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KENDRIYA VIDYALAYA SANGATHAN RAIPUR REGION25596856223000 HOTS for CLASS-XII MATHEMATICS Session -2015-16CHAPTER-1RELATION AND FUNCTIONLet f: R→R be defined by f(x)= x |x|State whether the function f(x) is onto.Let* be the binary operations on Z given by a* b = a+b+1 a, b ? Z. Find the identify element for * on Z, if any.State with reason whether the functions f : X→Y have inverse, where f(x) =1x x ? X. and X=Q - {o}, Y=Q.Where Q is set of rational number.Let Y= { n2: n ? N} be a subset of N and let “f ” be a function f : N→Y defined as f(x)=x2. Show that “f” is invertible and find inverse of “f”.Show that the function f : N→N given by f(x)=x- (-1)x is bijective.If f be the greatest integer function and g be the absolute value function ; find the value of (fog)(-3/2) + (gof)(4/3).Consider the mapping f :[0,2]→[0,2]defined by f(x)=4-x2. Show that f is invertible and hence find f-1.Give examples of two functions f N→N and g:Z→Z such that gof is injective but g is not injective.Give examples of two functions f:N→N such that gof is onto but f is not onto.Let f:R- -3/5→ R be a function defined as f(x)=2x5x+3, find the inverse of f.Show that the relation R defined by (a,b) R (c,d)a+d=b+ c on the set N×N is an equivalence relation.Let Q+ be the set of all positive rational numbers. ° Show that the operation * on Q+ defined by a*b = 12(a+b) is a binary operation. ° Show that* is commutative.° Show that * is not associative.Let A= N×N. Let * be a binary operation on A defined by (a,b)* (c,d) = (ab+bc,bd) a,b,c,d?N. Show that (i)* is commutative (ii)* is associative (iii) identity element w.r.t.* does not exist.Draw the graph of that function f(x)=x2 on R and Show that it is not invertible. Restrict its domain suitably so that f-1may exist, find f-1 and draw its graph.Show that the relation “ congruence modulo 2” on the set Z is an equivalence relation. Also find the equivalence class of 1.228600053975CHAPTER-2INVERSE TRIGONOMETRIC FUNCTIONS Prove that tanπ4+12cos-1ab+ tanπ4 - 12cos-1ab= 2ba. Solve tan-1x-1x+1+ tan-12x-12x+1= tan-12356Write tan-1(x+1+x2), x?R,in the simplest form.Solve that tan-1(x+1)+tan-1x+ tan-1(x-1)= tan-13Prove than α?2cosec? 12tan??αβ+β?2sec??(12tan??βα)= (α+ β)( α?+ β?)Solve for x: sin-12α1+α2+sin-12ββ2+1 = 2 tan-1xIf tan-1a+ tan-1b+ tan-1c=, prove that a+b+c =abcProve that cos(tan-1(sin[ cot-1x ]))= x2+1x2+2What is the principal value of cos-1cos87 ?If tan-1x+tan-1y + tan-1 z = π2 , prove that xy+yz+zx=1Show that 4 tan-115= tan-1120119If sin-1x+sin-1y+sin-1z= 3π2, then find the value of x100+y100+z100-1x101+ y101 + z101If cos-1xa+cos-1yb=β,prove that x2a2-2xyabcosβ+y2b2=sin2βIf x+1x=2, Find the value of sin-1xFind the value of sin (2 sin-10.8). 2381250434975CHAPTER-3MATRICESIf 06-5xxx-3is symmetric, find x.If A=xyz-xis such that A2=I,then find the value of 2 – x2– yzIf A =-4215133-22 then find f(A) when f(x) = x2– 2x+3. If A= i00i find A4n , n ? NGiven an example of a square matrix which is both symmetric as well as skew symmetric.If A and B are symmetric matrices, then show that AB+BA is also a symmetric matrix but AB – BA is skew symmetric matrix. Show that all the positive integral powers of symmetric matrix are Symmetric.Find the matrix A satisfying the matrix equation1223A4735=1001If A=ab01, a ≠ 1, Prove by induction that Ananban-1a-101 for all positive integer n.Find x if [x-5-1]102021203x41 = OBy using elementary row transformation, find A-1 where A=2-332233-22If A= 0100and I = 1001prove that (aI + bA)3 = a3I + 3a2bA.If A and B are two matrices such that AB=B and BA=A find A2 + B2If A= [aij]mxn is a skew-symmetric matrix, what is the value of aii for every i ?A= 3-4-12, find the matrix B such that AB = ICHAPTER-4DETERMINANTSIf a,b,c are non-zero real numbers, then find the inverse of matrix A=a000b000cIf A=100020003 then what is theadj(adjA)?If A is a square matrix of order 3 such that |Adj A|=64, then find |A|Find the value(s) of , if the matrix 2cos132cos is singular, where 0 <<.Evaluate the determinantlogab 11logbaIf 2+3-1+3+12--3 -3+43= A4 + B3 +C2+D +E, then find the value of EThe value of a third order determinant is 12. Find the value of the square of the determinant formed by the cofactor.Let A be a skew symmetric matrix of odd order, then what will be |A|If f(x)= cosx -sinx0sinxcosx0001, then show that f(x)-1 = f(-x)Prove the following by using the properties of determinants(b+c)?a?a?b?(c+a)?b?c?c?(a+b)?=2abc(a+b+c)?Using properties of determinants , solve for x. a+xa-xa-xa-xa+xa-xa-xa-xa+x=0Ifl ,m,n are in A.P. then , find value of 2x+45x+78x+l3x+46x+89x+m4x+67x+910x+nIf abaα+bb c bα+caα+b bα+c 0 and α is not a root of the equation ax2+bx+c=0, then show that a,b,c are in G.P.Let xkxk+2xk+3ykyk+2yk+3zkzk+2zk+3 = (x-y)(y-z)(z-x) (1x+ 1y+ 1z),then find k.Let A = 231-111-1-1-1, find A-1. Hence solve the following system of equations 2x-y-z=73x+y-z=7x+y-z=3Given that A=-444-7135-3-1 and B=1-111-2-2213 Find AB and use it to solve the system of equations x – y+z=4, x – 2y – 2z=9, 2x+y+3z=1.Prove that (a+1)(a+2)a+2 1(a+2)(a+3)a+3 1(a+3)(a+4)a+4 1= – 2Using the properties of the determinants, prove that mc1mc2mc3nc1nc2nc3pc1pc2pc3=mpnm-nn-pp-m12.Evaluate1aa2bc1bb2ca1cc2ab20. Two schools A and B decided to award prizes to their students for three values Honesty (x), punctuality (y) and obedience (z). School A decided to award a total Rs 15000 for the three values to 4, 3 and 2 students respectively, while school B decided to award Rs 19000 for the three values to 5, 4 and 3 students respectively. If all the three prizes together amount Rs 5000, thenRepresent the above situation by a matrix equation and form a linear equation using matrix multiplication.Which value you prefer to be rewarded most and why?237172557923CHAPTER-5CONTINTUITY AND DIFFERENIABILITYShow that the function f(x) =Sin x+Cos x is continuous at x= .Show that the logarithmic function is continuous.Let f(x) = (x – a)Cos1x-a for x ≠ a and let f(a) = 0. Show that f is continuous at x = a but not derivable there at.Let f(x) =x.|x| for all x ? R. Discuss the continuity and differentiability.Examine for continuity and differentiability of the following functions:- f(x) = xSin1xx>00,ifx≤0 at x=0Given that : If f(x) is continuous at x = 0, find the values of a.f(x) = 1-cos4xx2if x<0aif x=0x16+x-4if x>0If function f(x) = 3ax+b,if x>111if x=15ax-2bif x<1Discuss for continuity of the function at x = 0f(x) = Sin 3xtan2xif x<032if x=0log (1+3x)e2x-1if x>09. Find all points of discontinuity of f wherefx=sinxx, if x<0x+1 , if x≥010) Show that the function fx= 1+x, if x≤25-x, x>2is not differentiable at x = 211) Is the functionfx=x-1x-1, x≠1-1, x=1Continuous at x=1?12) Show that the function f is continuous at x = 0 for all values of a. Also find the value of a for which f is derivable at x = 0 when fx=x2, x≥0ax, x<013) Examine the continuity of the function fx= tan-1(3x3-2x+1)14) If fx=sin 2x, 0<x<π6ax+b, π6< x<1Is continuous and differentiable. Find a & b15) Find whether the function fx=e1x-1e1x+1, x≠00, x=0is continuous?16) Find whether the function fx=x4-5x2+4(x-1)(x-2),x≠1,26,x=112,x=2is continuous? 17) Find the value of derivative at x = 2 of the function fx=x-1+x-318. Find the derivative of the following w.r.t.x.1) y=log (11+x) .2) y=sin(xx).3) y=xsiny.4) xy=ex-y5) y=ex2.6) y=(sin-1x)2.7) y=sin-1.2x+11+4x8) y=sin-1 (a+bcosxb+acosx)9) y=btan-1 [ xa+tany/x].10) y=tan-1x1+1-x211) y=sin-1 [x21-x2+x1-x412) y = Cos (xx)13) y = e-ax2logsinxx=sin3tcos2t, y=cos3tcos2t, find dydx and d2 ydx2.20. If xpyq=(x+y)pq then show that dydx=yx21. Differentiate (sinx)x w.r.t. xsinx23. If x=asin2t(1 +cos2t) & y=bcos2t(1 -cos2t) Show thatdydx=baat t=π424. Differentiate cos-13cosx-4sinx5 w.r.t.x.25. Differentiate sin2x w.r.t. ecosx.26.Show that y=c1ex + c2e-x is the general solution of d2 ydx2-y=0 27. prove that the solution of y = x dydx+a dxdy is y=cx+a/c.28. if y=xlog(xa+bx), Prove that d2ydx2=1xaa+bx229. Differentiate y=log7(log x) w.r.t.x.30. Differentiate y=sin cosx w.r.t.x.31. Differentiate y=a+a+x w.r.t.x.32. Differentiate y=tan-1 x+a1- ax, w.r.t.x.33. Differentiate y=logtan Π4+x2w.r.t.x.34. Verify Rolle’s Theorem for f(x)= log (x2+2)- log3 on [-1,1]35. Verify Rolle’s Theorem for f(x)= Sin4x+ cos4x in0,Π236. Verify Rolle’s Theorem for f(x)= e-xSinx in [0, Π]37). Verify LMV Therorem for f(x) = Sinx-Sin2x on[0,π]38).Find a point on the Parabola y=(x-3)2 where the tangent is parallel to the chord joining (3,0) and (4,1)2428875106680CHAPTER -6APPLICATIONS OF DERIVATIESShow that the rate of change of the perimeter of a square is 4 times the rate of change the length of its sides. Using differentials ,find the approximate value of log? 4.01,given that log?4=1.3863 The pressure p and the volume v of a gas are connected by the relation pv=1.4=constant. Find the percentage error in p corresponding to a decrease of1/2 % in v. If there is an error of2% in measuring the length of a simple pendulum, then find the percentage error in its time period. While measuring the side of an equilateral triangle, an error of 5% is made. Find the percentage error in its area.For what value of x is the rate of increase of x3-5x2+8 is twice the rate of increase of x? If the rate of change of area of a circle is equal to the rate of change of its diameter, find the radius. The side of an equilateral triangle is increasing at the rate of 1/3cm/sec. Find the rate of increase of its perimeter.Find “a” for which f(x) = a (x+sin x)+a is increasing.Let g(x) = f(x) + f (2a-x) and f ’’(x)>0 for all x?[0,2a] then g(x) increasing or decreasing on [0,a]?Let f(x) = tan-1 {g(x)}, where g(x) is monotonically increasing for 0<x<π/2, then find f(x) is increasing or decreasing on(0,π/2).Find whether the function f(x) =tan-1(sinx+Cos x) on[0,π/4] is either strictly increasing or strictly decreasing..For what value of ‘λ’ for which the function f(x)= cos x-2λx is monotonic decreasing.Find the value of ‘a’ for which function f(x) = logaxis increasing on R.If the slope of tangent to curve y=x3+ ax + b at (1,-6) is -1.Find a& b.If x+y=k is normal to the curve y2=12x, then find the value k.Find the point at the curves x2=y and y2=x cut orthogonally. Find whether the function fx=x1+|x|is increasing or decreasingIs the function f(x)=2x is strictly increasing on R?Find the angle of intersection of the curves xy=a2 and x2-y2=2a2.Find the condition for which the curve y=aex and y=be-x cut orthogonally.Find the slope of tangent of curve y=3x2+4x at the point whose abscissa is-2?What is the slope of Normal to curve y=2x2+3 Sin x at x=0?If the function f(x)=x2– kx+5 is increasing on[2,4] then find the value of k.Find the interval for which the function f(x)=xx is decreasing.A man 2 meters high walks at a uniform speed 6 meters per minute away froma lamp- post,5 meters high.Find the rate at which the length of its shadow increases.A kite is 120m high and 130m string is out.If the kite is moving away horizontally at the rate of 52m/sec find the rate at which the string is being paid out.An inverted cone has a depth of 10cm and a base of radius 5cm.Water is poured into it at the rate of3/2cc per minute.Find the rate at which the level of water in the cone is rising when the depth is 4cm.The time T of complete oscillation of a simple pendulum of length l is given by the eq. T=2πlg , where g is constant. What is the percentage error in T when l is increased by 1%?Find the approximate value of tan(46) if it is given that 1°= 0.01745A man is walking at the rate of 4.5km/hr. towards the foot of the tower 120m high. At what rate is he approaching the top of the tower when he is 50m away from the tower?Find the rate of change of the curved surface of aright circular cone of radius r and height h with respect to the change in radius.Find the angle between the parabola y2=4ax and x2=4by at their point of intersection other than origin.If y=alogx+bx2+x has its extreme values at x=-1 & x=2, then find a & b. Show that a local Minimum value of fx=x+1x, x≠0 is greater than a local maximum value.Find the Absolute maxima and Absolute minimum values of the function fx=12-x2+x3 on [-2,-25]Determine the Maximum andMinimum Values of the functiony=2cos2x-cos4x,0≤x≤πFind the local minimum value of f(x)=3+x,x?RA given quantity of metal is to be cast into a solid half circular cylinder (i.e. with rectangular base and semicircular ends).Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its circular ends isA window has the shape of a rectangle surrounded by an equilateral triangle.If the perimeter of the window is 12m, find the dimensions of the rectangle that will produce the largest area of the window.Show that the isosceles triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.A cylinder box is to be made, which is open at the top and has a given surface area. Souvenirs of different life values are to be stared in the box, so we would like to have maximum volume of the box. What should be the dimensions of cylinder box? Name some of the values which are important to each person.The total cost cxof planting x plants in a garden is given bycx=0.005x3-0.02x2+30x+6000. Find the marginal cost, when 200 trees are planted. Do you think plantation helps in saving the environment?2343150226695CHAPTER-7INTEGRALSIndefinite Integrals1. Evaluate I= ?x5a3+x3 dx2. Evaluate I=?Sec2x tan2xe2sec2xdx3. Evaluate I=?ax1-a2xdx4. Evaluate I=?555x55x5xdx5. Evaluate I=?Sec-1xdx6. Evaluate I= ?sin(log x)dx7. Evaluate I=xx2+1(x2+4)dx8. Evaluate I=1(sinx)34(cosx)54dx9. Evaluate I =?tan x tan2x tan3xdx10. Evaluate I= ?sin2xa+bcosx2dx11. Evaluate I=?secx-1 dx12. Evaluate I=?tanx+tan3x2+3 tan2x dx13. Evaluate I=?dxsinx+3cosx 14 Evaluate I=?ex(1x-2x2+2x3)dx15. Evaluate I=?1(sinx-2cosx)(2sinx+cosx)dx16. Evaluate I=?sinxsin4xdx17. EvaluateI=? sin-1xa+xdx18. Evaluate I=?dx1+x+x2+x319. Evaluate I=?x(tan-1x)2dx20. Evaluate I=?1sinx+secxdx21. Evaluate I= ?xcosα+1(x2-2xcosα+1)3/2dx22. EvaluateI=?1+x21-x2dx23. Evaluate I=?1sin4x+cos4xdx24. Evaluate I= ?1x+1x2+2x+2dx25. Evaluate I=?x2x sinx+cosx2dx26. Evaluate I= ?x2x4+x2+1dx27. Evaluate I= ?tanx dx28. Evaluate I=?13+sin2xdx29. Evaluate I= ?sinx+αsinx-αdx30. Evaluate I=?dx(ex+1)331. Evaluate I= ?x2+1[logx2+1-2logx]x4dxEvaluate: I=?secx1+cosecxdx.Evaluate I= ?1sinx+sin2x dx.Evaluate I= ?exsin2xdxEvaluate I=?x(logx)2dxEvaluate I= ?1sinx+tanxdxDEFINITE ITEGRALSEvaluate: -11x21+ x2 dxEvaluate 02 x2-x dxEvaluate 01 log?(1 x-1)dxEvaluate 0π2xsinx+cosxdxEvaluate -π4π4x+π42-cos2xdx42. Evaluate 0π2sin2x1+sinxcosxdx43. Evaluate 1ee │logex│dx44.Evaluateπ3π21+cosx(1-cosx)32dx45. Evaluate 01.5 [x]dx 46. Evaluate -π2π2 │sinx│dx47. Evaluate -33│x+2│x+2dx48. Evaluate -132 │x sinπx│dx49. Evaluate 01log1+x1+x2dx50. Evaluate 14 (x-1+x-2+x-3)dx51. Evaluate as a limit of sum -11 e-5xdx52. Evaluate 0acot-11-ax+x2adx53. Prove that 02πxsin2nxsin2nx+cos2nxdx=π254. Evaluate 0∞ logx+1x11+x2dx55. Evaluate 0πx1-cosαsinαdx236220027305CHAPTER-8APPLICATION OF INTEGRALSDraw the graphs of the curves y=sinx and y=cosx,0≤x≤ π2.Find the common area between the above curves with the X-axis.Find the area bounded by the lines x+2y =2; y-x=1and 2x+y=7.Find the area bounded by the line y=x and the curve y=x3.Find the area bounded by the lines y=1+│1+x│, x=-2 , x=3 and y=0.Find the area enclosed between the curve y =x and the line y=x.Find the area bounded by the curve y=e│x│ and the line y=3 with X- axis.Find the area bounded by the curve y=│tanx│and the line y=3.Find the area included between the curve y =x-[x] and the line x=3 with X &Y axis.Find the area enclosed between the curve y=│sin x│ and the line y=12 within the interval 5π6,7π6.Find the common area between the curve y=5-x2and the lines y =│x-1│.1295400146685CHAPTER -9DIFFERENTIAL EQUATIONSSolve dxdy+cosxcosy=0.Find the degree and order of the differential equation 1+3dydx2/3 =4d3ydx3Find the differential equation of the family of curves given byx2+y2=2axFind the integrating factor of the differential equation xdydx-y-2x3= 0Verify that yx=c is a solution of the differential equation (ydx-xdy)y=0Verify that y = (1+ex)(1-ex) is solution of the differential equation dydx=ex[1-ex1-e2x]Find the equation of the family of curve whose x and y intercepts of the tangent at any point p are respectively double the x and y co-ordinates of the same point p respectively.The line normal to a given curve at each point (x,y)on the curve passes through the point (2,0).If the curve contains the point (2,3), find its equation.Prove that the curve with the property that all its normal passes through a constant point is a circle.A population grows at the rate of 8% per year.How long does it takes for the population to double?Solve: (1+e2x)dy+(1+y2)exdx=0 , given that y=1 when x=0 .Solve: (1+y2)dx=(tan-1y-x)dy.Solve the differential equation e-2xx-yxdxdy=1,x≠0Prove that the solution of the differential equationdydx=x2y21-y61-x6 isy31-x6-x31-y6=constant.Solve:xlogxdydx+y = 2x logxSolve:(x+y+1)dydx =1 Solve: xdydx=y(logy-logx+1)Solve (xx2-y2-y2)dx+xydy=0.A bank pays interest by continuous compounding that is by treating the interest rate as the instantaneous rate of change of the principal.Suppose that in an account the interest at 8%per year compounded continuously.Calculate the percentage increase in such an account over one year.(Take e0.08=1.08333 approximately.)Solve the differential equation d2xdy2=1+siny,given that x=0 and dxdy=0 when y=0.Solve the differential equation d2ydx2= xex,given that y=0 and dydx=0 when x=0.Solve (xdx-ydx)sinyx=(ydx+xdy)xcosyxSolve the differential equation 1-y2dx=(sin-1y-x)dy.Show that the differential equation (x-y)dydx= x+2y is homogenous and solve it.Find a particular solution of the differential equation dydx+ycotx=4xcosecx x≠0given that y=0 when x=π2.1695450207645CHAPTER-10VECTORSFind a unit vector parallel to XY – plane and perpendicular to the vector 4i-3j+kIf a=26,b= 7 anda ×b=35,find a?bWrite number of unit vectors perpendicular to i+j and j+k.If G is the centroid of the triangle ABC.Show that GA+GB+GC= 0.If a is a non-zero vector of magnitude a then find the value of λ if λ a is a unit vector.Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.Show that the straight line joining the mid points of non-parallel sides of a trapezium is parallel to the parallel sides and half of their sum.Find all the values of λ such that (x,y,z)(0,0,0)and (i+j+3k)x+(3i-3j+k)y+(-4i+5j)z=λ(xi+yj+zk)Prove that the middle point of the hypotenuse of a right angled triangle is equidistant from its vertices.In a triangle AOB,angle AOB=900.IfPand Q are the points of trisection of AB,show that OP2+OQ2=59AB2.For any vectora,show that a×i2 + a×j2 + a×k2 = 2a2.If A,B,C,D are four points such that AB=m(2i-6j+2k),BC=i -2j and CD=n(-6i+15j-3k).Find the conditions of the scalars m,n such that CD intersects AB at the same point E. Also find the area of the triangle BCE.Let OA=a, OB=10a+2b and OC=b where O is origin.Let p denote the area of the quadrilateral OABC and q denote the area of the parallelogram with OA and OC as adjacent sides. Prove that p=6qProve that the lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent.Points F and E are taken on the sides BC and CD of a parallelogram ABCD such that BF:FC= :1 and DE:EC=λ :1The straight lines FD and AR intersect at the point O.Find the ratio of |FO|:|OD|.ABCD is a quadrilateral such that AB = b, AD=d, AC=mb+pd. Show that the area of quadrilateral ABCDE is 12m+pb×d.The vector -i+?+k bisects angle between the vectors c and 3i+4j.Determine unit vector along c . If a and b are two unit vectors and θ is the angle between them, then show thatSinθ2=12a -b 22. If a , b , c are the position vectors of three non collinear points A, B, C respectively. Prove that a × b +b × c + c × a is perpendicular to plane ABC.23. For any three vectors a , b , c prove that a + b , b + c , c + a = 2a b c 24. Prove that a × b × c + b × c × a +c × a × b =0 25. Prove that a × b b × c c × a = a b c 21314450173990CHAPTER-11THREE DIMENSIONAL GEOMETRYFind the direction of angles of the line joining points. (-1,-5,-10) and the point of intersection of the line x-23 =y+1 4=z-212and the plane x-y+z=5 with x,y,z axes.Find the perpendicular distance of a vertex of a cube from its one of the diagonal, not passes through the vertex.Find the distance of the point (-2, 3,-4) from the linex+23=2y+34=3z+45 measured parallel to the plane 4x+12y-3z+1=0Separate the equation xy+yz=0 into two planes and find out whether the plane are || or to each other.If A(1,2,3) and B (3,6,11) are images to each other w.r.t. a plane.Find the vector equation of the plane mirror.Find the value of λ if the plane mirror is to 2x-3y+λz-5=0.Find k,if the plane 2x-4y+z-7=0 contains the line x-4=y-2=z-k2Find the point on the line x+23 =y+12 =z-32 at a distance 32from the point (1,2,3).Find the Direction Cosines of the line joining the images of the point (1,2,3) w.r.t xy and yz planes.A line makes the same angle θ with each of the X and Z axes. If the angle β,which it makes with Y axis such that sin2β=3sin2θ, then find the value of θ.Prove that the two planes x-2y+2z=6 and 3x-6y+6z=2 are parallel.Also find the distance between the planes.find the intercept on the line x-12=y+13=z-1 between the two planes.What is the direction cosines of a line equally inclined to the axes?What is the equation of Y axis in vector and Cartesian form in three dimensional space?If the projection of the line segment on X,Y and Z axes are respectively 4,32 and 1then find the length of the line segment.Find the distance of the point (2,3,4) from the plane 3x+2y+2z+5=0 measured parallel to the line x+33 =y-26=z2Find the equation of the line passing through the point 2,3,2and parallel to the line r = -2? + 3? +λ(2i-3j+6k)and also find the distance between them.Show that the equation of the plane which meets the axes in A,B and C and the centroid of triangle ABC is the point (u, v, w) isxu+yv+zw=3Find the vector equation of plane which is at a distance of 5 units from the origin and which has -1, 2, 2as the direction ratios of a normal to it.A line makes angles α,β,γ and δ with the four diagonals of a cube prove that (i)sin2α+sin2β+sin2γ+sin2δ=83(ii) cos2α=-23Show that the angle between any two diagonals of a cube is π2- cosec-1(3)If a point A(1,2,3)move towards and reaches a linex-63=y-72=z-7-2 in shortest distance and the point A move towards and reaches a line x0=y-2-3=z+33in shortest distance. Find the distance between the two new locations of A.Find the shortest distance between the given linesx-12= y+13=2, x+15= z-21=2If the angle between the line x= y-12= z-3λ and the plane x + 2y + 3z = 4 is cos-1 514 , then find the value of λ.1114425153670CHAPTER-12LINEAR PROGRAMMING PROBLEMSFind whether the maximum value of the objective function Z=-x+2y exists or not,subject to the following constraints.x2x+y5x+2y6 and y0Find whether the minimum value of the objective functionZ=-50x+20y exists or not,subject to the following constrains2x-y-5x +y32x-3y12x0,y0Maximize Z=2x+3ySubject to the constraintsx+y2x+2y3x0,y0Kellogg is a new cereal formed of a mixture of bran and rice that contains at least 88 gms of protein and at least 36 mg of iron. Knowing that bran contains 80 gms if protein and 40 mg of iron per kg, and that rice contains 100 gms of protein and 30 mg of iron per kg,find the minimum cost of producing this new cereal if bran costs Rs.5/-per Kg and rice Costs Rs.4/- per Kg.A brick manufacturer has two depots A and B with stock 30,000 and 20,000 bricks respectively.He receives orders from 3 buildings P,Qand R for 15,000,20,000 and 15,000 bricks respectively. The costs of transporting 1,000 bricks to the building from the depot (in Rs.)are given below.From/toPQRA402030B206040How should the manufacturer to fulfill the orders so as to keep the cost of transportation minimum.Solve it graphically.Find the constraints of the L.P.P if its graphical representation is given below and hence maximize Z=3x+9yA manufacturer produces two products A and B during a given period of time. These products require four different operations,viz.Grinding,Turning,Assembly and Testing.The requirement in hours per unit of manufacturing of the product is given below.OperationABGrinding12Turning31Assembly43Testing54The available capacities of this operation in hours for the given time are:Grinding30Turning60Assembly200Testing200Profit on each unit of A is Rs.3 ,and Rs.2 for each unit of B. Formulate the problem as LPP.Constrain of a L.P.P represents the graph given below.Write the constrains and Minimize Z=6x+7y9. A retired person has Rs 70,000 to invest in two types of bonds. First type of bond yields an annual income of 8% on the amount invested and the second type of bond yield 10 % per annum. As per the norms, he has to invest minimum Rs 10,000 in first type and not more than Rs 30,000 in second type. How should he plan his investment so as to get maximum return after one year of investment? Do you think that a person should start saving at an early age of his retirement? Can you name some avenues? 10. A dietician wishes to mix two types of food. X and Y in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C, Food X contains 2 unit/Kg of vitamin A and 1 unit/Kg of vitamin C, While food Y contains 1 unit/Kg of vitamin A and 2 unit/Kg of vitamin A and 1 unit/Kg of vitamin C. It costs at Rs 5 per Kg to purchase the food X and Rs 7 per Kg to purchase food Y. Determine minimum cost of the mixture. What is your opinion about healthy diet? Name few ingredients; necessary for a healthy diet.1581785275590Chapter -13PROBABILITY1) Find the minimum number of tosses of a pair of dice so that the probability of getting the sum of digits on the dice equal to 7 or at least one toss is greater than 0.95, givenlog102=0.3010 &log103=0.47712) The sum of mean and variance of a binomial distribution is 15 and their product is 54, find the distribution.3) If A and B are events such that p(A∪B)=34,p(A∩B)=14, p( A )= 23 , find p A ∩B.4) Two dice are rolled one after the other .Find the probability that the number on the first is smaller than the number on the second.5) A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that“ at the end of eleven steps, he is one step away from the starting point”.6) Three numbers are chosen at random without replacement 1,2, 3,……10. Find the probability that the minimum of the chosen numbering is 3 or their maximum is 7.7.) In a bolt factory three machines A, B, and C, where A produces one- fourth, C produces two-fifth of the products. Production of defective products in% by A, B, Care respectively 5, 4 and 2. An item is drawn at random and found to be difficult. What is the probability it was produced by either A or C.8.) Two persons A and B throw a pair of dice alternately beginning with A. If cosαrepresentsthe probability that B gets a doublet and wins before A gets a total of 9 to win. Find α9.) A bag contains 6 red and 5 blue and another bag contains 5 red and 8 blue balls.A ball is drawn at random from the first bag and without noticing its colour is put in the second bag. A ball is then drawn from the second bag. Find the probability that the ball drawn from the second bag is blue in colour.10) A, B, and C throw a die alternatively till one of them gets any number “more than 4”and wins the game. Find their respective probabilities of winning if A starts the game followed by B and C.11) One letter has to come from “LONDON” or “CLIFTON”. Only ON is seen on the post mark, find the probability of this letter from LONDON.12 Three stamps have been selected from 21 stamps which are marked from 1 to 21.Find the probability the numbers on selected stamps are in A.P.13 A bag contains 3 red balls bearing one of the 1, 2, 3(one number on one ball) and two black balls bearing the numbers 4 or 6. A ball is drawn and its number is noted and the ball is replaced in the bag. Then another ball is drawn and its number is noted. Find the probability of drawing:2 on the first draw and 6 on the second draw.A number ≤2 on the first draw and 4 on the second draw.A total of 5.14. In an examination, an examinee either guesses or copies or knows the answer of multiple choice questions with four choices. The probability that he makes a guess is1/3 and the probability that he copies the answer is 1/6.The probability that his answer is correct given that he copied it is 1/8.Find the probability that he knew the answer to the question, given that he correctly answered it.15. In a class having 60% boys,5% of the boys and 10% of the girlshave an I.Q. more than 150. A student is selected at random and found to have an I.Q. of more than 150. Find the probability that the selected student is a boy.16. Find the probability distribution of the numbers of kings drawn when 2 cards are drawn one by one without replacement from a pack of 52 playing cards.17. A bag contains 5 white, 7 red and 8 black balls. If 5 balls are thrown one by one with replacement, find the probability distribution that exactly 5 red balls drawn. 18. A speaks truth in 60% of the cases and B in 90% of cases. In what percentage of cases are they likely to contradict each other in stating the same fact? Which value A is lacking and should improve upon?19. There is a group of 100 people who are patriotic, out of which 70 believe in non-violence. Two persons are selected at random out of them. Write the probability distribution for the selected person who is non-violent? Also find the mean of distribution. Explain the importance of non-violence in patriotism.20. India plays two matches. The probability of India getting points 0, 1 & 2 are 0.45, 0.05 & 0.50 respectively. Find the probability of India getting at least 7 points in serves.18040352660651800225294640ANSWERS/HINTSChapter 11) f is not onto2)e = -1 3)No inverse 6) 27)f-1x=4-x210) f-1x=3x2-5x14)f-1x=xChapter 22) x=43, -384) x = -16) x=α+β1-αβ9)6π714) π215) 0.96Chapter 31) -32) 03)30-4-126 4) Identity matrix of order 2. 6) Null matrix11) A-1=-1520-31-10-2-1213) A+B14) Zero15)B=121232Chapter 41) A-1=1a0001b0001c2) 12963) ±84) θ=π65) Zero6) 21 7) 207368) Zero11) x=0, 3a12) Zero15) x=2, y=-1, z=-219) Zero 20) (i) 432543111xyz=150001900050004x+3y+2z=150005x+4y+3z=19000x+y+z=5000(ii) Equally, to each value, as each value has its own importance in life.Chapter – 54. Continuous at x = 0 Derivable only at x = 05) Continuous6) a = 8 7) a = 3, b = 28) Continuous at x = 09) No point of discontinuity 11) Discontinuous12) Continuous for all values of a13) yes continuous for all x ∈ R 14) a = 1, b = 32, -π/615) ( 0, ∞ )16) Continuous at R – { 1 , 2 }17) F-1(2) = 018) x + 1[ cos xx { xx(1 + log x) } ]y/[x(1- xcosy)]x-yx( 1+logx )2xex22sin-1x1-x2(2x+1log2)/(1+ 4x)–(b2-a2)/(b+acosx)1a-yx2+y2Sec2 yb - xx2 + y2121- x211- x2+2x1- x4cosx(x)x[logcos xx - xtan xx{ xx(1 + logx) } ]–e-ax2logsinx [ax2cotx + 2axlogsinx] 19). –coxt/sint [(cos2t – 3sin2t)/ 3cos2t – sin2t)]21) dy1dy2 = (sinx)x[x cotx+logSinx]xsinx[ Sinxx+cosxlogx ]24) 125) - 2cosx.e- cos x 29) log 7 exloge x30) Coscosx. 12cosx. -Sinx2x31) -14a+a+x.a+x32) 1 / [2x(1+x) ]33) Sec x 38) x = 72? (3 , 4)Chapter 61) 1.38383) 0.74) 1%5) 10%6) x = 3, 1/37) 1/11 8) 1 cm/sec.9) a> 010) Decreasing11) Increasing12) Strictly increasing13)≥12 14) a>1 15) a = - 4 , b = - 316) K = 917) (0,0) 18) Strictly Increasing 19) yes20) 900 or π/221) ab = 122) -8 23) -1/3 24) k ∈( ∞ ,2 )25) (0, 1/e)26) 4m/min27) 2028) [3/8 π] cm/sec 29) ? 30)1.0349031) –[45/26]km/hrs32) ds/dr ={ П(2r2+ h2)/r2+ h2 }33) θ = tan-13(ab)13a2/3+b2/334) a = 2, b = -1/2 35) Max value 178 at x = 10, Abs mini value 18 at x = 636) Max value 3/2 at π/6, 5 π/6 Min value -3 at π/237) minimum value 338) π : (π +2) 39) [12/(6-3)]m. & [(18 - 63)/ (6-3)]m.41) Radius of base = height of cylindrical box. Honesty, Respectful, punctual, observant 42) 622, yes.Chapter 71) 215(a3 + x3)52 – 29a3(a3+x3)32 + c2) -12 e-sec2x + c3)1loga sin-1ax + c 4) 1(log5)3555x+ c5) xsec-1x-logx+x2-1+c6)X2sinlogx-coslogx+c 7)16logx2+1x2+4+c8)4cot-14x+c9) logsec3x3-logsec2x2-logsecx+ c 10) -2b2loga+bcosx+aa+bcosx+c11) –logcosx+12+cos2x+cosx+c 12) 16log2+3tan2x+c13)12logcosecx+π3-cotx+π3+c14)ex1x-1x2+c15) 12logtanx-22tanx+1+c16) 142log1+2sinx1-2sinx-18log1+sinx1-sinx+c 17) axatan-1xa-xa+tan-1xa+C18) 12log1+x-12log1+x2+tan-1x+c 19) tan-1x2x22-tan-1x.x+log1+x2+tan-1x22+c20) 123logsinx-cosx+3cosx-sinx+3+tan-1sinx+cosx+c 21) cosec2αx2-2xcosα+1x2-xcos2α-2cosα22) I=x+3logx-4-24logx-5+30logx-6+c23) tan-1tan-1x-12tanx +c24)logx+1(x2+2x+2)25) xsecx-1xsinx+cosx-tanx+c26) 12tan-1x2-1x3+14logx2-x+1x2+x+1+c27) put t-1t=u, t+1t=v=duu2+2+duu2+2= 12tan-1tanx-12tanx+122logtanx-2tanx+1tanx+2tanx+1+c28) 122tan-13tanx+122+c29) -sin-1cosxcosα+logsin2x+sin2x-sin2α+c30) let t=ex+1-logt+1t+12t2+logt-1+c-logex+1+1ex+1+12ex+12+x+c31) -13log1+1x21+1x232-491+1x232+c32) Log1+secx+c33) 12logcosx+1+ 16logcosx-1-16log2cosx+1+c 34)25exsin2x+cos2x+c35) (logx)2x22-logxx22+x24+c36) logcosecx-cotx+141+cosx1-cosx-12(1+cosx)+c37) 2-π238)1621539) Zero40)π42log2+12-141) 318π242) π3343) 2-2e44) 145) 146) 247) 448) 1π2+3π49) π2log250)19251) 15 (e5-e-5)52) 2atan-1a-log?(1+a2)54) πlog255) π(π-α)sinαChapter 81) 2-√22) 63) ? 4)13.55) 1/6 6)2π3-log4 8) 3/2 9)π6+(3-2)10)52sin-115+ sin-125-12Chapter 91)Sinx + log(Siny)=c2) Order = 3; Degree = 33) 2xydxdy=y2-x24) 1x 5)dydx=-yx6) ex(1-e2x)(1-ex)7) Equation of the family of curve is xy = c 8) (x-2)2+ y2=99)252log2 years10) tan-1y+tan-1ex= π2 11) xetan-1y= etan-1ytan-1y-1+c 12)ye2x=e-2xxe2xdx+c=2x+c13) (x-a)2+(y-b)2=2c14) y31-x6-x31-y6=sin3c15) 2-logxx-1x+c16) x=cy-y-217) y=atan-1x+ya+c18)x2+y2+xlogcx=019) P1-P0P0 ×100=8.33%20) x=y22-siny+y21) y=xex-2ee+x+222) secyx=cxy23) x=sin-1y-1+cesin-1y25) ysinx=2x2-π22Chapter 101) Vector parallel to XY – plane will be of the form ai +bj . If it is perpendicular to 4i-3j+k, then 4i-3j+k.ai +bj=0=> b = 4a3∴the vector is ai+4a3j=a33i+4j∴The unit vector=a3(3i+4j )a332+42= ±153i+4j2) a×b=35 i.e absinθ= 526∴cosθ= 1-2526=126a.b=abcosθ=73) 24) GA+GB+GC=OA-OG+OB-OG+OC-OG=OA+OB+OC-3OG=a+b+c-3( a+b+c )3=05) a=1 ? a=1 ? =±1a6) Let ABC be the given triangle. Let AD, BE, CF be the medians. The required sum of vectors isAB+BD+BC+CE+CA+AF=AB+BC+CA+BD+CE+AF=AC+CA+12BC+12CA+12AB=AC+CA=07) OA=a , OB=b , OC=c ,OC=d AB=mc-d, OE= a+d2 ,OF= c+b2 ,EF= m+12DC8)1-3-41-(3+)531-=012) Let EB=pAB, CE=qCD∵EB+BD+CD=0 ?p=12m, q=13n Then area of ? BCE= 12EB×BD= 126 18) c=-1115i-1015j-215kChapter 111) cos-1313,cos-1413, cos-112132) a23 where'a' be the side3)4580154) y = 0, x + z = 0 and they are perpendicular to each other,5) x+2y+4z-38 = 0 & = 16) k = 77) 5617,4317,111178) D.C.'s are 110, 0, -3109) cos-13510) i169, ii14911) ±13, ±13, ±1312) Equation of Y-axis r =j is vector form &x0 = y1 = z0 = is Cartesian form.13)19.2514) 7 units.15) r =2i+3j+2k+μ 2i-3j+6k&5807, 17) r .-i+2j+2k=1520) 4621) 359 22) λ = 23Chapter 12Maximum value does not existMinimum value does not existThe objective function can be made as large as possible as we please. So the problem has unbounded solutions.Minimum cost of cereal is Rs. 4 & 60 paise.Minimum transportation cost is Rs. 1,200 when0,20,000,10,000 bricks are transported from the depot A and 15,000,0,5,000 bricks are transported from the depot B to the building P,Q and R respectively.Constraints arex+y≥10x+3y≤60x-y≤0x,y≥0Maximum value of Z = 180 when x = y = 15 7) Maximize Z = 3x + 2y subject to x + 2y ≤30;3x + y ≤ 60; 4x + 3y ≤ 200; 5x + 4y ≤ 200X, y ≥ 0 8)Maximum value: 14.9) Rs 40,000 in 8% bonds and Rs 30,000 in 10% bonds for a maximum return of Rs 6200. One should start saying at early age of his retirement. Saving bonds, NSC, Mutual funds etc.10) x=2, y=4, Rs.38, We must take balanced diet for good health. Wheat, Rice, Fruits, nuts etc.Chapter 131) 172)23+13273) 5/124) 5/125) 46262556)11/407)41/498) cos-1479) 93/15410) 919, 619, 41911) 12/1712) 10/13313) i125ii225iii42514) 24/2915) 3/7 16) Probability distribution is x012P(x)18822132221122117) p(x = 5 ) = 5c5 72057205= 720518) 42%, He lacks honesty and truthfulness.X012 Mean = 1.4P(X)29/330140/330161/33019) 2440940632543Non-Violence helps in presenting yours views in a calm and better atmosphere without distributing other activities. 20) 0.0875 ................
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