iitjee 2006
iitjee 2006 iitjee 2006
IIT-JEE (2006)(Memory Based Question Paper)MathematicsSection I:Multiple Choice Questions with onecorrect answer.Q.1 If t 1 = (tanθ) tanθ , t 2 = (tanθ) cotθ ,t 3= (cotθ) tanθ , t 4= (cotθ) cotθ and letθ ∈ (0, 4π ) then(a) t 4< t 2< t 1< t 3(b) t 4< t 1< t 3< t 2(c) t 4< t 3< t 2< t 1(d) t 2< t 1< t 3< t 4Ans. [d]Q.2 The axis of parabola is along the line y =xand the distance of vertex from origin is 2and that from its focus is 2 2 . If vertex andfocus both lie in the first quadrant, so theequation of parabola is –(a) (x–y) 2 = 8 (x+ y– 2)(b) (x–y) 2 = 4 (x+ y– 2)(c) (x –y) 2 = (x + y– 2)(d) (x– y) 2 = (x – y– 2)Ans. [a]Q.3x⎛⎜⎜ sin x0 ⎜⎝Lim→( )1xsin x ⎞⎛ 1 ⎞ ⎟+ ⎜ ⎟ ⎟⎝ x ⎠ , for x > 0⎟⎠(a) 0 (b) –1(c) 2 (d) 1 Ans. [d]Q.4 Given an isoscels triangle, whose one angleis 120° and inradius istriangle is–3 . So the area ofQ.5 Let a, b, c be sides of a triangle and any twoof them are not equal and λ∈R. If the rootsof the equation. tx 2 + 2 (a+b+c)x+3λ(ab+bc+ca) = 0 are real, then(a) 34 < λ < 35(c) λ > 35(b)(d) λ < 341 5 < λ 0. When 0< θ < 2π⎛13π ⎞(a) ⎜ , 2π ⎟⎝ 48 ⎠(c)Q.7∫ 3x(a)(c)⎛ π 5π⎞⎜ , ⎟⎝ 8 6 ⎠⎛ π ⎞ ⎛ π 5π⎞(b) ⎜0,⎟ ∪ ⎜ , ⎟⎝ 8 ⎠ ⎝ 6 6 ⎠⎛ π ⎞ ⎛ 5π ⎞(d) ⎜0,⎟⎝ 6∪ ⎜ , 2π ⎟⎠ ⎝ 6 ⎠2x − 1dx is equal to4 22x − 2x + 14 22 x − 2x + 122x(b)4 22 x − 2x + 12x+ c4 22 x − 2x + 1+ c3x+ cAns. [d](a) 4π (b) 12 + 7 3(c) 7+12 3 (d) 12 – 7 3Ans. [b](d)Ans. [a]4 22x − 2x + 1x+ cCAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049214PAPER - IIT-JEE-2006
Q.8 If ω = α+iβ where β ≠ 0 and z ≠ 1 satisfies⎛ ω − ωz⎞the condition that ⎜ ⎟⎝ 1−z ⎠is purely real,then set of z is –(a) {z : z = z } (b) {z : |z| = 1, z ≠1}(c) {z : z ≠ 1} (d) {z : |z| = 1}Ans. [b]Q.9 A plane passes through (1, –2, 1) and isperpendicular to two planes 2x – 2y + z = 0and x – y + 2z = 4. The distance of theplane from point (1, 2, 2) is –(a) 2 2 (b) 0(c) 1 (d) 2 Ans. [a]Q.10 If f ′′ (x) = –f(x) and g(x) = f′(x) and22⎛ ⎛ x ⎞⎞⎛ ⎛ x ⎞⎞F(x) =⎜f⎜⎟2⎟ +⎜g⎜⎟⎝ ⎝ ⎠⎠2⎟⎝ ⎝ ⎠⎠and given thatF(5) = 5, then F (10) is(a) 15 (b) 0(c) 5 (d) 10 Ans. [c]Q.11 Let a = î + 2 ĵ +kˆ , b = î – ĵ +kˆand c = î + ĵ – kˆ . A vector in the planeof a and b whose projection on c is of1length unit is –3(a) 4 î + ĵ – 4 kˆ(c) 2 î + ĵ – 2 kˆ(b) 4 î –ĵ + 4 kˆ(d) 3 î +ĵ – 3 kˆAns. [b]Q.12 If r, s, t are prime numbers and p, q are thepostive integers such that the LCM of p,q isr 2 t 4 s 2 , then the number of ordered pair (p,q)is –(a) 224 (b) 225(c) 252 (d) 256Ans.[b]Section II:Multiple Choice Questions with one ormore correct answer(s).Q.13 If f(x) = min{1, x 2 , x 3 }, then(a) f ′(x) > 0 ∀ x ∈ R(b) f(x) is continuous ∀ x ∈ R(c) f(x) is not differentiable for two values of x(d) f(x) is not differentiable but continuous∀ x ∈ R Ans. [b, d]Q.14 If a hyperbola passes through the focus of2 2x ythe + = 1 and its transverse and25 16conjugate axes coincide with the major andminor axis of ellipse, and product ofeccentricities is 1, then(a) Focus of hyperbola is (5, 0)(b) Focus of hyperbola is (5 3 , 0)2 2x y(c) The equation of hyperbola is − = 19 25(d) The equation of hyperbola is2 2x y− = 19 16Ans. [a, d]Q.15 The equation(s) of common tangent(s) to theparabola y = x 2 and y =–(x–2) 2(a) y = – 4 (x–1) (b) y = 0(c) y = 4(x–1) (d) y = – 30x–50Ans. [b, c]Q.16 Internal angle bisector of ∠A of triangle ABC,meets side BC at D. A line drawn through Dperpendicular to AD intersects the side ACat P and the side AB at Q. If a, b, c representthe sides of ∆ABC then.(a) AD =(b) PQ =2bcb + c4bcb + ccos 2Asin 2A(c) The triangle APQ is isosceles(d) AP is HM of b and cAns.[a, b, c, d]CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049215PAPER - IIT-JEE-2006
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Q.8 If ω = α+iβ where β ≠ 0 and z ≠ 1 satisfies⎛ ω − ωz⎞the condition that ⎜ ⎟⎝ 1−z ⎠is purely real,then set of z is –(a) {z : z = z } (b) {z : |z| = 1, z ≠1}(c) {z : z ≠ 1} (d) {z : |z| = 1}Ans. [b]Q.9 A plane passes through (1, –2, 1) and isperpendicular to two planes 2x – 2y + z = 0and x – y + 2z = 4. The distance of theplane from point (1, 2, 2) is –(a) 2 2 (b) 0(c) 1 (d) 2 Ans. [a]Q.10 If f ′′ (x) = –f(x) and g(x) = f′(x) and22⎛ ⎛ x ⎞⎞⎛ ⎛ x ⎞⎞F(x) =⎜f⎜⎟2⎟ +⎜g⎜⎟⎝ ⎝ ⎠⎠2⎟⎝ ⎝ ⎠⎠and given thatF(5) = 5, then F (10) is(a) 15 (b) 0(c) 5 (d) 10 Ans. [c]Q.11 Let a = î + 2 ĵ +kˆ , b = î – ĵ +kˆand c = î + ĵ – kˆ . A vector in the planeof a and b whose projection on c is of1length unit is –3(a) 4 î + ĵ – 4 kˆ(c) 2 î + ĵ – 2 kˆ(b) 4 î –ĵ + 4 kˆ(d) 3 î +ĵ – 3 kˆAns. [b]Q.12 If r, s, t are prime numbers and p, q are thepostive integers such that the LCM of p,q isr 2 t 4 s 2 , then the number of ordered pair (p,q)is –(a) 224 (b) 225(c) 252 (d) 256Ans.[b]Section II:Multiple Choice Questions with one ormore correct answer(s).Q.13 If f(x) = min{1, x 2 , x 3 }, then(a) f ′(x) > 0 ∀ x ∈ R(b) f(x) is continuous ∀ x ∈ R(c) f(x) is not differentiable for two values of x(d) f(x) is not differentiable but continuous∀ x ∈ R Ans. [b, d]Q.14 If a hyperbola passes through the focus of2 2x ythe + = 1 and its transverse and25 16conjugate axes coincide with the major andminor axis of ellipse, and product ofeccentricities is 1, then(a) Focus of hyperbola is (5, 0)(b) Focus of hyperbola is (5 3 , 0)2 2x y(c) The equation of hyperbola is − = 19 25(d) The equation of hyperbola is2 2x y− = 19 16Ans. [a, d]Q.15 The equation(s) of common tangent(s) to theparabola y = x 2 and y =–(x–2) 2(a) y = – 4 (x–1) (b) y = 0(c) y = 4(x–1) (d) y = – 30x–50Ans. [b, c]Q.16 Internal angle bisector of ∠A of triangle ABC,meets side BC at D. A line drawn through Dperpendicular to AD intersects the side ACat P and the side AB at Q. If a, b, c representthe sides of ∆ABC then.(a) AD =(b) PQ =2bcb + c4bcb + ccos 2Asin 2A(c) The triangle APQ is isosceles(d) AP is HM of b and cAns.[a, b, c, d]CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049215PAPER - IIT-JEE-<strong>2006</strong>
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