FIITJEE Solutions to IIT - JEE 8 (Paper, Code 4) Time: hours M. Marks: 4 Note: (i) The question paper consists of parts (Part I : Mathematics, Part II : Physics, Part III : Chemistry). Each part has 4 sections. (ii) Section I contains 9 multiple choice questions. Each question has 4 choices (A),, and, out of which only one is correct. (iii) Section II contains 4 questions. Each question contains STATEMENT and STATEMENT. Bubble (A) if both the statements are TRUE and STATEMENT- is the correct eplanation of STATEMENT- Bubble if both the statements are TRUE but STATEMENT- is NOT the correct eplanation of STATEMENT- Bubble if STATEMENT- is TRUE and STATEMENT- is FALSE. Bubble if STATEMENT- is FALSE and STATEMENT- is TRUE. (iv) Section III contains sets of Linked Comprehension type questions. Each set consists of a paragraph followed by questions. Each question has 4 choices (A),, and, out of which only one is correct. (v) Section IV contains questions. Each question contains statements given in columns. Statements in the first column have to be matched with statements in the second column. The answers to these questions have to be appropriately bubbled in the ORS as per the instructions given at the beginning of the section. Marking Scheme: (i) For each question in Section I, you will be awarded Marks if you have darkened only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases, minus one ( ) mark will be awarded. (ii) For each question in Section II, you will be awarded Marks if you darken only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases, minus one ( ) mark will be awarded. (iii) For each question in Section III, you will be awarded 4 Marks if you darken only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases, minus one ( ) mark will be awarded. (iv) For each question in Section IV, you will be awarded 6 Marks if you have darken ALL the bubble corresponding ONLY to the correct answer or awarded mark each for correct bubbling of answer in any row. No negative mark will be awarded for an incorrectly bubbled answer. Mathematics PART I SECTION I Straight Objective Type This section contains 9 multiple choice questions. Each question has 4 choices (A),, and, out of which ONLY ONE is correct.. An eperiment has equally likely outcomes. Let A and B be two non-empty events of the eperiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is (A), 4 or 8, 6 or 9
IIT-JEE8-PAPER-- 4 or 8 5 or P(A B) = 4 p = p/5 p is an integer 5 p = 5 or.. The area of the region between the curves y = (A) + sin cos and y = t dt ( ) + t t + 4t dt ( ) + t t + sin cos 4t + t t ( ) ( ) bounded by the lines = and = 4 π is t + t t dt dt π /4 = = π /4 + sin sin d cos cos + tan tan + tan tan d = d tan + tan tan /4 tan 4t d = dt as tan t tan ( + t ) t =. π. Consider three points P = ( sin(β α), cosβ), Q = (cos(β α), sinβ) and R = (cos(β α + θ), sin(β θ)), where < α, β, θ < 4 π. Then (A) P lies on the line segment RQ R lies on the line segment QP Q lies on the line segment PR P, Q, R are non-collinear P ( sin(β α), cosβ) (, y ) Q (cos(β α), sinβ) (, y ) and R ( cosθ + sinθ, y cosθ + y sinθ) cosθ + sin θ ycosθ+ ysin θ We see that T, cosθ+ sin θ cosθ+ sin θ and P, Q, T are collinear P, Q, R are non-collinear. e 4. Let I = d, J = 4 e + e + 4 e e + (A) log C 4 + e + e + e d. Then, for an arbitrary constant C, the value of J I equals 4 e + e + e + e + log C + e e + 4 e e + e + e + log + C log 4 + C e + e + e e + e(e ) (z ) J I= d = dz 4 4 e + e + where z = e z + z +
IIT-JEE8-PAPER-- dz z e + e = = ln c + e + e + z+ z e e + J I = ln c +. e + e + 5. Let g() = log(f()) where f() is a twice differentiable positive function on (, ) such that f( + ) = f(). Then, for N =,,,, g N+ g = (A) 4 + + +... + 9 5 ( N ) 4 + + +... + 9 5 ( N ) 4 + + +... + 9 5 ( N ) 4 + + +... + + 9 5 ( N ) + (A) g( + ) = log(f( + )) = log + log(f()) = log + g() g( + ) g() = log g ( + ) g () = g + g = 4 4 g + g + = 9.. 4 g N+ g N = (N ) Summing up all terms Hence, g N+ g = 4 + + + 9 (N ). 6. Let two non-collinear unit vectors âandb ˆ form an acute angle. A point P moves so that at any time t the position vector OP (where O is the origin) is given by âcost+ bsint ˆ. When P is farthest from origin O, let M be the length of OP and û be the unit vector along OP. Then, (A) û = â + and M = ( + aˆ b) / â + â û = â and M = ( + aˆ b ˆ ) / û = and M = ( + aˆ b) / â û = â and M = ( + aˆ b ˆ ) / (A) OP = aˆ cos t + bsin ˆ t ( cos t sin t cos t sin t a ˆ b ˆ ) / ( costsintaˆ b ˆ ) / ( sintaˆ b ˆ ) / = + + = + = +
IIT-JEE8-PAPER--4 OP ( aˆ ) / = + when, t ma û = û =. π = 4 π π 7. Let the function g: (, ), be given by g(u) = tan (e u π ). Then, g is (A) even and is strictly increasing in (, ) odd and is strictly decreasing in (, ) odd and is strictly increasing in (, ) neither even nor odd, but is strictly increasing in (, ) u π g(u) = tan ( e ) = tan e tan e cot e g( ) = g() g() is odd and g () > increasing. u u u u u = tan e cot e 8. Consider a branch of the hyperbola y 4 y 6 = with verte at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is (A) + ( ) (y + ) Hyperbola is = 4 a =, b = e = Area = b ( ) ( ) a(e ) = = a + Area =. 9. A particle P starts from the point z = + i, where i =. It moves first horizontally away from origin by 5 units and then vertically away from origin by units to reach a point z. From z the particle moves units in the direction of the vector ˆ i ˆ π + j and then it moves through an angle in anticlockwise direction on a circle with centre at origin, to reach a point z. The point z is given by (A) 6 + 7i 7 + 6i z ( + i) z (6 + 5i) z ( 6 + 7i). 7 + 6i 6 + 7i
IIT-JEE8-PAPER--5 SECTION II Reasoning Type This section contains 4 reasoning type questions. Each question has 4 choices (A),, and, out of which ONLY ONE is correct.. Consider L : + y + p = L : + y + p + =, where p is a real number, and C : + y + 6 y + =. STATEMENT : If line L is a chord of circle C, then line L is not always a diameter of circle C. and STATEMENT : If line L is a diameter of circle C, then line L is not a chord of circle C. (A) STATEMENT is True, STATEMENT is True; STATEMENT is a correct eplanation for STATEMENT STATEMENT is True, STATEMENT is True; STATEMENT is NOT a correct eplanation for STATEMENT. STATEMENT is True, STATEMENT is False STATEMENT is False, STATEMENT is True Circle ( + ) + ((y 5) = 4 Distance between L and L 6 radius < statement () is false But statement () is correct.. Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation + p + q = and α, β are the roots of the equation a + b + c =, where β {,, }. STATEMENT : (p q) (b ac) and STATEMENT : b pa or c qa (A) STATEMENT is True, STATEMENT is True; STATEMENT is a correct eplanation for STATEMENT STATEMENT is True, STATEMENT is True; STATEMENT is NOT a correct eplanation for STATEMENT. STATEMENT is True, STATEMENT is False STATEMENT is False, STATEMENT is True Suppose roots are imaginary then β = α and =α β= not possible β β roots are real (p q) (b ac) statement () is correct. b α c =α+ and =, α + β = p, αβ = q a β β a If β =, then α = q c = qa(not possible) b b also α+ = p = b = ap(not possible) a a statement () is correct but it is not the correct eplanation.
IIT-JEE8-PAPER--6. Suppose four distinct positive numbers a, a, a, a 4 are in G.P. Let b = a, b = b + a, b = b + a and b 4 = b + a 4. STATEMENT : The numbers b, b, b, b 4 are neither in A.P. nor in G.P. and STATEMENT : The numbers b, b, b, b 4 are in H.P. (A) STATEMENT is True, STATEMENT is True; STATEMENT is a correct eplanation for STATEMENT STATEMENT is True, STATEMENT is True; STATEMENT is NOT a correct eplanation for STATEMENT. STATEMENT is True, STATEMENT is False STATEMENT is False, STATEMENT is True b = a, b = a + a, b = a + a + a, b 4 = a + a + a + a 4 Hence b, b, b, b 4 are neither in A.P. nor in G.P. nor in H.P.. Let a solution y = y() of the differential equation dy y y d = satisfy y() =. π STATEMENT : y() = sec sec 6 and STATEMENT : y() is given by = y (A) STATEMENT is True, STATEMENT is True; STATEMENT is a correct eplanation for STATEMENT STATEMENT is True, STATEMENT is True; STATEMENT is NOT a correct eplanation for STATEMENT. STATEMENT is True, STATEMENT is False STATEMENT is False, STATEMENT is True d dy = y y sec = sec y + c sec = sec + c π π π c = = 6 6 π sec = sec y + 6 π y= sec sec 6 cos = cos + π y 6 cos = cos cos y = y =. y
IIT-JEE8-PAPER--7 SECTION III Linked Comprehension Type This section contains paragraphs. Based upon each paragraph, multiple choice questions have to be answered. Each question has 4 choices (A),, and, out of which ONLY ONE is correct. Paragraph for Question Nos. 4 to 6 a+ Consider the function f : (, ) (, ) defined by f() = + a+ 4. Which of the following is true? (A) ( + a) f () + ( a) f ( ) = ( a) f () ( + a) f ( ) = f () f ( ) = ( a) f () f ( ) = ( + a) (A) 4a( + a+ ) 4a( )(+ a)( + a+ ) f() = 4 ( + a + ) 4a 4a f() = f = ( + a) ( a) ( + a) f () + ( a) f ( ) =. ( ) 5. Which of the following is true? (A) f() is decreasing on (, ) and has a local minimum at = f() is increasing on (, ) and has a local maimum at = f() is increasing on (, ) but has neither a local maimum nor a local minimum at = f() is decreasing on (, ) but has neither a local maimum nor a local minimum at = (A) a( ) f() = ( + a + ) Decreasing (, ) and minima at = 6. Let g() = e f () t dt + t which of the following is true? (A) g () is positive on (, ) and negative on (, ) g () is negative on (, ) and positive on (, ) g () changes sign on both (, ) and (, ) g () does not change sign on (, ) f(e )e g() = + e Hence positive for (, ) and negative for (, ).
IIT-JEE8-PAPER--8 Paragraph for Question Nos. 7 to 9 Consider the line L : + y + z + = =, L : y + z = = 7. The unit vector perpendicular to both L and L is (A) ˆi+ 7j ˆ+ 7kˆ 99 ˆi+ 7j ˆ+ 5kˆ 5 ˆi 7j ˆ+ 5kˆ 5 7i ˆ 7j ˆ kˆ 99 i j k = i 7j+ 5k i 7j+ 5k Hence unit vector will be. 5 8. The shortest distance between L and L is (A) 4 5 7 7 5 ( + )( ) + ( )( 7) + ( + )(5) 7 S. D = =. 5 5 9. The distance of the point (,, ) from the plane passing through the point (,, ) and whose normal is perpendicular to both the lines L and L is 7 (A) 75 75 75 75 Plane is given by ( + ) 7(y + ) + 5(z + ) = + 7y 5z + = + 7 5+ distance = =. 75 75
IIT-JEE8-PAPER--9 SECTION IV Matri-Match Type This contains 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 eample. If the correct match are A-p, A-s, B-r, C-p, C-q and D-s, then the correctly bubbled 4 4 matri should be as follows: p q r s A B C D p q r s p q r s p q r s p q r s. Consider the lines given by L : + y 5 = L : ky = L : 5 + y = Match the Statements / Epressions in Column I with the Statements / Epressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 4 matri given in the ORS. Column I Column II (A) L, L, L are concurrent, if (p) k = 9 One of L, L, L is parallel to at least one of the other two, if L, L, L form a triangle, if (q) k = 6 5 (r) k = 5 6 L, L, L do not form a triangle, if (s) k = 5 (A) (s); (p, q); (r); (p, q, s) + y 5 = and 5 + y = intersect at (, ) Hence 6 k = k = 5 for L, L to be parallel = k = 9 k for L, L to be parallel k = k = 6. 5 5 6 for k 5, 9, they will form triangle 5 6 for k = 5 k = 9, they will not form triangle 5
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