FILL THE ANSWER HERE

Transcription

1

2 HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) ( ) at = is If a triangle has two sides of length and and has maimum area and and s be respectively area & semi-perimeter of the triangle then - (A) = s = s = s none of these. If A = a 0 and n 0 a lim A n n 0 0, then is equal to - (A) 0 none of these 5. The equation of the tangents to the curve (+ )y = at the point of its intersection with the curve ( + )y = are given by - (A) + y = ; y = + y = ; = + y = ; y = + y = ; = cos, 0 6. If for the function ƒ () =, LMVT is applicable in [, ], then (m, c) is - m c, 0 (A),,, 7. The angle of intersection between the curves y = (A) tan cot, 5 t dt and -ais is (where 0) - cot If A 0 0 and A pi = qa + ra, then the value of p + q + r is - sin 5 (A) 6 none of these FILL TH ANSWR HR. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D

3 9. Tangents are drawn to y = cos from the point P(0, 0). Points of contacts of these tangents will always lie on - (A) y y + y = y = cos, 0. Let ƒ () =. If ƒ () has local minima at = then complete set of values of 'a' are - a, (A) (, ] (, ) (, ) [, ). If A = is an orthogonal matri then the value of + y is equal to - y z (A) 0. If the quadratic equation (m + ) + (n + ) + (p + ) = 0, where m, n, p I, has real roots then- (A) both roots are rational both roots are irrational both roots are positive both roots are of opposite sign. The number of values of K for which, the equation ƒ () = + K = 0 has two real distinct roots in the interval (, ) is - (A) 0. If and are roots of the equation + = 0, then equation whose roots are + and ( + ) is - 5. If (A) + = 0 + = 0 + = 0 + = 0 0 ab ac a b 0 bc a c cb 0 = a p b q c r, then 0 + p + q + r is equal to - (A) Let ƒ() and g be the inverse of ƒ, then area bounded by the curve y = g() and -ais from = to =, is - (A) sq. units 7 sq. units 9 sq. units can not be determined 7. The surface area of a spherical balloon, being inflated, changes at a rate proportional to time t. If initially its radius is units and after seconds it is 5 units, then radius after seconds is - (A) 5 units 5 units 9 units 7 units 9. A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D

4 8. The solution of the differential equation y y = y is - (A) = A y + A y + A = A y + A = A y + A y none of these 9. ƒ '() Let ƒ () be a differentiable function satisfying the equation ƒ( ) e R. If ƒ '() =, then the number of solutions of the equation ƒ () = ƒ '() is - (A) none of these 0. Let ƒ () be a second degree polynomial function such that n(ƒ()) 0 R & the equation ƒ '() + 786ƒ () = 0, has no real roots. If g() = e 786 ƒ (), then - (A) g() is an increasing function g() is a decreasing function g() is an even function the graph of g() cuts -ais eactly once.. The number of real roots common between the two equations = 0 and + +7+=0 is - (A) 0. The area of the smaller portion above -ais bounded by the curves y = 8 and is - (A) ( ) y, 6 6. If the slope of tangent to curve y = e cos possess local maima at = a, then 'a' equals - (A) 0. The set of values of 'a' for which the function ƒ () = (a )( + n5) + (a 7) sin does not posses critical points is - (A), (, ) (, ) [, ) (, ) If A = and B = 5, and A = B, then the number of value(s) of is - (A) no value 6. Number of different values of satisfying the equation ( + ) + ( ) = 7 ( ) are- (A) 6 infinite 8. A B C D 9. A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D

5 If A = 0 0 and () = , then (A) = 5 5 (A) If A =, then A 50 is (A) If ( + ) = + then sum of the roots of () = 0 is : (A) If and are roots of the equation a + b + c = 0 then roots of the equation a( + ) b( + )( ) + c( ) = 0 are - (A),,, none of these. If A and det(an I) = n, n N then is - (A) r r. If the matri M r is given by Mr r r, r =,,,..., then the value of det(m ) + det(m ) det(m 008 ) is - (A) (008) (007). The equation (0p q + r)(p q 5r)(5p q r) = 0, (qpr 0) has atleast - (A) real roots real roots 6 real roots data insufficient. If AA T = I and det(a), then - (A) every element is equal to its cofactor every element is equal to additive inverse of its cofactor every element and its cofactor are multiplicative inverse of each other. none of these 5. If A is an idempotent matri then (I + A) 0 is equal to - (A) I + A I + 0A I + 0A I + 0A 7. A B C D 8. A B C D 9. A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D

6 6. If A = sin sin sin and B = cos cos cos where,, R and C = A 5 + B A B (A + B ) + 0A B (A + B ) then C = (A) I 5I I (A + B ) 7. The set of values of 'a' for which the quadratic a + ( a) is negative for eactly three integral values of, is - (A) (0, ) (0, ] [, ) [, ) 8. If ƒ () 0 R and area bounded by the curve y = ƒ (), = 0, = a and -ais is tan a, then the number of solutions of the equation ƒ () = tan is - (A) 0 infinitely many 9. Let ƒ () = + a + b with a b and suppose the tangent lines to the graph of ƒ at = a and = b have the same gradient. Then the value of ƒ () is equal to - (A) 0 0. A solid rectangular brick is to be made from cu feet of clay. The brick must be times as long as it is wide. The width of brick for which it will have minimum surface area is a. Then a is - (A) 9 / Consider the function ƒ () = 0 0, and identify the correct statement - (A) Mean value theorem is applicable in [, ] Mean value theorem is not applicable in [ 5, ] 5 Mean value theorem is applicable in [, ] and its c is Mean value theorem is applicable in [, ] and its c is. If y = e (K + ) is a solution of differential equation 5 d y d dy y 0, then k = d (A) 0. If ƒ is a differentiable function for all real and ƒ '() 5, R. If ƒ () = 0 and ƒ (5) = 5, then ƒ () - (A) A B C D 7. A B C D 8. A B C D 9. A B C D 0. A B C D. A B C D. A B C D. A B C D

7 . The angle of intersection of = y and + 6y = 7 at (, ) is - (A) 5 5. Which of the following statements is true for the general cubic function ƒ() = a + b + c + d (a 0) I. If the derivative ƒ '() has two distinct real roots then cubic has one local maima and one local minima. II. If the derivative ƒ '() has eactly one real root then the cubic has eactly one relative etremum. III. If the derivative ƒ '() has no real roots, then the cubic has no relative etrema (A) only I & II only II and III only I and III all I, II, III are correct. 6. The order and degree of the differential equation dy d y 7 0 are a and b, then a + b is - d d (A) The differential equation of the family of curves represented by y = a + b + ce (where a, b, c are arbitrary constants) is - (A) y''' = y' y''' + y'' = 0 y''' y'' + y' = 0 y''' + y'' y' = 0 8. The solution of differential equation ( + )dy = (y y )d is - (A) y n n c y y n n c y y n n c y y n n c y 9. Number of real roots of the equation e 5 is - (A) can not be determined 50. The positive value of the parameter 'a' for which the area of the figure bounded by the curve y = cosa, 5 y = 0, = and = is greater than are - 6a 6a (A) 0, 6 (, ), 5. A boat leaves the dock at PM and travels due south at a speed of 0 km/h. Another boat has been heading due east at 5 km/hr and reaches the same dock at PM. At what time the two boats were closest to each other - (A) ::6 PM :5:00 PM :5:5 PM ::0 PM. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D 50. A B C D 5. A B C D

8 5. The solution of the differential equation y..( y ) ( y ). y dy dy d d is - y (A) log y + tan = k y log y tan = k log y tan ( y ) = k log y + tan ( y ) = k 5. Area bounded by y = + 6 5, y = + and y = 5, for >, is - (A) 7 /6 7/6 none of these 5. The general solution of the differential equation tan y y (A) tan = y + tan y = y + 7 dy is - d ce tan y = + ce tan = + ce ce y 55. The angle between the tangents drawn to the curve y = y at the points where it meets with y = y is - (A) tan Area bounded by y + y 5y + = 0 is equal to - (A) tan dy 57. The solution of differential equation y.e y ny d is - (A) n = e y + cy e + ny = c ny = e + c n = e y + c 58. If (A) d dy dz, then z in terms of & y can be epressed as - tan( y) cot( y) sin( y) z C sin( y) z C cos( y) z C None of these 59. Area bounded by curve y =, -ais & the lines = and = is - (A) 9 5. A B C D 5. A B C D 5. A B C D 55. A B C D 56. A B C D 57. A B C D 58. A B C D 59. A B C D

9 60. Two villages X & Y are on the same side of a straight river at distances 'a' & 'b' respectively from river. A pumpset is to be installed by the river side at a point P. If the villages are situated at a distance 'c', then - (A) minimum value of XP + PY is c minimum value of XP + PY is c ab minimum value of XP + PY is c ab minimum value of XP + PY is c ab ab 6. The hands of an accurate clock have lengths and, then distance between the tips of the hands when the distance is increasing most rapidly is - (A) If the function ƒ () = cos a + b be increases along the entire number line, the range of values of a is given by - (A) a < b b a a a 6. If are the roots of the equation = 0 and + = 0, then is equal to - (A) If the roots of (p ) p + 5p = 0 are real and positive and p R, then p belongs to - (A) (, 5) (0, ) 0, 5, If the roots of m n 0 are real & distinct, then the roots of ( n)( m n) ( n m )( ) will be - (A) real & equal real & different imaginary 8 may be real or imaginary 66. The integral values of k for which the roots of k( +) + k ( + ) = 0 are rational is given by - (A) If one of the roots of a + b + c = 0 is greater than and the other is less than and if the roots of c + b + a = 0 are then - (A) < & & > & > > & 60. A B C D 6. A B C D 6. A B C D 6. A B C D 6. A B C D 65. A B C D 66. A B C D 67. A B C D

10 MULTIPL OBJCTIV TYP 68. Which of the following is true for the differential equation = (A) order of differential equation is degree of differential equation is degree of the differential equation is not defined degree of differential equation is d y d d y d 69. The general solution of the differential equation e d = e dy is - (A) y = K cos (e ) y = sin (e ) + C y = sin e + C y = cos e + C b, where b R Let A =, then - (A) A 5A + I = 0 A = 5I A A = adj A A is non-invertible matri 7. If the tangent to the curve y = t + at the point (t, t) cuts of the intercepts p, q on the & y-aes respectively, where A (p, q) is a point on the circle centred at origin and radius 6, then the value of 't' can be - (A) Which of the following statements are false - (A ) If ƒ () is increasing function in its domain and, belongs to the domain of the function, then > ƒ ( ) > ƒ ( ) If the function has local maima at = a, then it implies function is increasing for a < < a and function is decreasing in a < < a +, where '' can be sufficiently small. If ƒ () = a + b + c + d is monotonic function, then ƒ '() = 0 has no real roots. The order of the differential equation, representing family of cubic polynomials, having eactly one critical point is three If ƒ() a is a bounded function, then 'a' can be - (A) 6 7. The area bounded by the circles + y =, + y = and the pair of lines ( y ) y is equal to - (A) A B C D 69. A B C D 70. A B C D 7. A B C D 7. A B C D 7. A B C D 7. A B C D

11 75. If y = g() is a curve which is obtained by the reflection of y = ƒ () = then which of the following is/are true - (A) y = g() has eactly one tangent parallel to -ais y = g() has no tangent parallel to -ais The tangent to y = g() at (0, 0) is y = 0 e e about the line y =, g() has no etremum 76. The differential equation of the curve for which intercept cut by any tangent on y-ais is equal to the length of the sub normal - (A) is linear is homogeneous of first degree has separable variables is of first order 77. If eqautions (a + ) + b + c = 0 and + + = 0 have a common root where a, b, c N then - (A) b ac < 0 minimum value of a + b + c is 6 b < ac + 8c minimum value of a + b + c = 7 a b b c 78. b a a c is divisible by - a b a b b (A) (a b) (a b) a + b (a + b + c) 79. If A is an invertible matri then (adja) = (A) adj(a ) A det A A (deta)a 80. If AA T = I then identify the correct statement - (A) A T (A I ) = (A I ) T A is always invertible det(a I ) = 0 A is singular 8. If g() = 7 e R, then g() has - (A) local maimum at = 0 local minima at = 0 local maimum at = two local maima and one local minima 8. quation of common tangent(s) of y = and y = 8 is (are) - (A) y = + 6 y = + 6 y = + 6 y = 6 8. cos a, 0 The value of 'a' for which the function ƒ () has a local minimum at =,, is - (A) 0 / 8. The families of curves defined by the equations y = b and + y = a are orthogonal if - (A) a =, b = a =, b = 5 a =, b = a = 5, b = 75. A B C D 76. A B C D 77. A B C D 78. A B C D 79. A B C D 80. A B C D 8. A B C D 8. A B C D 8. A B C D 8. A B C D

12 85. If differential equation corresponding to family of curves y = A cos + B sin + C is given by d y d y dy ƒ() (cos ), where is real constant and ƒ () is some function in, then - d d d (A) = 0 = Number of solution of equation ƒ() = in (0,) are Number of solution of equation ƒ() = in (0,) are 86. If y satisfies the differential equation dy = y(y + )d & y() =, then - (A) lim y lim y 0 curve y = ƒ () is symmetric w.r.t. origin curve y = ƒ () is continuous R. k, 87. Let ƒ(). If ƒ() has a local minimum at =, then k can be equal to -, (A) If three quadratic equations BC AC AB ( AB AC) 0, ( BC AB) 0 and ( AC BC) 0 are given, where A, B, C are three points in a plane then - (A) if A, B, C are making a triangle then all equations have real & distinct roots. if A, B, C are collinear, then all equations have real & equal roots. if A, B, C are collinear, then two equations have real & distinct, one equation has real & equal roots. for every position of A, B, C all si roots are negative. 89. If a + b + c (a > & a, b, c are integers) is equal to p for two distinct integral values of, p prime, then a + b + c cannot be equal to p for - (A) = 0 = = = RASONING TYP 90. Consider A and B are square matrices of order such that AB = A and BA = B. Statement- : If (A B) 8 = k(a + B), then k = 0. and Statement- : (A B) is a null matri. (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. 85. A B C D 86. A B C D 87. A B C D 88. A B C D 89. A B C D 90. A B C D

13 b 9. Let B is square matri of order such that B = 0 and A = I B, X = y & b = b z b Statement- : System of linear equations AX = b has the solution X = b + Bb + B b. and Statement- : A = I + B + B. (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. 9. Four numbers a, b, c, d are such that the first three form an increasing geometric progression and last three form an arithmetic progression also a + d =, b + c = and three lines are defined as : L = a y + 5 = 0 L = + by 7 = 0 L = 9 5y + c = 0 Statement- : L, L, L are concurrent. and Statement- : L = L + L. (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. 9. Consider matri A, such that A = [a ij ], a ij I. Statement- : sgn(det(adja) 8) 0, for any set of value of a ij. and Statement- : det(adja) 0 for every set of values of a ij. {.} represents fractional part function. (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. n n 9. Statement- : n, where, > 0. and Statement- : If ƒ ' () > 0 & ƒ '' () < 0 D ƒ, then ƒ () increases with concavity downward and any chord of the curve lies below the curve. (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. 9. A B C D 9. A B C D 9. A B C D 9. A B C D

14 y z (z y ) z ( z ) y (y ) Statement- : If = y z (y z ) 6 6 z (z ) 6 6 y ( y ) y z (z y ) z ( z ) y (y ) equal to and. and = y z y z 5 6 y z then is Statement- : If elements of any determinant of order are replaced by their respective cofactors then the value of determinant thus formed is equal to. (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. 96. Statement- : and = n n i Statement- : i 0 = i i 0 (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. 97. Statement- : If a, b R and a < b, then there is atleast one real number c (a, b) such that c b a a b c and Statement- : If ƒ () is continuous in [a, b] and derivable in (a, b) & ƒ '(c) = 0 for atleast one c (a, b), then it necessarily implies that ƒ (a) = ƒ (b). (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. 95. A B C D 96. A B C D 97. A B C D

15 98. Statement- : If the tangent to the curve y = at point e P e, makes an angle with positive e -ais, then e cos <. and Statement- : For the curve y = c, dy 0 t R d t (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. 99. Statement- : and e 5. log log e log 5 Statement- : The function ƒ() = decreases in 0, & increases in, n. n (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. COMPRHNSION Paragraph for Question 00 to 0 Let ƒ () = +, R and a, b, c are roots of ƒ () = 0, P(, ƒ ()) is point of local minima Q(, ƒ ()) is point of local maima and R (, ƒ ()) is point of inflection in the graph of y = ƒ (). On the basis of above information answer the following : 00. ƒ(a) ƒ ''(a) ƒ(b) ƒ ''(b) ƒ(c) ƒ ''(c) is equal to - (A) (a b)(b c)(c a) (a b)(b c) (c a) cos (cos) cos 0. Triangle PQR is - (A) equilateral isosceles right angle none of these 0. Number of distinct real solutions of ƒ(ƒ()) = 0 are - (A) A B C D 99. A B C D 00. A B C D 0. A B C D 0. A B C D

16 Paragraph for Question 0 to 05 a + by + c = 0 b + cy + a = 0 c + ay + b = 0 is a system of linear equation then answer the following questions : 0. If a b c and a + b + c 0 then the system of linear equation have - (A) infinite solution all lying on a line entire y plane as solution unique solution no solution 0. If a = b = c and a + b + c 0 then the system of linear equation have - (A) infinite solution all lying on a line entire y plane as solution unique solution no solution 05. If a = b = c and a + b + c 0 then the system of linear equation have - (A) infinite solution all lying on a line entire y plane as solution unique solution no solution Paragraph for Question 06 to 08 Consider the system of linear equation y + bz = a + z = 5 + y = On the basis of above informations, answer the following questions. 06. If ab = and a then system of linear equations has - (A) no solution infinite solution unique solution finitely many solutions 07. If ab then system of linear equations has - (A) no solution infinite solution unique solution finitely many solutions 08. If a = & b = then system of equations has - (A) no solution infinite solution unique solution finitely many solutions Paragraph for Question 09 to Graph of ƒ () = a + b + c is shown adjacently, for which (AB) =, (AC) = and b ac =. On the basis of above informations, answer the following questions : 09. The value of a + b + c is equal to - (A) A B C D 0. A B C D 05. A B C D 06. A B C D 07. A B C D 08. A B C D 09. A B C D 5 A B C O

17 0. The quadratic equation with rational coefficients whose one of the roots is b + a c, is - (A) 6 + = 0 6 = = = 0. Range of g() = (a + ) + (b + ) (c ) when [, 0] is - (A) [ 0, 6] 9, 0 9, 6 9, Paragraph for Question to If a b then provided a & b are of same sign but if a and b are of opposite sign then a b,, a b. On the basis of above information, answer the following :. If 5 then belongs to - (A) [, ] [, ] 6,,. If < 7 < then belongs to (A),,,, If (, ] then 5 6 belongs to - (A) (, 7] (, 7] [, ) (, ) [7, ) Paragraph for Question 5 to Consider matri A 0 0 ; B Let C = AB, D = B I X = D + D + D D n. As n approaches to infinity matri X tends to matri Y. Let Y + Z = I. f() = trace of matri C. g() = f (), 0, where g() is an odd function. h(), 0 On the basis of above information, answer the following : 0. A B C D. A B C D. A B C D. A B C D. A B C D

18 7 5. The matri Z is - / / 0 0 (A) / / / 0 0 / 6. If the range of f() is [a, ) then the value of cos cos(a) is - (A) none of these 5 7. The range of the function sec h() is - (A),,,, none of these Paragraph for Question 8 to 0 Consider an open cylindrical reservoir whose cross-sectional view is as cm shown in the figure. The thickness of the wall of cylinder is cm as shown in the figure & the volume of the reservoir is 7 cm. Let & h be the radius and the height of the cylinder and h b c v() = ( a), where v() represents the volume of the material required to construct the cylinder epressed as a function of radius '' of the cylinder. On the basis of above informations, answer the following questions : 8. a.b c is - (A) prime number but not an even number even number but not a prime number even prime number irrational number 9. If the material cost to construct the cylinder is minimum, then which of the following relations must hold between & h - (A) = h = h = h none of these 0. If the function v() is redefined such that v : R 0 R, then number of solutions of the equation v() = 0 are - (A) Paragraph for Question to Consider the differential equation e (yd dy) = e (yd + dy). Let y = ƒ () be a particular solution to this differential equation which passes through the point (0, ) Let C y = log log (6 8 ), be another curve. On the basis of above information, answer the following questions : 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D 0. A B C D

19 . The range of the function g() = log (ƒ ()) is - (A) [, ) [, ) [0, ) none of these. The area bounded by the curve C, parabola = y + and the line = is - (A). If the area bounded by the curve y = ƒ (), curve C, ordinate = & the ordinate = a is, then the value of a is - e n e (A) n6 n n Paragraph for Question to 6 The slope of the tangent to the curve y =ƒ() at (0,0) is and curves at the points having equal abscissae cut the y-ais at same point, then On the basis of above information answer the following :. Area bounded by curves y and y between =0 and =e, is - (A) e+ e e y ƒ(t)dt. If the tangents to both 0 5. y lim 0 is - (A) 0 doesn't eist 6. Number of solutions of the equation y y = k (where k is constant), is - (A) 0 Paragraph for Question 7 to 9 Consider the function defined implicitly y + y = on various intervals on the real line. If y,, the equation implicitly defines a unique real valued differentiable function y = ƒ(). On the basis of above informations, answer the following questions. 7. The value of ƒ ''() is - (A) If y = g() is inverse of y = ƒ (), then g'( ) is - (A) 0. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 8

20 9. If y = h() is mirror image of y = ƒ () about the line y + = 0, then ƒ ''( 0 ) + h''( 0 ), where 0 9, is - (A) 0 ƒ( 0) (ƒ( 0) ) 0 Paragraph for Question 0 to Let ƒ () = a + a a + be a quadratic polynomial in, a be any real number. On the basis of above information, answer the following questions : 0. If one root of ƒ () = 0 is smaller than and other root is greater than, then the value of a belongs to - (A),,, none of these. If is a root of ƒ () = 0, then the sum of the series... a a is - (A). For a =, the minimum value of ƒ() is - (A) 0 Paragraph for Question to 5 A polynomial equation is given by + k + + k + = 0. On the basis of above information, answer the following questions :. Interval of k for which the equation has no imaginary roots can be - (A) k (, ) k (, ) [, ] k [, ]. Interval of k for which eactly roots of the equation are real can be - (A) k ( 0, 8) k (, ) k k (, 6) 5. Interval of k for which there are no real roots of the equation can be - (A) k [ 0,0] k [,] k (, ] k [0,) MATCH TH COLUMN 7 6. ()= 8 8 Column-I Column-II (A) If < < then () satisfies (P) () > / If < < then () satisfies (Q) () < 0 If < < 7 then () satisfies (R) () > 0 If > 7 then () satisfies (S) 0 < () < / 9. A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. (A) 9

21 7. Match the following for the system of linear equations + y + z =, + y + z =, + y + z = Column-I Column-II (A) = (P) unique solution (Q) infinite solutions, (R) no solution = (S) finite many solutions 8. Consider the matri A ; B 0 Let P be an orthogonal matri and Q = PAP T, R K = P T Q K P, S = PBP T & T K = P T S K P. Where K N. Column-I Column-II (A) 5 a K, where a K represents the element of (P) 9 K first row & first column in matri R K. bk, where b K represents the element of (Q) 0 K second row & second column in matri R K. K, where K represents the element of (R) 5 K first row & first column in matri T K. 0 yk, where y K represents the element of (S) K second row & second column in matri T K. 9. Column-I Column-II (A) The number of positive roots of the equation ( )( )( ) (P) + ( )( )( ) + ( )( )( ) + ( )( )( ) = 0 If the function g() = ƒ ƒ 6 R increases in the (Q) interval (a, ), where ƒ ''() > 0 R, then the value of a is If ƒ () = e [0,] & ƒ () ƒ (0) = ƒ '(c), where c (0, ) (R) then n(e c + ) is equal to m c, 0 If for the function ƒ () =, Lagrange's mean value e, 0 (S) theorem (LMVT) is applicable in [, ], then m + c is 7. (A) 8. (A) 0 9. (A)

22 0. Column-I Column-II (A) The length of the sub-tangent of the curve y (P) 6 7 at the point (,) is The slope of tangent to the curve = t +t 8, (Q) y = t t 5 at the point (, ) is A variable triangle is inscribed in a circle of radius 'R'. If (R) rate of change of a side is R times the rate of change of the opposite angle, then cosine of that angle is The number of solutions of the equation e + e = 8, is (S) 0 (T). Match the set of values of in column-ii which satisfy the equations in column-i. Column-I Column-II (A) {( + )} + = ( + ) (P) cos 6 ( )( + ) + = 0 (Q) sin = 0 (R) sin 6 + = 0 (S) cos 8. Match the set of values of in column-ii which satisfy the inequations in column-i. Column-I (A) (P) < 0 < (Q) = < (R) 0. (A) 9 (S). (A). (A) (T) Column-II

23 INTGR TYP / SUBJCTIV TYP. There are two possible values of A (say A & A ) in the solution of matri equation A 5 A 5 B D A A C F then find 7(A + A ) n (n ) (n ). If ƒ (n) = (n ) (n ) (n ) (n ) (n ) (n ) then find the value of ƒ (). ƒ (). ƒ (). 5. Given two curves : y = ƒ() passing through (0, ) and y = ƒ (t)dt passing through 0,. The tangents drawn to both the curves at the points with equal abscissas intersects on -ais. Find the value of nƒ (). 6. Consider ƒ () = a b + c + d, cot (cot ) where a = (cosec cosec6 ) b = c = (tan tan0) + + d = 5( sec sec5) If p represents the number of etremum of the function ƒ () and q represents the number of real roots of the equation ƒ () = 0, then find the value of p + q. 7. Let y = ƒ() be a differentiable curve satisfying then / 9 ƒ () d equals / cos 8. If difference between greatest & least value of function ƒ (t)dt t ƒ (t)dt, ƒ() (at t cos t)dt, a 0 [, ] 0 is 5 + sin cos cos, then value of a is 9. ABCD and PQRS are two variable rectangles such that P, Q, R and S lie on AB, BC, CD and DA respectively and perimeter '' of PQRS is constant. If maimum area of ABCD is, then to is equal

24 ANSWR KY. A. C. A. C 5. C 6. A 7. C 8. C 9. B 0. A. D. B. A. D 5. C 6. A 7. A 8. A 9. B 0. A. A. D. D. A 5. D 6. A 7. B 8. B 9. D 0. B. B. C. A. B 5. C 6. A 7. C 8. B 9. B 0. B. C. C. D. D 5. C 6. C 7. B 8. A 9. C 50. B 5. A 5. A 5. C 5. B 55. A 56. A 57. C 58. C 59. B 60. D 6. B 6. C 6. B 6. D 65. C 66. D 67. A 68. A,D 69. A,B,D 70. A,B,C 7. A,D 7. A,B,C 7. B,D 7. A 75. B,C,D 76. A,B,D 77. B,C 78. A,B,C 79. A,B 80. A,B 8. B,C,D 8. B,D 8. A,D 8. A,B,C,D 85. A,C 86. A,B,C 87. A,B 88. A,C,D 89. A,B,C,D 90. A 9. A 9. A 9. A 9. A 95. D 96. D 97. C 98. D 99. D 00. D 0. D 0. C 0. D 0. A 05. B 06. A 07. C 08. B 09. D 0. A. C. B. A. D 5. C 6. B 7. A 8. C 9. A 0. B. A. D. B. C 5. C 6. D 7. A 8. B 9. D 0. A. C. B. D. D 5. A 6. (A ) (R,S); (P,R); (Q); (P,R) 7. (A) (Q); (P,R); (P); (R) 8. (A) (R); (P); (S); (Q) 9. (A) (R), (Q), (P), (S) 0. (A) (T); (P); (Q); (T). (A) (P); (Q); (R,S); (Q). (A) (P,S); (P); (T); (Q)