âæ æ Ø çùîðüàæ Ñ (i) (ii) (iii) âöè ÂýàÙ çùßæøü ãñ Ð ë ÂØæ Áæ ÚU Üð ç â ÂýàÙ-Â æ ð 6 ÂýàÙ ãñ Ð ¹ Ç U ð ÂýàÙ 1-6 Ì çì Üƒæé- žæúu ßæÜð ÂýàÙ ãñ æñúu ÂýˆØ

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1 Series ONS ÚUæðÜ Ù. Roll No. SET-1 æðçu Ù. Code No. ÂÚUèÿææÍèü æðçu æð žæúu-âéçsì æ ð é¹-âëcæu ÂÚU ßàØ çü¹ð Ð Candidates must write the Code on the title page of the answer-book. ë ÂØæ Áæ ÚU Üð ç â ÂýàÙ-Â æ ð éçîýì ÂëcÆU 11 ãñ Ð ÂýàÙ-Â æ ð ÎæçãÙð ãæí è æðúu çî» æðçu Ù ÕÚU æð ÀUæ æ žæúu-âéçsì æ ð é¹-âëcæu ÂÚU çü¹ð Ð ë ÂØæ Áæ ÚU Üð ç â ÂýàÙ-Â æ ð 6 ÂýàÙ ãñ Ð ë ÂØæ ÂýàÙ æ žæúu çü¹ùæ àæém ÚUÙð âð ÂãÜð, ÂýàÙ æ ý æ ßàØ çü¹ð Ð â ÂýàÙ-Â æ æð ÂÉ Ùð ð çü 15 ç ÙÅU æ â Ø çîøæ»øæ ãñð ÂýàÙ-Â æ æ çßìúu æ Âêßæüq ð ÕÁð ç Øæ Áæ»æÐ ÕÁð âð ÕÁð Ì ÀUæ æ ð ßÜ ÂýàÙ-Â æ æð ÂÉ ð»ð æñúu â ßçÏ ð ÎæñÚUæÙ ßð žæú-âéçsì æ ÂÚU æð ü žæúu Ùãè çü¹ð»ðð Please check that this question paper contains 11 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 6 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 a.m. From a.m. to a.m., the students will read the question paper only and will not write any answer on the answer-book during this period.»ç æì MATHEMATICS çùïæüçúuì â Ø Ñ 3 ƒæ ÅðU çï Ì Ñ 100 Time allowed : 3 hours Maimum Marks : 100 1

2 âæ æ Ø çùîðüàæ Ñ (i) (ii) (iii) âöè ÂýàÙ çùßæøü ãñ Ð ë ÂØæ Áæ ÚU Üð ç â ÂýàÙ-Â æ ð 6 ÂýàÙ ãñ Ð ¹ Ç U ð ÂýàÙ 1-6 Ì çì Üƒæé- žæúu ßæÜð ÂýàÙ ãñ æñúu ÂýˆØð ÂýàÙ ð çü 1 çùïæüçúuì ãñð (iv) ¹ ÇU Õ ð ÂýàÙ 7-19 Ì Îèƒæü- žæúu I Âý æúu ð ÂýàÙ ãñ æñúu ÂýˆØð ÂýàÙ ð çü 4 çùïæüçúuì ãñ Ð (v) ¹ ÇU â ð ÂýàÙ 0-6 Ì Îèƒæü- žæúu II Âý æúu ð ÂýàÙ ãñ æñúu ÂýˆØð ÂýàÙ ð çü 6 çùïæüçúuì ãñ Ð (vi) žæúu çü¹ùæ ÂýæÚU Ö ÚUÙð âð ÂãÜð ë ÂØæ ÂýàÙ æ ý æ ßàØ çüç¹ Ð General Instructions : (i) (ii) (iii) (iv) (v) (vi) All questions are compulsory. Please check that this question paper contains 6 questions. Questions 1-6 in Section A are very short-answer type questions carrying 1 mark each. Questions 7-19 in Section B are long-answer I type questions carrying 4 marks each. Questions 0-6 in Section C are long-answer II type questions carrying 6 marks each. Please write down the serial number of the question before attempting it.

3 ¹ ÇU - SECTION - A ÂýàÙ â Øæ 1 âð 6 Ì ÂýˆØð ÂýàÙ æ 1 ãñð Question numbers 1 to 6 carry 1 mark each. 1. ØçÎ e N æñúu ãñ, Ìæð æ æù ææì èçá Ð If e N and , then find the value of. 3. ÂýæÚUç Ö SÌ Ö â ç ý Øæ C C 1C 1 çù Ù æãøêã â è ÚU æ ÂÚU Ü»æ Ñ Use elementary column operation C C 1C 1 in the following matri equation : æðçåu 3 ð âöè âö ß æãøêãæð è â Øæ, çáù æ ÂýˆØð ßØß 1, 3 ãñ, çüç¹ Ð Write the number of all possible matrices of order 3 with each entry 1, or 3. 3

4 4. â çõ Îé æ çsíçì âçîàæ çüç¹ Áæð çõ Îé æð, çáù ð çsíçì âçîàæ 3a ÌÍæ a 13b ãñ, æð ç ÜæÙð ßæÜð ÚðU¹æ¹ ÇU æð : 1 ð ÙéÂæÌ ð Õæ ÅUÌæ ãñð Write the position vector of the point which divides the join of points with position vectors 3a and a 13b in the ratio : 1. b b 5. Ù æ æ âçîàææð è â Øæ çüç¹ Áæð âçîàææð 5 i1 j1 k Ü Õ ãñ Ð a ÌÍæ b 5j1k ÎæðÙæð ÂÚU Write the number of vectors of unit length perpendicular to both the vectors i j a k and b 5j1k. 6. â â ÌÜ æ âçîàæ â è ÚU æ ææì èçá, Áæð ç, y æñúu z- ÿæ ÂÚU ý àæñ 3, 4 æñúu ÌÑ¹ ÇU æåuìæ ãñð Find the vector equation of the plane with intercepts 3, 4 and on, y and z-ais respectively. ¹ ÇU - Õ SECTION - B ÂýàÙ â Øæ 7 âð 19 Ì ÂýˆØð ÂýàÙ ð 4 ãñ Ð Question numbers 7 to 19 carry 4 marks each. 7. â çõ Îé ð çùîðüàææ èçá Áãæ ÂÚU çõ Îé æð A(3, 4, 1) æñúu B(5, 1, 6) âð ãæð ÚU ÁæÙð ßæÜè ÚðU¹æ XZ â ÌÜ æð ÂýçÌ ÀðUÎ ÚUÌè ãñð ßã æð æ Öè ææì èçá Áæð Øã ÚðU¹æ XZ â ÌÜ ð âæí ÕÙæÌè ãñð Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane. 4

5 8. â æ ÌÚU ÌéÖéüÁ è Îæð æâóæ ÖéÁæ i4 j5 k æñúu i1 j13 k ãñ Ð â ð ÎæðÙæð çß ææðz ð â æ ÌÚU Îæð æ æ âçîàæ ææì èçá Ð çß ææðz ð âçîàææð æ ÂýØæð» ÚU ð â æ ÌÚU ÌéÖéüÁ æ ÿæð æè Ü ææì èçá Ð The two adjacent sides of a parallelogram are i4 j5 k and i1 j13 k. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. 9. Âæâæ Èð Ùð ð ¹ðÜ ð ÃØç Ì ` 5 ÁèÌÌæ ãñ ØçÎ âð 4 âð ÕÇ è â Øæ ÂýæŒÌ ãæðìè ãñ ØÍæ ßã ` 1 ãæúu ÁæÌæ ãñð ßã ÃØç Ì 3 ÕæÚU Âæâæ Èð Ùð æ çù æüø ÜðÌæ ãñ Üðç Ù æúu âð ÕÇ è â Øæ ÂýæŒÌ ÚUÙð ÂÚU ¹ðÜ ÀUæðÇ ÎðÌæ ãñð ÙécØ mæúuæ ÁèÌè/ãæÚUè ÁæÙð ßæÜè ÚUæçàæ è ÂýˆØæàææ ææì èçá Ð ÍñÜð ð 4»ð Îð ãñ Ð ØæÎë ÀUØæ Îæð»ð Îð çõùæ ÂýçÌSÍæÂÙæ ð çù æüè» ü æñúu ÎæðÙæð âèð Î Âæ ü» üð â è Øæ ÂýæçØ Ìæ ãñ ç ÍñÜð ð âöè»ð Îð âèð Î ãñ? In a game, a man wins ` 5 for getting a number greater than 4 and loses ` 1 otherwise, when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the epected value of the amount he wins/loses. A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white? 5

6 10. sin 1(sin) cos æ ð âæâðÿæ ß ÜÙ èçá Ð ØçÎ y5 cos(log)13 sin(log) ãñ, Ìæð çâh èçá ç Differentiate sin 1(sin) cos with respect to. y y d y 1 d d d ãñð If y5 cos(log)13 sin(log), prove that y y d y 1 d d d 11. ØçÎ 5a sin t(11cos t) ÌÍæ y5b cos t(1cos t) ãñ, Ìæð t 5 ÂÚU p 4 d ææì èçá Ð If 5a sin t(11cos t) and y5b cos t(1cos t), find d at t p ØçÎ ß ý y 5a 3 1b ð çõ Îé (, 3) ÂÚU SÂàæü ÚðU¹æ æ â è ÚU æ y545 ãñ, Ìæð a ÌÍæ b ð æù ææì èçá Ð The equation of tangent at (, 3) on the curve y 5a 3 1b is y545. Find the values of a and b. 13. ææì èçá Ñ d 4 1 Find : d 4 1 6

7 14. æù ææì èçá Ñ p sin 1 0 sin cos d æù ææì èçá : cos p d 3 0 Evaluate : p sin 1 0 sin cos d 3 Evaluate : cos p d ææì èçá Ñ (311) 4 3 d Find : (311) 4 3 d 16. ß Ü â è ÚU æ æð ãü èçá Ñ y1 5y d d Solve the differential equation : y1 5y d d 7

8 17. çmìèø ÌéÍæZàæ ð ðâð ßëžææð ð é Ü æ ß Ü â è ÚU æ ææì èçá Áæð çùîðüàææ ÿææð æð SÂàæü ÚUÌð ãñ Ð Form the differential equation of the family of circles in the second quadrant and touching the coordinate aes. 18. ð çü ãü èçá Ñ sin 1 1sin 1 (1)5cos 1 ØçÎ y cos 1 1cos 1 5a a b ãñ, Ìæð çâh èçá ç y y cos a1 5 sin a a ab b ãñð Solve the equation for : sin 1 1sin 1 (1)5cos 1 If y cos 1 1cos 1 5a, prove that a b y y cos a1 5 sin a a ab b 19. ÅþUSÅU Ùð Îæð Âý æúu ð Õæ ÇU ( ÙéÕ Ï Â æ) ð ÏÙ çùßðçàæì ç ØæÐ ÂãÜð Õæ ÇU ð 10% ŽØæÁ æñúu ÎêâÚðU Õæ ÇU ð 1% ŽØæÁ ç ÜÌæ ãñð ÅþUSÅU æð `,800 é Ü ŽØæÁ ÂýæŒÌ ãé æð ÂÚU Ìé ØçÎ ÅþUSÅU Ùð ÎÜæ-ÕÎÜè ÚU ð Õæ ÇUæð ð ÏÙ Ü»æØæ ãæðìæ, Ìæð âð ` 100 ŽØæÁ ÂýæŒÌ ãæðìæð æãøêã çßçï âð ææì èçá ç ÅþUSÅU Ùð ç ÌÙæ ÏÙ çùßðçàæì ç ØæÐ ÂýæŒÌ ŽØæÁ æð ãðëâ ðá ççuøæ ð ÎæÙ ç Øæ Áæ»æÐ â ÂýàÙ ð æñù-âæ êëø ÎàææüØæ»Øæ ãñ? A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 1% interest. The trust received `,800 as interest. However, if trust had interchanged money in bonds, they would have got ` 100 less as interest. Using matri method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question? 8

9 ¹ ÇU - â SECTION - C ÂýàÙ â Øæ 0 âð 6 Ì ÂýˆØð ÂýàÙ ð 6 ãñ Ð Question numbers 0 to 6 carry 6 marks each. 0. Îæð Âý æúu ð ¹æÎ A æñúu B ãñ Ð A ð 1% Ùæ ÅþUæðÁÙ æñúu 5% È æsè æðçúu çâçu ãñ ÁÕç B ð 4% Ùæ ÅþUæðÁÙ æñúu 5% È æsè æðçúu çâçu ãñð ç ^è ð çsíçì ÂÚUèÿæ æ ð ÕæÎ ç âæù æð ææì ãé æ ç âð È âü ð çü âð 1 ç.»ýæ. Ùæ ÅþUæðÁÙ æñúu 1 ç.»ýæ. È æsè æðçúu çâçu è æßàø Ìæ ãñð ØçÎ A æ êëø ` 10 ÂýçÌ ç.»ýæ. æñúu B æ êëø ` 8 ÂýçÌ ç.»ýæ. ãñ Ìæð æüð¹ mæúuæ ÂçÚU çüì èçá ç âð ÂýˆØð Âý æúu è ç ÌÙè ¹æÎ ÂýØæð» ÚUÙè æçã ç âð è Ì ð Âæðá Ìˆßæð è æßàø Ìæ ÂêÚUè ãæð Áæ Ð There are two types of fertilisers A and B. A consists of 1% nitrogen and 5% phosphoric acid whereas B consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 1 kg of nitrogen and 1 kg of phosphoric acid for his crops. If A costs ` 10 per kg and B cost ` 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost ÀðU â ÌÚUæð ð 5 ¹ÚUæÕ â ÌÚð ÂýˆØæçàæÌ æúu æ âð ç Ü» ãñ Ð æúu â ÌÚðU žæúuæðžæúu ÂýçÌSÍæÂÙæ ð âçãì çù æüð», Ìæð ¹ÚUæÕ â ÌÚUæð æð çù æüùð è â Øæ æ ÂýæØç Ìæ Õ ÅUÙ ææì èçá Ð Õ ÅUÙ æ æšø ÌÍæ ÂýâÚU æ Öè ææì èçá Ð Five bad oranges are accidently mied with 0 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution. 9

10 . çõ Îé P, çáâ æ çsíçì âçîàæ i1 j1 k 3 4 ãñ âð â ÌÜ ( i j 1 k) r ÂÚU ¹è ð» Ü Õ ð ÂæÎ æ çsíçì âçîàæ ÌÍæ Ü ÕßÌ ÎêÚUè ææì èçá Ð ÌÜ ð P æ ÂýçÌçÕ Õ Öè ææì èçá Ð Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector i13 j14 k to the plane ( i j k) 1 r Also find image of P in the plane. 3. Îàææü ç çm æïæúuè â ç ý Øæ Áæð A5R{1} ÂÚU âöè a, b e A ð çü a*b5a1b1ab mæúuæ ÂçÚUÖæçáÌ ãñ ý çßçù ðø ÌÍæ âæã üø ãñð A ð * æ Ìˆâ ßØß ææì èçá ÌÍæ çâh èçá ç A æ ÂýˆØð ßØß ÃØéˆ ý æèø ãñð Show that the binary operation * on A5R{1} defined as a*b5a1b1ab for all a, b e A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible. 4. çâh èçá ç â çmõæãé ç æöéá, çáâ ð r ç æ Øæ æ ÌßëÌ ¹è æ»øæ ãñ, æ ØêÙÌ ÂçÚU æâ 6 3 r ãñð ØçÎ â æð æ ç æöéá ð æü ÌÍæ ÖéÁæ æ Øæð» çîøæ»øæ ãæð, Ìæð Îàææü ç ç æöéá æ ÿæð æè Ü çï Ì ãæð»æ ÁÕç Ù ð Õè æ æð æ 3 p ãæð»æð Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6 3 r. If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maimum, when the angle between them is 3 p. 10

11 5. çâh èçá ç ß ý y 54 æñúu 54y, â ß»ü ð ÿæð æè Ü æð ÌèÙ ÕÚUæÕÚU Öæ»æð ð Õæ ÅUÌð ãñ Áæð ç ÚðU¹æ æð 50, 54, y54 æñúu y50 mæúuæ ÂçÚUÕh ãñð Prove that the curves y 54 and 54y divide the area of square bounded by 50, 54, y54 and y50 into three equal parts. 6. âæúuç æ æð ð»é æï æðz æ ÂýØæð» ÚU çâh èçá ç DABC â çmõæãé ç æöéá ãñ ØçÎ cosA 11cosB 11cosC 50 cos A 1cosA cos B 1cosB cos C 1cosC Îé æùîæúu ð Âæâ ÌèÙ çßçöóæ Âý æúu ð ÂðÙ A, B æñúu C ãñ Ð èùê Ùð ÂýˆØð Âý æúu æ - ÂðÙ é Ü ` 1 ð ¹ÚUèÎæÐ ÁèßÙ Ùð A Âý æúu ð 4 ÂðÙ, B Âý æúu ð 3 ÂðÙ æñúu C Âý æúu ð ÂðÙ ` 60 ð ¹ÚUèÎð ÁÕç çàæ¹æ Ùð A Âý æúu ð 6 ÂðÙ, B Âý æúu ð ÂðÙ æñúu C Âý æúu ð 3 ÂðÙ ` 70 ð ¹ÚUèÎðÐ æãøêã çßçï âð ÂýˆØð Âý æúu ð ÂðÙ æ êëø ææì èçá Ð Using properties of determinants, show that DABC is isosceles if : cosA 11cosB 11cosC 50 cos A 1cosA cos B 1cosB cos C 1cosC A shopkeeper has 3 varieties of pens A, B and C. Meenu purchased 1 pen of each variety for a total of ` 1. Jeevan purchased 4 pens of A variety, 3 pens of B variety and pens of C variety for ` 60. While Shikha purchased 6 pens of A variety, pens of B variety and 3 pens of C variety for ` 70. Using matri method, find cost of each variety of pen. ãñð 11

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3 tan tan tan 2 x;? 2 x? Series ONS ÚUæðÜ Ù. Roll No. SET-1 æðçu Ù. Code No. ÂÚUèÿææÍèü æðçu æð žæúu-âéçsì æ ð é¹-âëcæu ÂÚU ßàØ çü¹ð Ð Candidates must write the Code on the title page of the answer-book. ë ÂØæ Áæ ÚU Üð ç â ÂýàÙ-Â