Page 1 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Determinants, Functions, Limits, Continuity , Differentiability Class Test - 7 B Time : 2hr 30 minutes M.M. 132 Read the carefully the instructions given below : 1. The test consists of 37 questions. The test contains four sections. Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section B : Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking. Section C : Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section D : Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking Student Name : __________________________________ Class : _____________________________ Marks Obtained : ________________________________ Date of Test : _________________________ Part A Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are (a) five (b) two (c) three (d) infinite Q.2 All solutions of the equation |x 2 â€“ x - 6| = x + 2 are (a) natural nos. (b) negative integer (c) rational nos. (d) irrational nos. Q.3 If in a triangle two angles are Tan -1 2 & Tan -1 3 then the measure of the 3 rd angle is (a) p / 4 (b) p / 2 (c) p / 3 (d) 3p / 4 _____ _____ Q.4 The identity Cos -1 x â€“ Cos -1 y = Cos -1 (xy + v1 â€“ x 2 v1 â€“ y 2 ) holds if (a) x 2 + y 2 = 1 (b) xy = 1 (c) x = y (d) x + y = 0 Q.5 If A, B, P & Q are sq. matrices of same order such that adj B = A & |P| = |Q| = 1 then adj (Q -1 B P -1 ) is given as (a) QAP (b) PAQ (c) P -1 B Q -1 (d) PBQ Q.6 If f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is (a) differentiable at x = 0 (b) continuous in (-1, 0) (c) differentiable in (-1, 1) (d) none Q.7 Let f(x) = [tan 2 x], where [.] denotes greatest integer function then (a) lim f(x) does not exist (b) f(x) is continuous at x = 0 x?0 (c) f(x) is not differentiable at x = 0 (d) f(0) = 1 Q.8 The function f(x) = max {1 â€“ x, 1 + x, 2}, x ? R is (a) continuous at all points except at one point (b) differentiable at all x (c) diff. at all points except two points in domain (d) discontinuous at x = 1& x = -1 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Page 2 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Determinants, Functions, Limits, Continuity , Differentiability Class Test - 7 B Time : 2hr 30 minutes M.M. 132 Read the carefully the instructions given below : 1. The test consists of 37 questions. The test contains four sections. Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section B : Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking. Section C : Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section D : Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking Student Name : __________________________________ Class : _____________________________ Marks Obtained : ________________________________ Date of Test : _________________________ Part A Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are (a) five (b) two (c) three (d) infinite Q.2 All solutions of the equation |x 2 â€“ x - 6| = x + 2 are (a) natural nos. (b) negative integer (c) rational nos. (d) irrational nos. Q.3 If in a triangle two angles are Tan -1 2 & Tan -1 3 then the measure of the 3 rd angle is (a) p / 4 (b) p / 2 (c) p / 3 (d) 3p / 4 _____ _____ Q.4 The identity Cos -1 x â€“ Cos -1 y = Cos -1 (xy + v1 â€“ x 2 v1 â€“ y 2 ) holds if (a) x 2 + y 2 = 1 (b) xy = 1 (c) x = y (d) x + y = 0 Q.5 If A, B, P & Q are sq. matrices of same order such that adj B = A & |P| = |Q| = 1 then adj (Q -1 B P -1 ) is given as (a) QAP (b) PAQ (c) P -1 B Q -1 (d) PBQ Q.6 If f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is (a) differentiable at x = 0 (b) continuous in (-1, 0) (c) differentiable in (-1, 1) (d) none Q.7 Let f(x) = [tan 2 x], where [.] denotes greatest integer function then (a) lim f(x) does not exist (b) f(x) is continuous at x = 0 x?0 (c) f(x) is not differentiable at x = 0 (d) f(0) = 1 Q.8 The function f(x) = max {1 â€“ x, 1 + x, 2}, x ? R is (a) continuous at all points except at one point (b) differentiable at all x (c) diff. at all points except two points in domain (d) discontinuous at x = 1& x = -1 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Q.9 The domain of the derivative of the function f(x) = tan -1 x if |x| = 1 (|x| - 1) if |x| > 1 is 2 (a) R â€“ {0} (b) R â€“ {1} (c) R â€“ {0, -1, 1} (d) R â€“ {-1, 1} Q.10 Given f ' (2) = 6 & f ' (1) = 4 , lim f(2h + 2 + h 2 ) â€“ f(2) is equal to h?0 f(h â€“ h 2 + 1) â€“ f(1) (a) 3 / 2 (b) 5 / 2 (c) 3 (d) â€“ 3 Q.11 Range of the expression f(x) = 3 â€“ 2 Cos x ? x ? R Cos x + 2 (a) [1/3, 1/5] (b) [-5, 3] (c) (0, 2/3] (d) [1/3, 5] Q.12 Let f(x) = Cos x x 1 2 Sin x x 2 2x then lim f ' (x) = Tan x x 1 x?0 x (a) 4 (b) 3 (c) 0 (d) -2 Q.13 If lim [(a - n)nx â€“ tan x] Sin nx = 0 , where n is a non-zero real no. then a is equal to x?0 x 2 (a) n (b) n + 1 (c) n + 1/n (d) n n + 1 n Q.14 If f(x) is differentiable and strictly increasing function, then the value of lim f(x 2 ) â€“ f(x) is x?0 f(x) â€“ f(0) (a) -1 (b) 1 (c) 0 (d) 2 Q.15 If y 2 = P(x) is a polynomial of degree 3 then 2 d (y 2 d 2 y ) equals dx dx 2 (a) p''' (x) + p' (x) (b) p'' (x) p''' (x) (c) p(x) p''' (x) (d) a constant Q.16 If ? (x) = x 2 â€“ 5x + 3 2x â€“ 5 3 3x 2 + x + 4 6x + 1 9 7x 2 â€“ 6x + 9 14x â€“ 6 21 = ax 3 + bx 2 + cx + d then (a) a ? 0, d = 141 (b) a = b = c = 0 (c) c = 7 (d) d = 29 PART B :- 4½ Marks Questions Q.17 The integer n for which lim (Cos x - 1) (Cos x - e x ) is equal to ? (a finite non zero no.) then is equal to x?0 x n (a) n = 4 (b) n = 3 (c) ? = -1/2 (d) ? = 1/2 Q.18 If |C| = 1 / 2 and f(x) is diff. at x = 0 given by f(x) = b Sin -1 (c + x) , -1/2 < x < 0 2 1 / 2 , x = 0 e ax / 2 â€“ 1 , 0 < x < 1 / 2 , then x (a) a = 1 (b) 16 b 2 = 4 â€“ c 2 (c) a = -1 (d) 64 b 2 = 4 â€“ c 2 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Page 3 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Determinants, Functions, Limits, Continuity , Differentiability Class Test - 7 B Time : 2hr 30 minutes M.M. 132 Read the carefully the instructions given below : 1. The test consists of 37 questions. The test contains four sections. Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section B : Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking. Section C : Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section D : Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking Student Name : __________________________________ Class : _____________________________ Marks Obtained : ________________________________ Date of Test : _________________________ Part A Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are (a) five (b) two (c) three (d) infinite Q.2 All solutions of the equation |x 2 â€“ x - 6| = x + 2 are (a) natural nos. (b) negative integer (c) rational nos. (d) irrational nos. Q.3 If in a triangle two angles are Tan -1 2 & Tan -1 3 then the measure of the 3 rd angle is (a) p / 4 (b) p / 2 (c) p / 3 (d) 3p / 4 _____ _____ Q.4 The identity Cos -1 x â€“ Cos -1 y = Cos -1 (xy + v1 â€“ x 2 v1 â€“ y 2 ) holds if (a) x 2 + y 2 = 1 (b) xy = 1 (c) x = y (d) x + y = 0 Q.5 If A, B, P & Q are sq. matrices of same order such that adj B = A & |P| = |Q| = 1 then adj (Q -1 B P -1 ) is given as (a) QAP (b) PAQ (c) P -1 B Q -1 (d) PBQ Q.6 If f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is (a) differentiable at x = 0 (b) continuous in (-1, 0) (c) differentiable in (-1, 1) (d) none Q.7 Let f(x) = [tan 2 x], where [.] denotes greatest integer function then (a) lim f(x) does not exist (b) f(x) is continuous at x = 0 x?0 (c) f(x) is not differentiable at x = 0 (d) f(0) = 1 Q.8 The function f(x) = max {1 â€“ x, 1 + x, 2}, x ? R is (a) continuous at all points except at one point (b) differentiable at all x (c) diff. at all points except two points in domain (d) discontinuous at x = 1& x = -1 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Q.9 The domain of the derivative of the function f(x) = tan -1 x if |x| = 1 (|x| - 1) if |x| > 1 is 2 (a) R â€“ {0} (b) R â€“ {1} (c) R â€“ {0, -1, 1} (d) R â€“ {-1, 1} Q.10 Given f ' (2) = 6 & f ' (1) = 4 , lim f(2h + 2 + h 2 ) â€“ f(2) is equal to h?0 f(h â€“ h 2 + 1) â€“ f(1) (a) 3 / 2 (b) 5 / 2 (c) 3 (d) â€“ 3 Q.11 Range of the expression f(x) = 3 â€“ 2 Cos x ? x ? R Cos x + 2 (a) [1/3, 1/5] (b) [-5, 3] (c) (0, 2/3] (d) [1/3, 5] Q.12 Let f(x) = Cos x x 1 2 Sin x x 2 2x then lim f ' (x) = Tan x x 1 x?0 x (a) 4 (b) 3 (c) 0 (d) -2 Q.13 If lim [(a - n)nx â€“ tan x] Sin nx = 0 , where n is a non-zero real no. then a is equal to x?0 x 2 (a) n (b) n + 1 (c) n + 1/n (d) n n + 1 n Q.14 If f(x) is differentiable and strictly increasing function, then the value of lim f(x 2 ) â€“ f(x) is x?0 f(x) â€“ f(0) (a) -1 (b) 1 (c) 0 (d) 2 Q.15 If y 2 = P(x) is a polynomial of degree 3 then 2 d (y 2 d 2 y ) equals dx dx 2 (a) p''' (x) + p' (x) (b) p'' (x) p''' (x) (c) p(x) p''' (x) (d) a constant Q.16 If ? (x) = x 2 â€“ 5x + 3 2x â€“ 5 3 3x 2 + x + 4 6x + 1 9 7x 2 â€“ 6x + 9 14x â€“ 6 21 = ax 3 + bx 2 + cx + d then (a) a ? 0, d = 141 (b) a = b = c = 0 (c) c = 7 (d) d = 29 PART B :- 4½ Marks Questions Q.17 The integer n for which lim (Cos x - 1) (Cos x - e x ) is equal to ? (a finite non zero no.) then is equal to x?0 x n (a) n = 4 (b) n = 3 (c) ? = -1/2 (d) ? = 1/2 Q.18 If |C| = 1 / 2 and f(x) is diff. at x = 0 given by f(x) = b Sin -1 (c + x) , -1/2 < x < 0 2 1 / 2 , x = 0 e ax / 2 â€“ 1 , 0 < x < 1 / 2 , then x (a) a = 1 (b) 16 b 2 = 4 â€“ c 2 (c) a = -1 (d) 64 b 2 = 4 â€“ c 2 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Q.19 If x ? [-p, p] then solution set of the inequality Sin x + Sin 2x > 0 is (a) (-p, -2p / 3) (b) (p, 3p / 4) (c) (0, 2p / 3) (d) (-3p / 4, -p / 2) Q.20 Solution set of the inequality |8x 2 + 25x + 12| = | |x 3 + 6x 2 + 5x - 12| - |x 3 â€“ 2x 2 â€“ 20x - 24| | is (a) [-4, -3] (b) [-3, -6] (c) [1, 6] (d) holds free for exactly 9 integers Q.21 The no. of possible values of t for which the system of equations (a - t)x + by + cz = 0, bx + (c - t)y + az = 0, cx + ay + (b - t)z = 0, has a non-trivial solution are ________.(say ?) and the product of these values is ____________ (say µ), then (a) ? = 2 (b) ? = 3 (c) µ = a b c (d) µ = a b c b c a c a b c a b b c a Q.22 Consider the f(x) = Sin -1 (Cot -1 x) then (a) domain of f(x) is [Cot 1, 8) (b) domain of f(x) is [0, Cot 1] (c) range of f(x) is (0, p / 2] (d) range of f(x) is (p / 4, p / 2] ___________ Q.23 Consider the definition f(x) = v3 Sec -1 x â€“ p __ (a) domain of f(x) is {-1} ? [1, 2] (b) range of f(x) is [0, v2p] â€“ {p / 2} __ (c) domain of f(x) is (- 8, -1] ? [2, 8) (d) range of f(x) is (p / 2, v2p] Q.24 The system of equations ax + by + cz = q - r ; bx + cy + az = r â€“ p; cx + ay + bz = p â€“ q is (a) inconsistent if a = b = c & p , q, r are distinct (b) consistent if p = q = r (c) consistent if a, b, c are distinct & a + b + c ? 0 (d) inconsistent if p = q = r PART C :- Passage I Consider the function f(x) = 2x + 1 , x < 0 g(x) = 3 â€“ 4x , x < 1 3x â€“ 4 , x = 0 , 5 + x , x = 1 Q.25 for x ? [1, 8) fog (x) = (a) 8x â€“ 7 (b) 11 + 3x (c) 7 â€“ 8x (d) 12 â€“ 5x Q.26 x ?(- 8, 3/4] fog(x) is defined as (a) 3x â€“ 11 (b) 7 â€“ 8x (c) 11 + 3x (d) 5 â€“ 12x Q.27 Range of fog(x) = (a) [-4, 8) (b) [3, 8) (c) R (d) [-4, 3] Q.28 for x ? [0, 5/3) , gof (x) = (a) 2x + 6 (b) 3x + 1 (c) 19 â€“ 12x (d) -1 â€“ 8x Q.29 Range of gof(x) is (a) [4, 8) (b) [-1, 8) (c) [-1, 19] (d) [19, 8) Passage II :- We define 6 bijection as follows :- 1. f : [p / 2, 3p / 2] ? [-1, 1] defines as f(x) = Sin x 2. g : [-p , 0] ? [-1, 1] , g(x) = Cos x 3. h : [0, p] â€“ {p / 2} ?R , h(x) = tan x SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Page 4 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Determinants, Functions, Limits, Continuity , Differentiability Class Test - 7 B Time : 2hr 30 minutes M.M. 132 Read the carefully the instructions given below : 1. The test consists of 37 questions. The test contains four sections. Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section B : Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking. Section C : Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section D : Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking Student Name : __________________________________ Class : _____________________________ Marks Obtained : ________________________________ Date of Test : _________________________ Part A Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are (a) five (b) two (c) three (d) infinite Q.2 All solutions of the equation |x 2 â€“ x - 6| = x + 2 are (a) natural nos. (b) negative integer (c) rational nos. (d) irrational nos. Q.3 If in a triangle two angles are Tan -1 2 & Tan -1 3 then the measure of the 3 rd angle is (a) p / 4 (b) p / 2 (c) p / 3 (d) 3p / 4 _____ _____ Q.4 The identity Cos -1 x â€“ Cos -1 y = Cos -1 (xy + v1 â€“ x 2 v1 â€“ y 2 ) holds if (a) x 2 + y 2 = 1 (b) xy = 1 (c) x = y (d) x + y = 0 Q.5 If A, B, P & Q are sq. matrices of same order such that adj B = A & |P| = |Q| = 1 then adj (Q -1 B P -1 ) is given as (a) QAP (b) PAQ (c) P -1 B Q -1 (d) PBQ Q.6 If f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is (a) differentiable at x = 0 (b) continuous in (-1, 0) (c) differentiable in (-1, 1) (d) none Q.7 Let f(x) = [tan 2 x], where [.] denotes greatest integer function then (a) lim f(x) does not exist (b) f(x) is continuous at x = 0 x?0 (c) f(x) is not differentiable at x = 0 (d) f(0) = 1 Q.8 The function f(x) = max {1 â€“ x, 1 + x, 2}, x ? R is (a) continuous at all points except at one point (b) differentiable at all x (c) diff. at all points except two points in domain (d) discontinuous at x = 1& x = -1 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Q.9 The domain of the derivative of the function f(x) = tan -1 x if |x| = 1 (|x| - 1) if |x| > 1 is 2 (a) R â€“ {0} (b) R â€“ {1} (c) R â€“ {0, -1, 1} (d) R â€“ {-1, 1} Q.10 Given f ' (2) = 6 & f ' (1) = 4 , lim f(2h + 2 + h 2 ) â€“ f(2) is equal to h?0 f(h â€“ h 2 + 1) â€“ f(1) (a) 3 / 2 (b) 5 / 2 (c) 3 (d) â€“ 3 Q.11 Range of the expression f(x) = 3 â€“ 2 Cos x ? x ? R Cos x + 2 (a) [1/3, 1/5] (b) [-5, 3] (c) (0, 2/3] (d) [1/3, 5] Q.12 Let f(x) = Cos x x 1 2 Sin x x 2 2x then lim f ' (x) = Tan x x 1 x?0 x (a) 4 (b) 3 (c) 0 (d) -2 Q.13 If lim [(a - n)nx â€“ tan x] Sin nx = 0 , where n is a non-zero real no. then a is equal to x?0 x 2 (a) n (b) n + 1 (c) n + 1/n (d) n n + 1 n Q.14 If f(x) is differentiable and strictly increasing function, then the value of lim f(x 2 ) â€“ f(x) is x?0 f(x) â€“ f(0) (a) -1 (b) 1 (c) 0 (d) 2 Q.15 If y 2 = P(x) is a polynomial of degree 3 then 2 d (y 2 d 2 y ) equals dx dx 2 (a) p''' (x) + p' (x) (b) p'' (x) p''' (x) (c) p(x) p''' (x) (d) a constant Q.16 If ? (x) = x 2 â€“ 5x + 3 2x â€“ 5 3 3x 2 + x + 4 6x + 1 9 7x 2 â€“ 6x + 9 14x â€“ 6 21 = ax 3 + bx 2 + cx + d then (a) a ? 0, d = 141 (b) a = b = c = 0 (c) c = 7 (d) d = 29 PART B :- 4½ Marks Questions Q.17 The integer n for which lim (Cos x - 1) (Cos x - e x ) is equal to ? (a finite non zero no.) then is equal to x?0 x n (a) n = 4 (b) n = 3 (c) ? = -1/2 (d) ? = 1/2 Q.18 If |C| = 1 / 2 and f(x) is diff. at x = 0 given by f(x) = b Sin -1 (c + x) , -1/2 < x < 0 2 1 / 2 , x = 0 e ax / 2 â€“ 1 , 0 < x < 1 / 2 , then x (a) a = 1 (b) 16 b 2 = 4 â€“ c 2 (c) a = -1 (d) 64 b 2 = 4 â€“ c 2 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Q.19 If x ? [-p, p] then solution set of the inequality Sin x + Sin 2x > 0 is (a) (-p, -2p / 3) (b) (p, 3p / 4) (c) (0, 2p / 3) (d) (-3p / 4, -p / 2) Q.20 Solution set of the inequality |8x 2 + 25x + 12| = | |x 3 + 6x 2 + 5x - 12| - |x 3 â€“ 2x 2 â€“ 20x - 24| | is (a) [-4, -3] (b) [-3, -6] (c) [1, 6] (d) holds free for exactly 9 integers Q.21 The no. of possible values of t for which the system of equations (a - t)x + by + cz = 0, bx + (c - t)y + az = 0, cx + ay + (b - t)z = 0, has a non-trivial solution are ________.(say ?) and the product of these values is ____________ (say µ), then (a) ? = 2 (b) ? = 3 (c) µ = a b c (d) µ = a b c b c a c a b c a b b c a Q.22 Consider the f(x) = Sin -1 (Cot -1 x) then (a) domain of f(x) is [Cot 1, 8) (b) domain of f(x) is [0, Cot 1] (c) range of f(x) is (0, p / 2] (d) range of f(x) is (p / 4, p / 2] ___________ Q.23 Consider the definition f(x) = v3 Sec -1 x â€“ p __ (a) domain of f(x) is {-1} ? [1, 2] (b) range of f(x) is [0, v2p] â€“ {p / 2} __ (c) domain of f(x) is (- 8, -1] ? [2, 8) (d) range of f(x) is (p / 2, v2p] Q.24 The system of equations ax + by + cz = q - r ; bx + cy + az = r â€“ p; cx + ay + bz = p â€“ q is (a) inconsistent if a = b = c & p , q, r are distinct (b) consistent if p = q = r (c) consistent if a, b, c are distinct & a + b + c ? 0 (d) inconsistent if p = q = r PART C :- Passage I Consider the function f(x) = 2x + 1 , x < 0 g(x) = 3 â€“ 4x , x < 1 3x â€“ 4 , x = 0 , 5 + x , x = 1 Q.25 for x ? [1, 8) fog (x) = (a) 8x â€“ 7 (b) 11 + 3x (c) 7 â€“ 8x (d) 12 â€“ 5x Q.26 x ?(- 8, 3/4] fog(x) is defined as (a) 3x â€“ 11 (b) 7 â€“ 8x (c) 11 + 3x (d) 5 â€“ 12x Q.27 Range of fog(x) = (a) [-4, 8) (b) [3, 8) (c) R (d) [-4, 3] Q.28 for x ? [0, 5/3) , gof (x) = (a) 2x + 6 (b) 3x + 1 (c) 19 â€“ 12x (d) -1 â€“ 8x Q.29 Range of gof(x) is (a) [4, 8) (b) [-1, 8) (c) [-1, 19] (d) [19, 8) Passage II :- We define 6 bijection as follows :- 1. f : [p / 2, 3p / 2] ? [-1, 1] defines as f(x) = Sin x 2. g : [-p , 0] ? [-1, 1] , g(x) = Cos x 3. h : [0, p] â€“ {p / 2} ?R , h(x) = tan x SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s 4. u : [-p / 2, p / 2] - {0}? R , u(x) = Cot x 5. v : [0, p] â€“ {p / 2} ? R â€“ (-1, 1) , v(x) = Sec x 6. w : [-p / 2 , p / 2] â€“ {0} ? R â€“ (-1, 1) , w(x) = Cosec x answer the following questions :- Q.30 Sin -1 x + Cos -1 x = x ? (-1, 1] (a) p (b) â€“p / 2 (c) p / 2 (d) â€“p Q.31 Domain of f -1 [h -1 (x)] is, where [x] starts for greatest integer = x (a) (- 8, tan 2) ? [0, 8) (b) (- 8, tan 1) (c) (tan 2, 0) (d) (tan 2, tan 1] Q.32 Sin -1 (Sin 16) + Tan -1 (Cot 4) (a) 12 + p (b) 16 - 4p (c) 16 + 3p/2 (d) 12 - 5p/2 Q.33 for x ? [-1, 1] â€“ {0}, Sin -1 x = ? + Sec -1 1 / x then ? = (a) p (b) â€“p (c) p / 2 (d) â€“p / 2 Q.34 Consider P(x) = h -1 (x) + f -1 (x) + v -1 (x) then the difference between greatest & smallest value of p(x) is (a) 3p / 2 (b) p / 2 (c) 5p / 2 (d) p PART D :- Match the columns Match the definitions of the function in the column I with set of points of continuity in the column II. Q.35 Column I ______ Column II (a) f(x) = [x] + vx â€“ [x] (p) { vx ; x ? I} (b) f(x) = [x] + [-x] (q) (0, 8) (c) f(x) = Cos log x (r) R â€“ Z (d) f(x) = x 2 + [x 2 ] (s) R Q.36 Column I Column II (a) The function f(x) = x 2 + 3x + 9 ; x = 1 (p) a = 3 bx + 2 ; x > 1 is diff. ? x ? R, then (b) The function f(x) = 1 / |x| ; |x| = 1 (q) b = 5 ax 2 + b ; |x| < 1, is diff. everywhere then (r) a = 35 / 9 (c) The function f(x) = ax 2 â€“ bx + 2 , x < 3 (s) b = 3 / 2 bx 2 â€“ 3 , x = 3 is diff. everywhere then (t) a = -1 / 2 Q.37 Column I Column II (a) The total no. of elements in the range of (p) 1 f(x) = [1 + Sin x] + [Cos x - 1] + [Tan -1 x] x ?[0, 2p] where [.] denotes step function is / are (q) 2 (b) only element which belongs to the range of [(Sin x) + |Cos x|] (r) 5 (c ) the total no. of solution of 8 Cos x = x is / are (s) 4 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Page 5 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Determinants, Functions, Limits, Continuity , Differentiability Class Test - 7 B Time : 2hr 30 minutes M.M. 132 Read the carefully the instructions given below : 1. The test consists of 37 questions. The test contains four sections. Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section B : Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking. Section C : Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong answer. Section D : Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking Student Name : __________________________________ Class : _____________________________ Marks Obtained : ________________________________ Date of Test : _________________________ Part A Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are (a) five (b) two (c) three (d) infinite Q.2 All solutions of the equation |x 2 â€“ x - 6| = x + 2 are (a) natural nos. (b) negative integer (c) rational nos. (d) irrational nos. Q.3 If in a triangle two angles are Tan -1 2 & Tan -1 3 then the measure of the 3 rd angle is (a) p / 4 (b) p / 2 (c) p / 3 (d) 3p / 4 _____ _____ Q.4 The identity Cos -1 x â€“ Cos -1 y = Cos -1 (xy + v1 â€“ x 2 v1 â€“ y 2 ) holds if (a) x 2 + y 2 = 1 (b) xy = 1 (c) x = y (d) x + y = 0 Q.5 If A, B, P & Q are sq. matrices of same order such that adj B = A & |P| = |Q| = 1 then adj (Q -1 B P -1 ) is given as (a) QAP (b) PAQ (c) P -1 B Q -1 (d) PBQ Q.6 If f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is (a) differentiable at x = 0 (b) continuous in (-1, 0) (c) differentiable in (-1, 1) (d) none Q.7 Let f(x) = [tan 2 x], where [.] denotes greatest integer function then (a) lim f(x) does not exist (b) f(x) is continuous at x = 0 x?0 (c) f(x) is not differentiable at x = 0 (d) f(0) = 1 Q.8 The function f(x) = max {1 â€“ x, 1 + x, 2}, x ? R is (a) continuous at all points except at one point (b) differentiable at all x (c) diff. at all points except two points in domain (d) discontinuous at x = 1& x = -1 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Q.9 The domain of the derivative of the function f(x) = tan -1 x if |x| = 1 (|x| - 1) if |x| > 1 is 2 (a) R â€“ {0} (b) R â€“ {1} (c) R â€“ {0, -1, 1} (d) R â€“ {-1, 1} Q.10 Given f ' (2) = 6 & f ' (1) = 4 , lim f(2h + 2 + h 2 ) â€“ f(2) is equal to h?0 f(h â€“ h 2 + 1) â€“ f(1) (a) 3 / 2 (b) 5 / 2 (c) 3 (d) â€“ 3 Q.11 Range of the expression f(x) = 3 â€“ 2 Cos x ? x ? R Cos x + 2 (a) [1/3, 1/5] (b) [-5, 3] (c) (0, 2/3] (d) [1/3, 5] Q.12 Let f(x) = Cos x x 1 2 Sin x x 2 2x then lim f ' (x) = Tan x x 1 x?0 x (a) 4 (b) 3 (c) 0 (d) -2 Q.13 If lim [(a - n)nx â€“ tan x] Sin nx = 0 , where n is a non-zero real no. then a is equal to x?0 x 2 (a) n (b) n + 1 (c) n + 1/n (d) n n + 1 n Q.14 If f(x) is differentiable and strictly increasing function, then the value of lim f(x 2 ) â€“ f(x) is x?0 f(x) â€“ f(0) (a) -1 (b) 1 (c) 0 (d) 2 Q.15 If y 2 = P(x) is a polynomial of degree 3 then 2 d (y 2 d 2 y ) equals dx dx 2 (a) p''' (x) + p' (x) (b) p'' (x) p''' (x) (c) p(x) p''' (x) (d) a constant Q.16 If ? (x) = x 2 â€“ 5x + 3 2x â€“ 5 3 3x 2 + x + 4 6x + 1 9 7x 2 â€“ 6x + 9 14x â€“ 6 21 = ax 3 + bx 2 + cx + d then (a) a ? 0, d = 141 (b) a = b = c = 0 (c) c = 7 (d) d = 29 PART B :- 4½ Marks Questions Q.17 The integer n for which lim (Cos x - 1) (Cos x - e x ) is equal to ? (a finite non zero no.) then is equal to x?0 x n (a) n = 4 (b) n = 3 (c) ? = -1/2 (d) ? = 1/2 Q.18 If |C| = 1 / 2 and f(x) is diff. at x = 0 given by f(x) = b Sin -1 (c + x) , -1/2 < x < 0 2 1 / 2 , x = 0 e ax / 2 â€“ 1 , 0 < x < 1 / 2 , then x (a) a = 1 (b) 16 b 2 = 4 â€“ c 2 (c) a = -1 (d) 64 b 2 = 4 â€“ c 2 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s Q.19 If x ? [-p, p] then solution set of the inequality Sin x + Sin 2x > 0 is (a) (-p, -2p / 3) (b) (p, 3p / 4) (c) (0, 2p / 3) (d) (-3p / 4, -p / 2) Q.20 Solution set of the inequality |8x 2 + 25x + 12| = | |x 3 + 6x 2 + 5x - 12| - |x 3 â€“ 2x 2 â€“ 20x - 24| | is (a) [-4, -3] (b) [-3, -6] (c) [1, 6] (d) holds free for exactly 9 integers Q.21 The no. of possible values of t for which the system of equations (a - t)x + by + cz = 0, bx + (c - t)y + az = 0, cx + ay + (b - t)z = 0, has a non-trivial solution are ________.(say ?) and the product of these values is ____________ (say µ), then (a) ? = 2 (b) ? = 3 (c) µ = a b c (d) µ = a b c b c a c a b c a b b c a Q.22 Consider the f(x) = Sin -1 (Cot -1 x) then (a) domain of f(x) is [Cot 1, 8) (b) domain of f(x) is [0, Cot 1] (c) range of f(x) is (0, p / 2] (d) range of f(x) is (p / 4, p / 2] ___________ Q.23 Consider the definition f(x) = v3 Sec -1 x â€“ p __ (a) domain of f(x) is {-1} ? [1, 2] (b) range of f(x) is [0, v2p] â€“ {p / 2} __ (c) domain of f(x) is (- 8, -1] ? [2, 8) (d) range of f(x) is (p / 2, v2p] Q.24 The system of equations ax + by + cz = q - r ; bx + cy + az = r â€“ p; cx + ay + bz = p â€“ q is (a) inconsistent if a = b = c & p , q, r are distinct (b) consistent if p = q = r (c) consistent if a, b, c are distinct & a + b + c ? 0 (d) inconsistent if p = q = r PART C :- Passage I Consider the function f(x) = 2x + 1 , x < 0 g(x) = 3 â€“ 4x , x < 1 3x â€“ 4 , x = 0 , 5 + x , x = 1 Q.25 for x ? [1, 8) fog (x) = (a) 8x â€“ 7 (b) 11 + 3x (c) 7 â€“ 8x (d) 12 â€“ 5x Q.26 x ?(- 8, 3/4] fog(x) is defined as (a) 3x â€“ 11 (b) 7 â€“ 8x (c) 11 + 3x (d) 5 â€“ 12x Q.27 Range of fog(x) = (a) [-4, 8) (b) [3, 8) (c) R (d) [-4, 3] Q.28 for x ? [0, 5/3) , gof (x) = (a) 2x + 6 (b) 3x + 1 (c) 19 â€“ 12x (d) -1 â€“ 8x Q.29 Range of gof(x) is (a) [4, 8) (b) [-1, 8) (c) [-1, 19] (d) [19, 8) Passage II :- We define 6 bijection as follows :- 1. f : [p / 2, 3p / 2] ? [-1, 1] defines as f(x) = Sin x 2. g : [-p , 0] ? [-1, 1] , g(x) = Cos x 3. h : [0, p] â€“ {p / 2} ?R , h(x) = tan x SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s 4. u : [-p / 2, p / 2] - {0}? R , u(x) = Cot x 5. v : [0, p] â€“ {p / 2} ? R â€“ (-1, 1) , v(x) = Sec x 6. w : [-p / 2 , p / 2] â€“ {0} ? R â€“ (-1, 1) , w(x) = Cosec x answer the following questions :- Q.30 Sin -1 x + Cos -1 x = x ? (-1, 1] (a) p (b) â€“p / 2 (c) p / 2 (d) â€“p Q.31 Domain of f -1 [h -1 (x)] is, where [x] starts for greatest integer = x (a) (- 8, tan 2) ? [0, 8) (b) (- 8, tan 1) (c) (tan 2, 0) (d) (tan 2, tan 1] Q.32 Sin -1 (Sin 16) + Tan -1 (Cot 4) (a) 12 + p (b) 16 - 4p (c) 16 + 3p/2 (d) 12 - 5p/2 Q.33 for x ? [-1, 1] â€“ {0}, Sin -1 x = ? + Sec -1 1 / x then ? = (a) p (b) â€“p (c) p / 2 (d) â€“p / 2 Q.34 Consider P(x) = h -1 (x) + f -1 (x) + v -1 (x) then the difference between greatest & smallest value of p(x) is (a) 3p / 2 (b) p / 2 (c) 5p / 2 (d) p PART D :- Match the columns Match the definitions of the function in the column I with set of points of continuity in the column II. Q.35 Column I ______ Column II (a) f(x) = [x] + vx â€“ [x] (p) { vx ; x ? I} (b) f(x) = [x] + [-x] (q) (0, 8) (c) f(x) = Cos log x (r) R â€“ Z (d) f(x) = x 2 + [x 2 ] (s) R Q.36 Column I Column II (a) The function f(x) = x 2 + 3x + 9 ; x = 1 (p) a = 3 bx + 2 ; x > 1 is diff. ? x ? R, then (b) The function f(x) = 1 / |x| ; |x| = 1 (q) b = 5 ax 2 + b ; |x| < 1, is diff. everywhere then (r) a = 35 / 9 (c) The function f(x) = ax 2 â€“ bx + 2 , x < 3 (s) b = 3 / 2 bx 2 â€“ 3 , x = 3 is diff. everywhere then (t) a = -1 / 2 Q.37 Column I Column II (a) The total no. of elements in the range of (p) 1 f(x) = [1 + Sin x] + [Cos x - 1] + [Tan -1 x] x ?[0, 2p] where [.] denotes step function is / are (q) 2 (b) only element which belongs to the range of [(Sin x) + |Cos x|] (r) 5 (c ) the total no. of solution of 8 Cos x = x is / are (s) 4 SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106 Classes in Mathematics for IIT-JEE/AIEEE +2 Sapra Classes M a t h e m a t i c s OBJECTIVE RESPONSE SHEET Name of the student :- _________________________ Class :- _________________ Date :- _____________ Topic :-________________________________ Marks Obtained :- _________________________ Darken the bubble corresponding to correct choice from Q.1 â€“ Q.34 A B C D A B C D 1 O O O O 18 O O O O 2 O O O O 19 O O O O 3 O O O O 20 O O O O 4 O O O O 21 O O O O 5 O O O O 22 O O O O 6 O O O O 23 O O O O 7 O O O O 24 O O O O 8 O O O O 25 O O O O 9 O O O O 26 O O O O 10 O O O O 27 O O O O 11 O O O O 28 O O O O 12 O O O O 29 O O O O 13 O O O O 30 O O O O 14 O O O O 31 O O O O 15 O O O O 32 O O O O 16 O O O O 33 O O O O 17 O O O O 34 O O O O Match the following :- Q.35 p q r s Q.36 p q r s t Q.37 p q r s A O O O O A O O O O O A O O O O B O O O O B O O O O O B O O O O C O O O O C O O O O O C O O O O D O O O O D O O O O O D O O O O _____________________________________________________________________________________ SAPRA CLASSES (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS) SCO- 43 , SEC â€“ 20 C , CHANDIGARH. 9041960872, SCF. 18 , SEC 15 , PANCHKULA, 98720-27106Read More