041/IX/SA2/06/B1 Class - IX MATHEMATICS Time : 3 to 3½ hours â×Ø : 3 âð 3½ æÅUð Maximum Marks : 80 ¥çÏ·¤Ì× ¥´·¤ : 80 Total No. of Pages : 13 ·é¤Ü ÂëcÆUæð´ ·¤è â´Øæ : 13 General Instructions : 1. All questions are compulsory. 2. 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 2 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, 4 questions of three marks each and 2 questions of four marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. 6. An additional 15 minutes time has been allotted to read this question paper only. âæ×æØ çÙÎðüàæ Ñ 1. âÖè ÂýàÙ ¥çÙßæØü ãñ´Ð 2. §â ÂýàÙ-Âæ ×ð´ 34 ÂýàÙ ãñ´, Áæð ¿æÚU ¹ÇUæð´ ×ð´ ¥, Õ, â ß Î ×ð´ çßÖæçÁÌ ãñÐ ¹ÇU - ¥ ×ð´ 10 ÂýàÙ ãñ´ ¥æñÚU ÂýØð·¤ ¹ÇU - Õ ×ð´ 8 ÂýàÙ ãñ´ ¥æñÚU ÂýØð·¤ ÂýàÙ 2 ¥´·¤æð´ ·ð¤ ãñ´, ¹ÇU - â ×ð´ 10 ÂýàÙ ãñ´ ¥æñÚU ÂýØð·¤ ÂýàÙ 3 ¥´·¤æð´ ·¤æ ãñ, ¹ÇU - Î ×ð´ 6 ÂýàÙ ãñ´ ¥æñÚU ÂýØð·¤ ÂýàÙ 4 ¥´·¤æð´ ·¤æ ãñÐ ÂýàÙ 1 ¥´·¤ ·¤æ ãñ, 3. ÂýàÙ â´Øæ 1 âð 10 Õãéçß·¤ËÂèØ ÂýàÙ ãñ´Ð çΰ »° ¿æÚU çß·¤ËÂæð´ ×ð´ âð °·¤ âãè çß·¤Ë 4. §â×ð´ ·¤æð§ü Öè âßæðüÂçÚU çß·¤Ë Ùãè´ ãñ, Üðç·¤Ù ¥æ´ÌçÚU·¤ çß·¤Ë 1 ÂýàÙ 2 ¥´·¤æð´ ×ð´, 4 ÂýàÙ 3 ¥´·¤æð´ ×ð´ ¥æñÚU 2 ÂýàÙ ¿éÙð´Ð 4 ¥´·¤æð´ ×ð´ çΰ »° ãñ´Ð ¥æ çΰ »° çß·¤ËÂæð´ ×ð´ âð °·¤ çß·¤Ë ·¤æ ¿ØÙ ·¤Úð´UÐ ßçÁüÌ ãñÐ 5. ·ñ¤Ü·é¤ÜðÅUÚU ·¤æ ÂýØæð» 6. §â ÂýàÙ-Âæ ·¤æð ÂɸÙð ·ð¤ çÜ° 15 ç×ÙÅU ·¤æ â×Ø çÎØæ »Øæ ãñÐ ¥æñÚU ßð ©æÚU-ÂéçSÌ·¤æ ÂÚU ·¤æð§ü ©æÚU Ùãè´ çܹð´»ðÐ 1 §â ¥ßçÏ ·ð¤ ÎæñÚUæÙ ÀUææ ·ð¤ßÜ ÂýàÙ-Âæ ·¤æð Âɸð´»ð SECTION - A 1. Question numbers 1 to 10 carry 1 mark each. The number of cubes of 3 cm side which can be cut out of a cuboid of dimensions l518 cm, b515 cm and h52 cm is. (A) 180 (B) 60 (C) 20 (D) Cant be cut 2. The linear equation 2y2350, represented as ax1by1c50, has. (A) a unique solution (B) infinitely many solutions (C) two solutions (D) no solution 3. If P (E) denotes the probability of an event E, then : (A) P(E) < 0 (B) P(E) > 1 (C) 0[P(E)[1 (D) 21[P(E)[1 4. The total surface area of a cone whose radius is 2 r and slant height (A) 5. 2pr (l1r) (B) 2pr ( l 1r) 2 (C) 2pr (l14r) l is : 2 (D) pr (l14r) The graph of the equation ax1by1c50 may be of the form : (A) (B) (C) (D) 6. Given three collinear points, then the number of circles which can be drawn through three points is : (A) Zero (B) One (C) Two (D) Infinite 7. In the given figure, ABCD is a parallelogram. If ∠ B 5 1008 , then (∠ A 1 ∠ C) is equal to : (A) 3608 041/IX/SA2/06/B1 (B) 2008 (C) 2 1808 (D) 1608 8. Any point on the line x1y50 is of the form : (A) 9. (B) (a, a) (C) (0, a) (D) (a, 0) In a class, there are x girls and y boys, a student is selected at random, then the probability of selecting a boy is : (A) 10. (2a, a) x/y (B) x/(x1y) (C) y/(x1y) (D) y/x A right circular cylinder just encloses a sphere of radius a cm, then the curved surface area of the cylinder is : (A) 4 p a2 cm (B) 2 p a2 cm2 (C) 3 p a2 cm (D) 4 p a2 cm2 SECTION - B Question number 11 to 18 carry 2 marks each. 11. A die is thrown 500 times, the frequency of outcomes 1, 2, 3, 4, 5 and 6 are noted in the following frequency distribution table : Outcome Frequency 1 90 2 70 3 75 4 95 5 88 6 82 Find the probability of occurrence of a prime number. 12. 13. Give one example of a situation in which (i) the mean is an appropriate measure of central tendency (ii) the mean is not an appropriate measure of central tendency but the median is an appropriate measure of central tendency Prove that equal chords of a circle subtend equal angles at the centre. OR Suppose you are given a circle. Give a construction to find its centre. 14. Find a value of k so that x521 and y521 is a solution of the linear equation 9kx112ky563 15. Give the equation of one line passing through (2, 14). How many more such lines are there and why ? 041/IX/SA2/06/B1 3 16. P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD. Show that ar (APB)5ar (BQC). 17. In the given figure, PQRS is a parallelogram and line segments PA and RB bisect the angles P and R respectively. Show that PA??RB. 18. Savitri had to make a model of a cylindrical kaleidoscope for her science project. She wanted to use chart paper to make the curved surface of the kaleidoscope. What would be the area of chart paper required by her, if she wanted to make a kaleidoscope of length 25 cm with a 3.5 cm radius ? SECTION - C Question number 19 to 28 carry 3 marks each. 19. If the diagonals of a parallelogram are equal, then show that it is a rectangle. 20. The following frequency distribution table gives the weights of 38 students of a class Weight in kg Number of students 30-35 10 35-40 5 40-45 15 45-50 5 50-55 1 55-60 2 Total 38 Find the probability that the weight of a student is (i) more than or equal to 45 kg (ii) less than 30 kg (iii) more than or equal to 30 kg but less than 60 kg OR 041/IX/SA2/06/B1 4 The following table shows the performance of two sections of students in a mathematics test of 100 marks : Marks Number of students 0-20 7 20-30 10 30-40 10 40-50 20 50-60 20 60-70 15 70 above 8 Total 90 Find (i) (ii) (iii) the probability that a student obtained less than 20% in the mathematics test. marks 60 or above marks more than or equal to 40 but less than 60 21. In the figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB5OD. If AB5CD then show that ar (DDOC) 5 ar (DAOB) 22. In the figure, AP ?? BQ ?? CR. Prove that ar (AQC) 5 ar (PBR) OR In the figure ar (DRC) 5 ar (DPC) and ar (BDP) 5 ar (ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums. 041/IX/SA2/06/B1 5 23. Find the mean for the weekly pocket money (in rupees) using the following data. Pocket money (in Rs.) Frequency 55 8 50 3 49 10 81 7 48 3 57 7 65 2 24. Express the linear equation 253x in the form ax1by1c50 and indicate the values of a, b and c. Also give the geometrical representation of above equation in two variables. 25. ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C . Show that (i) ABCD is a square. (ii) Diagonal BD bisects ∠ B as well as ∠ D 26. In the given figure, AB is a diameter of the circle ; CD is a chord equal to the radius of the circle. AC and BD when extended intersect at a point E. Prove that ∠ AEB 5 608 . 27. A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained. 28. The diameter of a metallic ball is 21 cm. What is the mass of the ball, if the density of the metal is 5 gm per cm3 ? OR The pillars of a temple are cylindrically shaped. If each pillar has a circular base of radius 20 cm and height 10 m, how much concrete mixture would be required to build 14 such pillars ? SECTION - D Question number 29 to 34 carry 4 marks each. 29. The taxi fare in a city is as follows : For the first kilometer, the fare is Rs. 8 and for the subsequent distance it is Rs. 5 per km. Taking the distance covered as x km and total fare as Rs. y, write a linear equation for this information, and draw its graph. OR If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance travelled by the body is 2 units. 041/IX/SA2/06/B1 6 30. If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal. 31. A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 17.6 m, find the cost of painting it, given the cost of painting is Rs. 5 per 100 cm2. 32. Prove that the parallelograms on the same base and between the same parallels are equal in area. 33. Construct a triangle with perimeter 10 cm and base angles 608 and 458. OR Construct a triangle ABC in which BC57.5 cm, ∠ B 5 458 and AC2AB52.5 cm. 34. The following table presents the number of literate females in a town : Age group Number of females 10-15 300 15-20 980 20-25 800 25-30 580 Draw a frequency polygon for the above data. -o0o- 041/IX/SA2/06/B1 7 30-35 290 35-40 50 ¹ÇU - ¥ 1. ÂýàÙ â´Øæ 1 âð 10 Ì·¤ ÂýØð·¤ ÂýàÙ 1 ¥´·¤ ·¤æ ãñÐ 3 âð×è ÖéÁæ ßæÜð æÙæð´ ·¤è â´Øæ çÁÙ·¤æð °·¤ æÙæÖ çÁâ·¤è çß×æ°´ âð×è ãñ´, ×ð´ âð ·¤æÅUæ Áæ â·¤Ìæ ãñÐ (A) (C) 2. 3. ÚñUç¹·¤ â×è·¤ÚUæ 6. 7. çÁâð ax1by1c50 60 ·¤æÅUæ ÁæÙæ â´Öß Ùãè´ ãñ ·ð¤ M¤Â ×ð´ ÃØÌ ç·¤Øæ »Øæ ãñ, ·ð¤ çÜ° Ñ ¥çmÌèØ ãÜ ãñ´ (B) ¥ÂçÚUç×Ì M¤Â âð ¥Ùð·¤ ãÜ ãñ´ (C) Îæð ãÜ ãñ´ (D) ·¤æð§ü ãÜ Ùãè´ ãñ ØçÎ ç·¤âè æÅUÙæ E P (E) ãñ, ÌÕ (B) P(E) > 1 ·¤è ÂýæçØ·¤Ìæ P(E) < 0 °·¤ àæ´·é¤, çÁâ·¤è çæØæ 2 r ß çÌØü·¤ ª¡¤¿æ§ü (A) 5. 2y2350, (B) (D) âð×è, b515 âð×è ¥æñÚU h52 (A) (A) 4. 180 20 l518 2pr (l1r) (B) 2pr ( Ñ (C) 0[P(E)[1 (D) 21[P(E)[1 l ãñ, ·¤æ âÂêæü ÂëDèØ ÿæðæÈ¤Ü ãñ Ñ 2 l 1r) 2 (C) 2pr (l14r) (D) pr (l14r) â×è·¤ÚUæ ax1by1c50 ·¤æ ¥æÜð¹ çÙÙ Âý·¤æÚU ·¤æ ãæð»æ Ñ (A) (B) (C) (D) ÌèÙ â´ÚðU¹èØ çÕÎé çΰ »° ãñ´, §Ù ÌèÙ çÕÎé¥æð´ âð »éÁÚUÙð ßæÜð ßëææð´ ·¤è â´Øæ ãñ Ñ (A) àæêØ (B) °·¤ (C) Îæð (D) ¥Ù´Ì çΰ »° ç¿æ ×ð´, ABCD °·¤ â×æ´ÌÚU ¿ÌéÖüéÁ ãñÐ ØçÎ ∠ B 5 1008 ãæð Ìæð (∠ A 1 ∠ C) ·¤æ ×æÙ ãñ Ñ (A) 3608 041/IX/SA2/06/B1 (B) 2008 (C) 8 1808 (D) 1608 8. ÚðU¹æ x1y50 (A) 9. (2a, a) (B) (a, a) (C) (0, a) (D) (a, 0) °·¤ ·¤ÿææ ×ð´ ÜǸ緤Øæð´ ·¤è â´Øæ x ¥æñÚU ÜǸ·¤æð´ ·¤è â´Øæ y ãñÐ §Ù×ð´ âð °·¤ çßlæÍèü ·¤æ ØÎëÀUØæ ¿ØÙ ç·¤Øæ »ØæÐ ¿éÙð »° çßlæÍèü ·ð¤ ÜǸ·¤æ ãæðÙð ·¤è ÂýæçØ·¤Ìæ ãñ Ñ (A) 10. ÂÚU çSÍÌ ·¤æð§ü çÕÎé çÙÙ M¤Â ·¤æ ãæð»æ Ñ y/x (B) x/(x1y) (C) y/(x1y) (D) x/y °·¤ ÜÕ ßëæèØ ÕðÜÙ, çæØæ a âð×è ßæÜð °·¤ »æðÜð ·¤æð ÂêæüÌØæ æðÚðU ãé° ãñ, ÌÕ ÕðÜÙ ·¤æ ß·ý¤ ÂëDèØ ÿæðæÈ¤Ü ãñ Ñ 4 p a2 (A) âð×è 2 p a2 (B) âð×è2 (C) 3 p a2 âð×è (D) 4 p a2 âð×è2 ¹ÇU - Õ ÂýàÙ â´Øæ 11 âð 18 Ì·¤ ÂýØð·¤ ÂýàÙ 2 ¥´·¤ ·¤æ ãñÐ 11. °·¤ Âæâð ·¤æð 500 ÕæÚU ©ÀUæÜæ ÁæÌæ ãñ ¥æñÚU çÙÙ ÕæÚ´UÕæÚUÌæ Õ´ÅUÙ âæÚUæè ×ð´ ÂçÚUææ×æð´ 1, 2, 3, 4, 5 ¥æñÚU 6 ·¤è ÕæÚ´UÕæÚUÌæ¥æð´ ·¤æð çÜ¹æ »Øæ Ñ ¬Á⁄UáÊÊ◊ ’Ê⁄Uê’Ê⁄UÃÊ 1 2 3 4 5 6 90 70 75 95 88 82 °·¤ ¥ÖæØ â´Øæ Âýæ# ãæðÙð ·¤è ÂýæçØ·¤Ìæ ææÌ ·¤èçÁ°Ð 12. 13. çÙÙ çSÍçÌ ÂÚU ¥æÏæçÚUÌ °·¤ ©ÎæãÚUæ ÎèçÁ° Ñ (i) ×æØ ãè ·ð¤ÎýèØ Âýßëçæ ·¤æ ©ÂØéÌ ×æ ãñÐ (ii) ×æØ ·ð¤ÎýèØ Âýßëçæ ·¤æ ©ÂØéÌ ×æ Ùãè´ ãñ, ÁÕç·¤ ×æØ·¤ °·¤ ©ÂØéÌ ×æ ãñÐ çâh ·¤èçÁ° ç·¤ °·¤ ßëæ ·¤è ÕÚUæÕÚU Áèßæ°´ ßëæ ·ð¤ ·ð¤Îý ÂÚU ÕÚUæÕÚU ·¤æðæ ¥´ÌçÚUÌ ·¤ÚUÌè ãñ´Ð ¥Íßæ ×æÙ ÜèçÁ° ¥æ·¤æð °·¤ ßëæ çÎØæ ãñÐ §â·ð¤ ·ð¤Îý ·¤æð ææÌ ·¤ÚUÙð ·ð¤ çÜ° °·¤ ÚU¿Ùæ ÎèçÁ°Ð 14. k ·¤æ ×æÙ ææÌ ·¤èçÁ° çÁâ·ð¤ çÜ° x521 ¥æñÚU y521 çÙÙ ÚñUç¹·¤ â×è·¤ÚUæ ·¤æ °·¤ ãÜ ãñ Ñ 9kx112ky563 15. çÕ´Îé (2, 14) âð »éÁÚUÌè ãé§ü °·¤ ÚðU¹æ ·¤æ â×è·¤ÚUæ çÜç¹°Ð ¥æñÚU Øæð´? 041/IX/SA2/06/B1 9 §â Âý·¤æÚU ·¤è ¥æñÚU ç·¤ÌÙè ÚðU¹æ°´ ãæð â·¤Ìè ãñ´, 16. P ¥æñÚU Q ·ý¤×àæÑ Îàææü§° ç·¤ ÿæð. â×æ´ÌÚU ¿ÌéÖüéÁ ABCD ·¤è ÖéÁæ¥æð´ DC ¥æñÚU AD ÂÚU çSÍÌ çÕÎé ãñ´ (¥æ·ë¤çÌ Îðç¹°)Ð (APB)5ÿæð. (BQC) 17. Îè »§ü ¥æ·ë¤çÌ ×ð´, PQRS °·¤ â×æ´ÌÚU ¿ÌéÖüéÁ ãñ ¥æñÚU ÚðU¹æ ¹ÇU PA ¥æñÚU RB ·ý¤×àæÑ ·¤æðææð´ P ¥æñÚU R ·¤æð â×çmÖæçÁÌ ·¤ÚUÌð ãñ´Ð Îàææü§° ç·¤ Ñ PA??RB 18. âæçßæè ·¤æð ¥ÂÙð çßææÙ ·ð¤ ÂýæðÁðÅU ·ð¤ çÜ° °·¤ ÕðÜÙæ·¤æÚU ·ð¤çÜÇUæðS·¤æð ·¤æ ×æòÇUÜ ÕÙæÙæ ÍæÐ ß㠧⠷ð¤çÜÇUæðS·¤æð ·¤è ß·ý¤ ÂëD ÕÙæÙð ·ð¤ çÜ° ¿æÅüU ·¤æ»Á ·¤æ ÂýØæð» ·¤ÚUÙæ ¿æãÌè ÍèÐ ØçÎ ßã 25 âð×è ÜÕæ§ü ¥æñÚU 3.5 âð×è çæØæ ·¤æ ·ð¤çÜÇUæðS·¤æð ÕÙæÙæ ¿æãÌè ãñ, Ìæð ©âð ¿æÅüU ·¤æ»Á ·ð¤ ç·¤ÌÙð ÿæðæÈ¤Ü ·¤è ¥æßàØ·¤Ìæ ãæð»è? ¹ÇU - â ÂýàÙ â´Øæ 19 âð 28 Ì·¤ ÂýØð·¤ ÂýàÙ 3 ¥´·¤ ·¤æ ãñÐ 19. ØçÎ °·¤ â×æ´ÌÚU ¿ÌéÖüéÁ ·ð¤ çß·¤æü ÕÚUæÕÚU ãæð´, Ìæð Îàææü§° ç·¤ ßã °·¤ ¥æØÌ ãñÐ 20. çÙÙ ÕæÚ´UÕæÚUÌæ Õ´ÅUÙ âæÚUæè °·¤ ·¤ÿææ ·ð¤ 38 çßlæçÍüØæð´ ·¤æ ÖæÚU ÎàææüÌè ãñ Ñ ÷Ê⁄U (Á∑§.ª˝Ê. ◊¥) ÁfllÊÁÕ¸ÿÊ¥ ∑§Ë ‚¥ÅÿÊ 30-35 35-40 40-45 45-50 50-55 55-60 10 5 15 5 1 2 ∑È ‹ 38 °·¤ çßlæÍèü ·¤æ ÖæÚU çÙÙ ãæðÙð ·¤è ÂýæçØ·¤Ìæ ææÌ ·¤èçÁ° Ñ (i) 45 ç·¤Üæð»ýæ× âð ¥çÏ·¤ Øæ ©â·ð¤ ÕÚUæÕÚU (ii) 30 ç·¤Üæð»ýæ× âð ·¤× (iii) 30 ç·¤Üæð»ýæ× âð ¥çÏ·¤ Øæ ©â·ð¤ ÕÚUæÕÚU Üðç·¤Ù 60 ç·¤Üæð»ýæ× âð ·¤× ¥Íßæ 041/IX/SA2/06/B1 10 çÙÙ âæÚUæè 100 ¥´·¤ ßæÜð »çæÌ ·ð¤ °·¤ ÅðUSÅU ×ð´ Îæð çßÖæ»æð´ ·ð¤ çßlæçÍüØæð´ ·ð¤ ÂýÎàæüÙ ·¤æð ÎàææüÌè ãñ Ñ •¥∑§ ÁfllÊÁÕ¸ÿÊ¥ ∑§Ë ‚¥ÅÿÊ 0-20 20-30 30-40 40-50 50-60 60-70 7 10 10 20 20 15 70 •ÊÒ⁄U •Áœ∑§ 8 ∑ȧ‹ 90 ÂýæçØ·¤Ìæ ææÌ ·¤èçÁ° ç·¤ °·¤ çßlæÍèü »çæÌ ÅðUSÅU ×ð´ Ñ 21. (i) 20% âð (ii) 60 Øæ ©ââð ¥çÏ·¤ ¥´·¤ Âýæ# ·¤ÚUÌæ ãñÐ (iii) 40 Øæ ©ââð ¥çÏ·¤ Üðç·¤Ù ·¤× Âýæ# ·¤ÚUÌæ ãñÐ 60 âð ·¤× ¥´·¤ Âýæ# ·¤ÚUÌæ ãñÐ ABCD ·ð¤ çß·¤æü AC ¥æñÚU BD ÂÚUSÂÚU OB5OD ãñÐ ØçÎ AB5CD ãñ, Ìæð Îàææü§° ç·¤ Ñ ar (DDOC) 5 ar (DAOB) Îè »§ü ¥æ·ë¤çÌ ×ð´, ¿ÌéÖüéÁ çÕ´Îé O ÂÚU §â Âý·¤æÚU ÂýçÌÀðUÎ ·¤ÚUÌð ãñ´ ç·¤ 22. Îè »§ü ¥æ·ë¤çÌ ×ð´, AP ?? BQ ?? CR ãñÐ çâh ·¤èçÁ° ç·¤ ar (AQC) 5 ar (PBR) ¥Íßæ Îè »§ü ¥æ·ë¤çÌ ×ð´, ar (DRC) 5 ar (DPC) ¥æñÚU ar (BDP) 5 ar (ARC) ABCD ¥æñÚU DCPR â×Ü´Õ ãñ´Ð 041/IX/SA2/06/B1 11 ãñÐ Îàææü§° ç·¤ ÎæðÙæð´ ¿ÌéÖüéÁ 23. çÙÙ ¥æ´·¤Ç¸æð´ ·ð¤ çÜ° ×æØ âæ#æçã·¤ ÁðÕ ¹¿ü (L¤ÂØæð´ ×ð´) ææÌ ·¤èçÁ° Ñ ¡’ πø¸ (UL§¬ÿÊ¥ ◊¥) ’Ê⁄Uê’Ê⁄UÃÊ 55 50 49 81 48 57 65 8 3 10 7 3 7 2 253x ·¤æð ax1by1c50 ·ð¤ M¤Â ×ð´ ÃØÌ ·¤èçÁ° ¥æñÚU a, b ¥æñÚU c ·ð¤ ×æÙ ÕÌ槰РâæÍ ãè, ©ÂÚUæðÌ â×è·¤ÚUæ ·¤æ Îæð ¿ÚU ßæÜð â×è·¤ÚUæ ·ð¤ M¤Â ×ð´ Øæç×ÌèØ çÙM¤Âæ ·¤èçÁ°Ð 24. ÚñUç¹·¤ â×è·¤ÚUæ 25. ABCD °·¤ ¥æØÌ ãñ çÁâ×ð´ çß·¤æü AC ÎæðÙæð´ ·¤æðææð´ A ¥æñÚU C ·¤æð â×çmÖæçÁÌ ·¤ÚUÌæ ãñÐ Îàææü§° ç·¤ (i) ABCD °·¤ ß»ü ãñÐ (ii) çß·¤æü BD ÎæðÙæð´ ·¤æðææð´ B ¥æñÚU D ·¤æð â×çmÖæçÁÌ ·¤ÚUÌæ ãñÐ 26. Îè »§ü ¥æ·ë¤çÌ ×ð´, AB ßëæ ·¤æ °·¤ ÃØæâ ãñ ¥æñÚU CD çæØæ ·ð¤ ÕÚUæÕÚU °·¤ Áèßæ ãñÐ AC ¥æñÚU BD ÕÉ¸æ° ÁæÙð ÂÚU °·¤ çÕÎé E ÂÚU ç×ÜÌè ãñ´Ð çâh ·¤èçÁ° ç·¤ ∠ AEB 5 608 . 27. ÖéÁæ¥æð´ 5 cm, 12 cm ¥æñÚU 13 cm ßæÜð °·¤ â×·¤æðæ çæÖéÁ ABC ·¤æð ÖéÁæ 12 cm ·ð¤ ÂçÚUÌ æé×æØæ ÁæÌæ ãñÐ §â Âý·¤æÚU Âýæ# ÆUæðâ ·¤æ ¥æØÌÙ ææÌ ·¤èçÁ°Ð 28. ÏæÌé ·¤è °·¤ »ð´Î ·¤æ ÃØæâ 21 âð×è ãñÐ ØçÎ §â ÏæÌé ·¤æ æÙß 5 »ýæ× ÂýçÌ âð×è3 ãñ, Ìæð §â »ð´Î ·¤æ ÎýÃØ×æÙ ææÌ ·¤èçÁ°Ð ¥Íßæ ç·¤âè ×´çÎÚU ·ð¤ ¹´Öð ÕðÜÙæ·¤æÚU ãñ´Ð ª¡¤¿æ§ü 10 ×èÅUÚU ãñ, Ìæð °ðâð 14 ¹´Öð ØçÎ ÂýØð·¤ ¹´Öð ·¤æ ¥æÏæÚU 20 âð×è çæØæ ·¤æ °·¤ ßëæèØ ÿæðæ ãñ ¥æñÚU ÕÙæÙð ·ð¤ çÜ° ç·¤ÌÙð ·´¤·ý¤èÅU ç×ææ ·¤è ¥æßàØ·¤Ìæ ãæð»è? ¹ÇU - Î ÂýàÙ â´Øæ 29 âð 34 Ì·¤ ÂýØð·¤ ÂýàÙ 4 ¥´·¤ ·¤æ ãñÐ 29. °·¤ Ù»ÚU ×ð´ ÅñUâè ·¤æ ç·¤ÚUæØæ çÙÙçÜç¹Ì ãñÐ ÂãÜð ç·¤Üæð×èÅUÚU ·¤æ ç·¤ÚUæØæ 8 L¤. ãñ ¥æñÚU ©â·ð¤ ÕæÎ ·¤è ÎêÚUè ·ð¤ çÜ° ÂýçÌ ç·¤Üæð×èÅUÚU ·¤æ ç·¤ÚUæØæ 5 L¤. ãñÐ ØçÎ ÌØ ·¤è »§ü ÎêÚUè x ç·¤Üæð×èÅUÚU ãæð, ¥æñÚU ·é¤Ü ç·¤ÚUæØæ y L¤. ãæð, Ìæð §â·¤æ °·¤ ÚñUç¹·¤ â×è·¤ÚUæ çÜç¹° ¥æñÚU ©â·¤æ ¥æÜð¹ ¹è´ç¿°Ð ¥Íßæ °·¤ ¥¿ÚU ÕÜ Ü»æÙð ÂÚU °·¤ çÂÇU mæÚUæ ç·¤Øæ »Øæ ·¤æØü ç´ÇU mæÚUæ ÌØ ·¤è »§ü ÎêÚUè ·ð¤ ¥Ùé·ý¤×æÙéÂæÌè ãæðÌæ ãñÐ §â ·¤ÍÙ ·¤æð Îæð ¿ÚU ßæÜð °·¤ ÚñUç¹·¤ â×è·¤ÚUæ ·ð¤ M¤Â ×ð´ ÃØÌ ·¤èçÁ° ¥æñÚU ¥¿ÚU ÕÜ 5 ×ææ·¤ Üð·¤ÚU §â·¤æ ¥æÜð¹ ¹è´ç¿°Ð ØçÎ ç´ÇU mæÚUæ ÌØ ·¤è »§ü ÎêÚUè 2 ×ææ·¤ ãæð, Ìæð ¥æÜð¹ âð ç·¤Øæ ãé¥æ ·¤æØü ææÌ ·¤èçÁ°Ð 041/IX/SA2/06/B1 12 30. 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