Lagos State University MAT 112 Calculus Past Paper PDF

Lagos State University Faculty Of Science Department of Mathematics Rain Semester Examination :: 2011/2012 Se...

Lagos State University Faculty Of Science Department of Mathematics Rain Semester Examination :: 2011/2012 Session MAT 112 - Calculus Time Allowed: 1 12 Hours QUESTION TYPE 2 REPRODUCED BY BUNDAY TUTORIAL Instruction: Answer All questions. Shade Option E where none of options A,B,C,D is Correct. 1. The first derivative of cosm ψ with respect to ψ is (A) −m cosm−1 ψ sin ψ (B) m cosm−1 ψ sinm ψ (C) −m sinm−1 ψ cosm ψ (D) m sinm ψ 2. Given that the profit function is 2x2 − 8x, the minimum profit is (A) 8 (B) -2 (C) -8 (D) 4 d2 y 3. At maximum point of y(x), dx2 is (A) > 0 (B) < 0 (C) = 0 (D) = 1 4. Find y in terms of x if y 0 = 3x2 − 6x + 2 and y = 7 when x = 0 (A) y = x3 − 3x2 + 2x + 7 (B) y = 3x2 − 6x + 7 (C) y = 3x2 + 2x + 7 (D) y = x3 − 3x2 + 2x 5. The rate of change of the area of a circle w.r.t its radius is (A) Diameter (B) Circumference (C) Area×radius (D) 2×diameter 6. Define y as an explicit function of x when x + y + y 2 = x2 (A) y = x2 − x − y 2 (B) y = −x or y = x − 1 (C) y = 1 − x or y = −x (D) Not possible π π 7. The area between the curve y = (sin x + cos x)2 , the x-axis and the ordinates at x = 0 and x = 2 is (A) 2 +1 (B) π2 + 2 (C) π2 + 3 (D) π2 + 4 dy 8. If x = a(t sin t + cos t − 1) and y = a(sin t − t cos t), find dx (A) tan t (B) sec2 t (C) 1 − cos t (D) sin t + cos t dx (A) 21 arcsin 13 (3x − 2) (B) 13 arcsin 12 (2 − 3x) (C) 13 arcsin 21 (3x − 2) (D) 34 arcsin 34 x R 9. √12x−9x 2 10. If y = sec θ, which of the following is correct? (A) y 00 = y(2y 2 + 1) (B) y 00 + y = y(2y 2 + 1) (C) y 00 + y 0 + y = 2y 3 (D) y 00 = y(2y 2 − 1) 11. If y = θn , where n ∈ Z+ , which of the following is not correct? (A) θ2 y 00 − n2 y 0 + ny = 0 (B)θy 0 = ny (C) θ2 y 00 = n(n − 1)y (D) y + y 0 = nθn (n−1 + θ−1 ) 12. Integrate cos2 21 x w.r.t x (A) x 2 + 21 sin x (B) 2 cos 12 x sin 21 x (C) x + sin x (D) 1 2 + 21 sin x dy 2x+3y −2x−3y 3x+2y −3x+2y 13. Find dx if 3xy + x2 + y 2 = 5 (A) 3x+2y (B) 3x+2y (C) 2x+3y (D) 2x+3y dy 4t 4 t 14. If x = t3 + t and y = 2t2 , then dx is (A) 3t2 +1 (B) 3t+1 (C) t2 +1 (D) 2t5 + t3 15. Integration is also known as (A) Pseudo-differentiation (B) Contra-differentiation (C) Dual-differentiation (D) Anti-differentiation 16. Is it true that if y = (ω 2 − 1)n , then (ω 2 − 1)y 0 − 2nωy = 0 (A) Yes (B) No (C) I don’t know x−q
x−q
1 1 1 ln x−p 1 x−p

17. Integrate x2 +(p+q)x+pq w.r.t x (A) ln x−p (B) p−q ln x−p (C) q−p x−q (D) p−q ln x−q d ln(ln(ln x))dx] (A) ln1x (B) ln(ln x) (C) ln(ln(ln x)) (D) x ln 1 R 18. Evaluate dx [ x R d R R d2 19. Simplify dx dx2 (cos x) (A) Very difficult (B) Not impossible (C) − cos x (D) cos x d2 20. If y = (x + 4)(11x + 1)(3x + 2), find (A) 198x + 226 (B) 198 + 226x (C) 198x − 78 (D) 99x2 + 226 − 78 dx2 21. Differentiate y = (x2 + 4)−1 w.r.t x (A) 12 arctan x2 (B) (x22x (C) (x−2x 1
+4)2 2 +4)2 (D) arctan x2 +4 22. The rate of change of velocity with time is (A) speed (B) acceleration (C) displacement (D) momentum R5 23. Evaluate 2 15x2 dx (A) 625 (B) x3 (C) 585 (D) 5x3 − 2x 24. Integrate 112 w.r.t x (A) 4 (B) 0 (C) 112x (D) 211 R 50 25. Evaluate 10 (3x2 + 4x + 1)dx(A) 128840 (B) 124088 (C) 128480 (D) 128884 26. Which of these is not correct? (A) [ f (x)]dx = [ f (x)dx] (B) ff0(x) dx = ln f 0 (x) (C) x dx R P PR R R R R R (x) ln x = ln(ln x) (D) 2f (x)dx = f (x)dx + f (x)dx √ 27. If an integrand involves x2 − a2 , which of the following substitution is best suitable? (A) x = a sin θ (B) x = a sec θ (C) x = a tan θ (D) x = a cos θ 28. Find the maximum value of y = x3 − 6x2 + 9x (A) 1 (B) 2 (C) 3 (D) 4 R 1 dx √ √ 29. 0 √3−2x (A) 1 (B) -1 (C) 3 − 1 (D) 1 − 3 30. Integrate xe3x w.r.t x (A) 31 xe3x − 91 e3x (B) 3e3x + 1 (C) 3xe3x + e3x (D) 91 xe3x − 13 e3x 31. ln xdx (A) x1 (B) x − ln x (C) x ln x − x (D) x ln x − x1 R 32. Differentiate 4x2 sin x − 3x2 cos x (A) 4x2 cos x − 3x2 sin x (B) (3x2 + 8x) cos x + (4x2 − 6x) sin x (C) (3x2 − 8x) cos x + (4x2 + 6x) sin x (D) (3x2 + 8x) sin x + (4x2 − 6x) cos x 1.5 33. The integral (5x + 3)0.5 evaluates to (A) (5x+3) (B) 32 (5x + 3)−0.5 (C) 10 3 2 (5x + 3)0.5 (D) 15 (5x + 3)1.5 R 1.5 dx 1 1 34. If y = x2 + 1, then dy is (A) 2 (B) 2x (C) 2x (D) Not possible x sin x dy x+sin2 x+sin x cos x x+cos2 x+sin x cos x x−sin2 x−sin x cos x x+sin x+sin x cos x 35. Given that y = cos x+sin x , then dx is (A) 1+sin 2x (B) 1+sin 2x (C) (cos x+sin x)2 (D) (cos x+sin x)2 R 2√3 dx π 1 1 π π π π 36. 2 x2 +4 (A) 4 (B) 2 arctan 2 x (C) 3 − 4 (D) 4 − 3 √
d 1+√x √ 1 √ 1 √1 √ √1 √ 37. Evaluate dx 1− x (A) 2 x(1+ x)2 (B) (1+ x)2 (C) (1+ x)2 x √ (D) (1− x)2 x f (x+h)−f (x) 38. If f (x) is a function of x, which of these best define the derivative of f w.r.t x (A) f 0 (x) (B) lim h (C) h→1 lim f (x+h)−f h (x) (D) lim f (x+h)−f h (x) x→0 h→0 3+x+2x2 39. Evaluate lim 2 (A) 1 (B) 2 (C) 3 (D) ∞ x→∞ 1+x d2 y 40. If dx2 = 0 at a point x, then x is a point (A) Fusion (B) Inflexion (C) Minimum (D) Maximum dy 41. If y = 1508, then dx is (A) 1 (B) 5 (C) 0 (D) 8 42. The second derivative of a function is used to determine its (A) Turning point (B) Value when it is squared (C) Minimum or Maximum points (D) Limits x+x2 +3x3 43. Evaluate lim x (A) 0 (B) 1 (C) 2 (D) 3 x→0 44. The equation y = x3 (x − 4) has number of inflexion (A) 1 (B) 2 (C) 3 (D) 4 45. Integrate ex sin x w.r.t x (A) ∞ (B) ex + cos x (C) 21 ex (cos x − sin x) (D) 12 ex (sin x − cos x) 46. tan5 x sec2 xdx is (A) 2 sec7 x (B) 16 tan6 x (C) sec2 x + tan6 x (D) 61 tan4 x sec2 x R dy 47. Find dx if y = 2x (A) 2x ln 2 (B) x2 ln 2 (C) 2x ln x (D) 2x ln 2x 48. Differentiate sin2 (x2 + 1) (A) 4x cos2 (x2 + 1) (B) 2 cos(x2 + 1) (C) 4x sin(x2 + 1) cos(x2 + 1) (D) 2x sin2 (x2 + 1) cos(x2 + 1) sin x π 49. Evaluate limπ (A) (B) Not Possible (C) ∞ (D) 0 x→ 2 cos x 2 d 1−x2 1−x2 −2 
 50. Find dx arcsin 1+x2 (A) Not possible (B) 2(1 + x) (C) 1+x2 (D) 1+x2