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Mathematics|, , local, , maximum, , ' kaa, is, , Second Derivative Test, defined on a, Letfbe a Function, , g p o l n t, , alflc)=0, , cis a, nt, , P(x)<0, if x e(c,c+ h), where, , e(c-h,c)and, a, , small but positive quantity., , local minimum, , 0X=cis a, , of flx), if, , point of local, , value of flc) is local, )X, , x, , =c, , is, , a, , maximum, , value of f., =0 and, minimum, , if f(c), , minimum, , and, (0) The test fails if f(c)=0, , changes sign in (c-h,c+ h) for any, dooes, x=cis neither a point of minimum nor, a, not, , if f(c) =0 and P"(c), , maximum, , point of local, , value of fc) is local, , positive, , but, , )=0but, , e!. Let fbe, interval /and c, , differentiable at c. Then:, , his, , Cif x elc-h,c) and F(x)>0, if e(c,c+h), where his, quantity., small, , 127, , (x), if, , of, , loca, , c, , |, Class12| Term-1, , value of f., In this, , fr(c)=0., , whether, , and find, the first derivative test, inflection., of, a, , quantity, then., , minima, , p o s i t i v e, , o o i n to f m a x i m u m ., , a, , <0. The, , P(c) >0. The, , case, we, , cis, , twice, , to, go back, , point of maxima,, , point, , or, , Objective TYPE QUESTIONS, a7., , Multiple Choice Questions, , x, The function flx)= log, , x, , c.(-), , y =x'-, , for which the function, , SoL (b) (0, a8. If y =x*(x -2),, , increasing is:, , c[1 ), , 5, , C.X o r x >2, Sol (c) x, , y=4x34x =4x(x -1), dx, ay =0 =, , Now., , x =0, x =1, , a, , dx, , xe(-0) u (0 1), , Since, , 02 The interval in which y, , x*e" is increasing is:, , =, , >0, , Sol (c) >0, , if:, , The function fx) =*is, , on, , its domain:, , d. information insufficient, , Sol (b) decreasing, a12. Let y, , +1, , =, , x*e", then the interval in which y increases w.rt, , is:, function, , a.(-o0, o), , decreasing, d. an, increasing function b. a, of these, None, d., C. an even, , c.(2,, , function, , (a) an increasing function, =, , x3, , a.-1<k<1, c.0<k<1, , 9kx2, , -, , +27x + 30is increasing, , -1<k <0, , the function fix), , =2x -, , kx +5 is increasing, , then k lies in the interval:, , b.(4,), , c(-8), , a 13. The, function fx) =1x +21+1x-1|is:, a. increasing in (l o)), b. increasing in [1), c. decreasing in (--1), Sol (a) increasing in (1 ), , b. k - 1 o r k>1, d, , on, , [1, 2],, , d. None of these, , a 14. The value of b for which the function flx) = sin x - bx +c, , is decreasing for x e R is given by:, c.b>1, b. b21, , a.b <1, , d. (8, o), , b.(-2,0), , d. (0. 2), , Sol (d) (0, 2), on, , SoL (a)-1<k<1, , a.(-, 4), , d. None of these, , C. constant, , 04fo)--1, , SoL (a), (-, 4), , b. (0, ), , a. increasing, b. decreasing, , Sol (c) x<-1or x>1, , 6. t, , tanx increases in the interval:, , -, , a.(-0), , Q11., , e function flx), R, then:, , 60x +41 in the, , Sol. (b) (0), , d. (0. 2), , d-1<x<-5, , 5., , -, , b. 0, d.20, , c.-, , Sol (d) (0. 2), , 3oL, , +21x, , interval(-o, 1, is:, , Q 10. logx, , b.(-2.0), , 2x-6x +5is an increasing function,, b.-1<x<1,, a.0<x<1, C.X-1or x>1, , or x>2, , a. <0, , (NCERT EXERCISE), , a.(-o,), c.(2,, , for which y,, , d. None of these, , -2x*, 9. The function flx)=, , and f is continuous in, , (x), (-0)and (0. 1), Therefore fis decreasing in (-, ]and fis increasingin[1-), <0 V, , x, , a.x<orx>3, , Sol.(c), , 3., , then the values of, , increases, are:, , b.(1,, d.(-o,0), , a, , 2+x, , b., (0.), d. None of these, , a.(-0), , valu of, 01. The, , 2x is increasing interval:, , -, , Sol (c) b>1, , d.bs1

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128 | MASTERMND CHAPTERWISE QUESTION BANK, Q 15. On which, , of the following intervals is the functionfgiven, , by flx)= x+sin x, a. (0.1), , -, , 1strictly decreasing?, , C.x21, , Sol (d) Given., , Sol (d) al values ofx, A sin x +6 cos X, , a 21., , f(x) = x +sin x 1, , P(x)=100x +cos x, In the, , interval, , (0. 1).cos x, , >0 and, , f(x)»0, , 100x, , Also., , strictly increasing in the interval (0. 1)., , cosx <0 and 100x, , a, , 100x391, , d.A>4, , >4, , a 22. The length of the longest interval, in which the funri., , 3 sin x-4 sin* x, , Sot(a, 23. The function, , a.(1. 2), , strictly increasing in the, , (c)In the interval, , acosx>0 and100x9, , b. (2.3), , interval }, , We have., , f(x)=2log (x -2) - x+ 4x +1, , rl)-z-2x +4, , strictly increasing in the interval O, strictly decreasing, , f(x)=3 cos, , x+, , r)-2, in the, , F(x)=-2X-)(x-3)(x-2), , 10cos3x +6 cos2x-3, , (x -2)?, , -2(x -1)(x, , xe-1)u(2.3), , decreasing in 5, d. None of, the above, , increasing in, , (a), , a17., , 2x, , -, , +5is an increasing function, if:, , 6x, , a.0<x<1, C.X-1or x >1, Sol, , b.-1<x<1, , Clearly, domain, Thus, f(x) is, a 24. The, , f(x) sin x cos x, the interval, decreasing in 0s xs 2n, is:, =, , -, , a., , a.9T 3n, , c, , SolL (d) None of these, , a19., , If flx), , =, , x lies in:, , x3, , -, , 6x, , a.(-,-1) n3-), c.(3o), , Sol (b) (1. 3), , +9x +3be, , a, , decreasing function, then, , function, , increases i, , decreases, , f(x)=Sin X-2x- x COs, , Sol (a), a 25., , (a, , 2), , above, , increases in o, , The function which is, a.cosecx, C. x, , y<2:, , in (a5]2, , d. None of, the, , b.(1.3), d. None of these, , f(x) is (2, -), increasing on (2. 3)., , b. increases in, , in which function is, , d. None of these, , of, , 2+ cos x, , d.-1< x <<--, , (c) x<-1or x >l, , a18. If, , -3) (x -2)>0, , (x-1)(x-2)(x-3)<0, , increasing in aJ, , Sol, , 2x-1)(x-3, x -2, , given, , a. increasing in, b., , d (2.4), , Clearly. f(x) is defined for all x > 2., , 0, , r(>ona, , function, 0sx>tis:, , c (1.3), , Sol (6), , 100x+cos X>0, , a 16. The, , flx)=2 log (x -2)- x* +4x +1 increas, , on the interval:, , Thus, function fis, , not, , unction, , is increasing, is:, d. T, , r(>oin, , Hence. function f is, intervals., , is increasing, if:, , b.A <1, , >1, , Sol (d), , :in (0 1), cos x >0, , interval, , 2 sin x +3 cos x, , C.A4, , >0, , x, , Thus, function fis, (6) In the, , Function f(X), a., , in0 cosx>0ie. (a 157).cos >0, , Thus. fis, , b. x S1, d. all values of x, , a.1sx s5, , d. None of these, , (a), , 20. The function f(x) = 1- x* - x is decreasing for, , Sol (a), cosec x, , neither, , decreasing nor incre, , b. tan x, , d.lx 1

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3ot, , 3 2 . The equation of the tangent to the curve x, , interval:, , b. (1), , a -o1), , d. None of, these, , cR, , 3at, , . a t the, , y, , 1299, , Term-1, , | Class 12, , Mathematics, , % The function f(x) =x? 2x is strictly decreasing in the, , (1+t2), , 2 is:, , pointt=, , 1+t, , The function f(x), , tan x, , =, , is, , strictly decreasing on:, , d. None of, , c(-. T), fx)=tanx, , 4x, , -, , 4x, , f, , a., , b. 3x+ 4y, , =0, , +a, , =0, , O, , -, , d. None of these, , tangent to the, , point (0, 1) is:, a. y +1 2x, C. y-1 2x, , 4, , -, , of the, , 3 3 . The equation, , curve, , the, y =e"" at, , b.1-y =2x, d. None of these, , Sol. (c) y -1=2x, , -3< (sec2x -4) <0, , x, , Thus for, , 3y, , Sol (c) 4x + 3y -12a = 0, , these, , (x) = sec?x, , Therefore.1<secsx <4, , +, , C. 4x +3y - 12a = 0, , X1< sec x<2, , When, , 4x, , 34. The, , (x) <0, , point, , paralle to, , is, the curve y =(x -3)' where the tangent, is:, and, (4, 1), the chord joining (3, 0), , on, , -, , Hence. fis strictly decreasing on, 28, The tangent to the curve given by x = e" cost, y = e' .sint, , Sol (, , att=makeswith X-axis an angle:, b., , a.U, , d., , ), , a 35. The two curves x3 -3xy, , +5 = 0and 3x y, , - y* - 7= 0, , a. cut at right angle, , b. touch each other, , C. Cut at an angle / 4, , d. cut at an angle r/3, , Sol (a) cut at right angle, dx, , a 36. The equation of the tangent to 4x2-9y, , =-e sin t + ef cos t., , dt, , a5y-3)-4, , ay =e cos t + e sin t, dt, , cost+sint 2, , Therefore, is:, , b.y=0, , C.X+ y=0, , is, , = -1, , COs X)x =0, , y-0=-1(x -0), , or, , X+y =0., he, 030., , point on the curve y2 = x, where the tangent makes, , an angle ofwith Xaxis is:, , Sol (), a 38. If the, , slope, , of the curve y, , then the values of, a. 1,-2, c. 1,2, , dx, , dy, dx, , 2ytan, , quation, , of normal at the, , X=a, , coss0, y =a sin 6 is:, d. X Cos +y sin 0 = a cos 268, .X, , COs -y sin e a Cos 26, XSin 0-y cos a COS 2, a. None of the above, ) X, COS -y sin 0 a cos 26, =, , =, , and b, , b- », are, , at the, , point, , 6 of the, , 2,, , d. None of these, , ax, , x, ab, , (b x), , ab, , (b-12 =2lgiven), curve, , through, , passes, , putting, , a, , =b, , -1in eq., , (b 1)7, , b, , a =2 1=1, , =lb =2, , the, , point (1. 1). therefore, , a =b -1, , b-1, , Hence,, , is, , respectively:, , (b - x)2, , curve, , On, , point (1, 1), , b. -1,2, , dx, , Since, the, , 031. The, , a, , ax, =, , dy (b x) a-ax:(-1), , d. (1. 1), , dy, , of these, , b, , 5, , b, , Sol. (b), , 10 12/ =0, , the curve y =e"+x* drawn at the point x = Ois:, , Sol. (c) We have, y=, , c(4.2), , +, , 10= 0 is:, , a 37. The distance between the point (1, 1) and the tangent to, a., , d. x -y =0, , So ()cos5 x.Therefore. slope of normal =, Hence, the equation of normal, , b2x-5y, , C.2x 5y -10+123 =0 d. None, Sol. (d) None of these, , cost sint, , 029. The equation of the normal to the curve y = sin x at (0, 0), , a.X =0, , =36 which is, , perpendicular to the straight line 5x + 2y -, , (1)., , we, , 2), get, , =2 and b 1

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an, , T h epointe, , the curve 9y*, , nakes equal, , x*, where the, , =, , to the, ercepts with the axes, normal, is:, , at which the, , tangent, , point on the curve y (x -2),, and (4,4)., is parallel to the chord joining the points (2, 0), , 53. Find a, , b.(4,1), , a.(3.1), , d.(5, 1), , c(6.1), , Sol. (a)(3. 1), , d. None of, these, , 54., , At, , what, , point, the slope of, , x +y-2x-3, , =t2 -1y =ti -t,, the tangent is parallel, to X-axis, where:, , b.t-, , the tangent to the, , curve, , 0is zero?, , b.(3.0). (1. 2), d. (1. 2). (1.-2), , a. (3.0).(-1.,0), , c.(-1.0). (1. 2), Sol (d) (1 2). (1.-2), 55. The, , =xand, , angle of intersection of the curves y, , is:, , a. tan, , d.t, , Ct=0, , 131, , Class 12| Term-1, , Mathematics, , x, , =y, , ctan(a tan"(, , SoL (c)Solving the given equations, we have, , b.0 0, , y =x and x =y»x* =X or x*, x(x3 -1) =0 x =0, x =1, Therefore. y=0 y =1, , d. 5, , Further, , y 3, For the curve x 5, is parallel to the X-axis when:, , cos 6,, , =, , = T, , =, , sin 0, 0ses 7, the tangent, , y=x, , 04. Thetangent to the parabola x* =2y at the point, , x*=y, , makes with the x-axis an angle of:, , dy, , dx 2y, , 2x, , At (0. 0). the slope of the tangent to the curve y, , =x is, , x, , =y is, , parallel, , to, , Y-axis, , and the tangent, , to the curve, , parallel to X-axis., , b. 45, . 60°, , Angle of intersection=, At (1.1). slope of the tangent to the curve y2 = x is equal to, , sol (b) 45, , 048. The, , equation, , x sin, , X, , of, , the, , the, , tangent, , curve, , and that of x = y is 2., , x*° at the origin is:, , tan6=, , x =0, a.x =0, , b. X = y, , Cy=0, , d. None of these, , [GX lo,0), , lim, h-0, , (NCERTEXERCISE), , m sin-0, , -0, , h, , Sol(o, , : Slope of the tangent at origin is ., , equation of the tangent at, , y-0 0(x-0), , (0, 0), , is, , a57. The abscissa ofthe point on the curve 3y, , y=0, , -y-2=0, he two curves x -3xy2 +2 =0 and 3x"y, intersect at an angle of:, , normal at which passes through, a. 1, , 6x - 5x3, the, , origin is:, , d5, , .2, , Sol (a)Let(x V)be the point on the given curve3y 6x -5x at, , ., , 4, , which the normal passes through the origin. Then we have, , eequation ofthe normal to the curve y, , =, , sin, , x, , at, , (0, 0), , dx., , a.X =0, , b.x +y=0, , ty 0, , d.x - y =0, , Sol (b), , x+y =0, 52. The, tangent to the parabola x* =2y, nakes with the x-axis an, , b. 45, , angle ot, C. 3 0, , =2-5x. Again the equation of the normal at, , passing, , is:, , SolL (b)45, , 2-0-tan"|, , curve makes equal intercepts with the axes are:, , h sin-0, , Hence, the, , 2, , a56 The points on the curve 9y' =*',where the normalto the, , Sol (c) We have., , a.0°, , 2y =1, dx, , and, , .30, , =0, , e. points of intersection are (0. 0) and (1. 1), , (0)5, , a.0, , -x, , 2-5xf, , through, , the, , origin, , gives, , - S i n c e x =1 satisfies the equation., , therefore, correct answer is (a)., at the, , d. 60, , pint, , a58. Find the points on the curve y =x* + 3x +4 at, which if we, draw tangents, then they will pass through (0, 0)., a.(-2.2).(2.14), b. (2,-1). (3. 14), , c(2. 14). (2,2), Sol. (a)(-2.2). (2,14), , d. None of these

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132, , MASTERMIND CHAPTERWISE QUESTION BANK, ay4 sin x COS X, , 59. The tangent to the curve y x' + 3x will, pass through the, point (0, -9) if it is drawn at the point:, =, , a. (0.1), , dx, , b.-3.0), d.(1.4), , c.(4,4), Sol. (b) (-3., 0), , On differentiating eq., , (2) w.r.t. x,, , we, , get, , dy-2, sin 2x, dx, , 60. The point on the curve y =x, where tangent makes an, angle of 45° with the X-axis is:, , 2sin, . -2sing-. m, .4.2), , d.(2,-2), , Hence,, , Sol.(b, , angle between two, , Q61. The normal to the curve 2y =3-x* at (1, 1) is:, =0, , or, , b. X+y+1=0, , Q 68. The values of a for which y =x* +ax + 25touches the axis, , d. x-y =0, , C.X-y+1=0, Sol. (d)x-y =0, , of x are:, , Q 62. The equation of the tangent to the curve y' = 4ax at the, , point (ot , 2at) is:, ty =X + at, , a., , b.ty = X - at, , C. tx + y = ats, , d. None of these, , a..t5, , b.t 10, , C.t2, , d.t1, , dy 0, , Sol. (b), , 2x+a =0ie., x = -5, , a63. Tangents to the curve x2+y2 =2at the points (1, 1) and, , (-1, 1) are, , Thereforea-25-0, , b. perpendicular, c. intersecting but not at right angles, , a. parallel, , 0=t10, , Hence, the values of a are t 10, , d. None of these, , a 69. The, , Sol (b) perpendicular, 64. The equation of the tangent to the curve y, , (4- x*)"° at, , =, , x =2is:, , a.x = -2, , b. x =2, , C. y =2, , d.y=-1, , y, a., , slope, , x2, , curves, , a, , 4, , xy=c°, , to, , intersect orthogonally is:, , b. a-b=0, , = -1, , c.a+b=0, , d. None of these, , Sol (b) Let the curves intersect at (x, y1). Therefore,, of the, , tangent, , to the, , curve, , x =a, , sint,, , a2 b dx, , dy_bx, dx, , b.cott, , tant, , d. None of these, , C. tan, , Q66. Tangents to the curve y x, a. parallel, b. intersecting obliquely but, =, , +, , 3x at x = - 1and x =1are:, , not at an, , xy =c, , Again, , dx, , angle of 45°, , .c. intersecting at right angles, 45°, d. intersecting at an angle of, , 2, curves, , y, , =, , 1 or a2-b2 =0, , Line3x-4y 7 at the point:, d. None of these, , a. (-3,-4), , b. (3,-4), , c.(3.4), , d. None of these, , Sol. (6) We have,, , C., , Sol (6) We have, y=2 sin x, , .(2), , y=COs 2x, (), , m=, , a 70. The tangent to the curve x +y2 =25 is parallel to, , a, , On differentiating eq., , x+y =0, , 2 sinx and, , y= cos2xat x =is:, , w.rt. x, we, , bx, , For orthogonality, mxm2 =-1, , Sol. (a) parallel, intersection of the, , ay, , Slope of tangent at the point of intersection (m), , Sol. (b) cot t, , a 67. The angle of, , 2x 2y dy=o, , atthe point t' is:, , =acost + log tan, , and, , condition for the, , a.b, , Sol (b) x = 2, , a 65. The, , is, , m-m2, tan1+, mm2 t tan', , t, , a.x+y, , curves, , get, , x+y=25, , 2x+2y-0, dx, , Now, slope of the line, , 3x-4y 7is m= 3, 4, , X, , the

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he, Since,, , Mathematics, , tangent parallel to the given line, is, , /3., , The, , slope of, , 2), , 25, , x+, Fomea (0,, , tangent to the, , curvex =t' +*, , -5atthe point (2, -1), , y=-2, 4, , 133, , Class12 | Term-1, , -, , 8, , is:, , d., Sol. (b), , X = t3, , fx-3 from eq (2). y - - 4, 076. The angle between the tangents to the, , fx-3from eg, (2), y=-(-3)=4, , curve, , at the points (a, 0) and (0, b) is:, The points are (3,-4) and (-3,4, , b5, , a., , ., , The equation, , of, , tangent, , the curve, , to, , where it crosses X-axis is:, , y(1+x*)=2- x,, , b.x-5y =2, d.5x+y 2, , ax+5y=2, c5x-y=2, , d. None of these, , 3, Sol. (b), , =2, al (a) x +5y, , The, , 4, , Q7. The, , points at which the tangents to the curve, , b.(2.34).(-2.0), , d. None of these, , C. touch each other, , d. (2.2).(-2.34, , (0.34).(-2. 0), Sal (6) (2. 2). (-2,34), , point (1, 2):, a. intersect orthogonally b. intersect at an angle, , y=x-12x +18are parallelto X-axis are:, a(2-2).(-2,34), , the, curves y=4x and x +y2 6x +1=0at, -, , two, , Sol (c) touch each other, 078. Ifa tangent to the parabola y, , 73. The angle at which the curves y= sin x and y = cos x, , the straight line y, , intersect in [O, 7} is:, , at tan 2N2, , c:tan, , =, , =, , 3x +5, then, , 8x makes an angle, , with, , the point of contact is:, , b. t tan 2, , al52, , 2, , d. None of these, , -2, , d. None of these, , Sol ( - 2, , Sol (a) We have, y = Sin x, , and, , (2), , V COS Xx, , Ihe two curves intersect at x = i n |0 t, , x-y=2a is:, a. 0, , b. 45, , C. 90°, , d. 30, , Q80. Ifthetangentto the curve x = at .y = 2at is perpendicular, , ay-cosx, , to X-axis, then its point of contact is:, , dx, , dy, , (say), , b.(0. o), , b. 16. 16, , a. 16, 14, , dy-sinx, , c.(0.0), , d.(a.0), , C.64,4, , d. 32. 8, , Sol (b) 16, 16, , dx, , sav), angle between the two curves, , a.(a, a), Sol (c) (0.0), , Q 81. Amongst all pairs of positive numbers with product 256,, find those whose sum is the least., , On differentiating eg. (2) w.r.t. x, we get, , Q 82. If the length of three sides of a trapezium other than base, are equal to 10 cm, then find the area of trapezium when it, , is maximum., , is, , m-m2, , t, , and, , Sol (c) 90, , On differentiating eq. (1) wrt.x,we get, , EnICe, the, , a79. The angle of intersection of the curves xy =a, , tan1+ mm2), , a.75 3/2 cm, , b. 75 3 cm2, , c.90 3 cm, , d. 48 cm, , Sol. (b) 753 cm2, , 74. The, t, tangent to the, X-axis at:, , a.(0.1), (2.0), , =t tan 2v2, curve, , y, , =, , e" at the piont (0, 1), , -0, d.(0,2), , meets, , a 83. For all real values of x, the minimum value of, , x+, x2, **, 1+x+x2, , is, a.0, , Sol (o), , (NCERT EXERCISE), b.1, , C.3

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134 | MASTER MIND CHAPTERwISE QUESrION BANK, Q 84. The maximum value of, , b5, , [x(x 1) +1],, , 0sxs 1 is:, (NCERT EXERCISE), , c.1, , So, at x = y or given function is maximum., , So. putx =, , d.O, , in (sin x + COS X), , Maximum value of function, , 5ol. (c)1, , oncos', , a 85. Find all the, , function, , points of local maxima and local minima of the, f(x)= (x 1) (x + 1)2:, -, , a. 1,-1,-1/5, , b. 1,-1, , c. 1,-1/5, , d.-1,-1/5, , Sol. (a) 1.-1,-1/5, , a 92. The area of a right-angled triangle of given hypotenuse i, maximum when the triangle is:, , Q 86. Divide 20 into two, parts such that the product of, and the cube of the other maximum. The, two, , a. 10. 10, , b. 12, 8, d. None of these, , C. 15, 5, , one, , part, , parts are:, , Sol. (c) isosceles, a 93. Find the point at which the function f, , Sol (c)15.5, , Q 87., , can, , and point of inflexion respectively., , be, , a. 2.-1, c2-1, , 33, , 4tr3, , SoL ( ) 3, , b.2.17, d-2-, , Sol (o)2-1, , (88-90):, , Find the maximum value of the, , following, , a94. Find the points of, f x ) - x * - 8x3, , minima for the function, x2 + 105:, , local, , Q 88. f(x) =3x + 6x +8, x E R:, a. 2, , 2, a.C, C. -5, , b. 5, d. does not exist, , C.-8, , Q 95. It is given that at x, 1, the function x* 62x +ax +9, attains its maximum value on the interval [0, 2]. Find the, value of a., =, , a 89. fx)= sin 3x +4 xe, b. -4, d. 3, , a. 5, , C, , a. 100, , a 90. fx) = sin (sin x)for all xeR:, b. sin 6, , d. -sin3, , SoL (c)sin 1, , 2, , /2, , b., , d.O, , d. 160, , a 96. The function flx) x*, = 5x, a. one minima and two, maxima, b. two minima and one maxima, C. two minima and two, maxima, , +5x3-1 has:, , one minima and one maxima, , SolL, , (d) one minima and one maxima, X, , = sin x + cos x, , Let, , and, , C. 140, , a 97. The function flx)= *+has:, , SolL (6), , Then,, , b. 120, , -, , C. sin 1, , Q91. The maximum value of (sin x+ cos x)is: (NCERTEXEMPLAR), 1, , -, , Sol. (b) 120, , Sol (a) 5, , a., , .-3, , d. Both b and c, , Sol. (b)-3, , Sol (d) does not exist, , a. sin 1, , given bv, , fx)=(x-2)° (x+1)* has local maxima, local minima, , Find the volume of the, largest cylinder that, inscribed in a sphere of radius r cm., a. t r, b. 4Ttrh, c. 4Tr3, , Direction, functions., , d. equilateral, d. None of these, , a. scalene, c. isosceles, , local maxima at x 2 and local minima at x -2, b. local minima at x 2. and, local maxima at x -2, C. absolute maxima at x, 2 and absolute minima at x, *, d. absolute minima at x 2 and, absolute maxima at x =*, Sol (b) local minima at x 2, and, local maxima at x -2, a. a, , dy, =coSx, -sin x, , =, , =, , =, , dx, , =, , d'y, , =, , - sin x cOS X, , =, , dx, , Now, for maximum or minimum value of y.=0, dx, , c o S x - sin x = 0, , Sin X =1, , =, , a 98. If, , sin x = cos x, , tan x = tan, , X=, SolL, , 4, , -a, We see that at x =, , the value o f i s negative., , dx, , then, 4x+2x+1, , a..3/4, c. 4/3, , COS X, , -sin-cos4, , fx)=, , its maximum value is:, , 8/3, , b., , d. 3/8, , (c), , For, , ie., , 4x2, , value, , =, , f to, , be, , maximum, 4x2, , +2x +1=4, , 2x, , +1should, , t, , be, , minim, , um, , gving the minimum, , x, , of 4xd +2x +1=, , Hence maximum value of f, , +, , =, , _

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Mathematics|, , cond derivat, derivative, , second, , Letfhavet h e n c is a point of:, r')>Q, , Both, , l, , and, , a, , f', , (c) =0 and, , d. None of, these, , b, , cos° e attains, , maximum,, , a, , when=, , d. tan, , tan'a, , tan, , b., , a.tan, Sol (a) tanP, , minima, , (b), , Q 108. sin^ e, , b. local minima, , a.local maxima, , c, , atc such that, , local, , Vq, , inimum value of fif fx) = sin xin, 109., , c-1, , Sol, , Saol ( c ) - 1, , at, , x, , =, , x+0 +6), , =, , imum, , distance, , b. 2,-4, a. 2,-1/2, (a) Given. y = a log x, dy +, , from, , the, , circle, , value of y, , d.-3. 2, , C.3-1/2, +, , bx*, , +, , X, , 2bx +1, , d. None of these, , c(2, 4), sol (a) (2,-4), The distance of, , value of, , dy, , the point on y =x* +3x + 2x which is, 2x, , line y, , d.-, , poles, , a., , 5, , minimum, maximum or, , +2b, , x2+1-0o, , we, , get, , b=-, , From eqs. (1)., o, , =1-2b = 1 - 2 x, , =, , is minimum., , b., , m, , m, , a = 2. b = -1/2, , Hence, a110. A, , can, , to hold, , plans to construct a cylindrical, cost of constructing top, metre of liquid. If the, , manufacturer, , cubic, the, the cost of constructing, and bottom of the can is twice, can?, economical, side, what are the dimensions of the, , one, , d. 14 m, , C. 10 m, , most, , Sol (c) 10 m, , of a right triangle is at distance, 0104. A poit on the hypotenuse, of the triangle. The minimum length, a and b from the sides, of the hypotenuse is:, b. a2/3 b2/3, , a.(a+b2/2, , d.(a2/3 +b2/3)3/2, , c(a2/3 +b2/3)2, , a. 4T, 167, , sot ( ) ), , Sol (d)(o2/3 +b2/3)3/2, , and vertical sides is to be, 105. An open tank with a square base, a given, sheet so as to hold, constructed from a metal, least, be, will, material, of the, , quantity of, , water., , Q 111. Find the, , having a, a.r, , The cost, , C.equal to its width, Sol (b) half of its width, , of, , a, , cylinder,, , which is open at the top,, , given surface area, greatest volume and of radius, d. 3tr/2, C.r/2, b. 2r, , r., , height of the cylinder of maximum volume that, can be inscribed in a sphere of radius a., , b. half of its width, d. None of these, , a. twice of its width, , height, , Sol. (a)r, , when depth of the tank is:, , Find, , at x =2, , .(2), , +3 =0, , points, , at, , =, , that RP+ RQ, , y is, , Adding eqs. (1) and (2)., , A and B, 22, m, and, AB, 20, 16, m, then, m, BQ=, respectively. If AP, R, AB, the, on, from, point A such, find the distance of a point, be two vertical, , -1 then, , +2b(-1)+1-0, , = 0ie., , 6b, , BQ, , X, , a+8b+2 0, , or, , b., , at, , dx Jx-2, , 1 is:, , -, , ie., , - a -2b +1=0, , or, , If the, , a103, Let AP and, , -, , or minimum, , Ldx J - -1, , b. (18, -12), , nearest to the, , is maximum, , 1are:, , a (2,-4), , 102, , If the, , and b., , -, , coordinates of the point on the parabola y2 = 88x, , is at, , or, , maximum, , is, + x, log x + bx2, of y, values of a, the, 1and x =2 find, =a, , If the value, minimum, , .0, d. None of these, , a, , which, , 135, , Term-1, Class12|, , Q 112. Find the, , 20, , a. 2a/3, , the points of local, , maxima, , and, , local minima, , SoL (b), , respectively for the function:, Q113., , ftx)= sin 2x - x, where-sxS5, , d.a/5, , C.a/3, , is to be cut into two pieces. One of, the pieces is to be made into a square and the other into a, , A wire of length 36 m, , circle. What should be the lengths (in m) of the two pieces,, a-, , so that the combined area of the square and the circle is, , (), 107., , 1f y, , minimum?, 36T, , ax -, , b, , =(x 1 ) (x -, , has a, , turning point, , a.1,2, (d) 1. 0, , b. 2.1, , 36T, , 4), , C., , the value of a and b respectively., , SoL, , P(2, -1), , c.0.1, , d. 1, 0, , 36r, , dT4 n+4, , then find, , Sol., , 144n, , TT +4 T +4, S6T, , 144, , (5)+4' T+ 4, , b. 36r, T+4, , 144, t + 4, , 36 125, , +4'T+, , 4

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136, , MASTERMWND CHAPTERWISE QuESTION BANK, , Q114. Find the, , area, , of the, , perimeter 18 metres., a. 93 m2, , b. 83 m, , largest isosceles triangle having, C. 43 m2, , d. 7, , Sol (a)93 m2, , Q 123. The maximum value of|, b. e, , a. e, , Q115. A beam of length is, supported, uniform load per unit length, the, distance, end is, , x from the, , at one, , end. If, , Wis, bending moment M at a, , given by, , M, , wx2., , Sol. (c) ee, , Q 124. Forthe functiony =x +, , b. X = -1is a point of minimum, C. maximum value > minimum value, d. maximum value < minimum value, , Sol. (d) maximum value < minimum value, , resistance R of two resistors R and R, , (R.R2 0) is given by, R +R,, , C (a, , =, , a. R> R2, , 1, , constant), then, , b. R <R2, , Sol. (c) R = R2, , a 125. The maximum value of the function y = x(x - 1), is:, , R, R, , obtained if:, , a117. The two, , a. x =1is a point of maximum, , d.W/, , Q116. Find the combined, , If, , a.0, , maximum resistance R is, , c.R1 R2, , (b), , We, , y=x(x 12, , have, , - (x-12.1+ 2x(x -1), , d. None of these, , dx, , = 3x - 4x +1=(x - 1) (3x -1), , positive numbers, whose sum is 15 and the sum of, , a., , d. None of these, , C.--4, , Sol, , For maximum or minimum. put=0, , whose squares is minimum are:, , dx, , b. 3 1, , SoL, , d, , C. ee, , the, , Find the distance of, point on the beam from the supporting, end at which the, bending moment has the maximumm, value., a. l /2W, b.2W/l, c./W, , Sol (a)/ 2W, , (NCERT EXEMPLAR), , m2, , (x-1)3x, , -1)=0, , 15, , Q118. Find both the maximum and minimum values, , of 3x, , ay=6x -4, , Now, , respectively, , dx, , 8x +12x -48x +1 on the interval [1, 4]., , a. -63. 257 b. 257,-40 C. 257, -63, Sol (c) 257, -63, , =-2<0, , d. 63,-257, , So. y is maximum when x =and maximum value is, , Q 119. An open box with a square base is to be made out ofa, given quantity of a cardboard of area c square units. The, , 3, , maximum volume of the box is (in cubic units):, c3, , 2, , b6, , c. 4c/5, , d. 6c2, , SoL. ()6, , So. y is minimum when x = 1, , a 120. The volume of the greatest cylinder which can be inscribed, in a cone of height h and semi-vertical angle a is (in cubic, , a 126., , If the function, value of f(x) is:, e, , 27h tan? a, , 5, , d.th, tan a, 36, , tan^a, , Sol. (c) Given, , b . 12, , . 0, , P(x) =2x e2 -2xe-2x =2x(1- x) e, minima, put f" (x) 0, , c. 16, , (NCERT EXEMPLAR), 1. 32, , a 122. flx) = ** has a stationary point at, , C.X =1, Sol. (b) x =, e, , =, , 2x(1-x)e- =0, X =0,1, , Now,, , Sol. (b) 12, , e, , fx) = xe, , For maxima and, , 0 121. Maximum slope of the curve y = -x* +3x2 +9x-27 is:, , =, , x>g then the maximum, , 2e, , b . T h tan?a, , Sol. (a)h' tan?a, , a. X, , f(x) =xex,, , a., , units):, C. rh, , dy-20, dx* Jx =1, , Also, , (NCERT EXEMPLAR), , f"(x)=2x(-1) e-2* +2(1-x) e2x, , f(0)=0+2e =2>0, , b. X, , and, , d.x =ve, , Thus, maximum value is, , Hence. (c), , -2-2x(1-x)e, minima, , f"()= -2e2 +0-0=-, , 0 maMIima, e2, , f(1) =1.e2, , is the correct, answer., , =, , e

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Term-1, , Class12, Mathematics, , fired from the ground level rises x metre, A, , missile, , yertically upwards, nwards in, in, height, , On, , b. 125 m, d. 190 m, , c / 6 0m, X, , (a), , On putting, , bothsides, , sidoe, , COS X, , w.r.t. t, we get, , X, =100-2t)=100-25t, dt, maximum, , height, velocity, , X, , At, , +Cos 2x, , cos, , = 0, , dt, , f (x), , =, , 4 45x, , 100 x, , 16, , = 200 m, , -, , one.of the, , a.fx)has, , maximum, , at x =1, , has, , maximum, , at, , x, , =, , no maximum or, , points of local, , 36 and f", , maximum or, , minimum,, , 42x, , (x), , =12x, , we must, , -, , Hence., , 42, , have, , Q 131., , (a) is the, , If the two, , "()=12 -42=-30 <0, , Sol. (b) We have,, , 0129. The minimum value of, , =60, , d. x = 20 y = 40, , fx)-x"(60-x), f(x)=3x(60 - x) - x3, , For maxima or minima, put f" (x) = 0, , .1/e, , 3. e, , d. None of these, , 3x(60- x) - x3 = 0, , f(x) *, , x*(180-3x - x) =0, , =, , Log, , P(x)=ogx-1, , X = 45, , log x), , :x *0), , F(45)<0 and f(45) >0, , maximum or minimum. f(x), logx =1, X=e, , =0, , Hence, local maxima is at, , So, , (log x2 -(log x -1) 2 logX, Now, f"(x) = X, X, , X, , =, , x*,, , xe, , [-2,31 Then, the, , f(x) =cos TX 10x +3x2+x3, (x) =-T sin tx 10+6x, 3x, f(x)= 3(x 1) +(7 n sin nx), +, , ., , ==e, , alue of function, , 2, , +, , b. -15, d. None of these, , Sol. (b) We have., , 33/3, , 45., , 45 and y =15, , c.3-2T, , 130. The maxir, , =, , absolute minimum value of, f(x) is:, a., , P(), =-0, 0,fx)is minimum at x=e., inimum value of f(x) =fle), , x, , a 132. Letf(x) =côs x+10x +3x, , (log x, , 4, , +y, , xy =(60 -x) (x*) x e (0. 60), Let, , - is:, , d.1, , a 3/2, , x, , y 60x, , logX, , xERis:, , numbers x and y such that, , x + y =60, , X=6., , aximum, , answer., , C.X = 30 y = 30, , So.f(x) has local maximum at x =1and local minimum at, , For, , correct, , positive, , F(6) 72 -42 =30»0, , (a) Let, , co ), , and x®y is maximum, then:, b.X = 45. y =15, a. x =15. y = 45, , X =16, , SoL, , maxima, , 0, , )3.3, , minimum, , F(x)=0, 6(x -7x+6) =0, and, , sin 2x, x)=-sin-2sin, -2, , Maximum value, f, , +, , -, , -sin x, , 6, , f(x) =2x-21x+36x -30, , P(x)=6x, , Clearly., , = T, X =:, , at x =1, , cfx)has maximum, , At, , 0, , -, , following is correct?, , sol (c)We, , =0cos, , x=, , At, , correct answer., , have., , 0, , f'(x)=, , Now, , is the, Hence. (a), f flx)=2x"- 21x +36x- 30, then which, , d.fx) has, , =, , 2x, , COs, , +, , 4, , X, , b.flx), , 0, , X, , 100-25t=0, , t, , get, , we, w.r.t. x,, , =, , X), , sin2x, , x+, , cos x, , =, , cos, , (1+, , x, , = 2cos|2coi)-0, , differentiating, , on, , differentiating, , P(x), , 25, , 100t, , =, , =sin, , f(x), , is:, , reached, , 2 0 0m, , Wehave,, , f(x)-sin, , Given,, , (a), , ts, where x= 100t 4 2 . The, , ximum, a, , So., , +, , flx)=, d., , sin, , 37/5, 4, , +, , x, , (1+ cos x)., , +, , -, , f(x)is strictly increasing on, >for all x e-2.3), (-2,3), Absolute minimum, value of f(x), f(-2) =cos 2T -20+12, 8, -15, =, , =

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138, , MASTER MND CHAPTERwISE QUESTiON BANK, , Q 133. Two towns A and B are 60 km, apart. A school is to be built, to serve 150 students in town A and 50, students in town B., If the total distance to be travelled, by 200 students is to be, as small as, possible, then the school be built at:, a. town B, b. 45 km from town A, C. town A, d. 45 km from town B, SolL (c) Given that, AB =60, , A, , 60-x, , Sol (d)Let oa and B be the roots of the 8venequation. so that, , a+B=a-2 and aß =- (a -1), , Dwill be least and, , ds, , Now,, , 50(60 x) =100x + 3000, =, , S=(a+B)-2aß, , ds=20-2, , >B, , equal to 3000, if x, , Then,, , do, , Let the, , at A., , S=a +B, , = (0 -2) +2(0 -1)=a*-20 +2, , school be at a distance x from A, (with 150, thenthe distance travelled by 200 studentsis students)., D 150x+, , Let, , 0, ie., school is built, , a 134. If the sum of the, lengths of the hypotenuse and another, side of a right-angled, triangle is given, the area of the, triangle is maximum when the angle between these sides, is:, , 2a-2 =0, , a =1, , da, , d'5 2>0.Ya, , do? > 0 Va, , Also., , Hence, Sis minimum when a =1, , Q 136. If y = x* +ax +b has a minimum at x=3 and the, , minimum value is 5, then the values of a and b are, b.a=-6, b =14, a.a =6, bs -14, d.a-14,, b =6, C.Q =14, b=-14, Sol (b)a =-6, b =14, 0137. The greatest value, , of the function flx), , b., , a. 0, , xe" in [0, -oja:, , =, , d. None of these, , C.e, , e, , SoL, , (c) Let ABC be a right angled triangle in which side, BC x (say) and, hypotenuse AC yy, =, , =, , Given,, , x+y=k, , Sol (b), , (say), , (constant), , y=k-x, , least value of the function ftx)=cosx, , 138. The, , in the, , interva, , A, , d. None of these, , Sol (a, a 139. The, , X, , Now, the area of AABC is given by, , greatest value of the function flx) tanx-logx, =, , A-BCAB-x - x, , -x-x, u-A, , Let, , d, , For maximum, , or, , -log3, , kR -2kx), kx-3x) and, , dx, , minimum of u., , u,, , X, , Sol (log3, , ;Mk -6x), , 140. The maximum, , put=0, , X =:, , When, , a. 0, , x, , 0, , k du6 x o } - P e v e ), , 3 dx2, , ., , value, of the function y x(x-1), is:, 4, =, , C.-4, , Q141., , f, , fx)= x* -5x +5x-10, , minimum at, equal to:, 0,1, , x, , =, , . 1,0, , y=k-x =, , Sol (b) 1.3, a 142. The, , Now coseA, , d. None of these, , 4, , SoL ()7, , .e., A is maximum when x =-, , and when, , og3, , p and, , x, , =, , 1 which are, , Hence. the required angle is, a 135. The value of a, so that the sum of the squares of the roots, least, (a2) x - a, value, is:, , ofthe equation x, a. 2, , c. 3, , +1 =Oassume the, , b., d. 1, , d. None of, these, , s, , G, , maximum, , b. 1.3, d. None of these, , points on the curve xy, , origin, are, , has, , and, , =q respectively, then (.9), , nearest to

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and z =x+my, where, k, l, constants, then the minimum value and m, of z is:, a. vmlk, ,k, , 143., , b. 2/mlk, d. None, of, , c. 2mlk, $oL (b) 2vmlk, , slope of, a 144., minimum at x, the, , the, , D. T, , a. O, , tangent, , e, , cos x is, , d., , None of these, , (Statement-l1)., , reason, Mark the Correct Choice as:, a. If both Statement- and Statement-ll are true and, , Statement-lis the correct explanation of Statement-, , b.e2, d. None of, these, , c.1, Sol () 1, , b. If both Statement-l and Statement-ll are true but, Statement-ll is not the correct explanation of, , Statement-., , maximum value of x ds + (x -2)2/3, b. 2, C.22/3, , C. If Statement- is true but Statement-l is false., , jis:, , d. If Statement-l is false and Statement-ll is true., , d. None of these, , Sol (b) 2, , a147,, , =, , -**, , a..e, , The, , Assertion and Reason Type Questions, In the following questions, a, statement of assertion (Statement-) is followed by a statement of, , the value of a is:, , Sol (b), , a. 0, , positive, , these, , to the, curve y, , C.2T, , o145. The minimum value of e, , are, , Directions (Q. Nos. 154-164):, , 0sas 27, then, , =, , On the interval [0, 1], the function x45 (1-x5 takes, maximum value at the point:, , Q 154. Statement I: Both sin x and cos x are decreasing functions, , its, , b., , a. 0, , Statement II: Ifa differentiable function decreases in (a, b,, then its derivative also decreases in (a, b), , Sol (), Sol., , q148. The function flx), , =, , 2x*, , -, , x, , and, , cos x, , decrease in, , T., , Statement l: Statement ll is false., , a155. Statement I: The function, x>a, , 0149. The maximum value of sin x cosx is:, C, , 4, , We know that, sin, , So, Statement I is true., , Sol (c) one maxima and one minima, , a, , (c) Statement :, , 3x2 -12x+ 4 has:, (NCERT EXEMPLAR), , a. two points of local maximum, b. two points of local minimum, c.one maxima and one minima, d. no maxima or minima, , for all, Sol., , ST, , x=Sx)=2sin 3x+3 cos 3x, , NCERTAEXEMPLAR), (NCERT, , a. maximum, , x>, , Oand the, , sum, , an, , are, , increasing, , of two increasing functions in any, , x(e, , increasing function, , for, , x, , +, , e* is, , >0., , + e*) is an increasing function., , Statement Il: Here. xe, not always increasing., , b. minimum, , xe*, , interval (a, b) is an increasing function in, (a, b), (a) Statement tx* is an increasing function and ex, , also, , is:, , x(e +e*) is increasing for all, , Statement l1: The function x*e* and, , (NCERT EXEMPLAR), d. 22, , Sol (b)5, 0150. At, , 139, , Mathematics| Class 12| Term-1, , If x y =, , 0, , is an, , increasing function butx'e*, , is, , C. zero, , a 156. Statement l:, If, , d. neither maximum nor minimum, 3oL (d) neither, maximum nor minimum, , 0151. If xisreal, the, a.-1, , Sol. (c) 1, , 132, , minimum value of, , x, , .135, , SolL (b) 0, , -, , 8x +17 is:, (NCERT EXEMPLAR), , a.1, , Sol (a) 1, , Statement, , d. 2, , ae+be, , sincreasing, , c e + d e - sincreasina, , function f(x) is, , -, , 18x+96x in, , and, , (NCERT EXEMPLAR), , d. 160, , f(x), , is an, , if, , f (x)>, , 0 for, , (cex de *)?, increasing function, , f(x)>0, , bc), 2add e - * z >0, , (ce, , >0, , 2(ad bc) >0 » ad > bc » be < ad, f:R->, differentiable and strictly, increasing, function throughout its domain., , 157. Let, (NCERT EXEMPLAR), , increasing, , f(x)= 2(ad-bc), , Sol. (d), , b..0, , b. 2, these, d. None of, , ll: A, , allx, , b. 0, , 153. The mini, inimum value of 2-3+27 is:, C.227, , =, , function of x, then bc> ad., , toaalest value of the polynomial x, TO, 9] is:, a. 126, , the function flx), , R be, , Statement I: If If(x)I is also strictly, increasing function,, f(x)= O has no real roots., , then

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140, , MASTERMIND CHAPTERWISE QUESTION BANK, Statement ll: At, , -, f(x), , or, , = - 2 e + xe", , may approach to 0, but, , For, , put f"(x), , maximum,, , x =l, , cannot be equal to zero., , Sol., , (b) Suppose f(x) =0, for all, , has, , real root say x =a, then f(x) <0, becomes strictly decreasing on, , a, , Thus, I f(x)|, , x <a., , (-0)which is a, , f()=-1<0, , and, , So, both the, , contradiction., , correct, , +x* +3x + sin x. Then,fis an increasing, , The, Q 162. Statement !:, , graph y, , (a) is, , he, , I, , answer., , +ax, , *, , function., , ifa<3b., , Statement l: If f(x) < Q, then flx) is a decreasing, , function, y, Statement ll: A, , function., , >0or<0for all xeR., , =, , f{x), , +bx, , dy 0 or <Ofor all, , chas extrenm, , +, , emum,, , has an, , dx, dx, Sol. (a) For an extremum, , Sol. (b) f(x) =3x* + 2x + 3+ coS x, , is., , Statement, , the, , Hence., , II, , true and Statement, , S t a t e m e n t s are, , for, Correct explanation, , a158. Statement I: Let f:R-> Rbe a function such that, , fx)= x, , =0, , extremum,, , i#, , eR, , x, , dx, , dx, , = 3x +2ax +b>0, , +COS X >0D, , ax, , »D, 3x+ 2ax +b>0, , 32a-1scos xsi, a 163., , f(x) is an increasing function., If f(x) <0 then f(x) is a decreasing function., Q 159. Statement I: The tangent at, , x, , =, , 1to the, , x, , curve, , -, , -3x-2x -1, , f(x), , Statement l: x = 2is the only critical point of f(x), Statement ll is true., , Q 164. Statement : The absolute, , Sol (a) Given., , curve, , again, , at, , x, , =-1, , curve y', Q 160. Statement I: The points on the, lies, x-axis, to, is, parallel, which the tangent, line., , function y, , =, , flx), , =, , * +, , =, , sin, , x, , 5x2+4x, , x, , at, , 21+X +2x1+x 2.1+X, -, , on a, , straight, , (x)=0» 5x +4x = 0, 21+X, , Now,, , , if tangent is, , 5x+ 4x =0, , x(5x+4)=0x =0Alsa,, , = X + sin x, , 0-both liein-, , stationary points,, Further,, FO)-0, , 2y=1+ COS X, , dx, Since, tangent is parallel to x-axis, , dx, , =, , xe, , has maximum at, , Statement Il: f (1)= Oand f ' (1)< 0., Sol (a) Given, f(x) = xe, P(x) = e, , and, , P'(x)=-e, , and both a, , -)-a, , = x, , Q 161. Statement : flx), , therefore. 0, , -25 J5 "25.5, , Oy=0cos x = -1 sin x =0, From eq (). y, , xin-1, , P(x)=x(1+ x)2+ +X-2x, , dx, y, , Iffattains its absolute maximum, , f(x)= x*1+X, , parallel to x-axis, then, , Sol (d) Given., , minimum values, , Differentiating eq. (1) w.rt. x, we get, , Hence. (d) is the correct answer., , dy=0, or, dx, , maximum and, , The given function is differentiable for all, , true., Statement lis false and Statement l is, , Statement I1: For, , greatest, , or minimum value at xo,then f (x) =0., , =1, , X =-11, , a, , =2 the, , x, , Ans. (d) 5-4(x -2)3 attains greatest value at x =2, , be any interior point of I., , x - x- x+1=0, Thus,, , nor, , attains neither least, , Statement l: Let fbe a differentiable function on land x, , Equation of tangent isy =1, x+2, Solving with the curve, x3 -x-, , tangent meets the, , then at, , flx) =5-44x-2)°,value, , x, , -10, , The, , <3b, , offtx)- x*1+ in|-1 areand respectively, , SoL (d) When x = 1 then y = 1, , ., , o, , value., , tangency., , Now, , Statement I: Let, , function, , x+2again meets the curve at x = -2., with, Statement Il: When an equation of a tangent solved, at point of, the curve, repeated roots are obtained, y =x*, , -, , 40-4-3.b<0, , <0, , =, , 1, , Therefore,, , + xe, , the, , absolute maximum value, absolute minimum value is, 0., The, , - xe"*, , -e, , x, , point of maxima is, , and, , is, , and the, , points of minima are(r-1,0

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Mathematics|, , Case Study, , Based, , Term-1, , Class 12, , 1441, , QUESTIONS, , e Study 1, interval (a, b), , [a, b] and, , on, , Letf, be continuo, , then:, , open, opeis, strictly increasing, , in, , strictly decreasing, , in, , la, b] if fx), , (a. b), , (, , E(a. b), fis a Constant function, , the, Based on, , above, , >, , iff' (x), , la, b], , iff' (x) =0, , in, , 04., , on, , the, , 0 for each, , la, b], , xE (a, b), following questions:, , differentiable, , <0 for each, for each, , infornmation, give the answer of the, , 01. The function fix) = cos(o, , The function, , a. only Z, , b. only R, d. only R, , b. (0, ), , Sol Let x, and x2 be any two numbers in R,, where, , f(x) = e2*, Given., Then, we have xi << az 2 x <2X2, 2x1<e2x2, , f(x)<fx2), So, option () is correct., Q5. The function flx), , f(x) = - sin x, , then, , =, , log(sin(x))is strictly increasing on:, , a(), , f(x)cosx, , Sol. Given,, , d. None of these, , In interval (r, 27)., f(x)>0, SolL Given, f(x) = log sin x, , Therefore. f(x) is strictly increasing on (T. 2)., , (a), , is, , then f(x)=, , correct., , 02 The function f(x) = 3x + 17 is strictly increasing in:, , c. R, b. Re, Sol Let x and x2 be any two numbers in R,, , a.R, , on(a), , f(x)=3x +17, , So, option (a) is correct., , fx)=3x+17, f(x2)=3x2 +17, , Case Study 2, , Now. we see that f(x1) < f(x2)., , shape of a toy, fx)=8x - 4x+3., The, , Hence, fis strictly increasing on R., , Alternate Method:, f(x) = 3x +17,, Given,, On differentiating w.r.t. x, we get, P(x)=3>0 in every interval ofR., , So, option (c) is correct., 03. The function, f(x) sin(x), b., , Un, , is:, , point?, , =, , b., , a, , 8x-4x2, dx, , sin, , x, , e, , we have f' (x) >0, , a) cosx>is, :, , cosx, , Hence,f is strictly increasing, 30,, , option (a), , is, , correct., , in, , +3} =0, , 8x(4x-1) =0, , x, , X = 0, , ( x ) = cos x, , each, , 2, , d. None of these, Sol For the critical point of f(x). f" (x) = 0, , differentiating w.r.t. x, we get, , Since. for, , 1, , C., , T, , f(x), , as, , answer of the following questions:, , d. None of the, , above, 30L The, given function is, , given, , 01. Which value from the following may be abscissa of critical, , Strictty decreasing in |0 :, , cStrictily increasing in, , is, , To make the toy beautiful 2 sticks which are, perpendicular to each other were placed at a, point (4, 5) above the toy., Based on the above information, give the, , Thus, the function is strictly increasing on R., , a.Strictly increasing in 0., , xcOSx = cot x, , P(x)>0, Therefore. f(x) is strictly increasing, , Then, we have, , =, , sin x, , In intervat 0., , d. Ze, , where X X2, , and, , on:, , Hence. f is strictly increasing on R, , d. (0. 2), , So. option, , is strictly increasing, , c.R, , is strictly increasing in:, , a.(, 2n), , =e", , f(x), , or, , x, , 32x3-8x, x =0, , or, , =0, , 4x2-1=0, , *=0 or, , At x 0 . f(0) = 8{0)* -4(0) +3 =0-0+3=3, , positive in, , 0 2}, , first, , quadrant), At, , X =:, , =8x

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142 I MASTERMND CHAPTERWISE QUESTION BANK, , Critical points are (0.3) and+, Thus, abscissa of critical points, , are, , Therefore. f(x)is decreasingin, 0 and, , a, , t, , So, option (b) is correct., , a2. Find the, stick., a., , So, option (b) is correct., , slope of the normal based on the position, , 2016, , b.-2016, , 2016, Sol. Given, f(x) =8x° -4x2+3, , d, , of the Case Study3, , The equation of a straight line passing through a iven, , point X0,yo, , 2016, , y-Yo, , f(x)=(8x4x+3) =32x-8x =8x(4x2-1), , Now., , having finite slope m is given hDy, , m(r - Xo)., , Slope of the tangent to the curvey=f(x) at the, , nt, , OX, , o,Yois given by=fo)., , Slope of normal = -, , and, , F(x)8x(4x-1), slope, , of normal based, , on, , the, , Based on the above information, give the answer of the, , the, , position of the stick i. e. (45), , following questions:, 01. Slope ofthe tanget to the curve y = 3x" -4x atx = 4is:, , 8x44(4)-, , 32 x63 2016, , . .167, , 3. What will be the equation, of the tangent at the critical, point if it passes through (4, 5)?, a. x+2016 y = 10084, b. y 2016x-8059, C.X = 8059 y+2016, d. None of these, SoL, , b. 764, d. 716, , . 761, , So, option (d) is correct., , SoL Given, curve y =3x-4x, On differentiating w.r.t. x, we get, , dy 12x4, , dx, , Slope of the curve ==12x3-4, , Equation of tangent at critical point if it passes through (4,5), , dx, , So, , (y-5)= (x-4)x slope of tangent on the position of stick..(), , 12(4P -4=12x64-4, , x=4, , 768-4 = 764, , Now,, , slope of tangent on the position of stick ie, (4, 5), =f(4) =8x4[4(4)* -1)=32 x63 =2016, , From eq. (1), , point whose x coordinate is 2:, , (y-5)- (x-4)x2016, y-5 2016x- 8064, , So, option (b) is correct., Q2. Slope of the tangent to the curve y = x* -x+1at the, , y=2016x-8059, , So,option (b) is correct., , .11, b. 9, C.3, d. 12, Sol Given, curve y = x -x+1, , 4. Find the second order derivativeofthe function at x =, a., , 2, , Since, the tangent to the curve at the point whose x, , Coordinate is 2., , b.-4, d. 6, , C.-2, , So, the, , P(x)=32x3-Bx, , SoL, , P)Atx=, , Now,, , 32x-8x)-96x-8, , dX, , -8 96x-8 =6-8 -2, , y=(2-2+1=8-1=7, , point of contact is (2.7)., , d, , ., which is the, , -x+1)=3x2-1, =3(2)-1=3x4-1=12-1=11, , required slope of the tangent, , So. option (c) is correct., , So. option (a) is correct., , a5. At which ofthe following intervals willf) be decreasing?, , 03., , a (--1/2)u(1/2,) b.(-1/2)u(01/2), d(-1/2,0)u(/2. ), .(0.1/2)u(1/2,-), Sol, f(x) =8x-4x+3, ., , P(x)=32x3-8x, , Putting, f (x) =0, we et., 32x-8x = 0, , 8x(4x-1) =0, x =0-and, , which divides real line into four intervals, , -GeiJan, , Slope of the normal to the curvex, at6, is:, , =, , 1-a sin6,y, , 2, , ., , 2b, , SoL Given, x, , equation of curve., , =1-asin0 and y, , 26, , bcos e, differentiating w.r.t.0, we get, and y, de a COSe, de -2bcose.sine, dy dy, d, =, , On, , de dx, Sine, , =, , -2bcose. sin0x, , 1, -a cose, , =, , b cos"b