IIT-JEE 2010 - Career Point

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so, h.at 2 = a 2 t 4 + 4a 2 t 2 Sol. Let P be (x 1 , y 1 ), at 2 [at 2 + (4a – h)] = 0 If a > 0 ⇒ 4a – h < 0 Q A P ⇒ h > 4a ...(i) Now point (at 2 ,2at) must lie inside the circle, on solving a 2 t 4 + 4a 2 t 2 – a 2 < 0 we get, h < a ( 5 + 2) ...(ii) from (i) & (ii) 4a < h < a ( 5 + 2) 5. Find the sum of the terms of G.P. a + ar + ar 2 + ..... + ∞ where a is the value of x for which the function 7 + 2x log e 25 – 5 x – 1 – 5 2– x has the greatest value and x t dt r is the Lt →0∫ 0 2 x tan( π + x a Sol. S = 1− r , x ) 2 To get the greatest value f´(x) = 2log e 25 – 5 x – 1 log 5 + 5 2– x log 5 f´(x) = 4 log e 5 – 5 x– 1 log e 5 + 5.5 1 – x log e 5 ⇒ f´(x) = 0 put 5 x – 1 t(> 0) t 2 – 4t – 5 = 0 ⇒ t = 5 ⇒ 5 x –1 = 5 ⇒ x = 2 to evaluate r : x t dt 1 r = Lt →0∫ x 0 2 x tan( π + x ) = π 2 1 2π since a = 2, r = ⇒ sum of G.P. = π π −1 6. The tangent and normal at a point P on the ellipse 2 x y = 1 meets the y-axis at A and B 2 a b respectively. Find the angle subtended by AB at the points of intersection of the circle (through A,S,B) and the ellipse. S being one of the foci. + 2 2 N B Points of intersection of tangent and normal at P ⎛ 2 b ⎞ ⎛ 2 a y ⎞ points with y-axis and A ⎜ ⎟ 0 , , B⎜ 1 ⎟ ⎝ y 0 , y1 − 2 , 1 ⎠ ⎝ b ⎠ S : (ae, 0) slope (SA). slope (SB) = –1 ⇒ ∠ASB = 90º (PA and PB are tangent and normal) P must lie on the circle with AB as diameter. Hence the point of intersection of the ellipse and the circle is P. Due to symmetry the angles made by AB at P,Q,M, N are all 90º. M Do you know • 100 years ago: The first virus was found in both plants and animals. • 90 years ago: The Grand Canyon became a national monument & Cellophane is invented. • 80 years ago: The food mixer and the domestic refrigerator were invented. • 70 years ago: The teletype and PVC (polyvinylchloride) were invented. • 60 years ago: Otto Hahn discovered nuclear fission by splitting uranium, Teflon was invented. • 50 years ago: Velcro was invented. • 40 years ago: An all-female population of lizards was discovered in Armenia. • 30 years ago: The computer mouse was invented. • 20 years ago: First test-tube baby born in England, Pluto’s moon, Charon, discovered. • 10 years ago: First patent for a geneticallyengineered mouse was issued to Harvard Medical School. S XtraEdge for IIT-JEE 50 DECEMBER 2009
MATH MONOTONICITY, MAXIMA & MINIMA Mathematics Fundamentals Monotonic Functions : A function f(x) defined in a domain D is said to be (i) Monotonic increasing : ⎧x1 < x 2 ⇒ f (x1) ≤ f (x 2 ) ⇔ ⎨ ∀ x 1 , x 2 ∈ D ⎩x1 > x 2 ⇒ f (x1) ≥ f (x 2) y x x O O ⎧x1 < x 2 ⇒ f (x1) > / f (x 2) i.e., ⇔ ⎨ ∀ x 1 , x 2 ∈ D ⎩x1 > x 2 ⇒ f (x1) < / f (x 2) (ii) Monotonic decreasing : ⎧x1 < x 2 ⇒ f (x1 ) ≥ f(x 2 ) ⇔ ⎨ ∀ x 1 , x 2 ∈ D ⎩x1 > x 2 ⇒ f (x 1) ≤ f(x 2 ) y x x O O ⎧x1 < x 2 ⇒ f (x1) < / f (x 2) i.e., ⇔ ⎨ ∀ x 1 , x 2 ∈ D ⎩x1 > x 2 ⇒ f (x1) / > f(x 2) A function is said to be monotonic function in a domain if it is either monotonic increasing or monotonic decreasing in that domain. Note : If x 1 < x 2 ⇒ f(x 1 ) < f(x 2 ) ∀ x 1 , x 2 ∈ D, then f(x) is called strictly increasing in domain D and similarly decreasing in D. Method of testing monotonicity : (i) At a point : A function f(x) is said to be monotonic increasing (decreasing) at a point x = a of its domain if it is monotonic increasing (decreasing) in the interval (a – h, a + h) where h is a small positive number. Hence we may observer that if f(x) is monotonic increasing at x = a then at this point tangent to its graph will make an acute angle with x- axis where as if the function is monotonic decreasing there then tangent will make an obtuse angle with x- axis. Consequently f´(a) will be positive or negative y y according as f(x) is monotonic increasing or decreasing at x = a. So at x = a, function f(x) is monotonic increasing ⇔ f´(a) > 0 monotonic decreasing ⇔ f´(a) < 0 (ii) In an interval : In [a, b], f(x) is monotonic increasing ⇔ f´(x) ≥ 0⎫ ⎪ monotonic decreasing ⇔ f´(x) ≤ 0⎬ ∀ x ∈ (a, b) constant ⇔ f´(x) = 0⎪ ⎭ Note : (i) In above results f´(x) should not be zero for all values of x, otherwise f(x) will be a constant function. (ii) If in [a, b], f´(x) < 0 at least for one value of x and f´(x) > 0 for at least one value of x, then f(x) will not be monotonic in [a, b]. Examples of monotonic function : If a functions is monotonic increasing (decreasing ) at every point of its domain, then it is said to be monotonic increasing (decreasing) function. In the following table we have example of some monotonic/not monotonic functions Monotonic increasing Monotonic decreasing Not monotonic x 3 1/x, x > 0 x 2 x|x| 1 – 2x |x| e x e –x e x + e –x log x log 2 x sin x sin h x cosec h x, x > 0 cos h x [x] cot hx, x > 0 sec h x Properties of monotonic functions : If f(x) is strictly increasing in some interval, then in that interval, f –1 exists and that is also strictly increasing function. If f(x) is continuous in [a, b] and differentiable in (a, b), then f´(c) ≥ 0 ∀ c ∈ (a, b) ⇒ f(x) is monotonic increasing in [a, b] f´(c) ≥ 0 ∀ c ∈ (a, b) ⇒ f(x) is monotonic decreasing in [a, b] XtraEdge for IIT-JEE 51 DECEMBER 2009
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