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1 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com CHAPTER THREE MARKS QUESTIONS.. Find the inverse of the matri :. For an non-singular matri A, show that A T A T 4 CHAPTER SIX MARKS QUESTIONS.. Find the rank of the matri Find the rank of the matri Find the rank of the matri Find adjoint of A if A = and verif A ( adja) ( adj A) A A. I 5 5. Verif that ( A ) T ( AT ) if A If A and 0 B, verif that ( AB) B A. 7. S.T the adjoint of A is A itself. 8. For A show that A A Solve b matri inversion method. +=, +=8 0. Solve the following non-homogeneous equations of unknowns using determinants : z 5, z, z 4. Eamine the consistenc of the sstem of equations using rank method z 7, z 8, z 6 visionto00.weebl.com th-maths-ke.weebl.com

2 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com CHAPTER TEN MARKS QUESTIONS.. A bag contains tpes of coins namel Re., Rs. & Rs.5. There are 0 coins amounting to Rs.00 in total. Find the number of coins in each categor.. A small seminar hall can hold 00 chairs. different colours (red, blue, green) of chairs are available. The cost of a red chair is Rs.40. Cost of Blue chair is Rs.60 and the cost of a green chair is Rs.00. The total cost is Rs.5,000. Find atleast different solution of the number of chairs in each colour to be purchased.. Show that the equations ++z=6, ++z=4, +4+7z=0 are consistent and solve them. 4. Verif whether the given sstem of equations is consistent. If it is consistent, solve +5+7z=5, ++z=9, + z=0.(rank Method) 5. Eamine the consistenc of the equations b rank method. If it is consistent, then solve + z =, + z =, + z = 6. Eamine the consistenc using rank method. +7z=5, + z=, +9 47z=. 7. Solve b matri determinant method + + z = 7, + z = 4, + z = 8. Solve b determinant method., 5, 0 9. Solve b matri inversion method + z = 9, + + z = 6, + z = 0. For what values of of the following equations ++z=0; 4++z=0; ++z=0 have a (i) trivial solution (ii) Non-trivial solution z 4 z z visionto00.weebl.com th-maths-ke.weebl.com

3 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com. For what values of K, the sstem of equations K++z=, +K+z=, ++Kz= have (i) unique solution, (ii) more than one solution, (iii) no solution.. Investigate for what values of λ, μ the simultaneous equations ++z=6, ++z=0, ++λz=μ have (i) no solution (ii) a unique solution and (iii) an infinite number of solutions.. Discuss the solutions of the sstem of equations for all values of λ, ++z=, + z=, λ++4z=. CHAPTER SIX MARKS QUESTIONS.. Diagonals of a rhombus are at right angles. Prove b vector method. [. Prove that,, ] [ ] a b b c c a a b c. Prove a Sin A b Sin B c SinC b vector method. 4. Prove that [ a b, b c, c a ] 0 5. Find the vector and Cartesian equations of the sphere whose centre is (,, ) and which passes through the point. (5, 5, ). 6. Find the Vector and Cartesian equation of the sphere on the join of points A,B having position vectors i 6 j 7 k, i 4 j k respectivel as a diameter. Find also the centre and radius of the sphere. 7. Obtain the vector and Cartesian equation of the sphere whose centre is (,,) and the radius is the same as that of the sphere r ( 8 i 6 j 0k ) 5 8. Obtain the vector and Cartesian equation of the sphere whose centre is (,,) and the radius is the same as that of the sphere r ( 8 i 6 j 0 k ) 5 9 Find the centre and radius of the sphere r r. ( 8 i 6 j 0 k ) 50 0 visionto00.weebl.com th-maths-ke.weebl.com

4 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com CHAPTER TEN MARKS QUESTIONS.. Prove that Sin (A+B) = SinA CosB + CosA SinB. Prove that sin(a B) sinacosb cosasinb. Prove that Cos(A B)=CosA CosB + SinA SinB. 4. Prove that Cos(A+B)=CosA CosB SinA SinB. 5. Altitudes of a triangle are concurrent-prove b vector method 6. If a i j k, b i 5 k, c j k then prove that a ( b c ) ( a. c ) b ( a. b ) c 7. Verif ( a b ) ( c d ) [ a b d ] c [ a b c ] d, if a i j k, b i k, c i j k, d i j k 8. Find the vector and Cartesian equation of the plane through the point (,, ) and parallel to the lines z, 4 z 9. Find the vector and Cartesian eqn. of the plane through the point (,,) & perpendicular to the planes ++4z+5=0, +z+=0. 0. Find the vector and Cartesian eqn. of the plane through the points (,,), (,,) and perpendicular to the plane ++z=5. Find the vector and Cartesian eqn. of the plane through the points (,,) & (,,) perpendicular to the plane +4z 5=0. Derive the equation of plane in the intercept form.. Find the vector and Cartesian equation the plane containing the line z and passing through the point (,, ) 4. Find the vector and Cartesian eqn. of the plane through the points (,, ), (,4,) & (7,0,6) visionto00.weebl.com th-maths-ke.weebl.com

5 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com 5 Find the vector and Cartesian eqn. of the plane through the points with 7 i k, position vectors i 4 j k, i j k, 6. Find the vector and Cartesian eqn the plane containing the line z and parallel to the line z 7. S.T the lines intersection. z, z intersect and find their point of 8. ST the lines intersection. z, 0 4 z 0 intersect & find their point of CHAPTER SIX MARKS QUESTIONS.. Show that the points ( 79i),( 7i ),( i) representing the comple numbers form a right angled triangle on the Argand diagram.. Find the square root of 8 6i. Find the square root of ( 7+4 i ) 4. State and prove the triangle inequalit of comple numbers. 5. Solve the equation =0 if i is a root. 6. Solve the equation =0 if one root is i n 7. If n is a positive integer, prove that ( i) n ( i) n cos : n N m n 8. If cos isin, cos isin prove that m n cos( m n ) n 4 9. If n N, Prove that ( i ) n ( i ) n n n cos CHAPTER TEN MARKS QUESTIONS. visionto00.weebl.com th-maths-ke.weebl.com

6 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com. Find all the values of values is. i. Find all the values of ( i ) 4 and hence prove that the product of the. If, β are the roots of the eqn. +4=0 then P.T n n n i n Sin & deduct that If, β are the roots of the eqn. p+(p +q )=0, q p tan S.T ( ) n ( ) n qn Sinn Sinn 5. P represents the variable comple no. of z. Find the locus of P, if I m z iz 6. Solve: = 0 7. Solve: = 0 8. Solve: = 0 9. If & β are the roots of the equation +=0, Cot=+, Show that ( ) n ( ) n Sinn Sin n CHAPTER 4 SIX MARKS QUESTIONS.. The headlight of a motor vehicle is a parabolic reflector of diameter cm and depth 4cm. Find the position of bulb on the ais of the reflector for effective functioning of the headlight. visionto00.weebl.com th-maths-ke.weebl.com

7 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com. Find the equation of the hperbola if the asmptotes are 80 & 0 and (5, ) is a point on the hperbola.. Find the equation of two tangents that can be drawn from the point (,) to the hperbola =6. 4. Prove that the tangent at an point to the rectangular hperbola forms with the asmptotes a triangle of constant area CHAPTER 4 TEN MARKS QUESTIONS.. Find the ais, verte, focus, directri, length and equation of latus-rectum and draw their graph for the parabola +8 6+=0. A cable of suspension bridge is in the form of a parabola whose span is 40 mts. The road wa is 5 mts below the lowest point of the cable. If an etra support is provided across the cable 0 mts above the ground level, find the length of the support if the height of the pillars are 55 m.. On lighting a rocket cracker it gets projected in a parabolic path and reaches a maimum height of 4 mts when it is 6 mts awa from the point of projection. Finall it reaches the ground mts awa from the starting point. Find the angle of projection. 4. Assume that water issuing from the end of a horizontal pipe 7.5 m above the ground, describes a parabolic path. The verte of the parabolic path is at the end of the pipe. At a position.5 m below the line of the pipe, the flow of water has curved outward m beond the vertical line through the end of the pipe. How far beond this vertical line will the water strike the ground? visionto00.weebl.com th-maths-ke.weebl.com

8 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com 5. A grider of a railwa bridge is in the form of parabola with span 00ft and the highest point on the arch is 0ft above the bridge. Find the height of the bridge at 0 ft, to the left or right from the mid-point of the bridge. 6. A ladder of length 5m moves with its ends alwas touching the vertical wall and the horizontal floor. Determine the equation of the locus of a point P on the ladder, which is 6m from the end of the ladder in contact with the floor. 7. A kho-kho plaer in a practice session while running realises that the sum of the distances from the two kho-kho poles from him is alwas 8m. Find the equation of the path traced b him if the distance between the poles is 6m. 8. The orbit of the planet Mercur around the Sun is in elliptical shape with sun at a focus. The semi-major ais is of length 6 million miles and the eccentricit of the orbit is Find (i) how close the mercur gets to sun? (ii) the greatest possible distance between mercur and sun. 9. The ceiling in a hall wa 0ft wide is in the shape of a semi ellipse and 8ft high at the centre. Find the height of the ceiling 4ft from either wall if the height of the side wall is ft. 0. Find the equation of the rectangular hperbola which has for one of its asmptotes the line + 5=0 passes through the points (6,0), (,0).. A satellite is travelling around the earth in an elliptical orbit having the earth at a focus and of eccentriccit / The shortest distance that the satellite gets to the earth is 400 kms. Find the longest distance that the satellite gets from the earth. visionto00.weebl.com th-maths-ke.weebl.com

9 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com. An arch is in the form of a semi-ellipse whose span is 48 feet wide. The height of the arch is 0 feet. How wide is the arch at a height of 0 feet above the base?. The arch of a bridge is in the shape of semi-ellipse having a horizontal span of 40 ft and 6 ft high at the centre. How high is the arch, 9 ft from the right or left of the centre. 4. A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle of radians with the ais of the orbit find (i) the equation of the comet s orbit, (ii) how close does the comet come nearer to the sun? (Take the orbit as open rightward). 5. Find the equation of the hperbola if its asmptotes are parallel to + = 0 and +8=0, (,4) is the centre of the hperbola and it passes through (,0). 6. Find the ais, verte, focus, directri, length and equation of latusrectum and draw their graph for the parabola 6 =0 CHAPTER 5 SIX MARKS QUESTIONS.. Obtain Maclaurin s series epansion for log ( ). Obtain Maclaurin s series epansion for tan. Verif Rolle s theorem for the function f()=e sin, 0 π 4. Verif Lagrange s law of mean for the function f ( ) 5,, 5. Obtain the Maclaurin s series epansion for arc tan or tan. e visionto00.weebl.com th-maths-ke.weebl.com

10 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com 6. Evaluate : (cos ec ) Lim 0 7. Evaluate : Lim (cos ec ) 0 8. Find the intervals on which f()= + 0 is increasing or decreasing. 9. Prove that sin < < tan, ϵ(0, π ) 0. (b) Find the critical numbers of / 5( 4 ). Determine the points of infleion in an, of the function. CHAPTER 5 TEN MARKS QUESTIONS.. Prove that the sum of the intercepts on the co-ordinate aes of an tangent to the curve a cos 4, a sin 4, 0 is equal to a. Show that the equation of the normal to the curve = a Cos, = a Sin at = 0 is Cos Sin = a cos.. A car is travelling from west to east at 50 km/hr and car B is travelling from south towards north at 60 km/hr. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0. km and car B is 0.4 km from the intersection? 4. Gravel is being dumped from a conveor belt at a rate of 0 ft /min and its coarsened such that it forms a pile in the shape of a cone whose base diameter and height are alwas equal. How fast is the height of the pile increasing when the pile is 0 ft high? 5. Find the equation of tangent and normal to the ellipse = a Cos, = a Sin at the point = 4 visionto00.weebl.com th-maths-ke.weebl.com

11 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com 6. Evaluate lim (tan ) cos 7. Find the local minimum and maimum values of f()= A farmer has 400 feet of fencing and want to fence of a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? 9. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r 0. The top and bottom margins of a poster are each 6 cms and the side margins are each 4 cms. If the area of the printed material on the poster is fied at 84 cms, find the dimension of the poster with the smallest area.. Find the intervals of concavit and the points of inflection of the function 4 CHAPTER 6 SIX MARKS QUESTIONS.. Find the approimate value of 6. using differentials.. Find dt derivatives. dw if e w where =t and =t b using chain rule for partial. Using chain rule find w 4. Find u and w v if for partial derivatives. dw if w=log( + ) where =e t, =e t dt w sin where =u+v and =u v b using chain rule 5. If u is a homogeneous function of and of degree n, prove that u u u ( n ) visionto00.weebl.com th-maths-ke.weebl.com

12 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com u sin. 6. If u log (tan tan tan z) then prove that CHAPTER 6 TEN MARKS QUESTIONS.. If w uev where u, v log find w & w. (i) Using Euler s Theorem, prove that, tan u, if u u u sin (ii) Using Euler s Theorem, Prove that, u u sin u, If u tan. Verif Euler s Theorem for f (, ) 4. Verif u u for u sin cos Use differentials to find Trace the curves.(i) (ii) (iii) CHAPTER 7 SIX MARKS QUESTIONS.. Evaluate : log(tan )d 0 /. Evaluate : (i) d / 6 cot (ii) / d / 6 tan. Using second fundamental theorem on integrals, evaluate n 4. Evaluate : ( ) d 5. Evaluate: ( 0 ) 0 d 0 sin cos d 0 visionto00.weebl.com th-maths-ke.weebl.com

13 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com 6. Evaluate : e 4 d 0 7. Find the area of the region bounded b the line 5 5=0, =, =4 and -ais. 8. Find the area of the region bounded 4,, and -ais. 9. Find the volume of the solid generated when the region enclosed b =, = and =0 is revolved about -ais. 0. Find the area of the circle with radius a units.. Find the volume of the solid generated when the region enclosed b =+, =, =, =0 is revolved about the -ais.. Find the volume of the solid generated when the region enclosed b a =( a) is revolved about -ais, a>0. CHAPTER 7 TEN MARKS QUESTIONS.. Find the area between the line = + and the curve =. a b. Find the area of the region bounded b the ellipse. Find the area of the loop of the curve a ( a), a 0 4. Find the perimeter of the circle with radius a b using integration. 5. Prove that the curved surface area of a sphere of radius r intercepted between two parallel planes at a distance a and b from the centre of the sphere is r ( b a ) and hence deduce the surface area of the sphere. ( a) 6. Find the length of the curve a 7. Find the volume of a right circular cone with radius r & height h a b. visionto00.weebl.com th-maths-ke.weebl.com

14 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com d d. Solve: CHAPTER 8 SIX MARKS QUESTIONS.. Solve: ( )d + ( )d=0, if it passes through the origin. d d. Solve: sin( ) 4. d d e d if it cuts the -ais. 5. Solve : d d d 6. Find the equation of the curve passing through (,0) and which has slope at (,) 7. Solve: (D +9)=sin 8. Solve : ( D D ). 9. Solve: (D +)=0 when =0, = and when =π/, = 0. Solve: (D D+)=e +5e. Solve: d d tan d d. Solve. cot cos CHAPTER 8 TEN MARKS QUESTIONS.. Solve: (i)(d +7D+)=e (ii) (D ) Cos Sin. Radium disappears at a rate proportional to the amount present. It 5% of the original amount disappears in 50 ears. How much will remain at the end of 00 ears. [Take A 0 as the initial amt]. The rate at which the population of a cit increases at an time is proportional to the population at that time. If there were,0,000 people in visionto00.weebl.com th-maths-ke.weebl.com

15 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com the cit in 960 and,60,000 in 990. What population ma be anticipated in 00. [log ; e.5 ] e 4. The sum of Rs.000 is compounded continuousl at the nominal rate of interest 4% per annum. In how man ears will the amount be twice the original principle? (Log e = 0.69) 5. A cup of coffee at temperature 00 o C is placed in a room whose temperature is 5 o C and it cools to 60 o C in 5 minutes. Find its temperature after a further interval of 5 minutes. d 6. Solve: ( ) d 7. Solve: ( e ) d e ( ) d 0 given that =, where =0. 8. Solve: ( ) d d 0 9. Solve: d+d=e sec d CHAPTER 9 SIX MARKS QUESTIONS.. Show that [(~ p) ( q)] pis a tautolog.. Show that q p & q p are not equivalent.. State and prove cancellation laws on groups. 4. Prepare a truth table for ( p q) (~ q) (ii) (p q) ( (p q)) (iii) ~ ( p q) (~ p) (~ q) (iv) ((~ p) q) ( p (~ p)) is a tautolog. 5. Show that (p q)= (( p) ( q)) 6. Prove that ( Z, ) is an infinite Abelian group. 7. Show that the cube roots of unit forms a finite abelian group under multiplication. Eg.9.4 visionto00.weebl.com th-maths-ke.weebl.com

16 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com 8. Prove that (C,+) is an infinite Abelian group. 9. Show that (R {0}, ) is an infinite abelian group. Here denotes usual multiplication. CHAPTER 9 TEN MARKS QUESTIONS.. Show that the set G of all positive rational forms a group under the composition * defined b ab a b (a, b G). S.T the set of G of all rational numbers ecept forms an Abelian group with respect to the operation * given b a b=a+b+ab, a, b G.. Prove that the set of four function f,f,f,f4 on the set of non-zero comple numbers C {0} defined b (z ) z, f (z) 4, z C {0} z f f (z) z, f (z), z forms and Abelian group with respect to composition of functions. 4. Show that Z [0 ], forms a group Show that the set [ ], [ ],[ 4 ],[ 5 ],[ 9 ] forms an Abelian group under multiplication modulo. 6. Show that the set G of all matrices of the form group under matri multiplication. 7. Show that the set G a b / a, bq respect to addition. 9. Show that the set G { / n Z} is an abelian group under multiplication. visionto00.weebl.com th-maths-ke.weebl.com where R {0} is a is an infinite abelian group with 8. Show that the set M of comple numbers z with the condition z = forms a group with respect to the operation of multiplication of comple numbers. n

17 questions Please visit visionto00.weebl.com and for Maths Ke th-maths-ke.weebl.com CHAPTER 0 SIX MARKS QUESTIONS.. In a continuous distribution the p.d.f. of X is f () ( ), 0 4 Find 0, otherwise mean & variance of the distribution.. In a Poisson distribution if P(X=)=P(X=) find ( X 5) P [ e ]. Find the probabilit mass function, and the cumulative distribution function for getting s when two dice are thrown. 4. A continuous random variable X has p.d.f f() =, 0. Find a and b such that (i) P(X a) = P(X>a) and (ii) P(X>b)= Find the probabilit distribution of the number of 6 in throwing dice once. 6 Find the epected value of the number on a die when thrown. CHAPTER 0 TEN MARKS QUESTIONS.. The number of accidents in a ear involving tai drivers in a cit follows a Poisson distribution with mean equals to. Out of 000 tai drivers, find approimatel the number of drivers with (i) no accident in a ear, (ii) more than accidents in a ear. [e =0.0498]. 7. Obtain K,, of the normal distribution whose probabilit distribution function is given b F() = ke 4-8. The total life time of 5 ear old dog of a certain breed is a Random Variable 0, for 5 whose distribution function is given b F()= 5. Find the, for 5 probabilit that such that a five ear old dog will live (i) beond 0 ears (ii) less than 8 ears (iii) anwhere between to 5 ears. visionto00.weebl.com th-maths-ke.weebl.com