜æ ·¤æ ÿæð˜æÈ¤Ü ™ææÌ ·¤èçÁ° ÁÕç·¤ ÐAOC5408 ãñÐ (p5 22 7 ÜèçÁ°) ¥æ·ë¤çÌ 6 In fig. 6, find the area of the shaded region, enclosed between two concentric circles of radii 7 cm and 14 cm where ÐAOC5408. (Use p5 22 7 ) Figure 6 15. ØçÎ Îæð â×æ´ÌÚU ŸæðçɸØæð´ ·ð¤ ÂýÍ× n ÂÎæð´ ·ð¤ Øæð»æð´ ×ð´ (7n11) : (4n127) ·¤æ ¥ÙéÂæÌ ãñ, Ìæð ©Ù·ð¤ m ßð´ ÂÎæð´ ×ð´ ¥ÙéÂæÌ ™ææÌ ·¤èçÁ°Ð If the ratio of the sum of first n terms of two A.P’s is (7n11) : (4n127), find the ratio of their mth terms.
  9. 930/1 P.T.O. 16. x ·ð¤ çÜ° ãÜ ·¤èçÁ° Ñ ( )( ) ( )( ) 2 1,2,3 1 2 2 3 1 1 ≠5 2 2 2 2 1 , 3 x x x x x Solve for x : ( )( ) ( )( ) 2 1,2,3 1 2 2 3 1 1 ≠5 2 2 2 2 1 , 3 x x x x x 17. °·¤ àæ´€Ãææ·¤æÚU ÕÌüÙ, çÁâ·ð¤ ¥æÏæÚU ·¤è ç˜æ’Øæ 5 âð×è ÌÍæ ª¡¤¿æ§ü 24 âð×è ãñ, ÂæÙè âð ÂêÚUæ ÖÚUæ ãñÐ ©â ÂæÙè ·¤æð °·¤ ÕðÜÙæ·¤æÚU ÕÌüÙ, çÁâ·¤è ç˜æ’Øæ 10 âð×è ãñ, ×ð´ ÇUæÜ çÎØæ ÁæÌæ ãñÐ ÕðÜÙæ·¤æÚU ÕÌüÙ ×ð´ ç·¤ÌÙè ª¡¤¿æ§ü Ì·¤ ÂæÙè ÖÚU ÁæØð»æ? ( p5 22 7 ÜèçÁ° ) A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel. (Use p5 22 7 ) 18. 12 âð×è ÃØæâ ßæÜæ °·¤ »æðÜæ, °·¤ Ü´Õ ßëžæèØ ÕðÜÙæ·¤æÚU ÕÌüÙ ×ð´ ÇUæÜ çÎØæ ÁæÌæ ãñ, çÁâ×ð´ ·é¤ÀU ÂæÙè ÖÚUæ ãñÐ ØçÎ »æðÜæ Âê‡æüÌØæ ÂæÙè ×ð´ ÇêÕ ÁæÌæ ãñ, Ìæð ÕðÜÙæ·¤æÚU ÕÌüÙ ×ð´ ÂæÙè ·¤æ SÌÚU 5 3 9 âð×è ª¡¤¿æ ©ÆU ÁæÌæ ãñÐ ÕðÜÙæ·¤æÚU ÕÌüÙ ·¤æ ÃØæâ ™ææÌ ·¤èçÁ°Ð A sphere of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by 5 3 9 cm. Find the diameter of the cylindrical vessel. 19. °·¤ ÃØç€ˆæ °·¤ ÁÜØæÙ ·ð¤ ÇñU·¤, Áæð ÂæÙè ·ð¤ SÌÚU âð 10 ×è. ª¡¤¿æ ãñ, âð °·¤ ÂãæÇ¸è ·ð¤ çàæ¹ÚU ·¤æ ©óæØÙ ·¤æð‡æ 608 ÌÍæ ÂãæÇ¸è ·ð¤ ÌÜ ·¤æ ¥ßÙ×Ù ·¤æð‡æ 308 ÂæÌæ ãñÐ ÂãæǸè âð ÁÜØæÙ ·¤è ÎêÚUè ÌÍæ ÂãæÇ¸è ·¤è ª¡¤¿æ§ü ™ææÌ ·¤èçÁ°Ð A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 608 and the angle of depression of the base of hill as 308. Find the distance of the hill from the ship and the height of the hill.
  10. 1030/1 20. ÌèÙ çßçÖóæ ç‷𤠰·¤ âæÍ ©ÀUæÜð »°Ð çِ٠·ð¤ ¥æÙð ·¤è ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° (i) ·ð¤ßÜ 2 ç¿žæ (ii) ·¤× âð ·¤× Îæð ç¿žæ (iii) ·¤× âð ·¤× Îæð ÂÅUÐ Three different coins are tossed together. Find the probability of getting (i) exactly two heads (ii) at least two heads (iii) at least two tails. ¹‡ÇU - Î SECTION - D Âýà٠ⴁØæ 21 âð 31 Ì·¤ ÂýˆØð·¤ ÂýàÙ 4 ¥´·¤ ·¤æ ãñÐ Question numbers 21 to 31 carry 4 marks each. 21. ç·¤âè ÚUæ’Ø ×ð´ ÖæÚUè Õæɸ ·ð¤ ·¤æÚU‡æ ãÁæÚUæð´ Üæð» ÕðƒæÚU ãæð »°Ð 50 çßlæÜØæð´ Ùð ç×Ü·¤ÚU ÚUæ’Ø âÚU·¤æÚU ·¤æð 1500 Åñ´ÅU Ü»æÙð ·ð¤ çÜ° SÍæÙ ÌÍæ ·ñ¤Ùßâ ÎðÙð ·¤æ ÂýSÌæß ç·¤Øæ çÁâ×ð´ ÂýˆØð·¤ çßlæÜØ ÕÚUæÕÚU ·¤æ ¥´àæÎæÙ Îð»æÐ ÂýˆØð·¤ Åñ´ÅU ·¤æ çÙ¿Üæ Öæ» ÕðÜÙæ·¤æÚU ãñ, çÁâ·ð¤ ¥æÏæÚU ·¤è ç˜æ’Øæ 2.8 ×è. ÌÍæ ª¡¤¿æ§ü 3.5 ×è. ãñÐ ÂýˆØð·¤ Åñ´UÅU ·¤æ ª¤ÂÚUè Öæ» àæ´·é¤ ·ð¤ ¥æ·¤æÚU ·¤æ ãñ çÁâ·ð¤ ¥æÏæÚU ·¤è ç˜æ’Øæ 2.8 ×è. ÌÍæ ª¡¤¿æ§ü 2.1 ×è. ãñÐ ØçÎ ÅñU´ÅU ÕÙæÙð ßæÜð ·ñ¤Ùßæâ ·¤æ ×êËØ ` 120 ÂýçÌ ß»ü ×è. ãñ, Ìæð ÂýˆØð·¤ çßlæÜØ mæÚUæ ·é¤Ü ÃØØ ×ð´ ¥´àæÎæÙ ™ææÌ ·¤èçÁ°Ð §â ÂýàÙ mæÚUæ ·¤æñÙ âæ ×êËØ ÁçÙÌ ãæðÌæ ãñ? ( p5 22 7 ÜèçÁ° ) Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively offered to the state government to provide place and the canvas for 1500 tents to be fixed by the government and decided to share the whole expenditure equally. The lower part of each tent is cylindrical of base radius 2.8 m and height 3.5 m, with conical upper part of same base radius but of height 2.1 m. If the canvas used to make the tents costs ` 120 per sq.m, find the amount shared by each school to set up the tents. What value is generated by the above problem ? (Use p5 22 7 )
  11. 1130/1 P.T.O. 22. çâh ·¤èçÁ° ç·¤ ç·¤âè Õæs çÕ´Îé âð ßëžæ ÂÚU ¹è´¿è »§ü SÂàæü ÚðU¹æ°¡ ÜÕæ´§ü ×ð´ â×æÙ ãæðÌè ãñÐ Prove that the lengths of the tangents drawn from an external point to a circle are equal. 23. 4 âð×è ç˜æ’Øæ ·¤æ °·¤ ßëžæ ¹è´ç¿°Ð ©â ßëžæ ÂÚU Îæð SÂàæü ÚðU¹æ°¡ ¹è´ç¿° çÁÙ·ð¤ Õè¿ ·¤æ ·¤æð‡æ 608 ãñÐ Draw a circle of radius 4 cm. Draw two tangents to the circle inclined at an angle of 608 to each other. 24. ¥æ·ë¤çÌ 7 ×ð´ Îæð â×æÙ ç˜æ’Øæ ·ð¤ ßëžæ, çÁٷ𤠷ð¤‹Îý O ÌÍæ O' ãñ´ ÂÚUSÂÚU çÕ´Îé X ÂÚU SÂàæü ·¤ÚUÌð ãñ´Ð OO' ÕɸæÙð ÂÚU O' ·ð¤‹Îý ßæÜð ßëžæ ·¤æð çÕ´Îé A ÂÚU ·¤æÅUÌæ ãñÐ çÕ´Îé A âð O ·ð¤‹Îý ßæÜð ßëžæ ÂÚU AC °·¤ SÂàæü ÚðU¹æ ãñ ÌÍæ O'D ^ AC ãñÐ DO' CO ·¤æ ×æÙ ™ææÌ ·¤èçÁ°Ð ¥æ·ë¤çÌ 7 In Fig. 7, two equal circles, with centres O and O', touch each other at X.OO' produced meets the circle with centre O' at A. AC is tangent to the circle with centre O, at the point C. O'D is perpendicular to AC. Find the value of DO' CO . figure 7
  12. 1230/1 25. x ·ð¤ çÜ° ãÜ ·¤èçÁ° Ñ , x x x x ≠1 5 2 2 2 1 1 1 1 2 4 1, 2, 4 1 2 4 Solve for x : , x x x x ≠1 5 2 2 2 1 1 1 1 2 4 1, 2, 4 1 2 4 26. Öêç× ·ð¤ °·¤ çÕ´Îé X âð °·¤ ª¤ŠßæüÏÚU ×èÙæÚU PQ ·ð¤ çàæ¹ÚU Q ·¤æ ©óæØÙ ·¤æð‡æ 608 ãñÐ °·¤ ¥‹Ø çÕ´Îé Y, Áæð çÕ´Îé X âð 40 ×è. ª¤ŠßæüÏÚU M¤Â ×ð´ ª¡¤¿æ ãñ, âð çàæ¹Ú Q ·¤æ ©óæØÙ ·¤æð‡æ 458 ãñ ×èÙæÚU PQ ·¤è ª¡¤¿æ§ü ÌÍæ ÎêÚUè PX ™ææÌ ·¤èçÁ° ( 3 51.73 ÜèçÁ° ). The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 608. From a point Y, 40 m vertically above X, the angle of elevation of the top Q of tower is 458. Find the height of the tower PQ and the distance PX. ( Use 3 51.73 ) 27. °·¤ âèÏè ÚðU¹æ ×ð´ çSÍÌ ƒæÚUæð´ ÂÚU 1 âð 49 Ì·¤ ·¤è ⴁØæ°¡ (·ý¤×æÙéâæÚU) ¥´ç·¤Ì ãñ´Ð Îàææü§° ç·¤ §Ù ¥´ç·¤Ì ⴁØæ¥æð´ ×ð´ °·¤ °ðâè ⴁØæ X ¥ßàØ ãñ ç·¤ X âð ÂãÜð ¥æÙð ßæÜð ƒæÚUæð´ ÂÚU ·¤è ¥´ç·¤Ì ⴁØæ¥æð´ ·¤æ Øæð», X ·ð¤ ÕæÎ ¥æÙðßæÜè ¥´ç·¤Ì ⴁØæ¥æð´ ·ð¤ Øæð» ·ð¤ ÕÚUæÕÚU ãñÐ The houses in a row are numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses proceeding the house numbered X is equal to sum of the numbers of houses following X.
  13. 1330/1 P.T.O. 28. ¥æ·ë¤çÌ 8 ×ð´ °·¤ ç˜æÖéÁ ABC ·ð¤ àæèáü A(4, 6), B(1, 5) ÌÍæ C(7, 2) ãñÐ °·¤ ÚðU¹æ¹´ÇU DE ÖéÁæ¥æð´ AB ÌÍæ AC ·¤æð ·ý¤×àæÑ çÕ´Îé¥æð´ D ÌÍæ E ÂÚU §â Âý·¤æÚU ·¤æÅUÌæ ¹è´¿æ »Øæ ãñ ç·¤ 5 5 AD AE 1 AB AC 3 ãñÐ DADE ·¤æ ÿæð˜æÈ¤Ü ™ææÌ ·¤èçÁ° ÌÍæ ©â·¤è DABC ·ð¤ ÿæð˜æÈ¤Ü âð ÌéÜÙæ ·¤èçÁ°Ð ¥æ·ë¤çÌ 8 In fig. 8, the vertices of DABC are A(4, 6), B(1, 5) and C(7, 2). A line-segment DE is drawn to intersect the sides AB and AC at D and E respectively such that 5 5 AD AE 1 AB AC 3 . Calculate the area of DADE and compare it with area of DABC. Figure 8
  14. 1430/1 29. ⴁØæ¥æð´ 1, 2, 3 ÌÍæ 4 ×ð´ âð ·¤æð§ü ⴁØæ x ØæÎë‘ÀUØæ ¿éÙè »§ü ÌÍæ ⴁØæ¥æð´ 1, 4, 9 ÌÍæ 16 ×ð´ âð ·¤æð§ü ⴁØæ y ØæÎë‘ÀUØæ ¿éÙè »§ü ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° x ÌÍæ y ·¤æ »é‡æÙÈ¤Ü 16 âð ·¤× ãñÐ A number x is selected at random from the numbers 1, 2, 3 and 4. Another number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability that product of x and y is less than 16. 30. ¥æ·ë¤çÌ 9 ×ð´, O ·ð´¤Îý ßæÜð ßëžæ ·¤æ °·¤ ç˜æ’ع´ÇU OAP ÎàææüØæ »Øæ ãñ çÁâ·¤æ ·ð¤‹Îý ÂÚU ¥Ì´çÚUÌ ·¤æð‡æ u ãñÐ AB ßëžæ ·¤è ç˜æ’Øæ OA ÂÚU Ü´Õ ãñ Áæð OP ·ð¤ ÕɸæÙð ÂÚU çÕ´Îé B ÂÚU ·¤æÅUÌæ ãñÐ çâh ·¤èçÁ° ç·¤ ÚðU¹æ´ç·¤Ì Öæ» ·¤æ ÂçÚU×æ r      pu u 1 u 1 2tan sec 1 180 ãñÐ ¥æ·ë¤çÌ 9 In Fig. 9, is shown a sector OAP of a circle with centre O, containing ∠ u. AB is perpendicular to the radius OA and meets OP produced at B. Prove that the perimeter of shaded region is r      pu u 1 u 1 2tan sec 1 180 Figure 9
  15. 1530/1 P.T.O. 31. °·¤ ×æðÅÚU ÕæðÅU, çÁâ·¤è çSÍÚU ÁÜ ×ð´ ¿æÜ 24 ç·¤×è/ƒæ´ÅUæ ãñ, ÏæÚUæ ·ð¤ ÂýçÌ·ê¤Ü 32 ç·¤×è ÁæÙð ×ð´, ßãè ÎêÚUè ÏæÚUæ ·ð¤ ¥Ùé·ê¤Ü ÁæÙð ·¤è ¥Âðÿææ 1 ƒæ´ÅUæ ¥çÏ·¤ â×Ø ÜðÌè ãñÐ ÏæÚUæ ·¤è ¿æÜ ™ææÌ ·¤èçÁ°Ð A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32 km upstream than to return downstream to the same spot. Find the speed of the stream.
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