This sample note is about Simple Problems in Mathematics.
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The Learning Hall Mathematics Class IX-X sep 23, 2018 Factorize 1. 27p3- 1/216 - 9p2/2 + p/4 Expression A. = 27p3- 1/216- 9p2/2 + p/4 = (3p)3- (1/6)3- + Which is of the form — y3 — 3x2y + 3x2y = (x-y)3 Hence, expression = (3p-1/6)3 In a rhombus ABCD, O is any interior point such that OA=OC. Then prove that D, O & B 2. are collinear A.
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c O Given ABCD is a rhombus O is any interior point such that OA=OC RTP D, O, B are collinear Const. Let OD & 0B be joined Proof In AAOB&A COB (Given) AO = CO (Common) (Since ABCD is a rhombus) AB = BC Hence, A AOB = A COB (By SSS Congruency) Hence, Z_ AOB=/ COB Similarly, A AOD• A COD Z— AOD = Z COD Now, BOC + / ( By c.p.c.t) ( By c.p.c.t) (ii) AOB = 3600
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From (i) & (ii) we get, 2 (Z AOB+Z COD) ( AOB+Z COD) = 3600 = 1800 3. A. Hence, D, O, B are collinear The volume of a cuboid is polynomial p(x) = 12 2x-3; Find possible expression for dimension of the cuboid. Verify the result by taking x: 5 units Volume of a cuboid p(x) = 8x3+ 12 x2- = (2x-1) Hence, the possible dimensions are Verification For x=5, dimensions are as follows 9 units 2x-1 = 2x5-1= = 2x5 +1 - - 11 units - 13 units Volume = 9 x 11 x 13 = 1287 cu. units Now, given volume = 8(5)3+ 12 (5)2-2(5)-3 = 1000 + 300 - 10-3 = 1287 cu. units and
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