# Application Of Derivative-03-Exercise

25 de May de 2023

### Application Of Derivative-03-Exercise

• 1. SECTION-I OBJECTIVE LEVEL-I Multiple Choice Questions with Single Answer: 1. Sum of the square of the intercepts on the axes cut off by the tangent x1/3 + y1/3 = a1/3 , a R  at (a/8, a/8) is 2. Then possible value(s) ofa is/are (A) 4 (B) 2 (C) 6 (D) 3 2. The criticalpoints off(x) = 2 2 | x 1| x  are (A) 0, – 1 (B) 1, – 1 (C) –1, 0 (D) noneofthese 3. Iff(x) = a secx – b tanx, a > b > 0, thenthe minimumvalue off(x) is (A) 2 2 a b  (B) 2 2 2 a b  (C) 2 2 a b  (D) a – b 4. The range ofy = (arc cosx) (arc sinx) is (A) , 2 2          (B) , 2 2          (C) 2 2 , 2 16         (D) 2 0, 16        5. If f is continuous function in [1, 2] such that |f(1) + 3| < |f(1)| + 3 and |f(2) + 10| = |f(2)| + 10, (f(2)  0), then the function fin(1, 2) has (A) at least one root (B) no root (C) exactlyone root (D) noneofthese 6. Iff(x) = x + 1  xR and g(x) = ex  x [–2, 0], then the maximumvalue off(|x|) – g(x) is (A) 1 3 e  (B) 2 1 3 e  (C) 2 1 3 e   (D) 2 1 3 e 
• 2. 7. The rangeoffunction 1 f (x) sin xsin x 2          (A) , 2 2          (B) 0, 2        (C) [0, 1] (D) noneofthese 8. Let f(x) = x3 + ax2 + bx+ 5 sin2 x be an increasing functioninthe set ofrealnumbers R. Thena and bsatisfythe condition (A) a2 – 3b – 15  0 (B) a2 – 3b + 15  0 (C) a2 – 3b + 15  0 (D) a > 0 and b > 0 9. Ifthe tangent at point Ponthe curve y2 =x3 intersectsthe curve againat Qand thestraight linesOP, OQ make angles ,  with the positive x–axis where ‘O’ is the origin then tan tan   has the value equalto (A) –1 (B) –2 (C) 2 (D) 2 10. If a, b, c, d are real numbers such that 3a 2b 3 0, c d 2     Then the equation ax3 + bx2 + cx + d = 0 has. (A) at least one root in [–2, 0] (B) at least one root in [0, 2] (C) at least two roots in [–2, 2] (D) No root in [–2, 2]
• 3. LEVEL - II Multiple Choice Questions with one or more than one correct Answers: 1. The function 2 x y 1 x   decreasesin the interval (A) (–1, 1) (B) [1, )  (C) ( , 1]   (D) ( , )   2. For x > 1, y = log x – (x – 1) satisfies the inequality (A) x – 1 > y (B) x2 – 1 > y (C) y > x – 1 (D) x 1 y x   3. Suppose f '(x) exists for each xand h(x) = f(x) – {f(x)}2 + {f(x)}3 for everyreal number x then (A) his increasing whenever fis increasing (B) his increasing when ever fis decreasing (C) his decreasing when ever fis decreasing (D) nothing canbe said ingeneral 4. 2 3x 12x 1, 1 x 2 If f (x) 37 x , 2 x 3             then (A) f(x) is increasing on [–1, 2] (B) f(x) iscontinous on [–1, 3] (C) f '(2) doesn’t exist (D) f(x) has the maximumvalue at x = 2 5. The equations ofthe tangents to the curvey= x4 fromthe point (2, 0)not onthe curveare givenby (A) y = 0 (B) y – 1 = 5(x – 1) (C) 4096 2048 8 y x 81 27 3           (D) 32 80 2 y x 243 81 3           6. 3 2 x x 10x 1 x 0 Let f(x) sin x 0 x 2 1 cos x x 2                        thenf(x) has (A) localmaxima at x 2   (B) localminima at x 2   (C) absolute maxima at x = 0 (D) absolute maxima at x = –1 7. For the functionf(x) = cos(cos(sinx)) (A) greatest value = cos cos 1 (B) least value = 0 (C) greatest value = 1 (D) least value = cos 1
• 4. 8. If f (x) 0 x R     and g(x) = f(x2 – 2) + f(6 – x2 ) then (A) g(x) is anincreasing in[–2, 0] (B) g(x) is an increasing in [2, ) (C) g(x) has a local minima at x = –2 (D) g(x) has a local maxima at x = 2 9. If 1 1 f (x) tan x log x 2    then (A) the greatest value off(x) on 1 1 , 3 is log3 6 4 3               (B) the least value off(x) on 1 1 , 3 is log3 3 4 3         (C) f(x) decreases on (0, )  (C) f(x) increases on ( , 0)  10. The functionf(x) = 2log(x – 2) – x2 + 4x + 1 increases in the interval (A) (1, 2) (B) (2, 3) (C) 5 ,3 2       (D) (2, 4)
• 5. Multiple Choice Questions with Single Answer from other competitive exams: 1. Ifthe perimeter ofa rectangle is 100 cm, what willbe its sides such that its area is maximum? (A) 25 cm 25 cm (B) 10 cm, 40 cm (C) 15 cm, 35 cm (D) 20 cm, 30 cm 2. The minimumvalue of log x x inthe interval [2, )  is (A) log2 2 (B)zero (C) 1/c (D) does not exist 3. If l2 + m2 = 1, then maximumvalue of l + mis (A) 1 (B) 2 (C) 1/ 2 (D) 2 4. The radius ofa cylinder is increasingat the rate 2 m/s and its height is decreasing at the rate 3m/s. When the radius is 3 mand height is 5 m, the volume ofthe cylinder would change at the rate of (A) 3 87 m / s  (B) 3 33 m /s  (C) 27 m /s  (D) 3 15 m /s  5. Ifthe rate of change inthe circumference ofa circle is 0.3 cm/sec, then the rate ofchange in the area ofthe circle when the radius is 5 cmis (A) 1.5 sq.cm/sec (B) 0.5 sq.cm/sec (C) 5 sq. cm/sec (D) 3 sq. cm/sec 6. A particle moves along a straight line according to thelaw s = et (sin t – cost). The acceleration at anytime t is (A) et (cos t + sin t) (B) et (cos t – sin t) (C) 2et (cos t – sin t) (D) 2et (cos t + sin t) 7. Ify = x3 – ax2 + 48x + 7 is an increasing function for all realvalues ofx, then a lies in (A) [–14, 14] (B) [–12, 12] (C) [–16, 16] (D) [–21, 21] 8. The value of‘a’for whichthe function 1 f (x) asin x sin3x 3   has anextremumat x 3   is (A) 1 (B) –1 (C) 2 (D) 0 9. Rolle’s theorem holds for the function 3 2 x bx cx, 1 x 2     at the point 4/3, the values of b and c are (A) b = 8, c = –5 (B) b = –5, c = 8 (C) b = 5, c = –8 (D) b = –5, c = –8 10. Rolle’s theoremisnot applicable for the function f(x) = |x| in the interval[–1, 1] because : (A) f '(1) does not exist (B) f '( 1)  does not exist (C) f(x) is discontinuous at x= 0 (D) f '(0) does not exist 11. At what points ofthe curve 3 2 2 1 y x x , 3 2   tangent makes equalangles with axis (A) 1 5 1 , and 1, 2 24 6               (B)   1 4 , and 1,0 2 9       
• 6. (C) 1 1 1 , and 3, 3 7 2               (D) 1 4 1 , and 1, 3 27 3               12. Themaximumvalueofthefunction f(x) sin x cos x 6 6                   intheinterval 0, 2        occurs at (A) /12 (B) /6 (C) /4 (D) /3 13. Ifthe curvey= ax2 – 6x+ bpasses through(0, 2) and has its tangent parallelto x-axis at x 3/ 2  , then the value of a and b are respectively (A) 2 and 2 (B) –2 and – 2 (C) – 2 and 2 (D) 2 and –2 14. On the interval[0, 1], the function x25 (1 – x)75 takes its maximumvalue at the point (A) 0 (B) 1/4 (C) 1/2 (D) 1/3 15. The equations ofthe tangents at those points where the curve y= x2 – 3x + 2 meets x-axis are (A) x – y + 2 = 0 = x – y – 1 (B) x + y – 1 = 0 x – y = 0 (C) x – y – 1 = 0 = x – y (D) x – y = 0 = x + y 16. Suppose the cubic equation x3 – px + q = 0 has three distinct real roots where p > 0 and q > 0. Thenwhichoneofthe following holds? (A) The cubic has minima at both p 3 and p 3  (B) The cubic has maxima at both p 3 and p 3  (C) The cubic has minima at p 3 and maxima at – p 3 (D) The cubic has minima at – p 3 and maxima at p 3 17. How many real solutions does the equation x7 + 14x5 + 16x3 + 30x – 560 = 0 have? (A) 3 (B) 5 (C) 7 (D) 1 18. Let f(x) = 1 (x 1)sin if x 1 x 1 0 if x 1          . Thenwhich one ofthe following is true? (A) f is differentiable at x = 0 but not at x = 1 (B) f is differentiable at x = 1 but not x = 0 (C) fis neither differentiable at x = 0 nor at x = 1 (D) fis differentiable at x = 0 and at x= 1
• 7. SECTION-II SUBJECTIVE LEVEL-I 1. A particle moves so that the space described in time t is square root of a quadratic functionoft. Then, prove that the accelerationofthe particle varies as 3 1 s . 2. Find the angle between the tangents to the ellipse 2 2 2 2 x y 1 a b   and the circle x2 + y2 = ab at their intersectingpoints. 3. Find the intervals ofincrease &decrease ofthe functionf(x) = x – ln |x|. 4. Find the criticalpoints ofthe function f(x) = (x – 3)2/3 (3x– 1). 5. Show that f(x) = sin(x a) ,b a k ,k l sin(x b)       , has no points ofextrema. 6. Find the greatest and the least value ofthe functionf(x) = 2 2 (1 x )(1 2x )   definedon[–1, 1]. 7. Find the least value off(x) = a2 sec2 x + b2 cosec2 x, given ab  0. 8. (i) Define afunctionfas follows: 2 3 x x 1 2 f(x) 1 x 1 x            (A) Sketch the graphfor x in the interval[0, 2] (B) Show that ‘f’satisfiesthe conditionsofthe meanvaluetheoremoverthe interval[0, 2] and determine allthe meanvalues provided bythe theorem. 9. Let f(x) = (x– 3) (x– 4) (x– 5)(x– 6) thenprove that f (x)  = 0 has exactlythree realroots which lies in the intervals (3, 4), (4, 5) and (5, 6). 10. f: R  R defined byf(x) is anodd continuous functionand is strictlymonotonic  x R. Prove that their does not exist   R – {0} such that f( ) = 0.
• 8. LEVEL-II 1. IfP(x) is afunction such that |P(x)|  |3x–1 –2x–1 | where P(x) = n r r r 0 a x   , where an  0, then prove that the maximumvalue of n r r 1 ra   is (ln(3/2)). 2. If xf (x) f(x) 0 x 2 and f(2) 1       , then show that f(x) < 2 x x 2   . 3. Let fbe differentiable for allx. If f(1) = –2 and f (x)  2 for all x [1,6]  . Then prove that least value off(6) is 8. 4. Prove that the value’s ofPfor whichthe functionf(x) = (4P–3)(x+ log 5)+ 2 (P–7) 2 x x cot sin 2 2 does not possess criticalpoints is given by 4 , (2, ) 3           . 5. Show that (i) tan x x for 0 x x sin x 2     (ii) 3 1 x x tan x, for 0 x 1 6      (iii) sin x + tan x  2x  0  x  /2 (iv) 2x + x cos x  3 sin x 0  x  /2 6. IfRolle’s theoremis applicable in [a, b] for the function f(x) = 3x4 + 4x3 + 12 x2 + 24x+ 96, then prove that c > a + b + ab, where ‘c’is the Rolle’s critical point. 7. Let g(x) = x 2f 2       + f(2– x)and f(x) <0 x (0,2)   . Findthe intervals ofincreaseand decrease ofg(x). 8. For what values of a, m and b does the function f(x) = 2 3 x 0 x 3x a 0 x 1 mx b 1 x 2               satisfy the hypothesis ofmeanvalue theoremonthe interval[0, 2]. 9. Discussthe globalmaxima/minima offollowing functions inthe giveninterval. (i) f(x) = 3x4 – 2x3 – 6x2 + 6x + 1 in [0, 2] (ii) f(x) = sinx sin2x in (– ,  ). 10. If polynomial equation P(x) = an xn + an - 1 xn -1 + an - 2 x n - 2 + ........+ a1 x = 0 has a positive root x0 , thenprove that the equation 0 ) x ( P   has positive root less then x0 .
• 9. SECTION-III-A Matrix–Match Type This sectioncontains 2questions. Each question contains statements given in two column which have to be matched. Statements (A, B, C, D) in Column I have to be matched with statements (p, q, r, s) in Column II. The answers to these questions have to be appropriatelybubbles as illustrated inthe following example. If the correct matches areA–p, A–s, B–q, B–r, C–p, C–q and D–s, then the correctly bubbled 4 × 4 matrix should be as follows : p q r s p q r s p p p q q q r r r s s s A B C D 1. Let f(x) = (2x – 1) (2x – 2) and g(x) = 2 sin x + cos 2x in [0, ] Column-I Column-II (A) fincreases on (p) (log2 (3/2), ) (B) f decreases on (q) (- , log2 (3/2)) (C) g decreases on (r) 5 , 6 6         (D) g increases on (s) 5 0, , 6 6                 2. The greatest andleast values offunctions inList I is Column-I Column-II (A) x 2 x on [0, 4]  (p) 3 1 & 5 (B) x 1 on [0, 4] x 1   (q) 13 & 4 (C) 2 2 1 x x on [0, 1] 1 x x     (r) 8 and 0 (D) x4 – 2x2 + 5 on [–2, 2] (s) 3 and 1 5 
• 10. SECTION-III-B Linked Comprehension Type This section contains 3 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. I. Let f(x) = x3 + ax2 + bx + c be a cubic polynomial where a, b, c  R. Now f(x) = 3x2 + 2ax + b and let D = 4a2 – 12b be the discriminant ofthe equation f(x) = 0. If D > 0, f(x) = 0 has two real roots. ,  ( < ), then x =  will be point of local maxima and x =  willbe a point oflocalminima off(x), also Iff() . f( ) > 0, then f(x) = 0 would have just one realroot. f() . f( ) < 0, then f(x) = 0 would have three realand distinct roots. f() . f( ) = 0, then f(x) = 0 would have three real roots. 1. If the function f(x) = x3 – 9x2 + 24x + k has three real and distinct roots x1 , x2 , x3 where x1 < x2 < x3 . Then the possible value ofk willbe (A) k < –20 (B) k > 20 (C) 16 < k < 20 (D) –20 < k < –16 2. In the questionNo. 1, [x1 ] + [x3 ] is equalto {where [x] is greatest integer function} (A) 2 (B) 3 (C) 4 (D) 5 3. In the questionNo. 1, x2 lies in the interval (A) (–2, 0) (B) (0, 2) (C) (2, 4) (D) noneofthese 4. If f(x) = ax3 + bx2 + cx + d has it non–zero local minimum and maximum values at x = 2 and x = 1 respectively. If a be the root of the equation x2 – 2x – 15 = 0, then a is equal to (A) –3 (B) 5 (C) both a and b (D) noneofthese II. The function‘g’definedbyg(x) = f(x2 – 2x+ 8) +f(14 +2x– x2 ), wheref(x) is twice differentiable function, f "(x) 0  for allrealnumbers x. 5. The functiong(x)is increasing inthe interval (A) [–1, 1]  [2, ) (B) (–, –1]  [1, 3] (C) [–1, 1]  [3, ) (D) (–, –2]  [1, ) 6. The functiong(x) is decreasing inthe interval (A) [–1, 1]  [2, ) (B) (–, –1]  [1, 3] (C) [–1, 1]  [3, ] (D) (–, –2]  [1, 3]
• 11. SECTION-III-C (Assertion – Reason Type) Each question contains STATEMENT – 1 (Assertion) and STATEMENT – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. Instructions: (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 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. 1. Consider theirrational numbers: e , e Statement–1 : e < e because Statement–2 : f(x) = x1/x is decreasing in(e, ). (A)A (B) B (C) C (D) D 2. Consider the equation x3 – 3x + k = 0, k  R. Statement–1 : Thereisnovalueofkforwhichthegivenequationhastwo distinct rootsin(0, 1) because Statement–2 : Between two consecutive roots of f(x) = 0, (f(x) is a polynomial), f(x) = 0 must have one root. (A)A (B) B (C) C (D) D 3. Statement-1. : 1 f (x) n(1 x), x 1      x> 0 is a decreasing function. Statement-2. : f (x) < 0  x> 0 (A)A (B) B (C) C (D) D 4. Statement–1 : The localmaximumofthe function x cos xwilloccur between/4 and /3 because Statement–2 : The function x tanx isincreasing in 0, 4        and decreasing in , 4 2         (A)A (B) B (C) C (D) D
• 12. PROBLEMS OBJECTIVE 1. The slope of the tangent to a curve y = f(x) at [x, f(x)] is 2x + 1. If the curve passes through the point (1, 2), then the area bounded bythe curve, the x-axis and the line x= 1 is (A) 5/6 (B) 6/5 (C) 1/6 (D) 6 2. Ifthe normalto the curve y= f(x)at thepoint (3, 4) makesanangle 3 / 4  withthe positive x–axis, then f(3) = (A) – 1 (B) – 3/4 (C) 4/3 (D) 1 3. The triangle formed by the tangent to the curve f(x) = x2 + bx – b at the point (1, 1) and the coordinate axes, lies in the first quadrant. Ifits area is 2, then the value ofb is (A) – 1 (B) 3 (C) –3 (D) 1 4. The point(s) onthe curve y3 + 3x2 = 12ywhere the tangent is vertical, is(are) (A) 4 , 2 3         (B) 11 ,1 3          (C) (0, 0) (D) 4 ,2 3        5. The tangent to the curve y= ex drawnat the point (c, ec ) intersectsthe line joiningthe points (c – 1, ec-1 ) and (c + 1, ec+1 ) (A) onthe left of x= c (B) onthe right of x= c (C) at no point (D) at allpoints 6. The normalto the curve x = a(cos + sin), y= a(sin – cos) at anypoint  is such that (A) it makes a constant anglewith the x–axis (B) it passesthroughthe origin (C) it isat a constant distancefromthe origin (D) noneofthese 7. Which one ofthe followingcurves cut the parabola y2 = 4ax at right angles? (A) x2 + y2 = a2 (B) y = e-x/2a (C) y = ax (D) x2 = 4ay 8. If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 has (A) at least one root in[0, 1] (B) one root in [2, 3] and the other in [–2, –1] (C) imaginaryroots (D) noneofthese 9. In [0, 1] Lagranges meanvalue theoremis not applicable to (A) 2 1 1 x, x 2 2 f (x) 1 1 x x 2 2                  (B) sin x , x 0 f(x) x 1, x 0         (C) f(x) = x|x| (D) f(x) = |x| 10. Iff(x) = x logxand f(0) = 0, then the value of  for which Rolle’s theoremcan be applied in [0, 1] is
• 13. (A) –2 (B) – 1 (C) 0 (D) 1/2 11. The function f(x) = ln( x) ln(e x)    is (A) increasingon (0, ) (B) decreasing on (0, ) (C) increasing on (0, /e), decreasing on (/e, ) (D) decreasing on (0, /e), increasing on (/e, ) 12. Iff(x) = x sin x and g(x) = x tan x , where 0 < x  1, then in this interval (A) bothf(x) and g(x) areincreasing functions (B) bothf(x) and g(x) aredecreasing functions (C) f(x) isanincreasing function (D) g(x)isanincreasingfunction 13. The function f(x) = sin4 x + cos4 xincreases if (A) 0 < x < /8 (B) /4 < x < 3/8 (C) 3/8 < x < 5/8 (D) 5/8 < x < 3/4 14. Consider thefollowing statements inS and R S :Both sinxand cosxare decreasing functioninthe interval , 2         R : Ifa differentiable function decreases in an interval(a, b), then its derivative also decreasesin (a, b). Which ofthefollowing is true? (A) both S and R are wrong (B) both S and R are correct, but R is not the correct explanationofS (C) S is correct and R is the correct explanationfor S (D) S is correct and R is wrong 15. Iff(x) = xex(x – 1) , then f(x) is (A) increasing on [–1/2, 1] (B) decreasing on R (C) increasing on R (D) decreasing on [–1/2, 1] 16. The lengthofa longest intervalinwhichthe function 3sinx – 4sin3 xis increasing, is (A) 3  (B) 2  (C) 3 2  (D)  17. If f(x) = x3 + bx2 + cx + d and 0 < b2 < c, then in (–, ) (A) f(x) isa strictlyincreasing function (B) f(x) has a localmaxima (C) f(x) is a strictlydecreasing function (D) f(x)is bounded 18. The function defined byf(x) = (x + 2) e-x is (A) decreasingfor allx (B) decreasing in (-, -1) and increasing in(-1, ) (C) increasing for allx (D) decreasing in (-1, ) and increasing in (-, -1) 19. For allx(0, 1) (A) ex < 1 + x (B) loge (1 + x) < x (C) sinx> x (D) loge x> x 20. If y = a ln x + bx2 + x has its extremumvalues at x = –1 and x = 2, then
• 14. (A) a = 2, b = –1 (B) a = 2, b = – 1 2 (C) a = –2, b = 1 2 (D) noneofthese 21. On the interval[0, 1] the function x25 (1 – x)75 takes its maximumvalue at the point (A) 0 (B) 1/4 (C) 1/2 (D) 1/3 22. Let f(x) = | x |, for 0 | x | 2 1, for x 0       then at x= 0, f has (A) alocalmaximum (B) no localmaximum (C) alocalminimum (D)no extremum 23. Let f(x) = (1 + b2 ) x2 + 2bx + 1 and let m(B) be the minimumvalue off(x).As bvaries, the range ofm(B) is (A) [0, 1] (B) (0, 1/2] (C) [1/2, 1] (D) (0, 1] 24. The totalnumber oflocalmaxima and localminima ofthefunctionf(x) = 3 2/ 3 (2 x) , 3 x 1 x , 1 x 2            is (A) 0 (B) 1 (C) 2 (D) 3 Comprehension Passage Consider the function f : (–, )  (–, ) defined by 2 2 x ax 1 f (x) , 0 a 2 x ax 1        25. Whichofthefollowing is true? (A) (2 + a)2 f (1) + (2 – a)2 f (–1) = 0 (B) (2 – a)2 f (1) – (2 + a)2 f (–1) = 0 (C) f (1) f (–1) = (2 – a)2 (D) f (1) f (–1) = –(2 + a)2 26. Whichofthefollowing is true? (A) f(x) is decreasing on (–1, 1) and has a localminimumat x = 1 (B) f(x) is increasing on (–1, 1) and has a localmaximumat x= 1 (C) f(x) is increasing on (–1, 1) but has neither a localmaximmnor a localminimumat x = 1 (D) f(x) is decreasing on (–1, 1) but has neither a localmaximumnor a localminimumat x = 1 27. Let x e 2 0 f (t) g(x) dt 1 t     Whichofthefollowing is true? (A) g(x) is positive on (–, 0) and negative on(0, ) (B) g(x) is negative on (–, 0) and positive on(0, ) (C) g(x) changes signon both (–, 0) and (0, ) (D) g(x) does not change sign on (–, )
• 15. SUBJECTIVE 1. Find the equation ofthe normal to the curve y= (1 + x)y+ sin-1(sin2 x) at x = 0. 2. The curve y= ax3 + bx2 + cx + 5 , touches the x-axis at P(-2, 0) and cuts the y-axis at a point Q, where its gradient is 3. Find a, b, c. 3 If|f(x1 ) – f(x2 )| < (x1 – x2 )2 , for allx1 , x2 R. Find theequationoftangent to the curve y=f(x) at the point (1, 2). 4. Find allthe tangents to the curve y= cos (x+ y), - 2  x  2, that are parallelto the line x + 2y = 0. 5. If f(x) and g(x) are differentiable function for 0 x  1 such that f(0) = 2, g(0) = 0, f(1) = 6; g(1) = 2, then show that there exist c satisfying 0 < c < 1 and f (c) = 2g (c). 6. Find the coordinates ofthe point on the curve y= 2 x 1 x  , where the tangent to the curve has the greatest slope. 7. Use the function of f(x) = x1/x , x > 0, to determine the bigger ofthe two numbers e and e . 8. Show that 1 + x ln (x 2 2 x 1) 1 x     for all x  0 . 9. Show that 2sinx + tan x  3x where 0  x < /2. 10. Let f(x) = ax 2 3 xe , x 0 x ax x , x 0        . Where a is a positiveconstant. Find the intervalinwhichf(x)is increasing. 11. Using the relation 2(1 – cos x) < x2 , x  0 or otherwise, prove that sin x (tan x )  x,  x  0, 4        . 12. Prove that for x 0, 2        , sinx+ 2x  3x(x 1)   . Explain the identityifanyused in the proof. 13. Let xand y be two real variable such that x > 0 and xy = 1. Find the minimumvalue ofx + y.
• 16. 14. For allxin[0, 1], let thesecond derivative f(x) ofa functionf(x) exist and satisfy|f(x)| <1. If f(0) = f(1), that show that |f(x)| < 1 for allxin [0, 1]. 15. Let f(x) = sin3 x+ sin2 x, - /2 < x< /2. Find the intervals in which  shouldlies in order that f(x) has exactlyone minimumand exactlyone maximum. 16. Let f(x) = 3 2 3 2 (b b b 1) x , 0 x 1 (b 3b 2) 2x 3, 1 x 3                  . Find allpossible realvalues of b such that f(x) has the smallest value at x= 1. 17. Determine the points ofmaxima and minima ofthe function f(x) = 1 8 lnx – bx+ x2 , x> 0, where b  0 is a constant. 18. Ifp(x) be a polynomialofdegree 3 satisfying p(-1) = 10, p(1) = -6 and p(x) has maxima at x=-1 and p(x) has minimaat x= 1. Find the distance betweenthelocalmaxima and localminima ofthe curve. 19. AwindowoffixedperimeterP(includingthebaseofthearch)isintheformofarectanglesurmounted byasemicircle. The semicircular portionis fitted withcolouredglass while the rectangularpart is fitted withclear glass. transmits three times as much light per square meter as the coloured glass does. What is the ratio for the sides ofthe rectangle so that the window transmits the maximum light? 20. If ax2 + b c x  for all positive x where a > 0 and b > 0 show that 27ab2  4c3 . 21. Suppose p(x) = a0 + a1 x + a2 x2 + …. + an xn . If |p(x) |  |ex-1 – 1| for all x  0, prove that |a1 + 2a2 + … + nan |  1. 22. Let –1  p  1. Show that the equation 4x3 – 3x– p = 0 has a unique root in the interval[1/2, 1] and identifyit. 23. Using Rolle’s theorem, prove that there is at least one root in (451/100 , 46) of the polynomial P(x) = 51x101 – 2323 (x)100 – 45x + 1035. 24. For a twice differentiablefunction f(x), g(x) is define d as g(x) = f(x)2 + f(x) f(x) on [a, e]. Iffor a < b < c < d < e, f(a) = 0, f(b) = 2, f(c) = -1, f(d) = 2, f(e) = 0 then find the minimum number of zeros ofg(x). 25. Ifthe function f: [0, 4]  R is differentiable, then show that, (f(4))2 – (f(0))2 = 8f (A) f(B) for some a, b  (0, 4). 26. Let C1 and C2 be respectively, the parabolas x2 = y – 1 and y2 = x – 1. Let P be any point on C1 and Q be any point on C2 . Let P1 and Q1 be the reflections ofPand Q respectively, with respect to the line y = x. Prove that P1 lies on C2 , Q1 lies on C1 and PQ > min {PP1 , QQ1 }. Hence or otherwise, determine points P0 and Q0 on the parabolas C1 and C2 respectively such that P0 Q0  PQ for all pairs of points (P, Q) with P on C1 and Q on C2 .
• 17. ANSWERS SECTION-I OBJECTIVE LEVEL - I 1. (A) 2. (B) 3. (C) 4. (C) 5. (A) 6. (D) 7. (B) 8. (C) 9. (B) 10. (B) SECTION – I LEVEL - II 1. (B C) 2. (AB D) 3. (A C) 4. (A B C D) 5. (A C) 6. (AD) 7. (AD) 8. (AD) 9. (A B C) 10. (B C) Multiple Choice Questions with Single Answer from other competitive exams: 1. (A) 2. (D) 3. (B) 4. (A) 5. (A) 6. (D) 7. (B) 8. (C) 9. (B) 10. (D) 11. (A) 12. (A) 13. (A) 14. (B) 15. (B) 16. (C) 17. (D) 18. (A) SECTION – II SUBJECTIVE LEVEL–I 2. tan–1 a b ab        3. Intervalofincrease : ( ,0) (1, )    . Intervalofdecrease: (0, 1) 4. 29 x 3, 15  6. 3 and 0 2 2 7. (a + b)2 LEVEL–II 7. increasing in 4 0, 3       decreasing in 4 , 2 3       8. a = 3, m = 1, b = 4 9. (i) Global max. value = 21, Global min. value = 1 (ii) Globalmax. value 4 3 3  , Globalmin. value 4 3 3  
• 18. SECTION-III-A 1. (A – p), (B – q), (C – s), (D – r) 2. (A – r), (B – s), (C – p), (D – q) SECTION-III-B 1. (D) 2. (D) 3. (C) 4. (A) 5. (C) 6. (B) SECTION-III-C 1. (A) 2. (C) 3. (A) 4.(C) PROBLEMS OBJECTIVE 1. (A) 2. (D) 3. (C) 4. (D) 5. (A) 6. (C) 7. (D) 8. (A) 9. (A) 10. (D) 11. (B) 12. (C) 13. (B) 14. (D) 15. (A) 16. (A) 17. (A) 18. (D) 19. (B) 20. (B) 21. (B) 22. (A) 23. (D) 24. (C) 25. (A) 26. (A) 27. (B) SUBJECTIVE 1. x + y = 1 2. a = 1 3 ,b 12 4    , c = 3 3. y = 2 4. 2x + 4y -  = 0 , 2x + 4y + 3 = 0 6. (0, 0) 7. e 10. 2 a , a 3        13. 2 15.  3 3 ,0 0, 2 2               16. b(-2, -1)  (1, ) 17. min at x = 1 4   2 b b 1   , max at x =   2 1 b b 1 4   18. 4 65 19. 6 +  : 6 24. 0