Mathematics_Today_-_October_2016

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* Alok Kumar is a winner of INDIAN NATIONAL MATHEMATICS OLYMPIAD (INMO-91). He trains IIT and Olympiad aspirants. DEFINITION The ...

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Remarks t
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x-axis is       y = 0. t
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parallel to x-axis (or perpendicular to y-a ...

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line whose equation is to be determine with the given line y = mx + c are yy m m − 11 = ± xx− 1 tan tan α() ∓ α^     ...

SOME IMPORTANT RESULTS z Area of the triangle formed by the lines y = m 1 x + c 1 , y = m 2 x + c 2 and y = m 3 x + c 3 , is^1 2 ...

(c) x x y 11 y += 1 (d) none of these The equation of line passing through the point of intersection of the lines 4x – 3y – 1 = ...

(a) a fixed point (b) a variable point (c) origin (d) none of these If u ≡ a 1 x + b 1 y + c 1 = 0 and v ≡ a 2 x + b 2 y + c 2 ...

If (3, 2), (–4, 1) and (–5, 8) are vertices of triangle then (a) orthocentre is (4, 1) (b) orthocentre is (–4, 1) (c) circumcen ...

SOLUTIONS (b) : A line perpendicular to the line 5x – y = 1 is given by x + 5y – λ = 0 = L, (given) In intercept form : xy λλ ...

(a) : Any line through the middle point M(1, 5) of the intercept AB may be taken as xy− (^15) = − =r cosθθsin ...(i) where ‘r’ ...

(c) : Since equation of      
 

 

  diagonal 11x + 7y = 9 does not pass through origin, so ...

Line (i) is parallel to x-axis, ∴ab+=⇒ =−a b λλ 0 Put the value of λ in (i), we get yba b b a b 2 2 3 3 0 22 + ⎛ ⎝⎜ ⎞ ⎠⎟ ++ = ⇒ ...

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(c) : S Q ab ab a b SQ Q QS S = − (^222) ⇒ = +± + 2 () (a) : Note that sides are ()sinab− α 2 and ()cosab− α 2 A → q ; B → ...

FUNDAMENTAL PRINCIPLES OF COUNTING z Multiplication Principle If an operation A can be performed in ‘m’ different ways and a sec ...

z Number of ways of choosing r things out of n given things if p particular things must be included (p ≤ r) is n – pCr (^) – (^) ...

z The number of ways in which m + n things can be divided into two groups containing m and n things respectively =()!+ !! mn. mn ...