# DPP-53-58

STUDY INNOVATIONSEducator um Study Innovations

Subject : Maths DPP No. __5_3_ Course Name : Batch : CC Time: 30 Min. DPP No. – 1 1. Set of all values of a for which the inequality (a – 4) x2 – 2ax + (2a – 6) < 0 is satisfying for all real values of x, is (A*) (– , 2) (B) (– , 2)    (C) (2,12) (D) (– , 4) 2. Set of all values of x satisfying the inequation 2 sin2x – sin x – 3 > 0, is (A)  2n  2 , (2n  1)     (B)  2n   , 2n  11     3 3     6 6  (C*)  2n  4 , 2n  5   (D) No value of x    3 3  3. If ai > 0  i  N such that ai  1, minimum value (1 + a1) (1 + a2) (1 + a3) ........ (1 + an) is i  1 (A) 1 (B*) 2n (C) 2n (D) does not exist 4. The system of equations 2x + py + 6z = 8 x + 2y + qz = 5 & x + y + 3z = 4 has (i) No solution if p  2 (ii) a unique solution if p  2, q  3 (iii) Infinitely many solutions if p = 2, then (A) (i) and (ii) are correct (B*) (ii) and (iii) are correct (C) (i) and (iii) are correct (D) none is correct 5. Value of x satisfying the equation sin–1 6x + sin–1 6  x = – 2 is 1 (A) – 6 (B) 12 (C*) – 1 12 1 (D) 6 1 1 3  x2 6. Domain of definition of f(x) =   is  x  (A) R – {2} (B) (–, –3)  (2, ) (C) R – {0, 2} (D*) (–, –3)  (0, ) – {2} lim  1  2 (2x  3)  7. x2   x  2 x3  3x2  2x  = (A) 1 (B) – 1 (C*) – 1 (D) does not exist 2 3x ,  1  x  1 8. f(x) = 4  x , 1  x  4 is (A) not continuous at x = 1 (B*) continuous and but not differentiable at x = 1 (C) continuous at x = 1 (D) continuous and differentiable at x = 1 x 9. x  y  In a x  y , then dy dx = (A) 2 + x (B*) 2 – y x (C) 2 + y y (D) 2 – y x x 10. STATEMENT-1 : x  1  = [x] if {x} < 1  2  2 STATEMENT-2 : [nx] = n[x] if {x} < 1 n where [.], {.} stands for greatest integer and fraction part functions respectively and n is a natural number (A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 11. STATEMENT-1 : Perpendicular from origin O to the line joining the points A (c cos , c sin ) and B (c cos , c sin ) divides it in the ratio 1 : 1 STATEMENT-2 : Perpendicular from opposite vertex to the base of an isosceles triangle bisects it. (A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 12. STATEMENT-1 : The number of terms in the expansion of (x2 + x + 1)10 is 21. STATEMENT-2 : The number of terms in the expansion of (x + y + z)n is 1 (n + 1) (n + 2) 2 (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct ex

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### DPP-53-58

• 1. Subject : Maths DPP No. ____ Course Name : Batch : CC 53 Time: 30 Min. DPP No. – 1 1. Set of all values of a for which the inequality (a – 4) x2 – 2ax + (2a – 6) < 0 is satisfying for all real values of x, is (A*) (– , 2) (B) (– , 2)  (C) (2,12) (D) (– , 4) 2. Set of all values of x satisfying the inequation 2 sin2 x – 3 sin x – 3 > 0, is (A)              3 ) 1 n 2 ( , 3 2 n 2 (B)             6 11 n 2 , 6 n 2 (C*)             3 5 n 2 , 3 4 n 2 (D) No value of x 3. If ai > 0  i  N such that    n 1 i i 1 a , minimum value (1 + a1 ) (1 + a2 ) (1 + a3 ) ........ (1 + an ) is (A) 1 (B*) 2n (C) 2n (D) does not exist 4. The system of equations 2x + py + 6z = 8 x + 2y + qz = 5 & x + y + 3z = 4 has (i) No solution if p  2 (ii) a unique solution if p  2, q  3 (iii) Infinitely many solutions if p = 2, then (A) (i) and (ii) are correct (B*) (ii) and (iii) are correct (C) (i) and (iii) are correct (D) none is correct 5. Value of x satisfying the equation sin–1 6x + sin–1 6 3 x = – 2  is (A) – 6 1 (B) 12 1 (C*) – 12 1 (D) 6 1 6. Domain of definition of f(x) = 2 x 1 x 3 1         is (A) R – {2} (B) (–, –3)  (2, ) (C) R – {0, 2} (D*) (–, –3)  (0, ) – {2} 7. 2 x lim             x 2 x 3 x ) 3 x 2 ( 2 2 x 1 2 3 = (A) 1 (B) – 1 (C*) – 2 1 (D) does not exist
• 2. 8. f(x) =          4 x 1 , x 4 1 x 1 , 3x is (A) not continuous at x = 1 (B*) continuous and but not differentiable at x = 1 (C) continuous at x = 1 (D) continuous and differentiable at x = 1 9. y x a In y x x    , then dx dy = (A) 2 + y x (B*) 2 – y x (C) 2 + x y (D) 2 – x y 10. STATEMENT-1 :        2 1 x = [x] if {x} < 2 1 STATEMENT-2 : [nx] = n[x] if {x} < n 1 where [.], {.} stands for greatest integer and fraction part functions respectively and n is a natural number (A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 11. STATEMENT-1:Perpendicular from origin O to the line joining the points A (c cos , c sin ) and B (c cos , c sin ) divides it in the ratio 1 : 1 STATEMENT-2:Perpendicular from opposite vertex to the base of an isosceles triangle bisects it. (A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 12. STATEMENT-1:The number of terms in the expansion of (x2 + x + 1)10 is 21. STATEMENT-2:The number of terms in the expansion of (x + y + z)n is 2 1 (n + 1) (n + 2) (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B*) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 13. STATEMENT-1:If one A.M. ‘A’ and two G.M.'s p and q be inserted between any two numbers, then p3 + q3 = 2Apq STATEMENT-2:If x, y, z are in G.P., then y2 = xz (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B*) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
• 3. Subject : Maths DPP No. ____ Course Name : Batch : CC Time: 30 Min. DPP No. – 2 Comprehension # 1 The number of distinct real roots of 2x3 + 3x2 – 12x + d = 0 is n. 1. Set of all values of d if n = 1, is (A) (–, –7)  (20, ) (B) (–20, 7) (C*) (–, –20)  (7, ) (D) {–20, 7} 2. Set of all values of d if n = 3, is (A) (–, –7)  (20, ) (B*) (–20, 7) (C) (–, –20)  (7, ) (D) {–20, 7} 3. Set of all values of d if n = 2, is (A) (–, –7)  (20, ) (B) (–20, 7) (C) (–, –20)  (7, ) (D*) {–20, 7} Comprehension # 2 If f(x) =                                  x 2 if |) x cos | 1 ( 2 x if 2 b 2 x 0 if 5 6 b | x |tan a x 5 tan x 6 tan is continuous at x = 2  consider an infinite geometric progression whose first term is a + 1 and common ratio is 3 b 1  . Let it's sum be S. Further A is square matrix of order 3 × 3 such that aij =       j i , S j i , 2 S 4. The value of a + b is (A*) – 1 (B) 0 (C) 1 (D) 2 5. trace of A is equal to (A) 8 (B) – 4 (C*) 6 (D) – 6 6. If A (adj A) = 3 where 3 is a unit matrix of order 3, then  is equal to (A*) 8 (B) 64 (C) 16 (D) 6 MATCH THE COLUMN 7. Column -  Column -  (A) The minimum value of (p) 1 f(x) = |x – 4| + |x – 6| + |x – 2| is (B) The number of solution/solutions (q) 4 of |x| = |cosx| is/are (C) The value of x satisfying the equation (r) 0 |2x + 3| – |x – 1| = 5 is (D) The product of all the solutions (s) 2 of the equation (x – 2)2 – 3 |x – 2| + 2 = 0 is Ans. (A) (q), (B)  (s), (C)  (p), (D)  (r) 54
• 4. 8. Column –  Column –  (A) Number of solutions of tan–1 x = sin–1 (x + k) (k  R) is (p) 0 (B) Number of solutions of sin-1 x = sgn(x) is (q) 1 (C) Number of solutions of |1 – |1 – x2 || = 5 is (r) 2 (D) Number of solutions of the equation x . 2x = x + 1 is (s) 3 Ans. (A)  (q), (B)  (s), (C)  (r), (D)  (r) 9. If y = f(x) has following graph then match the column. Column –  Column –  (A) y = |f(x)| (p) (B) y = f(|x|) (q) (C) y = f(– |x|) (r) (D) y  | f ( |x| ) | (s) Ans. (A) (r), (B)  (p), (C)  (q), (D)  (s)
• 5. Subject : Maths DPP No. ____ Course Name : Batch : CC 55 Time: 30 Min. DPP No. – 3 1. The function f(x) = x2 e–2x , x > 0. If  is maximum value of f(x) then ] [ 1 2   is equal to {where [.] represents greater integer function} (A) 64 (B*) 128 (C) 256 (D) none of these 2. For the function f(x) = n (cos–1 log3 x) (A*) domain is [1/3, 3) (B) range is (– , n /2) (C) domain [1, 3) (D) range (– , n ) 3. If x  2, y  2, z  2 and 2 y x z 2 x z y 2 = 0, then the value of z 2 2 y 2 2 x 2 2      = (A) 1 (B*) 2 (C) 3 (D) 4 4. y = et sin t and x = et cos t and y = dt dy , y = 2 2 dt y d , x = dt dx , x = 2 2 dt x d , then (A) y = x (B) y = – 2x (C*) x = – 2y (D) x = – y 5. A straight line passing through the point P(2, 5) intersects the circle x2 + y2 = 4 is two different points Q and R. Then the value of PQ . PR is (A) 27 (B) 29 (C*) 25 (D) 5 6. Equation 3x2 – 8xy – 3y2 = 0 and x – 2y = 3 represents the sides of a triangle which is (A*) right angled but not isosceles (B) isosceles (C) equilateral (D) right angled and isosceles 7. There are two equal circles with centres at C1 and C2 , each is of radius 2 and both are touching the lines x = 2 and y = 3. x-coordinate of C1 is less than 2 and that of C2 is greater than 2. y-coordinate of C1 is greater than 3 and that of C2 is less than 3. Then length of a direct common tangent to these circles is (A) 4 (B*) 2 4 (C) 2 2 (D) 8 8. If r is an odd natural number greater than 1 and n is an even natural number, then the value of the expression n C2 . n Cr – n C3 . n Cr + 1 + .......... + (–1)n – r . n Cn – r . n Cn – 2 , considering all those terms which are defined, is (A*) 0 (B) n (C) 2n Cr – 2 (D) n Cr + 2 9. There are 10 speakers S1 , S2 , ........ S10 . Speaker S4 wants to speak before S9 but S4 can not be the first and S9 can not be the last speaker. If at least 2 speaksers are to speak after S4 and before S9 , then the number of ways in which the speaches can be arranged is (A) 12 (B*) 15 (C) 28 (D) 32 10. STATEMENT-1 : e is bigger then e . STATEMENT-2 : f(x) = x1/x is a decreasing function when x  [e, ) (A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
• 6. 11. STATEMENT-1 : The number of solutions of the equation sin–1 x + 2x = 3 is one. STATEMENT-2 : If f(x) is strictly increasing real function defined on R and c is a real constant, then number of solutions of f(x) = c is always equal to one. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C*) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 12. In both the statements [.] represents greatest integer function. STATEMENT-1 : The greatest value of sin        ] x [ 2 3 x 2 3 is sin 2 3 . STATEMENT-2 : The greatest value of [sin x] is 1, where x  R. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D*) Statement-1 is False, Statement-2 is True 13. In both the statements [.] represents greatest integer function. STATEMENT-1 : 0 x lim         x sin x = –1. STATEMENT-2 : 0 x lim        x x sin = 0. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D*) Statement-1 is False, Statement-2 is True
• 7. Subject : Maths DPP No. ____ Course Name : Batch : CC Time: 30 Min. DPP No. – 4 Comprehension Let L1 and L2 be the lines whose equation are 1 3 z 1 8 y 3 3 x       and 3 3 x   = 2 7 y  = 4 6 z  respectively. . A and B are two points on L1 and L2 respectively such that AB is perpendicular both the lines L1 and L2 . 1. Shartest distence between the lines L1 and L2 is (A) 30 (B) 2 30 (C*) 3 30 (D) none of these 2. Co-ordinates of the point Aare (A) (1, 8, 2) (B*) (3, 8, 3) (C) (–3, 8, 3) (D) none of these 3. Co-ordinates of the point B are (A*) (–3, –7, 6) (B) (2, 7, 6) (C) (1, 6, 3) (D) none of these Comprehension Let f(x) =        0 x , x ax x 0 x , ) x ( g x 3 2 , where g(t) = 0 x lim  (1 + a tan x)t/x , a is positive constant. 4. If a is even prime number, then g(2) = (A) e2 (B) e3 (C*) e4 (D) none of these 5. Set of all values of a for which function f(x) is continuous at x = 0 (A) (–1, 10) (B) (–, ) (C*) (0, ) (D) none of these 6. If f(x) differentiable at x = 0, then a belongs to (A) (–5, –1) (B) (–10, 3) (C*) (0, ) (D) none of these MATCH THE COLUMN 7. Column –  Column –  (A) Minimum value of , for which 61+x + 61–x , , 4x + 4–x are in (p) 1 A.P., is (B) Sn = 3 1 1 + 3 3 2 1 2 1   + ........... + 3 3 3 3 n ....... 3 2 1 n ..... 3 2 1         , (q) 3 then the greatest integer less than or equal to S100 is (C) If x = 2 + 3 , then        2 2 x 1 x + 4        x 1 x = (r) 7 (D) In a cyclic quadrilateral ABCD, tan A = 12 5 , cos B = – 5 3 , (s) 30 then D tan C cos D tan C cos  = Ans. (A)  (r), (B)  (p), (C)  (s), (D)  (q) 56
• 8. 8. Column –  Column –  (A) If f(x) = a ax ax 1 a ax 0 1 a 2   , then f(x) – f(–x) is a polynomial (p) 1 of degree (B) Number of points of discontinuity of f(x) = tan2 x – sec2 x in (q) 0 (0, 2) is (C) Fundamental period of f(x) = sin                 2 x x 2 is (r) 3 {.} represents fractional part function. (D) The integral part of a for which the point (1, a) lies in (s) 2 between the lines x + y = 1 and 2(x + y) = 3 Ans. (A)  (p), (B)  (s), (C)  (s), (D)  (q) 9. Column –  Column –  (A) The reflection of the point (t – 1, 2t + 2) in a line is (p) 0 (2t + 1, t), then the equation of the line has slope equals to (B) Let u  = i  + j  , v = i  – j  , w = k 3 j 2 i      . If n  is a (q) 3 unit vector such that u  . n  = 0, n . v   = 0, then the value of | n . w |   is (C) The number of solutions of the equation (r) 1 9 log 2 x log x 2 3 | 1 x |   = (x – 1)7 , is (D) Ladder of length 5m leaning against a wall is being (s) 2 pulled along the ground at 2cm/s. When the foot of the ladder is 4m away from the wall, if the top of the ladder slides down on the wall at  8 cm/s, then  = Ans. (A)  (r), (B)  (q), (C)  (s), (D)  (q)
• 9. Subject : Maths DPP No. ____ Course Name : Batch : CC Time: 30 Min. DPP No. – 5 1. If x2 – 2x + 4 = 0 and x2 – bx + c = 0 have atleast one common root then [b + c] = {where [.] represents greastest integer function and b & c are real} (A) 4 (B*) 6 (C) 8 (D) none of these 2. If 14a2 + 11b2 + 6c2 + 14 = 4ab + 12bc + 6ac + 2a + 4b + 6c, then a, b, c are in (A) G.P. (B) H.P. (C*)A.P. (D) none of these 3. If m and n are number of points (not end points of the domain) where y = sin–1 (3x – 4x3 ) is non-differentiable and discontinuous respectively, then [2m + 3n ] is equal to. {where [.] represents greastest integer function} (A) 4 (B*) 5 (C) 6 (D) none of these 4. If  = 0 x lim  x 6 sin x 1 x 2 tan 2 1         and m = 0 x lim        x x tan , then 8 + 3m = {where [.] represents greastest integer function} (A) 2 (B) 3 (C*) 5 (D) none of these 5. A fair coin is tossed 50 times. The probability of getting tails an odd number of times is (A) 3 1 (B*) 2 1 (C) 3 2 (D) 4 1 6. The equation of the plane, passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 8 = 0 and whose x-intercept is 4, is (A*) x –3y – 7z – 4 = 0 (B) 2x + 3y + 4z = 8 (C) x + y + z – 4 = 0 (D) none of these 7. If | a |  = 2, | b |  = 3 and | c |  = 4, then the maximum value of ] a c c b b a [          is (A) 625 (B*) 576 (C) 596 (D) 288 8. If A is the area of equalateral triangle inscribed in the circle x2 + y2 – 4x + 6y – 3 = 0, then [A] = {where [.] represents greastest integer function} (A*) 15 (B) 16 (C) 17 (D) 5 9. The function f(x) = 2 x 1 x  , where x > 0, decrease in the interval. (A) [0, ) (B*) [1, ) (C) [1/2, ) (D) none of these 10. STATEMENT-1 : Range of the function f(x) = 6 cos2 x + 8 sin x cos x is [–2, 8] STATEMENT-2 : – 2 2 b a   a cos  + b sin  2 2 b a  ,  , a, b R. (A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 57
• 10. 11. STATEMENT-1 : Lines x(cos  + sin ) + y(cos  – 2 sin ) – 3cos  = 0 passes through fixed point (2, 1) STATEMENT-2 :  L1 + L2 = 0 represents family of all straight lines passing through the point of intersection of L1 = 0 & L2 = 0 (A*) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 12. STATEMENT-1 : The largest integer n for which 35! is divisible by 3n is 15. STATEMENT-2 : n C2 = 2 ) 1 n ( n  (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B*) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 13. STATEMENT-1 : If Aand B are two symmetric matrices of the same order, thenAB – BA is skew symmetric. STATEMENT-2 : AB = –BA always, where A and B are matrices of same order. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 isTrue, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C*) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
• 11. Subject : Maths DPP No. ____ Course Name : Batch : CC Time: 30 Min. DPP No. – 6 Comprehension Let M be the set of all members of the family of circles x2 + y2 – 2x – 2by – 8 = 0 where b is a variable and {.} represents fraction part function. 1. If every member of set M passes through two fixed points (x1 , y1 ) and (x2 , y2 ), then x1 + x2 + y1 + y2 = (A) 0 (B) 1 (C*) 2 (D) none of these 2. If a member of set M cuts the circle x2 + y2 – 4x – 2y – 7 = 0 orthogenally, then {b} = (A) 0 (B*) 1/2 (C) 3/4 (D) none of these 3. If n is the number of those circles belonging to set M to whom y = x is tangent, then n = (A*) 0 (B) 1 (C) 2 (D) none of these Comprehension For points P(x1 , y1 ) and Q(x2 , y2 ), a new distance formula d(P, Q) is defined by d(P, Q) = |x1 – x2 | + |y1 – y2 | Let O(0, 0), A(1, 2), B(2, 3), C(4, 3) are four fixed points on x – y plane. 4. If 0  x < 1 and 0  y < 2, then locus of point (x, y), which is equi-distant from point O and point A, with respect to the defined distance formula, is a part of (A) x + y = 3 (B) x + 2y = 3 (C) 2x + y = 3 (D*) 2x + 2y = 3 5. Let S(x, y), such that S is equidistant from O and B, with respect to the defined distance formula and if x  2, 0  y < 3, then locus of S is (A) a line segment of finite length (B) y = 1/2 (C*) a ray of infinite length (D) none of these 6. Let T (x, y), a moving point is such that it is equidistant from O and C, with respect to the defined distance formula. If T lies in first quadrant, then graph of moving point T is (A*) (B) (C) (D) 58
• 12. 7. Column – I Column – II (A) If y = 100 x 12 x 100 x 12 x 2 2     , x  R, then greatest value of y is (p) 6 (B) If 1 4 . 2 6 . 3 8 . 4 10 ........... 30 62 . 31 64 = (64)x , then x is (q) 4 (C) If there are n integers between 1 and 1000 whose number (r) 3 of proper divisors is odd, then the value of       5 n , where [.] represents greatest integer function, is (D) If V1 and V2 are the maximum and minimum value of (s) 2 parallelopiped formed by the vectors k̂ ĵ a î   , k̂ a ĵ  and k̂ î a  respectively, where –1 < a < 1, then the value of (V1 + V2 )2 is equal to Ans. (A)  (q), (B)  (p), (C)  (p), (D)  (q) 8. Column – I Column – II (A) If m is absolute value of the difference of all integral (p) 8 roots of (3 – x)5 + (x – 5)5 + 32 = 0, then 3m is (B) If m & n are the largest natural numbers for which 100! (q) 7 is divisible by 2m and 5n , then n 1 n 3 m   is (C) If the domain at y = cos–1        4 x 2 1 is [a, b], then twice (r) 9 of b – a is equal to (D) If a chord of the circle x2 + y2 – 4x – 2y + k = 0 (s) 1 is trisected at the points       3 1 , 3 1 and       3 8 , 3 8 and  is the length of the chord then 2  is equal to Ans. (A)  (r), (B)  (s), (C)  (p), (D)  (q) 9. Column – I Column – II (A) Let D(20, 25), E(8, 16) and F(8, 9) be the feet of perpendicular (p) 3 from the verticesA, B and C of a triangle ABC to the respective opposite sides. If (a, b) is the orthocentre of the triangle ABC, then b – a is equal to (B) Let f(x + y) = f(x) . f(y) for all x, y  R and suppose that f is (q) 4 differentiable for all x and f(0) = 2. It f(a) = 3, then f(a) is equal to (C) If x2 + y2 = 2 and 2y + ay–3 = 0, then a is equal to (r) 5 (D) If         5 , 2 is the reflection of (1, 4) about the line y = x, (s) 6 then  –  is equal to Ans. (A)  (r), (B)  (s), (C)  (q), (D)  (p)
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