iitjee 2006
iitjee 2006 iitjee 2006
Q.17 A tangent at P(x,y) point on the curve y =f(x) intersects to the axes at A and B pointsrespectively such that AP : BP ≡ 1 : 3,given that f(1) = 1, then–(a) normal at (1, 1) is x + 3y = 4(b) equation of curve is 3y + xy′ = 0(c) curve passes through (2, 1/8)(d) equation of curve is xy′ – 3y = 0Ans. [b,c]Q. 18 f(x) is a cubic polynomial such that f(3) = 18,f(–1) = 2 and f(x) has local maximum atx = –1. If f ′(x) has local maximum at x =0, then(a) f(x) is increasing for x ∈ [1, 2, 5 ](b) the distance between (–1, 2) and (a, f(a))where x = a is the point of local minimumSection III :Questions based on comprehensionswith one correct answer.Passage – INow we define the definite integral using theformula ∫ b b − af (x) dx = (f(a) + f(b)), for morea2accurate result for c ∈ (a,b) ,b − a⎢⎣ 2b − c⎢⎣ 2⎡⎤ ⎡⎤F (c) = ( f(a) + f(c) ) ⎥⎦ + ( f(b) + f(c) ) ⎥⎦when c =a + b,2∫ b ⎛ b − a ⎞f (x) dx = ⎜ ⎟a ⎝ 4(f(a)+ f(b) + 2f(c))⎠is 2 5(c) f(x) has local minima at x = 1(d) the value of f(0) = 5Ans. [b c]Q.19 Let A is a vector parallel to line ofintersection of planes P 1and P 2throughorigin. P 1is parallel to the vectors 2 ĵ +3 kˆand 4 ĵ – 3 kˆ and P 2is parallel to ĵ – kˆπQ.21 Evaluate 2 sin x dx∫0π(a) (1 + 2 )8 2π(c) (1 2)4 +(b)π(18 + 2)(d)π(1 24 + 2)Ans [b]and 3 ĵ +3 kˆ than angle between vectorsA and 2 î + ĵ –2 kˆ is–π(a) 2(c)π6⎧ xe⎪Q.20 Let f(x) = ⎨ 2−ex−1⎪⎩ x −e(b) 4π3π(d)4Ans [b, d]0 ≤ x < 11
Passage – IITotal n urns each containing (n+1) balls suchthat the i th urn contains i white balls and(n + 1 – i) red balls.Now u ibe the event of selecting i th urn,i = 1, 2, 3 .... n and w denotes the event ofgetting a white ball.Q.24 If P (u i) ∝i, where i = 1, 2, 3 , ...... n, thenlimn → ∞P(w) is equal to(a) 32(b) 1Q.28 A variable circle touches to the line L andcircle C 1 externally such that both circlesare on the same side of the line, then thelocus of center of variable circle is –(a) ellipse(b) circle(c) Hyperbola (d) parabolaAns. [d]Q.29 A line M through A is drawn parallel to BD.Locus of point R, which moves such that itsdistances from the line BD and the vertex Aare equal, cuts to line M at T 2and T 3andAC at T 1, then area of triangle T 1T 2T 3is(c)34(d) 41Ans [a]Q.25 If P (u i) = c, where c is a constant thenP (u n/w) is equal to(a) 21 (sq.units)(c) 1 (sq.units)(b) 2(sq.units)(d) 34 (sq.units)Ans. [c]2(a)n + 1(c)1n + 1(b)nn + 11(d)2( n + 1)Ans [a]Passage – IVIf A =⎡10 1 ⎤⎢01 −1⎥, if U⎢⎣1 1 0 ⎥ 1, U 2and U 3are⎦column matrices satisfyingQ.26 If n is even and E denotes the event ofchoosing even numbered urn and alsoP(u i ) = n1 , then find the value of P (w/E)n + 2(a)2( n + 1)(b)n + 22n + 1n1(c)(d)n + 1n + 1Ans [a]Passage – IIILet C 1is a circle touching to all the sides ofsquare ABCD of side length 2 units internallyand C 2circle is passing through the verticesof square. A line L is drawn through A.Q.27 Let P is a point on C 1 and Q is on C 2 , then2 2 2 2PA + PB + PC + PD2 2 2 2QA + QB + QC + QD=(a) 0.75 (b) 0.5(c) 1.25 (d) 1Ans. [a]CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049217⎡1⎤⎡2⎤⎡3⎤A U 1 = ⎢0⎥, AU⎢⎣0⎥2 = ⎢3⎥, AU⎦ ⎢⎣0 ⎥ 3 = ⎢2⎥, and⎦ ⎢⎣1⎥⎦U is 3×3 matix whose columns are U 1, U 2,U 3then answer the following questions.Q.30 The value of |U| is –(a) 2 (b) 3(c) 6 (d) 12Q.31 Sum of the elements of U –1 is –(a) 1/12 (b) 1/6(c) 1/3 (d) 1/4Q.32 Find the value of –⎡3⎤[3 2 0] U ⎢2⎥⎢⎣0⎥⎦is -(a) 13 (b) 26(c) 12 (d) 24Ans. [b]Ans. [c]Ans. [a]PAPER - IIT-JEE-2006
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Passage – IITotal n urns each containing (n+1) balls suchthat the i th urn contains i white balls and(n + 1 – i) red balls.Now u ibe the event of selecting i th urn,i = 1, 2, 3 .... n and w denotes the event ofgetting a white ball.Q.24 If P (u i) ∝i, where i = 1, 2, 3 , ...... n, thenlimn → ∞P(w) is equal to(a) 32(b) 1Q.28 A variable circle touches to the line L andcircle C 1 externally such that both circlesare on the same side of the line, then thelocus of center of variable circle is –(a) ellipse(b) circle(c) Hyperbola (d) parabolaAns. [d]Q.29 A line M through A is drawn parallel to BD.Locus of point R, which moves such that itsdistances from the line BD and the vertex Aare equal, cuts to line M at T 2and T 3andAC at T 1, then area of triangle T 1T 2T 3is(c)34(d) 41Ans [a]Q.25 If P (u i) = c, where c is a constant thenP (u n/w) is equal to(a) 21 (sq.units)(c) 1 (sq.units)(b) 2(sq.units)(d) 34 (sq.units)Ans. [c]2(a)n + 1(c)1n + 1(b)nn + 11(d)2( n + 1)Ans [a]Passage – IVIf A =⎡10 1 ⎤⎢01 −1⎥, if U⎢⎣1 1 0 ⎥ 1, U 2and U 3are⎦column matrices satisfyingQ.26 If n is even and E denotes the event ofchoosing even numbered urn and alsoP(u i ) = n1 , then find the value of P (w/E)n + 2(a)2( n + 1)(b)n + 22n + 1n1(c)(d)n + 1n + 1Ans [a]Passage – IIILet C 1is a circle touching to all the sides ofsquare ABCD of side length 2 units internallyand C 2circle is passing through the verticesof square. A line L is drawn through A.Q.27 Let P is a point on C 1 and Q is on C 2 , then2 2 2 2PA + PB + PC + PD2 2 2 2QA + QB + QC + QD=(a) 0.75 (b) 0.5(c) 1.25 (d) 1Ans. [a]CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049217⎡1⎤⎡2⎤⎡3⎤A U 1 = ⎢0⎥, AU⎢⎣0⎥2 = ⎢3⎥, AU⎦ ⎢⎣0 ⎥ 3 = ⎢2⎥, and⎦ ⎢⎣1⎥⎦U is 3×3 matix whose columns are U 1, U 2,U 3then answer the following questions.Q.30 The value of |U| is –(a) 2 (b) 3(c) 6 (d) 12Q.31 Sum of the elements of U –1 is –(a) 1/12 (b) 1/6(c) 1/3 (d) 1/4Q.32 Find the value of –⎡3⎤[3 2 0] U ⎢2⎥⎢⎣0⎥⎦is -(a) 13 (b) 26(c) 12 (d) 24Ans. [b]Ans. [c]Ans. [a]PAPER - IIT-JEE-<strong>2006</strong>
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