IITJEE Mathematics 2007

IIT JEE –Past papersMATHEMATICS- UNSOLVED PAPER - 2007

SECTION – ISingle Correct Answer TypeThis section contains 9 multiple choice questions numbered 1 to 9. Each question has 4 choices (A), (B), (C) and (D), out of which only one is correct.

01ProblemA hyperbola, having the transverse axis of length 2 sinθ, is confocal with the ellipse 3x2 + 4y2 = 12. Then its equation isx2 cosec2θ − y2 sec2θ = 1 x2 sec2θ − y2 cosec2θ = 1x2 sin2θ − y2 cos2θ = 1 x2 cos2θ − y2 sin2θ = 1

Problem02The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c − 1, ec−1) and (c + 1, ec+1)on the left of x = c on the right of x = cat no pointat all points

Problem03A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Arg and plane is3eiπ/4 + 4i (3 − 4i) eiπ/4(4 + 3i) eiπ/4(3 + 4i) eiπ/4

Problem04Let f(x) be differentiable on the interval (0,∞) such that f(1) = 1, and for each x > 0. Then f(x) isa.b.c.d.

Problem05The number of solutions of the pair of equations2 sin2θ − cos2θ = 02 cos2θ − 3 sinθ = 0in the interval [0, 2π] iszero onetwo four

Problem06Let α, β be the roots of the equation x2 − px + r = 0 and α/2 , 2β be the roots of the equation x2 − qx + r = 0. Then the value of r is2/9 (p-q)(2q-p)2/9 (q-p)(2p-q)2/9 (q-2p)(2q-p)2 /9(2p-q)(2q-p)

Problem07The number of distinct real values of λ, for which the vectors are coplanar, isZeroone two three

08ProblemOne Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is1/2 1/32/5 1/5

SECTION – IIAssertion - Reason TypeThis section contains 4 questions numbered 10 to 13. Each question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Problem10Let the vectors represent the sides of a regular hexagon.STATEMENT -1 :becauseSTATEMENT -2 : anda. Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1b. Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1c. Statement -1 is True, Statement -2 is Falsed. Statement -1 is False, Statement -2 is True

Problem11Let F(x) be an indefinite integral of sin2x.STATEMENT -1 : The function F(x) satisfies F(x + π) = F(x) for all real x.becauseSTATEMENT -2 : sin2(x + π) = sin2x for all real x.a. Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1b. Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1c. Statement -1 is True, Statement -2 is Falsed. Statement -1 is False, Statement -2 is True

12ProblemLet H1, H2, …, Hn be mutually exclusive and exhaustive events with P(Hi) > 0, i = 1, 2, …, n. Let E be any other even withSTATEMENT -1 : P(Hi | E) > P(E | Hi) . P(Hi) for i = 1, 2, …, nbecauseSTATEMENT -2 : a. Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1b. Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1c. Statement -1 is True, Statement -2 is Falsed. Statement -1 is False, Statement -2 is True

Problem13Tangents are drawn from the point (17, 7) to the circle x2 + y2 = 169.STATEMENT -1 : The tangents are mutually perpendicular.becauseSTATEMENT -2 : The locus of the points from which mutually perpendicular tangents can be drawn to the givencircle is x2 + y2 = 338a. Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1b. Statement -1 is True, Statement -2 is true; Statement-2 is NOT a correct explanation for Statement-1c. Statement -1 is True, Statement -2 is Falsed. Statement -1 is False, Statement -2 is True

SECTION – IIILinked Comprehension Type This section contains 2 paragraphs P14-16 and P17-19. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for Question Nos. 1 4 to 16Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.

Problem14The ratio of the areas of the triangles PQS and PQR is1 : 21 : 21 : 41 : 8

Problem15The radius of the circumcircle of the triangle PRS is53 3 2

Problem16The radius of the incircle of the triangle PQR is438/32

Paragraph for Question Nos. 17 and 19Let Vr denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r − 1). Let Tr = Vr+1 − Vr − 2 and Qr = Tr+1 − Tr for r = 1, 2, …

17ProblemThe sum V1 + V2 + … + Vn is1/12 n (n 1)(3n2-n+1)1/12 n (n 1)(3n2 +n+ 2)1/2 n (2n2 –n+ 1)1/3 (2n3 -2n +3)

Problem18Tr is alwaysan odd numberan even numbera prime numbera composite number

Problem19Which one of the following is a correct statement?Q1, Q2, Q3, … are in A.P. with common difference 5Q1, Q2, Q3, … are in A.P. with common difference 6Q1, Q2, Q3, … are in A.P. with common difference 11Q1 = Q2 = Q3 = …

SECTION – IVMatrix-Match TypeThis section contains 3 questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in column I have to be matched with statements (p, q, r, s) in column II. The answers to these questions have to be appropriately bubbled as illustrated in the following example.If the correct match are A-p, A-s, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows:

Problem20Consider the following linear equationsax + by + cz = 0bx + cy + az = 0cx + ay + bz = 0Match the conditions / expressions in Column I with statements in Column II and indicate your answers by darkeningthe appropriate bubbles in 4 × 4 matrix given in the ORS.

Column I Column II(A) a + b + c ≠ 0 and a2 + b2 + c2 = ab + bc + ca (p) the equations represent planes meeting only at a single point.(B) a + b + c = 0 and a2 + b2 + c2 ≠ ab + bc + ca (q) the equations represent the line x = y = z.(C) a + b + c ≠ 0 and a2 + b2 + c2 ≠ ab + bc + ca (r) the equations represent identical planes.(D) a + b + c = 0 and a2 + b2 + c2= ab + bc + ca (s) the equations represent the whole of the three dimensionalspace.

Problem21Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS. Column I Column II(A) (p)(B) (q)(C) (r)(D) (s)

Problem22In the following [x] denotes the greatest integer less than or equal to x. Match the functions in Column I with the properties Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS. Column I Column II(A) x |x| (p) continuous in (−1, 1)(B) X (q) differentiable in (−1, 1)(C) x + [x] (r) strictly increasing in (−1, 1)(D) |x − 1| + |x + 1| (s) not differentiable at least at one point in (−1, 1)

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IITJEE Mathematics 2007

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