(ii) Ûéú ÖµÖÖ ÖÖÑ Ö Ûú ü»öë Ûú ÃÖ ÖÏ Ö- Ö Ö Öë 26 ÖÏ Ö Æïü
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1 SET 1 Series : SSO/1/C üöê»ö ÖÓ. Roll No. ÛúÖê ü ÖÓ. Code No. 65/1/1 Ö üßõöö Öá ÛúÖê ü ÛúÖê ˆ Ö ü- Öã ßÖÛúÖ Ûêú ÖãÜÖ- Öéšü Ö ü ¾Ö µö»öüöë Candidates must write the Code on the title page of the answer-book. Ûéú ÖµÖÖ ÖÖÑ Ö Ûú ü»öë Ûú ÃÖ ÖÏ Ö- Ö Ö Öë Öã üÿö Öéšü 8 Æïü ÖÏ Ö- Ö Ö Öë üö Æü Öê ÆüÖ Ö Ûúß Öê ü ü ÝÖ ÛúÖê ü Ö ²Ö ü ÛúÖê ûö Ö ˆ Ö ü- Öã ßÖÛúÖ Ûêú ÖãÜÖ- Öéšü Ö ü»öüöë Ûéú ÖµÖÖ ÖÖÑ Ö Ûú ü»öë Ûú ÃÖ ÖÏ Ö- Ö Ö Öë 6 ÖÏ Ö Æïü Ûéú ÖµÖÖ ÖÏ Ö ÛúÖ ˆ Ö ü»öüö ÖÖ Öãºþ Ûú ü Öê ÃÖê ÖÆü»Öê, ÖÏ Ö ÛúÖ ÛÎú ÖÖÓÛú ¾Ö µö»öüöë ÃÖ ÖÏ Ö- Ö Ö ÛúÖê ÖœÌü Öê Ûêú»Ö 15 Ö Ö ü ÛúÖ ÃÖ ÖµÖ üµöö ÝÖµÖÖ Æîü ÖÏ Ö- Ö Ö ÛúÖ ¾ÖŸÖ üþö Öæ¾ÖÖÔÆËü Ö Öë ²Ö Öê ÛúµÖÖ ÖÖµÖêÝÖÖ ²Ö Öê ÃÖê 10.0 ²Ö Öê ŸÖÛú ûö Ö Ûêú¾Ö»Ö ÖÏ Ö- Ö Ö ÛúÖê ÖœÌëüÝÖê Öî ü ÃÖ ¾Ö Ö Ûêú üöî üö Ö ¾Öê ˆ Ö ü- Öã ßÖÛúÖ Ö ü ÛúÖê Ô ˆ Ö ü ÖÆüà»ÖÜÖëÝÖê Please check that this question paper contains 8 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 minutes time has been allotted to read this question paper. The question paper will be distributed at a.m. From a.m. to 10.0 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. ÝÖ ÞÖŸÖ MATHEMATICS Ö ÖÖÔ üÿö ÃÖ ÖµÖ : ÖÞ êüü] [ ÖÛúŸÖ Ö ÓÛú : 100 Time allowed : hours ] [ Maximum Marks : 100 ÃÖÖ ÖÖ µö Ö ìü Ö : ÃÖ³Öß ÖÏ Ö Ö¾ÖÖµÖÔ Æïü (ii) Ûéú ÖµÖÖ ÖÖÑ Ö Ûú ü»öë Ûú ÃÖ ÖÏ Ö- Ö Ö Öë 6 ÖÏ Ö Æïü (iii) ÜÖÞ ü- Ûêú ÖÏ Ö ÃÖÓ. 1 6 ŸÖÛú ŸÖ»Ö Öã-ˆ Ö ü ¾ÖÖ»Öê ÖÏ Ö Æïü Öî ü ÖÏŸµÖêÛú ÖÏ Ö Ûêú»Ö 1 ÓÛú Ö ÖÖÔ üÿö Æîü (iv) ÜÖÞ ü-²ö Ûêú ÖÏ Ö ÃÖÓ ŸÖÛú üß ÖÔ-ˆ Ö ü I ÖÏÛúÖ ü Ûêú ÖÏ Ö Æïü Öî ü ÖÏŸµÖêÛú ÖÏ Ö Ûêú»Ö ÓÛú Ö ÖÖÔ üÿö Æïü (v) ÜÖÞ ü-ãö Ûêú ÖÏ Ö ÃÖÓ. 0 6 ŸÖÛú üß ÖÔ-ˆ Ö ü II ÖÏÛúÖ ü Ûêú ÖÏ Ö Æïü Öî ü ÖÏŸµÖêÛú ÖÏ Ö Ûêú»Ö 6 ÓÛú Ö ÖÖÔ üÿö Æïü (vi) ˆ Ö ü»öüö ÖÖ ÖÏÖ Óü³Ö Ûú ü Öê ÃÖê ÖÆü»Öê Ûéú ÖµÖÖ ÖÏ Ö ÛúÖ ÛÎú ÖÖÓÛú ¾Ö µö»ö ÜÖ 65/1/1 1 [P.T.O.
2 General Instructions : (ii) All questions are compulsory. Please check that this Question Paper contains 6 Questions. (iii) Questions 1 6 in Section-A are Very Short Answer Type Questions carrying 1 mark each. (iv) Questions 7 19 in Section-B are Long Answer I Type Questions carrying marks each. (v) Questions 0 6 in Section-C are Long Answer II Type Questions carrying 6 marks each (vi) Please write down the serial number of the Question before attempting it. ÖÏ Ö ÃÖÓܵÖÖ 1 ÃÖê 6 ŸÖÛú ÖÏŸµÖêÛú ÖÏ Ö 1 ÓÛú ÛúÖ Æîü ÜÖÞ ü SECTION A Question numbers 1 to 6 carry 1 mark each. 1. ÃÖ ü Ö a = ^i ^j Ûúß ü ÖÖ Öë Ûú ÃÖ ü Ö ÖÖŸÖ Ûúß Ö ÖÃÖÛúÖ Ö ü ÖÖÞÖ 7 ÛúÖ Ô Æîü 1 Find a Vector in the direction of a = ^i ^j that has magnitude 7 units.. µö ü a ŸÖ ÖÖ b ÖÖ ÖÛú ÃÖ ü Ö Æïü, ŸÖÖê a ŸÖ ÖÖ b Ûêú ²Öß Ö ÛúÖ ÛúÖêÞÖ ŒµÖÖ Æîü, Ö²Ö Ûú a b Ûú ÖÖ ÖÛú ÃÖ ü Ö Æîü? 1 If a and b are unit Vectors, then what is the angle between a and b so that a b is a unit vector?. µö ü Ûú êüüöö Ö ìü ÖÖÓÛú õööë Ûúß Ö ÖÖŸ ÖÛú ü ÖÖ Öë Ûêú ÃÖÖ Ö ÛúÖêÞÖ α, β, γ ²Ö ÖÖŸÖß Æîü, ŸÖÖê sin α + sin β + sin γ ÛúÖ ÖÖ Ö»Ö ÜÖ 1 If a line makes angles α, β, γ with the positive direction of coordinate axes, then write the value of sin α + sin β + sin γ.. µö ü A = 1 1, B = If A = 1 1 and B = 1 1 1, write the value of AB. 65/1/1 Æïü, ŸÖÖê AB ÛúÖ ÖÖ Ö»Ö ÜÖ 1
3 5. C ÛúÖ ¾Ö»ÖÖê Ö Ö Ûú üûêú ¾ÖÛÎúÖë Ûêú Ûãú»Ö xy = C cos x ÛúÖê Öºþ ÖŸÖ Ûú ü Öê ¾ÖÖ»ÖÖ ¾ÖÛú»Ö ÃÖ ÖßÛú üþö»ö ÜÖ 1 Write the differential equation obtained by eliminating the arbitrary constant C in the equation representing the family of curves xy = C cos x. 6. ¾ÖÛú»Ö ÃÖ ÖßÛú üþö 1 + dy = 7 d y dx dx Ûúß ÛúÖê ü ŸÖ ÖÖ ÖÖŸÖ ÛúÖ µööêýö»ö ÜÖ 1 Write the sum of the order and degree of the differential equation 1 + dy dx = 7 d y dx ÖÏ Ö ÃÖÓܵÖÖ 7 ÃÖê 19 ŸÖÛú ÖÏŸµÖêÛú ÖÏ Ö ÓÛú ÛúÖ Æîü ÜÖÞ ü ²Ö SECTION B Question numbers 7 to 19 carry marks each. 7. üöê ¾ÖªÖ»ÖµÖ A ŸÖ ÖÖ B Ö Öê ¾ÖªÖÙ ÖµÖÖë ÛúÖê ÃÖÆüµÖÖêÝÖ Ûúß ³ÖÖ¾Ö ÖÖ, ÃÖŸµÖ¾ÖÖ üÿöö ŸÖ ÖÖ ÃÖÆü Ö Ö߻֟ÖÖ Ûêú Ö滵ÖÖë Ûêú»Ö ÖÏ ŸÖ ¾ÖªÖ Öá ÛÎú Ö Ö: ` x, ` y ŸÖ ÖÖ ` z ÛúÖ Öã üãûúö ü êü ÖÖ ÖÖÆüŸÖê Æïü ¾ÖªÖ»ÖµÖ A Ö Öê ÛÎú Ö Ö:, 1, ¾ÖªÖÙ ÖµÖÖë ÛúÖê Ö Ö滵ÖÖë Ûêú»Ö Ûãú»Ö ` 1,100 Öã üãûúö ü þֺþ Ö êü ÖÖ ÖÖÆüŸÖÖ Æîü ¾ÖªÖ»ÖµÖ B Ö Öê ÛÎú Ö Ö: 1,, ¾ÖªÖÙ ÖµÖÖë ÛúÖê ` 1,00 Öã üãûúö ü þֺþ Ö êü ÖÖ ÖÖÆüŸÖÖ Æîü µö ü Ö ŸÖß ÖÖë Ö滵ÖÖë Ö ü ü ÝÖ Ûú- Ûú Öã üãûúö ü Ûúß Ûãú»Ö üö Ö ` 600 Æîü, ŸÖÖê ˆ Ö üöêœÿö Ã Ö ŸÖ ÛúÖê îü ÜÖÛú ÃÖ ÖßÛú üþö ²Ö ÖÖÛú ü Ûú Ö¾µÖæÆü ÃÖ ÖßÛú üþö Ûêú ºþ Ö Öë ¾µÖŒŸÖ Ûúß Ö (ii) ŒµÖÖ ÃÖ ÖßÛú üþö ÖÛúÖµÖ ÛúÖê Ö¾µÖæÆüÖë Ûêú ÖϵÖÖêÝÖ ÃÖê Æü»Ö ÛúµÖÖ ÖÖ ÃÖÛúŸÖÖ Æîü? (iii) Ö Ö ÛúÃÖ Öæ»µÖ Ö ü ÖÛúŸÖ Ö Öã üãûúö ü êü ÖÖ ÖÖÆüŸÖê Æïü Öî ü ŒµÖÖë? Two schools A and B decided to award prizes to their students for three values, team spirit, truthfulness and tolerance at the rate of ` x, ` y and ` z per student respectively. School A, decided to award a total of ` 1,100 for the three values to, 1 and students respectively while school B decided to award ` 1,00 for the three values to 1, and students respectively. If one prize for all the three values together amount to ` 600 then Represent the above situation by a matrix equation after forming linear equations. (ii) Is it possible to solve the system of equations so obtained using matrices? (iii) Which value you prefer to be rewarded most and why? 65/1/1 [P.T.O.
4 8. µö ü [x ] 1 0 x Ö¾µÖæÆü A = 1 1 = O Æîü, ŸÖÖê x ÖÖŸÖ Ûúß Ö ÛúÖ ¾µÖãŸÛÎú Ö ÖÖŸÖ Ûúß Ö 1 If [x ] 1 0 x = O, find x 1 Find the inverse of the matrix A = ÃÖÖ ü ÞÖÛúÖë Ûêú ÝÖãÞÖ Ö ÖÖí ÛúÖ ÖϵÖÖêÝÖ Ûú üûêú ÃÖ ü Ûúß Ö Ûú (a + 1) (a + ) a + 1 (a + ) (a + ) a + 1 = (a + ) (a + ) a + 1 Using properties of determinants, prove that (a + 1) (a + ) a + 1 (a + ) (a + ) a + 1 (a + ) (a + ) a + 1 = π/ 10. ÖÖ Ö ÖÖŸÖ Ûúß Ö : 0 dx 1 + tan x π/ Evaluate : 0 dx 1 + tan x 11. ÖÖŸÖ Ûúß Ö : x (x 1) (x + ) dx ÖÖŸÖ Ûúß Ö : x + x + 6x + 5 dx Find : x (x 1) (x + ) dx Find : x + x + 6x + 5 dx 65/1/1
5 1. 15 ²Ö»²ÖÖë Ûêú œìêü ü ÃÖê, ÖÃÖ Öë 5 ²Ö»²Ö ÜÖ üö²ö Æïü, ²Ö»²ÖÖë ÛúÖ Ûú Ö Öæ ÖÖ µöö ü ûµöö, ²Ö ÖÖ ÖÏ ŸÖà ÖÖ Ö ÖÖ Ûêú ÖÛúÖ»ÖÖ ÖÖŸÖÖ Æîü ÜÖ üö²ö ²Ö»²ÖÖë Ûúß ÃÖÓܵÖÖ ÛúÖ ÖÏÖ µöûúÿöö ²ÖÓ ü Ö ÖÖŸÖ Ûúß Ö Ûú ¾Ö ÖêÂÖ ÃÖ ÖõÖÖ ÛúÖê A ŸÖ ÖÖ B «üö üö þ֟ÖÓ Ö ºþ Ö ÃÖê Æü»Ö Ûú ü Öê Ûúß ÖÏÖ µöûúÿöö Ñ ÛÎú Ö Ö: 1 ŸÖ ÖÖ 1 Æïü µö ü üöê ÖÖë þ֟ÖÓ Ö ºþ Ö ÃÖê ÃÖ ÖõÖÖ ÛúÖê Æü»Ö Ûú ü Öê ÛúÖ ÖϵÖÖÃÖ Ûú üÿöê Æïü, ŸÖÖê ÖÏÖ µöûúÿöö ÖÖŸÖ Ûúß Ö Ûú (ii) ÃÖ ÖõÖÖ Æü»Ö ÆüÖê ÖÖŸÖß Æîü ˆ Ö Öë ÃÖê ŸÖ µöÿöö ÛúÖê Ô Ûú ÃÖ ÖõÖÖ Æü»Ö Ûú ü»öêÿöö Æîü From a lot of 15 bulbs which include 5 defectives, a sample of bulbs is drawn at random (without replacement). Find the probability distribution of the number of defective bulbs. Probability of solving specific problem independently by A and B are 1 and 1 respectively. If both try to solve the problem independently, find the probability that (ii) the problem is solved exactly one of them solves the problem. 1. λ ÛúÖ ÖÖ Ö ÖÖŸÖ Ûúß Ö ŸÖÖ Ûú ÖÖ ü Ø²Ö ãü A, B, C ŸÖ ÖÖ D Ö ÖÛêú Ã Ö ŸÖ ÃÖ ü Ö ÛÎú Ö Ö: ^i + 5^j + ^k, ^j ^k, ^i + λ^j + ^k ŸÖ ÖÖ ^i + ^j + ^k Æïü, ÃÖ ÖŸÖ»ÖßµÖ Æïü Find the value of λ so that the four points A, B, C and D with position vectors ^i + 5^j + ^k, ^j ^k, ^i + λ^j + ^k and ^i + ^j + ^k respectively are coplanar. 1. êüüöö Öë r = (^i + ^j + ^k) + λ(^i ^j + ^k) ŸÖ ÖÖ r = (^i ^j ^k) + µ(^i + ^j + ^k) Ûêú ²Öß Ö Ûúß µöæ ÖŸÖ Ö æü üß ÖÖŸÖ Ûúß Ö Find the shortest distance between the lines r = (^i + ^j + ^k) + λ(^i ^j + ^k) and r = (^i ^j ^k) + µ(^i + ^j + ^k) 65/1/1 5 [P.T.O.
6 15. Æü»Ö Ûúß Ö : tan 1 x + tan 1 x = π ÃÖ ü Ûúß Ö Ûú : tan = 5 sin cos 1 5 Solve : tan 1 x + tan 1 x = π Prove that : tan = 5 sin cos µö ü y = x cos 1 x 1 x log 1 x Æîü, ŸÖÖê ÃÖ ü Ûúß Ö Ûú dy dx = cos 1 x (1 x ) / If y = x cos 1 x 1 x log 1 x, then prove that dy dx = cos 1 x (1 x ) / 17. x Ûêú ÃÖÖ ÖêõÖ (sin x) x + sin 1 x ÛúÖ ¾ÖÛú»Ö Ö ÖÖŸÖ Ûúß Ö Find the derivative of (sin x) x + sin 1 x w.r.t. x 18. µö ü x = a sec θ, y = a tan θ Æîü, ŸÖÖê If x = a sec θ, y = a tan θ, find d y dx at θ = π d y dx ÛúÖ ÖÖ Ö θ = π Ö ü ÖÖŸÖ Ûúß Ö 19. ÖÖŸÖ Ûúß Ö : (x + 1)e x (x + 1) dx Find (x + 1)e x (x + 1) dx 65/1/1 6
7 ÜÖÞ ü ÃÖ SECTION C ÖÏ Ö ÃÖÓܵÖÖ 0 ÃÖê 6 ŸÖÛú ÖÏŸµÖêÛú ÖÏ Ö 6 ÓÛú ÛúÖ Æîü Question numbers 0 to 6 carry 6 marks each. 0. ÖÖ ÖÖ A = Æîü ŸÖ ÖÖ, A Öë (a, b) (c, d) = (a + c, b + d) «üö üö Ö ü³öö ÂÖŸÖ Ûú «ü Ö ÖÖ üß ÃÖÓ ÛÎúµÖÖ Æîü ÃÖ ü Ûúß Ö Ûú ÛÎú Ö ¾Ö Ö ÖêµÖ ŸÖ ÖÖ ÃÖÖÆü ÖµÖÔ Æîü A Öë ÛúÖ ŸÖŸÃÖ ÖÛú ¾ÖµÖ¾Ö ÖÖŸÖ Ûúß Ö 6 Let A = and be the binary operation on A defined by (a, b) (c, d) = (a + c, b + d). Show that is commutative and associative. Find the identity element for on A. 1. ÃÖ ÖÖÛú»Ö ÖÖë Ûêú ÖϵÖÖêÝÖ ÃÖê ÖÏ Ö Ö ÖŸÖã ÖÖÕ Ö Öë ¾Öé Ö x + y =, êüüöö y = x ŸÖ ÖÖ y- õö ÃÖê Ö êü õöê Ö ÛúÖ õöê Ö±ú»Ö ÖÖŸÖ Ûúß Ö 6 Find the area of the region in the first quadrant enclosed by the y-axis, the line y = x and the circle x + y =, using integration.. ¾ÖÛú»Ö ÃÖ ÖßÛú üþö x dy + y x + xy cot x = 0; x 0 ÛúÖ ¾Ö Ö ü Æü»Ö ÖÖŸÖ Ûúß Ö, üµöö Æîü Ûú Ö²Ö dx x = π Æîü, ŸÖÖê y = 0 Æîü 6 ¾ÖÛú»Ö ÃÖ ÖßÛú üþö x dy + (xy + y ) dx = 0, üµöö Æîü Ûú Ö²Ö x = 1 Æîü, y = 1 Æîü Find the particular solution of the differential equation x dy dx + y x + xy cot x = 0; x 0, given that when x = π, y = 0 Solve the differential equation x dy + (xy + y ) dx = 0 given y = 1, when x = 1. ŸÖ»ÖÖë x + y + z = 1 Öî ü x + y + z = 5 Ûúß ÖÏ ŸÖ êû ü Ö êüüöö ÃÖê ÆüÖêÛú ü ÖÖ Öê ¾ÖÖ»Öê ŸÖ ÖÖ ŸÖ»Ö x y + z = 0 Ö ü»öó²ö¾öÿö ŸÖ»Ö ÛúÖ ÃÖ ÖßÛú üþö ÖÖŸÖ Ûúß Ö ÖÏÖ ŸÖ ŸÖ»Ö Ûúß Ø²Ö ãü A(1,, 6) ÃÖê æü üß ÖÖŸÖ Ûúß Ö 6 Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and x + y + z = 5 which is perpendicular to the plane x y + z = 0. Then find the distance of plane thus obtained from the point A(1,, 6). 65/1/1 7 [P.T.O.
8 . Ûú Öî»Öê Öë»ÖÖ»Ö ŸÖ ÖÖ ÛúÖ»Öß ÝÖë ëü Æïü Ûú µö Öî»Öê Öë»ÖÖ»Ö ŸÖ ÖÖ 6 ÛúÖ»Öß ÝÖë ëü Æïü üöê ÖÖë Öî»ÖÖë Öë ÃÖê ÛúÖê Ô Ûú Öî»ÖÖ µöö ü ûµöö Öã ÖÖ ÖÖŸÖÖ Æîü Öî ü ˆÃÖ Öë ÃÖê µöö ü ûµöö ÝÖë ëü, ²Ö ÖÖ ÖÏ ŸÖà ÖÖ Ö ÖÖ Ûêú, ÖÛúÖ»Öß ÖÖŸÖß Æïü ÖÖê üöê ÖÖë»ÖÖ»Ö ÖÖ Ô ÖÖŸÖß Æïü ÖÏÖ µöûúÿöö ÖÖŸÖ Ûúß Ö Ûú üöê ÖÖë ÝÖë ëü ÖÆü»Öê Öî»Öê Öë ÃÖê ÖÛúÖ»Öß ÝÖ Ô Æïü 6 A bag contains red and black balls, another bag contains red and 6 black balls. One of the two bags is selected at random and two balls are drawn at random without replacement from the bag and are found to be both red. Find the probability that the balls are drawn from the first bag. 5. Ûú ÖÏÛúÖ ü Ûêú ÛêúÛú ÛúÖê 00 ÝÖÏÖ. Ö üö ŸÖ ÖÖ 5 ÝÖÏÖ. ¾ÖÃÖÖ (fat) Ûúß Ö¾Ö µöûúÿöö ÆüÖêŸÖß Æîü Ö²Ö Ûú æüãö êü ÖÏÛúÖ ü Ûêú ÛêúÛú Ûêú»Ö 100 ÝÖÏÖ. Ö üö ŸÖ ÖÖ 50 ÝÖÏÖ. ¾ÖÃÖÖ Ûúß Ö¾Ö µöûúÿöö ÆüÖêŸÖß Æîü ÛêúÛúÖë Ûúß ÖÛúŸÖ Ö ÃÖÓܵÖÖ ÖÖŸÖ Ûúß Ö ÖÖê 5 Ûú»ÖÖê Ö üö ŸÖ ÖÖ 1 Ûú»ÖÖê ¾ÖÃÖÖ ÃÖê ²Ö Ö ÃÖÛúŸÖê Æïü µöæü ÖÖ Ö»Öß Ö Ûú ÛêúÛúÖë Ûêú ²Ö ÖÖ Öê Ûêú»Ö µö Ö üö ÖÖí Ûúß Ûú Öß ÖÆüà üæêüýöß ˆ Ö üöêœÿö ÛúÖê îü ÜÖÛú ÖÏÖêÝÖÏÖ Ö Ö ÃÖ ÖõÖÖ ²Ö ÖÖÛú ü ÝÖÏÖ±ú Ûúß ÃÖÆüÖµÖŸÖÖ ÃÖê Æü»Ö Ûúß Ö 6 One kind of cake requires 00 g of flour and 5 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Make an L.P.P. of the above and solve it graphically. 6. ÖµÖŸÖÖÛúÖ ü Ö ÖÖ ü ¾Ö ÖµÖŸÖÖÛú ü üß¾öö üöë Ûúß Öß ÝÖÆü üß ŸÖ ÖÖ 75 Öß ÖµÖŸÖ Ö Ûúß Ûú ²Ö ÖÖ œüœûú Ö Ûúß ÓüÛúß ÛúÖ Ö ÖÖÔÞÖ Ûú ü ÖÖ Æîü µö ü ÓüÛúß Ûêú Ö ÖÖÔÞÖ Öë Ö ÖÖ ü Ûêú»Ö ` 100 ÖÏ ŸÖ ¾ÖÝÖÔ Öß ü ü Öî ü üß¾öö üöë Ûêú»Ö ` 50 ÖÏ ŸÖ ¾ÖÝÖÔ Öß ü ü ¾µÖµÖ ÖŸÖÖ Æîü, ŸÖÖê Ö ÖŸÖ Ö ÜÖ ÖÔ ÃÖê ²Ö Öß ÓüÛúß Ûúß»ÖÖÝÖŸÖ ÖÖŸÖ Ûúß Ö 6 Ûú ÃÖ ÖÛúÖêÞÖ Ö³Öã Ö Ûúß ³Öã ÖÖ Öë ÃÖê a Öî ü b æü üß Ö ü Ö³Öã Ö Ûêú ÛúÞÖÔ Ö ü Ã ÖŸÖ Ûú Ø²Ö ãü Æîü ÃÖ ü Ûúß Ö Ûú ÛúÞÖÔ Ûúß µöæ ÖŸÖ Ö»ÖÓ²ÖÖ Ô a + b Æîü A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is m and volume is 75 m. If building of tank costs ` 100 per square metre for the base and ` 50 per square metres for the sides, find the cost of least expensive tank. A point on the hypotenuse of a right triangle is at distances a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is a + b. 65/1/1 8
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