Title CONTINUITY & DIFFERENTIABILITY.pdf ✅

CHAPTER 5 CONTINUITY & DIFFERENTIABILITY Exercise 1: NCERT Based Topic-wise MCQs 5.1 & 5.2 INTRODUCTION AND CONTINUITY 1. The relationship between a and b, so that the function f defined by f(x) = { ax + 1, if x ≤ 3 bx + 3, if x > 3 is continuous at x = 3, is NCERT Page-149/N-106 (a) a = b + 2 3 (b) a − b = 3 2 (c) a + b = 2 3 (d) a + b = 2 2. All the points of discontinuity of the function f defined by f(x) = { 3, if 0 ≤ x ≤ 1 4, if 1 < x < 3 5, if 3 ≤ x ≤ 10 (a) 1,3 NCERT Page-149/N-106 (b) 3,10 (c) 1,3,10 (d) 0,1,3 3. Let f(x) = { 1−sin3 x 3cos2 x , x < π 2 p, x = π 2 q(1−sin x) (π−2x) 2 , x > π 2 If f(x) is continuous at x = π 2 , (p, q) = NCERT Page-149/N-108 (a) (1,4) (b) ( 1 2 , 2) (c) ( 1 2 , 4) (d) None of these 4. If f(x) = √4+x−2 x , x ≠ 0 be continuous at x = 0, then f(0) = (a) 1 2
(b) 1 4 (c) 2 (d) 3 2 5. If f(x) = { −x 2 , when x ≤ 0 5x − 4, when 0 < x ≤ 1 4x 2 − 3x, when 1 < x < 2 3x + 4, when x ≥ 2 , then (a) f(x) is continuous at x = 0 NCERT Page-149/N-106 (b) f(x) is continuous at x = 2 (c) f(x) is discontinuous at x = 1 (d) None of these 6. Let f(x) = (e x−1) 2 sin ( x a )log (1+ x 4 ) for x ≠ 0, and f(0) = 12. If f is continuous at x = 0, then the value of a is equal to NCERT/ Page-147/N-108 (a) 1 (b) -1 (c) 2 (d) 3 7. The value of λ, for which the function NCERT Page-149/N-106 f(x) = { λ(x 2 − 2x) if x ≤ 0 4x + 1 if x > 0 is continuous at x = 0, is : (a) 1 (b) -1 (c) 0 (d) None of these 8. Let f(x) = { 3x − 4, 0 ≤ x ≤ 2 2x + l, 2 < x ≤ 9 . If f is continuous at x = 2, then what is the value of l ? (a) 0 NCERT Page-149/N-106 (b) 2 (c) -2 (d) -1 9. f(x) = 1 1+tan x NCERT Page-149/N-108 (a) is a continuous, real-valued function for all x ∈ (−∞, ∞) (b) is discontinuous only at x = 3π 4 (c) has only finitely many discontinuities on (−∞, ∞) (d) has infinitely many discontinuities on (−∞, ∞) 10. If for p ≠ q ≠ 0, then function f(x) f(x) = √p(729+x) 7 −3 √729+qx) 3 −9 is continuous at x = 0, then: (a) 7pqf(0) − 1 = 0 NCERT Page 149/N − 108 (c) 21qf(0) − p 2 = 0 (b) 63qf(0) − p 2 = 0 (d) 7pqf(0) − 9 = 0 11. The function f: R → R defined by
f(x) = limn→∞ cos (2πx)−x 2nsin (x−1) 1+x 2n+1−x 2n is continuous for all x in: NCERT Page 149/N-108 (a) R − {−1} (c) R − {1} (b) R − {−1,1} (d) R − {0} 12. Let the function f(x) = { loge (1 + 5x) − loge (1 + αx) x ; if x ≠ 0 10 ; if x = 0 be continuous at x = 0. Then a is equal to : NCERT Page 147/N − 106 (a) 10 (b) -10 (c) 5 (d) -5 13. Let f, g:R → R be functions defined by NCERT Page-147/N-106 f(x) = { [x], x < 0 |1 − x|, x ≥ 0 and g(x) = { e x − x, x < 0 (x − 1) 2 − 1, x ≥ 0 where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly : (a) one point (b) two points (c) three points (d) four points 14. Let f: R → R be defined as f(x) = [ [e x ], x < 0 ae x + [x − 1], 0 ≤ x < 1 b + [sin (πx)], 1 ≤ x < 2 [e −x ] − c, x ≥ 2 where a, b, c ∈ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statement is true? NCERT Page N − 108 (a) There exists a, b, c ∈ R such that f is continuous of R (b) If f is discontinuous at exactly one point, then a + b + c = 1. (c) If f is discontinuous at exactly one point, then a + b + c ≠ 1. (d) f is discontinuous at atleast two points, for any values of a, b and c. 15. If f(x) = { 1−√2sin x π−4x , if x ≠ π 4 a, if x = π 4 continuous at π 4 , then a is equal to NCERT Page-149/N-106 (a) 4 (b) 2 (c) 1 (d) 1 4
16. The number of points at which the function f(x) = 1 x−[x] , [.] denotes the greatest integer function is not continuous is NCERT Page-155/N-108 (a) 1 (b) 2 (c) 3 (d) None of these 17. If f(x) = { √1+kx−√1−kx x , for − 1 ≤ x < 0 2x 2 + 3x − 2, for 0 ≤ x ≤ 1 is continuous at x = 0, then k is equal to (a) -4 (b) -3 (c) -2 (d) -1 18. Let f(x) = { x−4 |x−4| + a, x < 4 a + b, x = 4 x−4 |x−4| + b, x > 4 Then f(x) is continuous at x = 4 when (a) a = 0, b = 0 NCERT Page-149/N-108 (c) a = −1, b = 1 (b) a = 1, b = 1 (d) a = 1, b = −1 19. If f: R → R is defined by f(x) = { 2sin x−sin 2x 2xcos x , if x ≠ 0 a, if x = 0 then the value of a, so that f is continuous at 0 , is NCERT Page-149/N-108 (a) 2 (b) 1 (c) -1 (d) 0 20. Let f(x) = { x 3+x 2−16x+20 (x−2) 2 , x ≠ 2 k , x = 2 If f(x) is continuous for all x, then k = (a) 3 (b) 5 (c) 7 (d) 9 21. If f(x) = 1 1−x , then the points of discontinuity of the function f[f{f(x)}] are NCERT Page-155/N-108 (a) {0, −1} (b) {0,1} (c) {1, −1} (d) None
22. If f(x) = { x k cos (1/x), x ≠ 0 0 , x = 0 is continuous at x = 0, then (a) k < 0 (b) k > 0 (c) k = 0 (d) k ≥ 0 23. If f(x) = { 8 x−4 x−2 x+1 x 2 , x > 0 e x sin x + πx + λln 4, x ≤ 0 is continuous at x = 0. Then, the value of λ is (a) 4loge 2 NCERT Page-153/N-106 (b) 2loge 2 (c) loge 2 (d) None of these 24. Let f(x) = { 5 1/x , x < 0 λ[x], x ≥ 0 and λ ∈ R, then at x = 0 (a) f is discontinuous NCERT Page-153/N-106 (b) f is continuous only, if λ = 0 (c) f is continuous only, whatever λ may be (d) None of these 25. If f(x) = x 2 + sin x − 5, then NCERT Page-149/N-106 (a) f(x) is continuous at all points (b) f(x) is discontinuous at x = π. (c) It is discontinuous at x = π 2 (d) None of the above 26. If we can draw the graph of the function around a point without lifting the pen from the plane of the paper, then the function is said to be NCERT Page-147/N-106 (a) not continuous (b) continuous (c) not defined (d) None of these 27. If the function f(x) = { 1, x ≤ 2 ax + b, 2 < x < 4 7, x ≥ 4 is continuous at x = 2 and 4 , then the values of a and b are. NCERT Page-149/N-106 (a) a = 3, b = −5 (b) a = −5, b = 3 (c) a = −3, b = 5 (d) a = 5, b = −3 28. The point(s), at which the function f given by f(x) = { x |x| , x < 0 −1, x ≥ 0 is continuous, is/are : (a) x ∈ R NCERT Page-149/N-106 I CBSE Sample 2021-22 (b) x = 0 (c) x ∈ R − {0} (d) x = −1 and 1