Mathematics_Today_-_October_2016

[sin ] / / ∫ xdx= π π 2 32 (where[.] is greatest integer function) (a) –π (b) π/2 (c) –π/2 (d) π Matrix-Match Type Match the ...

(c) : Let I = cos cos 21 21 x x − dx ∫ + ⇒ =− − I ∫∫x =− x dx x x (cos) dx (cos) sin cos 12 12 2 2 2 2 ⇒ Ixdx xdx=−∫tan^22 ...

∴ I= ∫ t (^1) dt 3 1 34 / = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟+= − ⎛ ⎝⎜ ⎞ ⎠⎟ + 1 314 4 3 1 2 t^1414 c x x c / / / (a) : I x dx dx xx = + = ∫ ∫ + 1 ...

(a) : Put x = tanθ ⇒ dx = sec^2 θdθ Also asxx== 00 and == 1 4 ,,θθπ Therefore, tan sec / − ∫∫= 1 0 1 2 0 4 xdx θθθd π =ππ− = − ...

(a) : f ′(x) = f (x) + k ′ ∫ fx =∫ fx k ()dx dx () ⇒ log(f (x) + k) = x + C ⇒ f(x) = k 1 ex – k ∴ (^) fkk() 0 4 e (^13) 2 = ...

Adding (i) and (ii), we get 21 2 2 2 Idx= ⋅ = − − ∫ α πα π α / ⇒ I=π−α 4 (B) Let I= xxdx − + ∫ sin^2 π^1 α π ... (i) ⇒ I= −xx dx ...

(8) : 2 ≥ max.{|x – y|, |x + y|} ⇒ |x – y| ≤ 2 and |x + y| ≤ 2, which forms a square of diagonal length 4 units.  2 –2  2  ...

DEFINITION An equation involving the derivative(s) of dependent variable y w.r.t. independent variable x or equation of dependen ...

(iii) Reducible to variable separable : If the equation is dy dx =++fax by c(), then put ax + by + c = z. (iv) Homogeneous equat ...

On squaring, we get dy dx dy dx 2 2 (^22) ⎛ 1 ⎝⎜ ⎞ ⎠⎟ =+⎛⎝⎜ ⎞⎠⎟ Clearly, order of differential equation is 2 and degree is also ...

Integrating, we get tan− + + (^1) ∫ 2 = 1 y dt t c (Where t = ex) ⇒ tan–1y + tan–1t = c ⇒ tan–1y + tan–1(ex) = c When x = 0, y = ...

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⇒−^1 += 3 v^3 log |vx| c ⇒−x += y yc 3 3 3 log | | This is the required solution of the given differential equation. Let P be t ...

SECTION-I Single Correct Answer Type A continuous function f : [0, 1] → R satisfies the equation ∫f x dx() =+^1 ∫f ( ) ,x dx 3 ...

dfx dx xx 2 2 12 ()=+δδ() () and f(0) = f ′(0) = 0. The value of f(5) is (a) 2 (b) 5 (c) 7 (d) 10 The value of f ′(5) is (a) 2 ...

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and^11 n f x nxdx n fx nxdx a b a b ∫ ′()cos ≤⋅∫|()||cos |′ ≤^1 ∫ = − → 0 n Mdx Mb a a n b () where |f ′(x)| ≤ some M Hence, f x ...

Only One Option Correct Type The equation of the tangent to the curve y = e–|x| at the point where the curve cuts the line x = ...

Let fx x bbb bb x xx () , , = − + − + − ++ ≤ < −≤< ⎧ ⎨ ⎪ ⎩⎪ 3 32 2 1 32 01 231 3 If f(x) has least value at x = 1, then ...

JEE MAIN If f (x, y) is a polynomial of degree 3 such that f (0, 0) = f (±1, 0) = f (0, ± 1) = f (2, 2) = 0, then f (a, b) = 0 ...