POSSIBLE AREAS OR UNITS OF SELECTION OF QUESTIONS IN II P.U.ANNUAL EXAMINATION- OF KARNATAKA, INDIA ALGEBRA ELEMENTS OF NUMBER THEORY: a)Properties of Divisibility and congruences b)Use of property of congruence to find the unitdigit and remainder. Solving linear congruence, c) finding the number and sum of divisors d)Finding GCD of two numbers and representation of two as a linear combination of l and m and showing l and m are not unique.(ESSAY TYPE) STRESS MORE: on finding GCD , last digit, remainder, number and sum of divisors of number. MATRICES AND DETERMINANTS : a). Solving the simultaneous linear equations by Cramer’s Rule b). Solving the simultaneous equations by matrix method. c). Finding the Inverse, adjoint of a matrix. d). Finding characteristic equation, roots, e).Finding the inverse and verification using Caley –Hamilton Theorem.. f). Properties of Determinants and problems using properties (definite possible question) STRESS MORE : On solving equations by matrix method, cramers rule, finding inverse and determinants using properties. GROUPS: a). Proving a set (given) forms an Abelian group. b). Questions regarding Properties of groups, Theorems&problems on subgroups, STRESS MORE: On proving a given set forms a group under given binary operation. VECTORS: a). Questions on vector product, Cross product, Vector triple product, Scalar triple product. APPLICATON OF VECTORS like sine rule, projection rule, cosine rule , proofs of compound angle formulae, angle in a semicircle is right angle, diagonals of parallelogram bisect each other, medians of a traingle are concureent. STRESS MORE: On application of vectors, problems on vector triple product, cross product. Vector triple product TIPS: CONCENTRATE MORE ON CHAPTERS: MATRICES AND DETERMINANTS. AND VECTORS(MORE ALLOTMENT OF MARKS) ANALYTICAL GEOMETRY: CIRCLES: a). Any derivation on circles. concentrate more on Derivation of equation of tangent, condition of orthoganality, length of a tangent, radical axis is perpendicular to line of centers, condition for the line y=mx+c to be tangent to circle and point of contact. b). Frequently questions on circles is asking on finding the equation of circle by finding g, f&c; and also on orthogonal circles. STRESS MORE: On finding constants g, f and c using given conditions and problems on orthogonal circles CONIC SECTION: a). Any(13) derivation on conic section. (Definite) Concentrate more on Derivation of Parabola, ellipse, Hyperbola, condition for the line y=mx+c to be tangent to parabola, ellipse, Hyperbola, Equation of tangent and normal to parabola, ellipse, Hyperbola at (x 1, y 1 ). b). Finding the properties of standard forms and other forms of conics i.e. finding vertex, focus,directrix,etc c). Finding the conics by using the properties of conics.(Determination of conics) STRESS MORE : On Derivation (total 13) and Finding the properties of conics from the given equations of conic CONCENTRATE MORE ON: CONIC SECTION TRIGNOMETRY: INVERSE TRIGNOMETRIC FUNCTIONS: Problems using the concept of tan - x ! tan -1 y, sin -1 x ! sin -1 y etc Finding the value of x or solve for x . STRESS MORE: On problems using properties of Inverse funtions. GENERAL SOLUTION OF TRIG. EQUATIONS: General solution of problems of a cosx + b sinx =c , and solving trig equations using transformation formulae(product into sum or sum into product), COMPLEX NUMBERS: a).Finding the cube roots and fourth roots of complex numbers and representing them in argand diagram.or finding the continued product of roots . b). Statement and proving Demoivre’s theorem and problems using demoivre’s theorem. CONCENTRATE MORE ON: COMPLEX NUMBERS. CALCULUS: DIFFERENTIATION: a). Finding the derivatives of trigonometric functions, exponential functions, logarithmic functions, Inverse trigonometric functions, Derivatives of sinax, cosax, tanax.secax, cosecax, cotax, sec2x.cos2x, etc. Sin 2 x, cos 2 x, log ax, etc by first principles method. (Definite) b). Problems on second order Differentiation. Page 1 Page 1 http://pucpcmb.wordpress.com c). sub tangent, subnormal, length of the sub tangent and subnormal, Question on Maxima and Minima or Derivative as a rate measure , Angle of Intersection of two curves. STRESS MORE : On finding the derivative from first principles, and using Implicit, Parametric differentiation. Second order derivatives, Derivative as a rate measure, maxima and minima INTEGRATION: c). One question on Problems on Integrals of the form(Particular forms) 1 a+bcosx , 1 a+bsinx , 1 acosx+bsinx+c 1 acos 2 x+bcos 2 x+c 1 .a 2 !x 2 1 . x 2 −a 2 1 x . x 2 !a 2 1 a 2 !x 2 1 x 2 −a 2 .a 2 !x 2 , . x 2 −a 2 , px+q ax 2 +bx+c , px+q .ax 2 +bx+c , pcosx+qsinx acosx+b sinx , e x [f(x)+f'(x)] Integration by substitution and by parts, Integration by partial fractions, b). Evaluating the definite Integral using the properties. c). Finding the area bounded by the two curves or curve and line, finding the area of the circle, ellipse by integration method c). Solving the Differential equation by the method of separation of variables and equation reducible to variable separable form STRESS MORE: units a) and b LIKELY QUESTIONS according to Pattern of th II PUC Question paper (Essay Type/Long answer questions) NOTE: • Here some of the following possible questions on (1, 3 to 6 marks) are given for practice. The pattern and type of the question(as in Part A , Part C, Part D, Part E ) on the basis of the question given below are possible to be asked in the examination. Here likely questions in Part B (Each carries 2M) is excluded. • This likely questions does not imply that same questions will appear for examination. However some definite possible questions similar to the following problems, derivation will be asked . • order and arrangement of the questions given below may be different in examination . BEST OF LUCK. DEPT OF MATHEMATICS PART -A Answer all the ten questions: 1.Question on elements of number theory (LEVEL: Knowledge) Areas likely to be asked : Properties of Divisibility and congruences, unit digit , remainder, finding the number and sum of divisors , 1.If a/b and b/c the prove that a/c 2.Find the number of incongruent solutions of 6x ≡ 9(mod15) 3.If a/b and b/a then prove that a= ! b 4.If a ≡ b(mod m) and b ≡ c(mod m) the prove that a ≡ c(mod m) 5.If a ≡ b(mod m), c/m, c>0 show that a ≡ b(mod m) 6.If (c, a)=1 and c/ab then prove that c/b 7.Find the least positive integer x satisfying 2x+5 ≡ x+4(mod5) 8.Find the least +ve integer a if 73 ≡ a(mod 7) 9. The linear congruence 8x ≡ 23(mod 24) has no solution . Why? 10.Why is “congruence mod m” is an equivalence relation. 11.Find the least non negetive reminder when 76x204 is divided by 7 2. Matrices and Determinants: (LEVEL: Aptitude): Areas: Types of matrices, operation on matrices, symmetric , skew symmetric matrices, Multiplication of matrices, adjoint, characteristic roots etc • If A= | 3−x y−3 0 2 ¦ is a scalar matrix , find x and y • Evaluate ∣ 4996 4997 4998 4999 ∣ • Find x such that | x−1 2 0 3 ¦ has no multiplicative inverse. • Find the value of ∣ 1 3 3 2 3 3 2 3 3 3 2 3 3 3 4∣ • Solve for x in ∣ x x 3 x ∣ =4 Page 2 Page 2 http://pucpcmb.wordpress.com • If A= ¦ 2 3 4 5 ) B= ¦ 6 7 8 9 ) find AB' • Find x such that | 2 4x+5 x+2 1 ¦ is symmetric matrix.. • If ¦ 0 2x−6 3x−4 0 ) is a skew symmetric matrix, find x. • If the matrix | a 2 −3 b c 5 d e f ¦ is skew symmetric matrix , find the value of a+b+c+d +e+f. • Find the characteristic roots of the matrix | 1 4 2 3 ¦ (2M or 1M) • Evaluate ∣ 4996 4997 4998 4999 ∣ using properties of determinants (2M or 1M) 3.Groups:Level: knowledge •On Q -1 * is defined by a*b=a+b+ab find the identity element •Find the inverse of 3 in the group {1,3,7,9} under multiplication modulo 10 • If a∗b =a+b-5 ∀ a,b c I, find the identity. •In a group (G * ), Prove that (a -1 ) -1 =a for all a in G •Define subgroup of a group •Define binary operation •a*b= .ab on the set of all integers . Is * a binary operation? Justify your answer. •Define semi group •In the set of all non negative integers S if a*b=a b prove that * is not a binary operation. •On Q, a*b= ab/4 find the identity •On the set Q 1 the set of all rationals other than 1, * is defined by a*b=a+b-ab for all a, b ∈ Q , find the identity element. 4.VECTORS: Level: Understanding •Find the direction cosines of the vector 2i-j+2k •Find a unit vector in the direction of 2i+3j-k •If ¯a =(2,-1,3), ¯ b =(2,1,-2) find the magnitude of 2¯a+3 ¯ b •Find the magnitude of the vector secθi + tanθj-k •If the vectors (a, b) and (3,2) are parallel , what is the relation connecting a and b? •If ¯a =3i +2j+8k and ¯ b =2i +λj+k are perpendicular, then find the value of λ •If ¯a =3i-j , ¯ b =i+k find ¯a× ¯ b •Find the value of ´ i צ ´ j. ´ k ) •If the direction cosines of ¯a are 2/3, 1/3, and n , find n. •Define coplanar vectors •Find the direction ratios of line joining P(4,3,5) and Q(-2,1,-8) 5.CIRCLES: Level: skill •Find the length of intercept of the circle x 2 +y 2 -2x-7y-8=0 on the x axis. •Find the equation of circle with centre (3,-2) and touching x axis. •If the equation 3x 2 -py 2 +qxy-8x+6y-1=0 represents a circle , find p and q. •Verify whether the point (-2,-3) lies inside, outside or on the circle x 2 +y 2 +4x+6y-7=0. •Show that the circle x 2 +y 2 -4x-2y+1=0 touches y axis. •Find the length of tangent from (1,2) to the cricle x 2 +y 2 -x+2y+1=0 • Find the power of the point (-2,1) w.r.t. the circle x 2 +y 2 -3x +5y -7 = 0 • Find the equation of the tangent to the circle x 2 +y 2 =30 at ( 5, .5 ) on it. 6.Conic section: Level: skill •Find the angle between asymptotes of hyperbola x 2 a 2 − y 2 b 2 =1 •Find the eccentricity of the ellipse x 2 a 2 + y 2 b 2 =1 •Write the equation of the director circle of the hyperbola x 2 a 2 − y 2 b 2 =1 having 10 and 8 as the lengths of axes. •Find the vertex of the parabola 20x 2 +20x-32y+53=0 Page 3 Page 3 http://pucpcmb.wordpress.com •Find the equation of parabola with vertex at (0,0) , x axis as the X axis and passing through the point (2,4) •What is the sum of focal distances(difference of focal distances of ) any point on the ellipse(hyperbola) x 2 25 + y 2 1 =1 ¦ x 2 9 + y 2 4 =1 ) •Find the equation to the parabola given focus (-4,0) and directrix x=4. •Find the focal distance of the point (8,8) from the focus of y 2 =8x 7. Inverse trigonometric function (Level: U): •Show that cos -1 (-x)=π-cos -1 x for x>0. •Show that tan -1 x +cot -1 x =π/2 •Evaluate cos(cosec -1 5/4) •Find the value of cos -1 (cos(-585 0 )) •Find the value of sin -1 ( sin 240) •Find the value of cosec(cot -1 1) •Prove that cos -1 (12/13)= tan -1 (5/12) •Find sec[sin -1 (-1/2)-π/6] 8. Complex number: Level : knowledge ● Express the complex number .3 -i in polar form ● Find the imaginary part of 1 7+4i ● Find the multiplicative inverse of (5+6i) 2 ● Find the complex conjugate of (3-7i) 2 ● Find the modulus of 1+ .3 I ● Express 3+2i 4−3i in the form a+ib. ● Write the modulus of the function 1+cosθ+i sinθ, 0<θ<π ● Find the amplitude of 1+cos n 16 +i sin n 16 9. Differentiation and application of derivatives :Level: skill •Define derivative of y=f(x) at a point x=a. •If y=cosec 3 (1/x) find dy/dx •If x=ct y=c/t find dy/dx •Differentiate x 2 log 2 x w.r.t x •Find the derivative of log(secx+tanx) •Differentiate sin(1/x) w.r.t x •Differentiate 5 x 2 + 6 x +2 w.r.t x •If y=log e 1+sinhx then find dy/dx •Differenatiate e log sinx w.r.t x 10.Integration or definite integrals: Level: knowledge Evaluate ∫ 1 e 1 x dx Evaluate ∫ 1 .3 dx 1+x 2 Evaluate ∫ 1 2 x 2 dx Evaluate ∫ 1 2 ¦ x+1) 5/3 dx Evaluate ∫ 0 1 1−x 1+x dx Evaluate ∫ 2 3 dx x−1 Evaluate ∫ 0 n 3 tanx dx Evaluate ∫ x 3 .x 2 +1dx Evaluate ∫ 1 ∞ 1 x 2 dx Evaluate ∫ −n 2 n 2 sin 3 x dx PART -C ANSWER ANY THREE QUESTIONS: (AS PER PATTERN) 23. QUESTION IS ASKED ON ELEMENTS OF NUMBER THEORY: 1)Find the GCD of 189 and 243 and express it in the form of 189+243y where x and y are integers. Also show that this expression is not unique.(or x and y are not unique) 2) a)If (a, c)=1 and (b,c)=1 Prove that (ab, c)=1 b)If a ≡ b(mod m) and c ≡ d (modm) Prove that ac ≡ bd(modm) 3)a)Find the GCD of 595 and 252 b)Prove that 5 700 ≡ 6(mod 23) 4)Define congruent relation on Z and Prove that it is an equivalence relation . 5)a)Find the least non negative integers when 2 301 is divided by 7? b)If (a, b)=1 and (a, c)=1, Prove that (a, bc)=1 6) a)Find the number of all +ve divisors and the sum of all such +ve divisors of 432 b)Find the remainder when 71x 73 x 75 is divided by 23. Page 4 Page 4 http://pucpcmb.wordpress.com 7)a) Prove that the number of primes is infinite. b)Find the digit in the unit place of 7 123 . 8)a) If (210, 55)=210(5)+55(k) find k 3 b)If (a, b)=1, a/c and b/c prove that ab/c 2 9)a) Prove that smallest positive divisor of a composite number ' a' does not exceed .a 3 b)Find the remainder when 89 x111 x 135 is divided by 11 2 24. ON MATIRCES AND DETERMINANTS: 1)a) Solve the equation 2x+3y=5 and x=2y=3 using matrix method b)Solve for x : ∣ 1+x 2 3 1 x+2 3 1 2 x+3 ∣ =0 2)State cayley Hamilton theorem and verify it for A= ¦ 1 −3 4 −5 ) . Hence find A -1 . 3)Solve by matrix method: 3x+2y-z=4 , x-y+4z=11, 2x+y-z=1. 4)Prove that ∣ bc a a 2 ca b b 2 ab c c 2∣ =(a-b)(b-c)(c- a) 5)a)Prove that ∣ b+c c+a a+b c+a a+b b+c a+b b+c c+a ∣ =2 ∣ a b c b c a c a b ∣ b)Find x, y and z if | x 2 −3 5 y 2 1 −1 z ¦ | 3 −1 2 4 2 5 2 0 3 ¦ = | 5 3 3 19 −5 16 1 −3 0 ¦ 6)Prove that ∣ x x 2 y+z y y 2 z+x z z 2 x+y ∣ =(x-y)(y-z)(z-x)(x+y+z) 7)Find the inverse of the matrix : | −3 5 −1 4 −1 2 0 8 −2 ¦ 8)Prove that ∣ a 2 +1 ab ac ab b 2 +1 bc ac bc c 2 +1 ∣ =1+a 2 +b 2 +c 2 9)Solve by Cramer's rule : x-2y=0, 2x-y+z=4, 3x+y-2z=-3 10)If a+b+c=0 then solve ∣ a−x c b c b−x a b a c−x ∣ =0 11)Prove that ∣ a 2 bc ac+c 2 a 2 +ab b 2 ac ab b 2 +bc c 2 ∣ =4a 2 b 2 c 2 12)Show that ∣ −bc b 2 +bc c 2 +bc a 2 +ac −ac c 2 +ac a 2 +ab b 2 +ab −ab ∣ =(ab+bc+ca) 3 13) a) Show that ∣ x y y y x y y y x ∣ =(x+2y) (x- y ) 2 b)If the matrix | 2 x 3 4 1 6 −1 2 7 ¦ has no inverse, find x 14)State caley-Hamilton theorm and find A 3 if A= | 1 2 2 1 ¦ using Caley-Hamilton theorem 15)State Cayley-Hamilton theorem and find the inverse of the matrix | 1 3 2 −1 ¦ using the theorem. 16) a)Find the inverse of | a 0 0 0 b 0 0 0 c ¦ where a, b,c are real numbers b)If matrix A=[a ij ] = i+2j, i=1,2,3 and j=1,2 . Find the matrix A see other standard problems using properties of determinants. Page 5 Page 5 http://pucpcmb.wordpress.com 1. ON GROUPS : 1)a)Prove that the set of fourth roots of unity is an abelian group under multiplication. b)Prove that a group of order 3 is abelian. 2)Prove that set of all matrices of the form | x x x x ¦ where xεR and x#0 forms an abelian group w.r.t multiplication of matrices. 3)Prove that the set of all +ve rational numbers forms an abelian group w.r.t multiplication * defined by a*b=ab/6 and hence solve x*3 -1 =2. 4)If Q + is the set of all positive rational numbers, Prove (Q + , *) is an abelain group where * is defined by i) a*b = 2ab 3 ii) a*b = ab 2 iii) a*b = ab 5 a, b ЄQ + (each carries 5 M) 5)If Q 1 is the set of rationall numbers other than 1 with binary operation * defined by a*b=a+b-ab for all a,b εQ 1 , Show that (Q 1 ,*) is an abelian group and solve 5*x=3 in Q 1 . 6)If Q -1 is the set of all rational numbers except -1 and * is a binary operation defined on Q -1 by a*b= a+b+ab, Prove that Q -1 is an abelian group. 7)Define a subgroup and Prove that a non empty subset H of a group (G ) is a subgroup of (G ) if and only if ∀ a, b ∈H , a*b -1 ∈ H. 8)Define an abelina group and prove that the set of all integral powers of 3 is a multiplicative group. 9)Prove that the set of all complex numbers whose modulie are unity is a commutative group under multiplication. 10) a)Show that set of even integers is a subgroup of the additive group of integers. b)Prove that the identity element is unique in a group. 11)Show that the set {1,5,7,11} is an abelian group under x mod 12 and hence solve 5 -1 x 12 x=7 12)Show that set of all matrices of the form A α = | cos o −sino sino coso ¦ where α is a real number forms a group under matrix multiplication. 13)a)In a group (G, ) if a*b=a*c prove that b=c and if a*b=c*b prove that a=c (Cancellation laws) b)If H is a subgroup of G then show that identity element of H is the same as that G. 14)Prove that a nonempty subset H of a group G is a subgroup of G iff closure and inverse law are true and hence show that a set of even integers is a subgroup of additive group of integers. 15)Prove that set of all matrices of the form | x y −y x ¦ x#0, y#0 , xε R is a group under matrix multiplication. 2. ON VECTORS : 1)a)If ¯a =i+j-k, ¯ b =i-3j+k, ¯c =3i-4j+2k find ¦¯a× ¯ b)ׯc b)Simplify : (2 ¯a +3 ¯ b ) x (3 ¯a -2 ¯ b ) 2)a)The position vectors of A, B C respectively are i-j+k, 2i+j-k and 3i-2j-k. Find are of traingle ABC b)If ¯a =i+j+2k and ¯ b =3i+j-k, find the cosine of the angle between ¯a and ¯ b 3)a) Prove that |¯a− ¯ b , ¯ b−¯c ¯c−¯a¦ =0 b)The position vectors of the points A, B, C and D are 3i-2j-k, 2i+3j-4k, -i+j+2k and 4i+5j+k. If the four points lie on a plane, find λ. 4)a) In a regular hexagon ABCDEF, show that ¯ AD+ ¯ EB+ ¯ FC=4 ¯ AB b)Show that 2¯a=i צ¯a×i )+j צ¯a×j )+kצ¯a×k ) 5)a) Prove that | ¯a× ¯ b , ¯ bׯc ¯cׯa¦=| ¯a ¯ b¯c¦ 2 b)Find the sine of the angle between the vectors i-2j+3k and 2i+j+k=0 6)a)Find a unit vector perpendicular to each of the vectors 4i+3j+2k and i-j+3k b)Prove by vector method:In traingle ABC a= bcosC + CcosB (3M) 7) a) If ¯a i+j+k ¯ b =i+2j+3k and ¯c =2i+j+4k, find the unit vector in Page 6 Page 6 the direction of ¯aצ ¯ bׯc) b)If cosα, cosβ and cosγ are the direction ratios of the vector 2i+j-2k, Show that cos 2 α +cos 2 β+cos 2 γ=1. 8)a)Prove that |¯a+ ¯ b , ¯ b+¯c , ¯c+¯a¦=2|¯a ¯ b¯c¦ b)Find the projection of ¯a =i+2j+3k on ¯ b =2i+j+2k 9)a) Prove by vector method , In any traingle ABC a sin A = b sin B = c sin C b)For any three vectors a, b, c Prove that ¯aצ ¯ bׯc)+ ¯ bצ¯cׯa)+¯cצ¯a× ¯ b) =0 10) a)Prove cosine rule by vector method: a 2 =b 2 +c 2 +2bc cos A b)If ABC is an equilateral triangle of side a then prove that ¯ AB. ¯ BC+ ¯ BC . ¯ CA+ ¯ CA. ¯ AB= −3 2 a 2 11)a)Prove that sin(A+B) =sinAcosB +cosA.sinB by vector method b)If | ¯a+ ¯ b |=5 and ¯a is perpendicular to ¯ b . Find | ¯a− ¯ b | 12)a)Prove that | ¯a+ ¯ b ¯ b+¯c ¯c+¯a¦ =2 | ¯a ¯ b ¯c¦ b)Find the projection of ¯a =i+2j+3k on ¯ b =2i+j+2k 13. a) If ¯a=a 1 i+a 2 j+a 3 k show that ´ i צ¯a× ´ i )+ ´ j צ¯a× ´ j )+ ´ kצ¯a× ´ k)=2¯a b) Find the area of parallelogram whose diagonals are λi+2j-k and i-3j+2k respectively. Additional Questions: 14. In a triangle prove projection formula a=bcosC+ c cosB using vector method (3M) 15. Given the vectors ¯a =2i-j+k, ¯ b =i+2j-k ¯c =i+3j-2k . Find a vector perpendicular to ¯a and coplanar with ¯ b & ¯c (3M) II. Answer any two questions: 3. ON CIRLCES a)Questions carrying 3 Marks 1)Define orhoganality of two circles. Find the condition for the circles x 2 +y 2 +2g 1 x+2f 1 y+c 1 =0 and x 2 +y 2 +2g 2 x +2f 2 y +c 2 =0 to cut orthoganally. 2)Find the condition that the line y=mx+c may be tangent to the circle x 2 +y 2 =a 2 . Also find the point of contact 3)Derive the equation to the tangent to the circle x 2 +y 2 +2gx+2fy+c=0 at point (x 1 ,y 1 ) on it. 4)Define Radical axis, Show that radical axis is perpendicular to line joining the centres of two circles 5)Find the equation of the circle with centre on 2x+3y=7 and cutting orthogonally circles x 2 +y 2 -10x-4y+21=0 and x 2 +y 2 -4x-6y+11=0 6)Find the equation of the circle which passes through the point (2,3), cuts orthogonally the circle x 2 +y 2 -4x+2y-3=0 and length of the tangent to it from the point (1,0) is 2. 7)Find the equation of the circle passing through the points (5,3), (1,5) and (3,-1). 8)Show that general second degree equation in x and y x 2 +y 2 +2gx+2fy+c=0 always represnts a circle .Find its centre and radius. 9)Find the equation of the two circles which touch both co-ordinate axes and pass thorugh the point (2,1). 10)Define Power of the point w.r.t circle . Find the length of the tangent from an external point (x 1 , y 1 ) to the circle x 2 +y 2 +2gx+2fy+x=0. 11)Find the equation of the circle which passes through the point (2,3), cuts orthoganally the circle x 2 +y 2 -2x-4y-5=0 Page 7 Page 7 Questions carrying 5 Marks 12)Find the equation of the circle such that the lengths of tangents from the points (-1,0) , (0,2) and (2, -1) are 3, .10 and 3 .3 13)Find the equation of the circle, cutting the three circles x 2 +y 2 +4x+2y+1=0, 2x 2 +2y 2 +8x+6y-3=0 and x 2 +y 2 +6x-2y-3=0 orthogonally. 14)Find the equation of the cirlcle passing through (2,3) having the length of the tangent from (1,0) as 2 units and cutting x 2 +y 2 -4x+2y-3=0 orthogonally. b) Questions carrying 2 Marks 1)Show that the line 7x-24y-35=0 touches the cirlce x 2 +y 2 -2x-6y-6=0 2)Find the equation of the circle passing through the origin (4,0) and (0, -5) 3)Prove that the length of the tangent form any point on the cirlce x 2 +y 2 +2gx+2fy+c=0 to the circle x 2 +y 2 +2gx+2fy+d=0 is .d −c 4)A and B are points (6,0) and (0,8), Find the equation of the tangent at origin O the circum cirlce of traingle OAB. 5)Show that the cirlces x 2 +y 2 -4x-10y+25=0 and x 2 +y 2 +2x-2y-7=0 touch each other. 6)Find the equation of the cirlce two of whose diameters are x+y=6 and x+2y=4 and whose radius is 10 units. Questions (sample): 1) a) Fin the equation of the circle through origin and having portion of the line x+3y=6 intercepted between the co-ordinate axes as diameter 3 b)Find the points on the circle x 2 +y 2 =25 at which tangents are parallel to the x axis. 2)Prove that general second degree equation in x and y x 2 + y 2 + 2g x +2fy+c=0 always represents a circle . what is the equation of the circle if the centre of the circle lies on x axis. 3)a)Show that the four points (1,1) (-2,2), (-2,-8) and (-6,0) are concyclic 3 b)If x 2 +y 2 -2x+3y+k=0 and x 2 +y 2 +8x-6y-7=0 are the equations of the circles intersecting orthogonally find k. 4)Find the equation to the circle which passes throug the points (0,5) , (6,1) and has its centre on the line 12x+5y=25 5) a)Define orthogonal circles, Derive the condition for the two circles x 2 +y 2 +2g 1 x +2f 1 y +c 1 =0 and x 2 +y 2 +2g 2 x +2f 2 y+c 2 =0 to cut orthoganaly. b)Find the equation of the circle passing through the origin , (4,0)&(0,-5) 6)Find the equation of the circle passing through (-6,0) and having length of the tangent from (1,1) as √5 units and cutting orthoganally the circle x 2 +y 2 -4x-6y-3=0; 4. a)QUESTION ON CONIC SECTION: ( Questions carrying 3 M) 1)Find the centre and foci of the hyperbola 9x 2 -4y 2 +18x-8y-31=0 2)Find the equation of parabola having vertex (3,5) and focus (3,2) 3)Find the centre and foci of the ellipse 3x 2 +y 2 -6x-2y-5=0 4)If the normal at one end of latus rectum of the ellipse x 2 a 2 + y 2 b 2 =1 passes through other end of the minor axis the prove that e 4 +e 2 =1, where 'e ' is the eccentricity of the ellipse. 5)Find the eccentricity and equations to directrices of the ellipse 4x 2 +9y 2 -8x+36y+4=0 6)Find the equaton of tangent and normal to a)Parabola b)Ellipse c)Hyperbola at (x 1 , y 1 ) [Each carrying 3 marks]. 7)Find the equation of tangent and normal to a)parabola b)Ellipse c)Hyperbola at t (Each carrying 3M) Page 8 Page 8 8)Find the condition for the line y=mx+c to be tangent to the a)paraboal b)Ellipse c)Hyperbola in standard form.(Each carrying 3Marks) b) Questions carrying 2M each 1)Find the equations of the asymptotes of hyperbola 9x 2 -4y 2 =36. Also find the angle between them. 2)Find the equation of the tangent at any point (t) on the hyperbola y 2 =4ax. 3)Find the equation of the parabola whose vertex is (-2,3) and focus is (1,3) 4)Find the equation of tangent to the ellipse x 2 12 + y 2 6 =1 at (-2,2). 5)Find the equation to the parabola with vertex (-3,1) and directrix y=6. Questions carrying 5 Marks. 1)Find the condition for the line y=mx+c to be tangent to a)parabola b)Ellipse c)Hyperbola in standard form also find the point of contact. (Each carrying 5 Marks) 2)Find the equation of the parabola whose vertices's is on the line y=x and axis parallel to x axis and passing through (6,-2) and (3,4). 3)Find the equation of hyperbola in the standard form given that the distance between the foci is 8, and the distance between the directrices is 9/2. Also find its eccentriciy. 5. a)QUESTIONS ON INVERSE TRIGNOMETRIC FUNCTION : 1)If cos -1 x +cos -1 y +cos -1 z=π, then prove that x 2 +y 2 +z 2 +2xyz=1(other problems of this type) 2)Show that tan −1 3 4 +tan −1 3 5 +tan −1 8 19 = n 4 3)Solve for x: cos -1 x -sin -1 x =cos -1 x .3 4)Solve for x: sin -1 x +sin -1 2x = 2n 3 4)Find sin(cos -1 1/3 - sin -1 2/3) 5)If sin −1 ¦ 2x 1+x 2 ) +cos −1 ¦ 1−y 2 1+y 2 ) +tan −1 ¦ 2z 1−z 2 ) =2n Prove that x+y+z=xyz b)QUESTION ON GENERAL SOLUTION OF TRIGNOMETRIC EQUATIONS 1)Find the General solution of sin 2 θ = 2−.3 4 2)Find the General solution of tan5x tan2x=1 3)Find the general solution of tan 2 x -3secx+3=0 4)Find the General solution of .3 sinx +cosx = .2 5)Find the General solution of 2(sin 4 x+cos 4 x)=1. 6)Find the General solution tan5x tan2x=1 7) Find the general solution of cos 2x +cos3x=0 CALCULUS ANSWER ANY THREE QUESTIONS: 6. a)Differentiate i)trigonometric functions like sinx, cos x , tanx, cosec x, sec x, tanx, cotx, sin ax, cosax etc ii)e x , logx, , a x , x n , etc iii)sin 2 x, cos 2 x etc iv)sin -1 x, cos -1 x, tan -1 x etc, by first principles.(Each carries 3M) b)Some standard and Previous years Problems on Implicit, Parametric, Logarithmic, Derivatives of one function w.r.t antohter, Derivatives of inverse trignometric functions by substitution, Standard problems using chain rule. 7. a)Some standard problems on successive differentiation.(3M each) 1)If y=log(x+ . x 2 −1 ), Prove that (x 2 -1)y 1 +xy 1 =0 2)If y= e mcos −1 x Prove that (1-x 2 )y 2 -xy 1 -m 2 y=0 3)If y=sin(m sin -1 x) Prove that (1-x 2 )y 2 -xy 1 +m 2 y=0 4)If y=acos(logx)+bsin(logx), Prove that x 2 y 2 +xy 1 +y=0. 5)If y=cos(atan -1 x) show that (1+x 2 )y 2 +2x(1+x 2 )y 1 +a 2 y=0. Page 9 Page 9 b) Standard problems on Differentiation or Application of Differentiation(2M each) 1)P is a point on the line AB=8cms. Find the position of P such that AP 2 +BP 2 is minimum. 2)Find the minimum value of xe x 3)Find the angle between the curves x 2 +y 2 +3x-8=0 and x 2 +y 2 =5 at (1,2) 4)Find the equation to tangnet to the curve y=6x-x 2 where the slope of the tangent is -4. 5)Show that sinx(1+cosx) is maximum when x= n 3 8. a)Some standard problems on Application of Differentiation or Integration(3M each) 1). Show that the curves 2y=x 3 +5x and y=x 2 +2x+1=0 touch each other at (1,3). Find the equation to common tangnet. 2)Evaluate ∫ x 2 +1 x 4 +1 dx 3)Evaluate ∫ dx 5+4sinx dx 4)Evaluate ∫ x+1 x 2 +4x+5 dx 5)Prove that ∫ dx ¦ a 2 +x 2 ) 3/ 2 = x a 2 .a 2 +x 2 +C 6)Evaluate ∫ dx 5+4cosx b)Some standard problems on application of Differentiation.(2M each) 1). If the displacement 's' at time't' is given by s= .1−t , Show that the velocity is inversly proportional to displacement. 2).Find the range in which the funtion x 2 -6x+3 is a)increasing b)decreasing 3)If the displacement s metres of a particle at time 't' seconds is given by s=2t 3 -5t 2 +4t-3, then find the initial velocity. 4)When the breakes are applied to a moving car , the car travels a distance of 's' metres in time 't' seconds given by s=20t-40t 2 . When and where does the car stops. 5)If the law of motion is s 2 =at 2 +2bt+c then show that acceleration is a−v 2 s where v is velocity. 9. a)QUESTIONS CARRYING 3M EACH: Evaluating the integrals of particular types 1)Evaluate ∫ −a a . a−x a+x dx 2)Evaluate ∫ 2xinx+cosx 3sinx−2cosx dx 3)Differentiate tan -1 ¦ .1+x 2 −1 x ) w.r.t tan -1 ¦ 2x 1+x 2 ) 4)Differentiate tan -1 ¦ 1−x 1+x ) w.r.t sin -1 ¦ 2x 1+x 2 ) 5)If .1−x 2 +.1−y 2 =a¦ x−y) Prove that dy dx = .1−y 2 .1−x 2 6)If y= x 2 . x 2 −a 2 − a 2 2 cosh −1 ¦ x a ) then prove that dy dx = . x 2 −a 2 b)QUESTIONS CARRYING 2M EACH: 1)Evaluate ∫ 4 x 3 . x 2 dx 2)Differentiate (sin -1 x) x w.r.t x 3)Evaluate ∫ 1 .5+4x−4x 2 dx 4)Integrate sinx 13−9sin 2 x w.r.t x 5)Integrate x 2 4x 6 +1 w.r.t x 6)Integrate 1 .4x 2 −4x+2 w.r.t x Page 10 Page 10 10. Question is exclusively asked in AREA UNDER A CURVE : (5M EACH) 1)Find the area bounded by parabola y=11x-24-x 2 and the line y=x. 2)Find the area enclosed between the parabola y 2 =4ax and x 2 =4ay 3)Find the area of the ellipse x 2 a 2 + y 2 b 2 by integration. 4)Find the area enclosed between the parabola y 2 =4x and the line y=2x-4. 5)Find the area of the circle x 2 +y 2 =a 2 using integration. 6)Find the area between the curves x 2 =y and y=x+2 7)Find the area enclosed between the parabola y 2 =4ax and x 2 =4by PART D ANSWER ANY TWO: (Includes Question Numbers 35, 36, 37, 38. Problems asked on topics a)Conic section (6M) b)Complex numbers (6M) c)Application of Differentiation or Integration (6M) d)Vectors (6m or 4m) e)Matrices and Determinants(4M) f)Differential equation(4M) g)General solution of trigonometric equation (4m) h)Matrices and Determinants. 11. a)Total 6Marks Question ONE OF THE QUESTION IS asked on CONIC SECTION (All Derivations of Parabola , Ellipse, Hyperbola and others) 1)Define Ellipse and Derive standard equation to the ellipse x 2 a 2 + y 2 b 2 =1 2)Define Hyperbola as a locus of a point and Derive the equation of the same in the standrad form x 2 a 2 − y 2 b 2 =1 3)Find the condition for the line y=mx+c to be tangent to a)parabola b)Ellipse c)Hyperbola in standard form also find the point of contact. Hence deduce the condition for the line x cosα+ysinα=p to be a tangent to the a) ellipse b)Hyperbola (Each carrying 6 Marks) 4)Define a parabola and obtain its equation in the standard form 5) a)Obtain the equation of asymptotes of the hyperbola x 2 a 2 − y 2 b 2 =1 b)Prove that the locus of point of intersection of perpendicular tangnets to the parabola y 2 =4ax is the directrix x+a=0 (Each carrying 3M) 6)Show that locus of point of intersection of perpendicular tangents to a)Ellipse b)Hyperbola is the director circle a)x 2 +y 2 =a 2 +b 2 b)x 2 +y 2 =a 2 -b 2 7)Define asymptotes of Hyperbola. Find the equations of the asymptotes of the Hyperbola x 2 a 2 − y 2 b 2 =1 . What is meant by rectangular hyperbola. 8)Show that an equation 9x 2 +5y 2 -36x-50y-164=0 represents an ellipse, find its centre, eccentricity, length of latus rectum and foci. 9)Derive the equation of the hyperbola in the standard form x 2 a 2 − y 2 b 2 =1 , Write equation to the locus of the point of intersection of perpendicular tangents to the hyperbola x 2 a 2 − y 2 b 2 =1 and write the name of the equation.. 12. ON COMPLEX NUMBERS: (6M each) 1)If cos o +cos ß +cos y =0=sin o +sin ß +sin y , Prove that Page 11 Page 11 i) cos 3o +cos 3ß + cos 3y =3 cos( o + ß + y ) sin 3o + sin 3ß + sin 3y =3 sin( o + ß + y ) ii) cos 2 o +cos 2 ß +cos 2 y =sin 2 o +sin 2 ß +sin 2 y = 3 2 2)Find the fourth roots of the complex number -1+ .3 i and represent them in the Argand diagram. Also find the continued product of the roots. 3)State and Prove Demoivres theorem 4)Find all the fourth roots of the complex number (i- .3 ) 3 , represent them on an Argand plane. Also, find their continued product. 5)i) If z 1 and z 2 are any two non zero complex numbers, Prove that |z 1 z 2 |=|z 1 ||z 2 | and arg(z 1 z 2 )=arg z 1 +arg z 2 ii)Prove that 1+sin 0+icos 0 1−sin 0−icos 0 =i(tanθ+secθ) 6) If z 1 and z 2 are any two non zero complex numbers, Prove that |z 1 z 2 |=|z 1 ||z 2 | and arg(z 1 z 2 )=arg z 1 +arg z 2 and arg ¦ z 1 z 2 ) =argz 1 -argz 2 . Use these results to find that modulus and amplitude of ¦1+i )¦1−.3i ) 1−i . 7)Find all values of ¦ . 3+i) 1 3 and represent then on an Argand diagram, also find their continued product. 8)Show that the continued product of four values of ¦ 1 2 +i . 3 2 ) 3/4 is 1 and reprsent all values in the argand diagram 13. ON CALCULUS: ON DIFFERENTIATION OR APPLICATION OF DERIVATIVES OR INTEGRATION. (6M each)(Problems related to Derivative as a rate measure , angle of intersection between two curves, problems on maxima and minima) 1)Water is being poured into a right circular cone of base radius 15 cms and height 40 cms at the rate of 12π cc per minute. Find the rates at which the depth of water and radius of the water cone increase when the depth of water is 16 cms. 2)A metal cube expands on heating such that its side is increasing uniformly at 2 mm/sec. Find the rate at which its i)volume ii)Surface area and iii)diagonals are increasing when the side is 10 mm 3)A man 160 cm tall, is walking away from a source of light, which is 480 cm above the ground at 5 kms /hr. Find the rate at which i) his shadow lengthens b)tip of the shadow moves 4)Prove that the greates size rectangle than can be inscribed in a circle of radius 'a' is a square. 5)Prove that the rectangle of least perimeter for a given area is a square. 6)An inverted circular cone has depth 12 cms and base radius 9 cms. Water is poured into it at the rate of 1 1 2 cc/sec. Find the rate of rise of water level and the rate of increase of the surface area when the depth of water is 4 cm. 7)Define subtangent and subnormal to a curve and Prove that in the curve x m+n =a m-n y 2n , the power of subtangent varies as the nth power of subnormal. 8)Show that the curve x 2 A + y 2 B =1 and x 2 a 2 + y 2 b 2 =1 cut each other orhoganally if Page 12 Page 12 A-B =a-b 9)Prove that the portion of tangent to the curve x 2/3 +y 2/3 =a 2/3 intercepted between the coordinate axes is of constant length. 10)Show that a rectangle of maximum area that can be inscribed in a circle is a square. 11)Show that right circular cone of greatest volume which can be inscribed in a given sphere is such that three times its height is twice the diameter of the sphere. 12)A man 6 feet in height moves away at a uniform rate of 4m.p.h. From a source of light which is 20 feet above the ground. Find the rate at which the shadow lengthens and the rate at which the tip of his shadow is moving. 14. ON INTEGRATION: ON DEFINITE INTGRALS: (6M each) 1)Prove that ∫ 0 1 log¦1+x) 1+x 2 dx= n 8 log2 2)Prove that ∫ 0 2a f ¦ x) dx= ∫ 0 a f ¦ x)dx+ ∫ 0 2a f ¦ 2a−x) dx and evaluate ∫ 0 n xsinx 1+sinx dx 3)Prove that ∫ a b f ¦ x) dx= ∫ a b f ¦a+b−x) dx and hence evaluate ∫ 1 3 .4−x .4−x+. x dx 4)Evaluate ∫ 0 n x dx a 2 cos 2 x+b 2 sin 2 x 5)Prove that ∫ a b f ¦ x) dx= ∫ 0 a f ¦t ) dt and evaluate ∫ 0 1 log¦1+x) 1+x 2 dx 6)Prove that ∫ 0 n 2 log ¦sinx )dx = n 2 log(1/2) 7)Prove that ∫ −a a f ¦ x) dx=2 ∫ 0 a f ¦ x)dx if f(x) is even function =0 if f(x) is odd function . Hence evaluate ∫ −99 99 ¦ x 2 +3x 3 −7x)dx 8)Prove that ∫ 0 a f ¦ x) dx= ∫ 0 a f ¦a−x) dx and hence show that ∫ 0 n 4 log ¦1+tanx) dx 9)Evaluate ∫ 0 n x tanx sec x+tan x dx 10)Show that ∫ 0 n 2 log sinx dx = − n 2 log2 11)Prove that ∫ 0 a f ¦ x) dx = ∫ 0 a f ¦ a−x) dx and hence evaluate ∫ 0 n 2 . sinx . sinx+ . cosx dx In the above questions section (b) contains 4 marks questions on following topics merged with any questions given above. 1. Matrices and Determinants: Calyey Hamilton theorem, verifying cayley Hamilton theorem, finding Inverse using caley Hamilton theorem. Problems using properties of Determinants. 2. Vectors: Application of vectors, Scalar triple product, Vector triple product, Vector product. 3.General solution of Trignometric equation: 4.Differential equations: Finding the perticular solution, variable seperable forms, Reducing to variable seperable form. Page 13 Page 13 The question paper pattern in second PUC mathematics has changed from 2007-2008 for 100 marks instead of earlier 90 marks. Department has given some specific topics for giving questions in the last part of question paper i.e. PART E , Here is some likely questions on specific units for practice which may be asked in PART E . BEST OF LUCK. PART E: i)Questions in Part E , should be selected from the following topics, which are included in assignments/projects, confined to II PU syllabus . ii)There will two questions of 10 marks each. Each question will have three sub divisions. The first and second questions carry 4 marks each and the third question carries 2 marks. iii)Students will have to answer only one of the two questions ALGEBRA: (a) Problems on scalar product of type i)Show that ∣¯a+ ¯ b∣=∣¯a− ¯ b∣ U ¯a is perpendicular to ¯ b ii) Given ¯a+ ¯ b+¯c=0 | ¯a | | ¯ b | | ¯c | to find the angle between any two vectors etc Practice questions: 1. If ¯a , ¯ b and ¯c be three vectors such that ¯a+ ¯ b+¯c=0 and and ∣¯a∣ =3, | ¯ b |=5 and | ¯c |=7. find the angle between ¯a , and ¯ b . (4M) (Ans:60 0 ) 2.If ∣¯a+ ¯ b∣=∣¯a− ¯ b∣ show that ¯a and ¯ b are orthogonal.(4M) 3.If ¯a , ¯ b are two vectors such that ¯a =2, ¯ b =3 and ¯a . ¯ b =4 find ∣¯a− ¯ b∣ (2M) 4.Let ¯a , ¯ b and ¯c be three vectors such that ¯ a. ¦ ¯ b+¯c)+ ¯ b. ¦¯c+¯a)+¯ c.¦¯a+ ¯ b)=0 and ∣¯a∣ =1, | ¯ b| =4, | ¯c | =8 then find ¦¯a+ ¯ b+¯c) 2 (2M) 5. If ¯a , ¯ b and ¯c be three vectors such that ¯a+ ¯ b+¯c=0 , find the value of ¯a . ¯ b+ ¯ b.¯c+¯ c. ¯a (2M) 6. If | ¯a| is a unit vector and (x- ¯a ).(x+ ¯a )=8 , then find | ¯x |. (4M) 7. If ¯a , ¯ b and ¯a+ ¯ b are unit vectors find ∣¯a− ¯ b∣ (2M) 8. If ¯a , ¯ b and ¯c be three vectors whose magnitudes are respectively 3,4,5 and ¯a , is perpendicular to ¯ b+¯c , ¯ b is perpendicular to ¯c+¯a and ¯c is perpendicular to ¯a+ ¯ b the show that | ¯a+ ¯ b+¯c |= 5.2 9. If ¯a , and ¯ b are vectors of equal magnitudes prove that ¯a+ ¯ b is orthogonal to ¯a− ¯ b (2M) 10. For any two vectors ¯a , ¯ b prove that a) ∣¯a+ ¯ b∣ 2 = | ¯a| 2 + | ¯ b| 2 +2. ¯a . ¯ b b) ∣¯a− ¯ b∣ 2 = | ¯a| 2 + | ¯ b| 2 -2. ¯a . ¯ b c) ∣¯a+ ¯ b∣ 2 + ∣¯a− ¯ b∣ 2 =2( | ¯a| 2 + | ¯ b| 2 ) d) ∣¯a+ ¯ b∣ 2 - ∣¯a− ¯ b∣ 2 =4. ¯a . ¯ b (Each carries 2M) PUC PCMB 11. Prove by vector method cos(A+B) =cosA cosB – sin A sin B (4M) (b) To find the least +ve remainder, and the digit in the unit place of a given number using congruence, and to find the incongruent solutions of a linear congruence. 1.Find the digit in the unit place of a) 2 37 b)7 53 (2M each) 2.solve the following congruences: a)2x ≡ 3(mod 5) b)4x ≡ 7(mod 12) 3.Find the incongruent solution of 3x ≡ 9(mod 6) (3 incongruent solutions x=3,5,7) 4.Find the sum of all positive divisors of 360 5.Find all incongruent solutions of 2x ≡ 4(mod10) (2M) 6.Find the least non negetive remainder when 2 50 is divided by 7.(2M) 7.Find the reminder when 768x217x87 is divided by 11 (2M) 8.Prove that 3 10 ≡ 1(mod 31). 9.Find the reminder in the following divisions: a)3 50 by 7 b)5 225 by 3 c)2 125 by 11 d)2 575 by 13 e)2 100 by 19 (Each question carries 2M) 10. Solve 4x-3 ≡ -2x+6(mod 11) Page 14 Page 14 ANALYTICAL GEOMETRY: a) To find the length of the common chord of two intersectiong circles Method: Find the RA, Find centre(C1) and radius(r) of one of the either circles; Find the length of the perpendicular(p) from centre of one of the circle to RA, then find AM using pythogorean formula and length of chord = 2AM PRACTICE QUESTIONS: 1.Find the length of the common chord of intersecting circles (x-h) 2 +(y-k) 2 =a 2 and (x-k) 2 +(y-h) 2 =a 2 ans: length= .4a 2 −2¦ h−k) 2 2.Find the length of the common chord of two intersecting circles x 2 +y 2 +2gx+2fy+c=0 and x 2 +y 2 +2fy+2gy +c=0 length= .2¦ g+ f ) 2 −4c 3.Find the length of common chord of intersecting circles x 2 +y 2 -4x-5=0 and x 2 +y 2 -2x+8y+9=0 4.Find the length of the chord of the circle x 2 +y 2 -x+3y-10=0 intercepted by the line x+y+2=0. Ans: 4 .3 5.Find the length of the chord of the circle x 2 +y 2 -4x-2y-20=0 which is bisected at (2,3). Ans: 2.21 (EACH ABOVE QUESTIONS CARRIES 4M) 6.Find the length of the chord of the circle x 2 +y 2 -6x-15y-16=0 intercepted by x axis ,Ans :10(2M) 7.Find the length of the chord of the cirlc x 2 +y 2 -4x-8y+12=0 intercepted by the y axis . Ans: 4(2M) TRIGNOMETRY: a)To find the cube roots of a complex number and their representation in argand plane and to find their continued product. PRACTICE QUESTIONS: (EACH CARRIES 4M): .3 2.Find continued product of cube roots of 1+i .3 3.Find the fourth roots of complex number 1- .3 I and represent them in the argand diagram. 4.Find the fourth rootsof 3-3i b)Problems related to the cube roots of unity 1, using the properties of . 1.If 1, w, w 2 are the cube roots of unity then a)show that (1+w-2w 2 )(1+w 2 -2w)(w+w 2 -2)+27=0 (2M) b)Show that (1+ω+5 ω 2 )(1+5 ω+ ω 2 )(5+ ω+ ω 2 )=64 c)(2- ω)(2- ω 2 )(2- ω 10 )(2- ω 11 )=49 d)(a+b)(a+b ω)(a+b ω 2 )=a 3 +b 3 e)(1+ ω)(1+ ω 2 )(1+ ω 4 )(1+ ω 8 )........to 100 factors =1 (Each carries 2M) CALCULUS: (a)Finding the derivative of functions of the following type only Log a f(x), ii)sin(3x) 0 , tan(x/2) 0 etc(Here degree must be converted into radians) PRACTICE QUESTIONS: Find dy/dx if a) y=log .sinx b)y=log e 1+sinx c)y=log tanx when x is measured in degrees. d)y=log a−x a+x e)y=log 10 (logx) f)y=log x e g)y=cos(log(sinx)) f) y=sin(2x) 0 h) y=sin 3 5x 0 (Each carries 2M) (b)Applications of derivative in finding the maxima and minima of functions involving two dimensions only. Practice questions: (Each question carries 4M) 1.The perimeter of a rectangle is 100 meteres . Find the sides when the area is maximum. (4M)(Ans: 25,25) 2.Prove that among all the right angled traingles of given hypotenus, the isosceles traingle has the maximum area. 3.A wire of length 8cms is cut into two pieces. One piece is bent into the form of a square and the other piece is bent into the shape of circle. Show that the sum of the areas of the square and circle will be minimum when the side of the square is equal to the diameter of the circle. 4.Show that largest rectangle of given perimeter is a square. 5.Show that rectangle of maximum area that can be inscribed in a circle of given radius is a square. r p RA A B M C 1 1.Find all the fourth roots of -i PUC PCMB Page 15 Page 15 6.What is the largest size rectangle that can be inscribed in a semi circle of radius 1 so that two vertices lie on the diameter. 7.Prove that maximum rectangle inscribed in a circle of radius r is square of side r .2 8.Show that triangle maximum area that can be inscribed in a given circle is an equilateral triangle. (c)Indefinite Integrals of the type sec(ax), tan(ax) etc (sin 3 x, cosec 3 x, sec 3 x , cos 3 x etc problems confined to power 3 and 4 only of any trigonometric functions) PRATICE QUESTIONS: Inegrate the following w.r.t x a) sin 3 x, b) cos 3 x c) cosec 3 x, d) sec 3 x e)sec 3 2x f)cosec 3 2x g)sin 5 2x (Each carries 4M) e) cosec 2x f)sec2x g)tan 2 x h)cot 2 x i)tanax j)sec ax (Each carries 2M) For question c and d use integration by parts (d)Integrals of the type ∫ 0 a .x .a−x+.x etc PUC PCMB PRACTICE QUESTIONS: 1.Evaluate ∫ 1 3 . x .4−x+. x 2. Evaluate ∫ 0 4018 .x .4018−x+. x 3. ∫ 0 4018 2 x 2 x +2 4018−x dx 4. ∫ 0 2 ¦2 5 −5c 1 2 3 x 2 −5c 3 2 2 x 3 +5c 4 2 x 4 −x 5 ) dx (Hint: G.E= ∫ ¦2−x) 5 and use the properties( of definite integral)(2M) (e)Finding the order and degree of a differential equations having with fractional powers. Find the order ad degree of differential equation. 1. d 2 y d 2 x = | 1+ ¦ dy dx ) ¦ 3 2 2. a 2 d 2 y dx 2 = | b+ ¦ dy dx ) 2 ¦ 3 4 3. d 2 y d 2 x = . | 1+ ¦ dy dx ) 2 ¦ f)Finding the particular solution of a differential equations of first order and first degree only. Find the particular solution of following differential equation 1. dy dx =ytan2x when x=0, y=2 2. dy dx =2e x y 3 when c=0, y=1/2 3.xy dy dx =y+2 when x=2, y=0 4.(y 2 +y)dx +(x 2 +x)dy=0 given that x=1 when y=2 Example 1: PART -E: I. Answer any one question: 39. a)Find all the fourth roots of 1+i . 3 4 b)Find the length of the common chord of intersecting circles x 2 +y 2 -4x-5=0 and x 2 +y 2 -2x+8y+9=0 4 c)Find the remainder when 5 20 is divided by 7 2 40.a)Show that maximum rectangle that can be inscribed in a circle is a square 4 b)Evaluate ∫ cosec 3 x dx 4 c)Find the order and degree of Differential equation d 2 y d 2 x = | 1+ ¦ dy dx ) ¦ 1 2 Example -2: PART E: Answer any ONE question 1. a)If ¯a is a unit vector and ¦¯x−¯a). ¦¯x+¯a) =8 then find ∣¯x∣ 4 b)Find the equation of chord of circle x 2 +y 2 -2x+4y-17=0 bisected at (-1,2) 4 c) Find the derivative of sin(3x) 0 2 2. a)Show that maximum rectangle that can be inscribed in a circle is a square 4 b)Evaluate ∫ tan 4 x dx 4 Page 16 Page 16 c)Find the order and degree of the differential equation a 2 d 2 y dx 2 = | b+ ¦ dy dx ) 2 ¦ 3 4 2 Example-3: PART E:Answer any ONE question 1. a)Find all values of (1+i .3 ) 2/3 4 b)If a+b+c=0 and |a|=3, |b|=5 |c|=7 find the angle between a and b 4 c) Find the derivative of tan(x/2) 0 2 2. a)Of all the rectangles of constant area show that the square has the least perimeter 4 b)Evaluate ∫ sec 3 x dx 4 c)Find the unit digit of 7 129 Example-4: 2 Part E: Answer any ONE question 1. Page 17 Page 17 http://pucpcmb.wordpress.com finding the number and sum of divisors . ellipse by integration method c). ex [f(x)+f'(x)] Integration by acosxb sinx 2 1 1 .com . derivation will be asked . Second order derivatives. STRESS MORE : On finding the derivative from first principles. 5. • This likely questions does not imply that same questions will appear for examination.If a/b and b/c the prove that a/c ≡ x+4(mod5) 2. b). a)=1 and c/ab then prove that c/b 7. adjoint.If a ≡ b(mod m) and b ≡ c(mod m) the prove 8x ≡ 23(mod 24) has no solution . Evaluating the definite Integral using the properties. Part C. Why? 10. finding the area of the circle.Page 2 c). Finding the area bounded by the two curves or curve and line. 1. remainder.Find the least positive integer x satisfying 2x+5 2. Angle of Intersection of two curves.Why is “congruence mod m” is an equivalence that a ≡ c(mod m) relation. Here likely questions in Part B (Each carries 2M) is excluded. . Part E ) on the basis of the question given below are possible to be asked in the examination. Part D. Derivative as a rate measure. Question on Maxima and Minima or Derivative as a rate measure .If (c. unit digit . 2 ax bxc ax 2bxc pcosx qsinx . subnormal. BEST OF LUCK. DEPT OF MATHEMATICS PART -A Answer all the ten questions: 1. c). The linear congruence 4. operation on matrices. • order and arrangement of the questions given below may be different in examination . 3−x y −3 • If A= is a scalar matrix . Multiplication of matrices. characteristic roots etc multiplicative inverse. Integration by partial fractions.Find the least +ve integer a ≡ 9(mod15) if 73 ≡ a(mod 7) 3. sub tangent. 3 to 6 marks) are given for practice. and using Implicit. Matrices and Determinants: (LEVEL: Aptitude): Areas: Types of matrices. One question on Problems on Integrals of the form(Particular forms) 1 a 2± x 2 .If a/b and b/a then prove that a= ± b 9. 2 x −a pxq pxq .Question on elements of number theory (LEVEL: Knowledge) Areas likely to be asked : Properties of Divisibility and congruences. length of the sub tangent and subnormal. . symmetric . c>0 show that 11. abcosx absinx 1 1 2 2 acosxbsinxc a cos xbcos xc 1 1 1 1 2 2 2 2 2 2 2 a ±x x −a x x ±a a ±x 2 substitution and by parts. The pattern and type of the question(as in Part A . However some definite possible questions similar to the following problems. 2 • [ 0 2 ] find x and y • • • 4996 4997 Evaluate 4998 4999 x−1 2 Find x such that 0 3 ∣ ∣ Find the value of ∣ ∣ 1 3 3 3 32 33 32 33 3 4 x x =4 [ ] has no • Solve for x in ∣ ∣ x 3 http://pucpcmb.If a ≡ b(mod m). x 2−a 2 . maxima and minima INTEGRATION: c). c/m. Parametric differentiation. skew symmetric matrices.Find the number of incongruent solutions of 6x 8. Solving the Differential equation by the method of separation of variables and equation reducible to variable separable form STRESS MORE: units a) and b LIKELY QUESTIONS according to Pattern of th II PUC Question paper (Essay Type/Long answer questions) NOTE: Here some of the following possible questions on (1.Find the least non negetive reminder when 76x204 a ≡ b(mod m) is divided by 7 6.wordpress. •Verify whether the point (-2. •Find the vertex of the parabola 20x2 +20x-32y+53=0 http://pucpcmb. find the a*b=ab prove that * is not a binary operation. •Define coplanar vectors •Find the direction ratios of line joining P(4.3. find the value of a+b+c+d +e+f. •Define binary operation 4. the circle x2 +y2 -3x +5y -7 = 0 • • 6.1. Is * a binary operation? Justify your answer.2) are parallel . what is the relation connecting a and b? 5.-2) find the magnitude a b of 2 3 b a •Find the magnitude of the vector secθi + tanθj-k •If the vectors (a. identity.r.com . 1/3. b ∈ Q . [ 6 7 find AB' 8 9 2 4x5 is x2 1 • If the matrix ] [ a b d 2 −3 c 5 e f ] is skew • If 0 3x−4 2x−6 0 • symmetric matrix .-1. find x. G * is defined by a*b=a+b-ab for all a. b) and (3. •Define subgroup of a group find the identity element.2) to the cricle x2 +y2 -x+2y+1=0 Find the power of the point (-2.Groups:Level: knowledge •On Q-1 * is defined by a*b=a+b+ab find the identity element •Find the inverse of 3 in the group {1. •If a =3i +2j+8k and =2i +λj+k are b perpendicular. Find the characteristic roots of the matrix is a skew symmetric matrix. and n . • [ ] 1 4 2 3 Evaluate •a*b= (2M or 1M) ∣ 4996 4997 4998 4999 ∣ using properties of determinants (2M or 1M) 3..9} under multiplication modulo 10 ab on the set of all integers .-8) •Show that the circle x2 +y2 -4x-2y+1=0 touches y axis. 2 •Write the equation of the director circle of the x2 y2 − hyperbola =1 having 10 and 8 as the a2 b2 lengths of axes.-2) and touching x axis.Conic section: Level: skill •Find the angle between asymptotes of x2 y2 − hyperbola =1 a2 b2 •Find the eccentricity of the ellipse x2 y2 2 =1 2 a b Find the equation of the tangent to the circle x +y2=30 at ( 5.-3) lies inside.b I.1) w. •On Q. •Find the equation of circle with centre (3. •Define semi group •In the set of all non negative integers S if • If a∗b =a+b-5 ∀ a.t.3). outside or on the circle x2 +y2 +4x+6y-7=0.Page 3 • If A= 2 3 4 5 B= • Find x such that symmetric matrix. Prove that (a ) =a for all a in •On the set Q1 the set of all rationals other than 1. •If the direction cosines of are 2/3.CIRCLES: Level: skill •Find the length of intercept of the circle x2 +y2 -2x-7y-8=0 on the x axis. •Find the length of tangent from (1. =i+k find × b a b k •Find the value of i× j. a find n. find p and q.7.wordpress.VECTORS: Level: Understanding •Find the direction cosines of the vector 2i-j+2k •Find a unit vector in the direction of 2i+3j-k •If =(2.1.3.5) and Q(-2. =(2. then find the value of λ •If a =3i-j . •If the equation 3x2 -py2 +qxy-8x+6y-1=0 represents a circle . 5 ) on it. a*b= ab/4 find the identity -1 -1 •In a group (G * ). 0) and directrix x=4.wordpress.r.(or x and y are not unique) 2) a)If (a.t x •If y=log e then find dy/dx •Find the derivative of log(secx+tanx) •Differenatiate elog sinx w.com . Complex number: Level : knowledge ● Express the complex number 3 -i in polar form ● ● ● ● the value of sin-1 ( sin 240) •Find the value of cosec(cot-11) •Prove that cos-1(12/13)= tan-1 (5/12) •Find sec[sin-1(-1/2)-π/6] •Find ● ● Express Find the imaginary part of 1 74i 32i in the form a+ib.t x •Differentiate 2 •If x=ct y=c/t find dy/dx x x 2 1+sinhx •Differentiate x log2x w. QUESTION IS ASKED ON ELEMENTS OF NUMBER THEORY: 1)Find the GCD of 189 and 243 and express it in the form of 189+243y where x and y are integers. x axis as the X axis and passing through the point (2.4) x2 y2 =1 9 4 •Find the equation to the parabola given focus (-4. h ttp://pucpcmb. 5)a)Find the least non negative integers when 2301 is divided by 7? b)If (a. Differentiation and application of derivatives :Level: skill •Define derivative of y=f(x) at a point x=a. 0<θ<π ● Find the amplitude of 1 Evaluate ∫ dx 1 x 2 e dx Evaluate ∫ 2 1 1x 2 3 Evaluate 5 /3 3 ∫ tanx dx 0 Evaluate ∫x 1 1 2 dx Evaluate ∫ x1 1 3 dx Evaluate ∫ x x 1 dx 2 2 − 2 3 Evaluate 1−x Evaluate ∫ dx 0 1x dx Evaluate ∫ 2 x−1 1 ∫ x 2 dx 1 ∞ Evaluate ∫ sin3 x dx PART -C ANSWER ANY THREE QUESTIONS: (AS PER PATTERN) 23.r.t x 3 •If y=cosec (1/x) find dy/dx 5 6 2 w.t x 10. •What is the sum of focal distances(difference of •Find the focal distance of the point (8.0) .Page 4 •Find the equation of parabola with vertex at (0. •Show that tan-1x +cot-1x =π/2 •Evaluate cos(cosec-15/4) •Find the value of cos-1(cos(-5850)) 8. c)=1.Integration or definite integrals: Level: knowledge Write the modulus of the function 1+cosθ+i sinθ. •Differentiate sin(1/x) w. Inverse trigonometric function (Level: U): •Show that cos-1(-x)=π-cos-1x for x>0. b)=1 and (a. bc)=1 6) a)Find the number of all +ve divisors and the sum of all such +ve divisors of 432 b)Find the remainder when 71x 73 x 75 is divided by 23. c)=1 and (b. 4−3i Find the multiplicative inverse of (5+6i)2 Find the complex conjugate of (3-7i)2 1+cos +i sin 16 16 Find the modulus of 1+ 3 I 9.8) from focal distances of ) any point on the the focus of y2 =8x x2 y2 ellipse(hyperbola) =1 25 1 7.c)=1 Prove that (ab. c)=1 b)If a ≡ b(mod m) and c ≡ d (modm) Prove that ac ≡ bd(modm) 3)a)Find the GCD of 595 and 252 b)Prove that 5700 ≡ 6(mod 23) 4)Define congruent relation on Z and Prove that it is an equivalence relation .r. Prove that (a.r. Also show that this expression is not unique. find x 14)State caley-Hamilton theorm and find A3 if 1 2 A= using Caley-Hamilton theorem 2 1 15)State Cayley-Hamilton theorem and find the 1 3 inverse of the matrix using the 2 −1 theorem. 3)Solve by matrix method: 3x+2y-z=4 .y )2 [ ] [ ] [ ] 8)Prove that ∣ a 1 ab ac 2 ab b 1 bc 2 ac bc c 1 =1+a2 +b2 +c2 2 see other standard problems using properties of determinants. 55)=210(5)+55(k) find k 3 b)If (a.wordpress. ON MATIRCES AND DETERMINANTS: 1)a) Solve the equation 9)Solve by Cramer's rule : 2x+3y=5 and x=2y=3 using matrix x-2y=0. y and z if [ x 2 −3 5 y 2 1 −1 z ][ = ] ] [ 5 3 3 19 −5 16 1 −3 0 6)Prove that ∣ x y z x y z y 2 z x z 2 x y 2 ∣ ∣ =(x-y)(y-z)(z-x)(x+y+z) 7)Find the inverse of the matrix : [ −3 5 −1 4 −1 2 0 8 −2 ] http://pucpcmb.2 . b)Find the digit in the unit place of 7123 . i=1. . 0 0 c b. 4)Prove that a) 5)a)Prove that ∣ ∣ bc a a 2 ca b b 2 ab c c2 =(a-b)(b-c)(c- ∣ ∣ a bc acc 2 2 a ab b ac 2 ab b bc c2 2 2 −bc b 2bc c 2bc a 2ac −ac c 2ac a 2 ab b2 ab −ab ∣ ∣ ∣ bc ca ab ca ab bc ab bc ca =2 ∣ 13) a) Show that ∣ ∣ a b b c c a c a b 3 −1 2 4 2 5 2 0 3 b)Find x. 8)a) If (210. 2x+y-z=1. Find the matrix A b)If the matrix ∣ ∣ [ ] x y y y x y y y x =(x+2y) (x.com 2 x 3 4 1 6 has −1 2 7 no inverse.3 and j=1. a 0 0 16) a)Find the inverse of 0 b 0 where a. Hence find A-1 .c are real numbers b)If matrix A=[aij] = i+2j.Page 5 7)a) Prove that the number of primes is infinite.2. b)=1. 2x-y+z=4. a/c and b/c prove that ab/c 2 9)a) Prove that smallest positive divisor of a composite number ' a' does not exceed b)Find the remainder when 89 x111 x 135 is divided by 11 a 3 2 24. x-y+4z=11. 3x+y-2z=-3 method a− x c b 1 x 2 3 10)If a+b+c=0 then solve c b−x a b)Solve for x : =0 1 x2 3 b a c−x ∣ 1 2 x3 ∣ ∣ ∣ =0 11)Prove that =4a2b2c2 12)Show that =(ab+bc+ca)3 2)State cayley Hamilton theorem and verify it for A= 1 −3 4 −5 . 4)If Q + is the set of all positive rational numbers. ON GROUPS : 1)a)Prove that the set of fourth roots of unity is an abelian group under multiplication. Find are of traingle ABC b)If =i+j+2k and =3i+j-k. =3i-4j+2k find × × a c a b c b b)Simplify : (2 +3 ) x (3 -2 ) a a b b 2)a)The position vectors of A. find λ. b ЄQ+ (each carries 5 M) 5 5)If Q1 is the set of rationall numbers other than 1 with binary operation * defined by a*b=a+b-ab for all a.b εQ1. 11)Show that the set {1. 10) a)Show that set of even integers is a subgroup of the additive group of integers. B C respectively are i-j+k.r. 7)Define a subgroup and Prove that a non empty subset H of a group (G ) is a subgroup of (G ) if and only if ∀ a. C and D are 3i-2j-k.t multiplication of matrices. xε R is a group under matrix multiplication.*) is an abelian group and solve 5*x=3 in Q1. 8)Define an abelina group and prove that the set of all integral powers of 3 is a multiplicative group. b)Prove that the identity element is unique in a group.5. 4)a) In a regular hexagon ABCDEF. Prove (Q +. find the unit vector in a c b . 2. 3)Prove that the set of all +ve rational numbers forms an abelian group w. ON VECTORS : 1)a)If =i+j-k. ) if a*b=a*c prove that b=c and if a*b=c*b prove that a=c (Cancellation laws) b)If H is a subgroup of G then show that identity element of H is the same as that G. 2i+3j-4k.Page 6 1. 2)Prove that set of all matrices of the form [ ] x x x x ab 2 where xεR and x#0 forms an abelian group w. 6)If Q-1 is the set of all rational numbers except -1 and * is a binary operation defined on Q-1 by a*b= a+b+ab. 15)Prove that set of all matrices of the form [ x −y y x ] x#0.r. 2i+j-k and 3i-2j-k. y#0 . B. find the cosine of the angle between and a a b b 3)a) Prove that [ − . b)Prove that a group of order 3 is abelian. Show that (Q1. 14)Prove that a nonempty subset H of a group G is a subgroup of G iff closure and inverse law are true and hence show that a set of even integers is a subgroup of additive group of integers.11} is an abelian group under x mod 12 and hence solve 5-1 x12 x=7 12)Show that set of all matrices of the form Aα = [ cos −sin sin cos ] where α is a real number forms a group under matrix multiplication. *) is an abelain group where * is defined by i) a*b = 2ab ii) a*b = 3 iii) a*b = ab a. -i+j+2k and 4i+5j+k. show that AD EB FC =4 AB a a a a b)Show that 2 =i× ×i j× × jk × ×k 5)a) Prove that [ × . 13)a)In a group (G. c − ] =0 a b b− c a b)The position vectors of the points A. c × ]=[ ]2 a b b× c a abc b)Find the sine of the angle between the vectors i-2j+3k and 2i+j+k=0 6)a)Find a unit vector perpendicular to each of the vectors 4i+3j+2k and i-j+3k b)Prove by vector method:In traingle ABC a= bcosC + CcosB (3M) 7) a) If i+j+k =i+2j+3k and =2i+j+4k.t multiplication * defined by a*b=ab/6 and hence solve x*3-1 =2.7. b ∈H . Prove that Q-1 is an abelian group. =i-3j+k. a*b-1 ∈ H. 9)Prove that the set of all complex numbers whose modulie are unity is a commutative group under multiplication. If the four points lie on a plane. 3).3). a) If =a1 ia 2 ja3 k show that a × a × k × × k =2 i× i j× j a a b) Find the area of parallelogram whose diagonals are λi+2j-k and i-3j+2k respectively.3). BC BC . Find | − | a b b a b 12)a)Prove that [ a b b)Find the projection of a 13. 8)a)Prove that [ .Find its centre and radius.-1). ]=2 [ ] a b b c a ab c b)Find the projection of =i+2j+3k on =2i+j+2k a b 9)a) Prove by vector method . 9)Find the equation of the two circles which touch both co-ordinate axes and pass thorugh the point (2. Also find the point of contact 3)Derive the equation to the tangent to the circle x2 +y2 +2gx+2fy+c=0 at point (x1 .sinB by vector method b)If | |=5 and a is perpendicular to . Find the condition for the circles x2 +y2 +2g1x+2f1y+c1 =0 and x2 +y2 +2g2x +2f2y +c2 =0 to cut orthoganally.t circle . CACA. (1. In a triangle prove projection formula a=bcosC+ c cosB using vector method (3M) 15. Show that cos2α +cos2 β+cos2γ=1. Given the vectors a =2i-j+k. 10)Define Power of the point w. In any traingle ABC a b c = = sin A sin B sin C b)For any three vectors a. AB= 2 11)a)Prove that sin(A+B) =sinAcosB +cosA. Find a vector perpendicular c b to ] b c c a a =i+2j+3k on b =2 [a ] b c =2i+j+2k a and coplanar with b & c (3M) II. 2)Find the condition that the line y=mx+c may be tangent to the circle x2 +y2 =a2 .r. Answer any two questions: 3.Page 7 the direction of a b× × c b)If cosα. Show that radical axis is perpendicular to line joining the centres of two circles 5)Find the equation of the circle with centre on 2x+3y=7 and cutting orthogonally circles x2 +y2 -10x-4y+21=0 and x2 +y2 -4x-6y+11=0 6)Find the equation of the circle which passes through the point (2. c Prove that × c c × × × =0 a b× b× a c a b 10) a)Prove cosine rule by vector method: a2 =b2 +c2 +2bc cos A b)If ABC is an equilateral triangle of side a then prove that −3 a 2 AB . 4)Define Radical axis.y1 ) on it. =i+2j-k =i+3j-2k . b. 7)Find the equation of the circle passing through the points (5. cosβ and cosγ are the direction ratios of the vector 2i+j-2k. cuts orthoganally the circle x2 +y2 -2x-4y-5=0 . cuts orthogonally the circle x2 +y2 -4x+2y-3=0 and length of the tangent to it from the point (1. c . 11)Find the equation of the circle which passes through the point (2. Find the length of the tangent from an external point (x1 . Additional Questions: 14. y1 ) to the circle x2 +y2 +2gx+2fy+x=0. ON CIRLCES a)Questions carrying 3 Marks 1)Define orhoganality of two circles.1). 8)Show that general second degree equation in x and y x2 +y2 +2gx+2fy+c=0 always represnts a circle .0) is 2.5) and (3. Find the equation of the tangent at origin O the circum cirlce of traingle OAB.0) and (0.0)&(0. (6.Page 8 Questions carrying 5 Marks 12)Find the equation of the circle such that the lengths of tangents from the points (-1.-8) and (-6.0) .0) and having length of the tangent from (1. 5)Find the eccentricity and equations to directrices of the ellipse 4x2 +9y2 -8x+36y+4=0 6)Find the equaton of tangent and normal to a)Parabola b)Ellipse c)Hyperbola at (x1 . -1) are 3. -5) 3)Prove that the length of the tangent form any point on the cirlce x2 +y2 +2gx+2fy+c=0 to the circle x2 +y2 +2gx+2fy+d=0 is d −c 4)A and B are points (6.3) having the length of the tangent from (1. 10 and 3 3 13)Find the equation of the circle.5) . what is the equation of the circle if the centre of the circle lies on x axis.0) are concyclic 3 b)If x2 +y2 -2x+3y+k=0 and x2 +y2 +8x-6y-7=0 are the equations of the circles intersecting orthogonally find k.2).1) as √5 units and cutting orthoganally the circle x2+y2-4x-6y-3=0. 3)a)Show that the four points (1.0) and (0.0) as 2 units and cutting x2 +y2 -4x+2y-3=0 orthogonally. 14)Find the equation of the cirlcle passing through (2. Derive the condition for the two circles x2 +y2 +2g1x +2f1y +c1 =0 and x2 +y2 +2g2x +2f2y+c2 =0 to cut orthoganaly. Questions (sample): 1) a) Fin the equation of the circle through origin and having portion of the line x+3y=6 intercepted between the co-ordinate axes as diameter 3 2 2 b)Find the points on the circle x +y =25 at which tangents are parallel to the x axis. 5)Show that the cirlces x2 +y2 -4x-10y+25=0 and x2 +y2 +2x-2y-7=0 touch each other. (4.5) and focus (3.-5) 6)Find the equation of the circle passing through (-6. 7)Find the equation of tangent and normal to a)parabola b)Ellipse c)Hyperbola at t (Each carrying 3M) . (0.2) and (2.1) (-2. b)Find the equation of the circle passing through the origin . (-2. 4. y1 ) [Each carrying 3 marks]. b) Questions carrying 2 Marks 1)Show that the line 7x-24y-35=0 touches the cirlce x2 +y2 -2x-6y-6=0 2)Find the equation of the circle passing through the origin (4. 2x2 +2y2 +8x+6y-3=0 and x2 +y2 +6x-2y-3=0 orthogonally. cutting the three circles x2 +y2 +4x+2y+1=0. 4)Find the equation to the circle which passes throug the points (0.1) and has its centre on the line 12x+5y=25 5) a)Define orthogonal circles.8).2) 3)Find the centre and foci of the ellipse 3x2 +y2 -6x-2y-5=0 x2 y2 =1 passes through other 4)If the normal at one end of latus rectum of the ellipse a2 b2 end of the minor axis the prove that e4 +e2 =1. 6)Find the equation of the cirlce two of whose diameters are x+y=6 and x+2y=4 and whose radius is 10 units. a)QUESTION ON CONIC SECTION:(Questions carrying 3 M) 1)Find the centre and foci of the hyperbola 9x2 -4y2 +18x-8y-31=0 2)Find the equation of parabola having vertex (3. where 'e' is the eccentricity of the ellipse. 2)Prove that general second degree equation in x and y x 2 + y 2 + 2g x +2fy+c=0 always represents a circle . cosec x. 3)Find the equation of hyperbola in the standard form given that the distance between the foci is 8. 7.3) and focus is (1. cos x . Also find the angle between them.3) x2 y2 4)Find the equation of tangent to the ellipse =1 at (-2.sin 2/3) 5)If sin −1 2x 1− y 2z cos−1 tan−1 =2 Prove that x+y+z=xyz 2 2 2 1x 1 y 1− z 2 b)QUESTION ON GENERAL SOLUTION OF TRIGNOMETRIC EQUATIONS 2− 3 1)Find the General solution of sin2 θ = 4 2)Find the General solution of tan5x tan2x=1 3)Find the general solution of tan2x -3secx+3=0 4)Find the General solution of 3 sinx +cosx = 2 5)Find the General solution of 2(sin4x+cos4x)=1. a)Differentiate i)trigonometric functions like sinx. tan-1x etc. 2)Find the equation of the tangent at any point (t) on the hyperbola y2 =4ax. by first principles. a)QUESTIONS ON INVERSE TRIGNOMETRIC FUNCTION: 1)If cos-1x +cos-1y +cos-1 z=π. tanx. Parametric. tanx. Derivatives of one function w.r. −1 .t antohter.(Each carries 3M) b)Some standard and Previous years Problems on Implicit.4). Also find its eccentriciy. ax.(3M each) 1)If y=log(x+ x 2−1 ). Questions carrying 5 Marks. cotx. Logarithmic. xn . 1)Find the condition for the line y=mx+c to be tangent to a)parabola b)Ellipse c)Hyperbola in standard form also find the point of contact. then prove that x2 +y2 +z2 +2xyz=1(other problems of this type) 3 8 −1 3 tan −1 tan−1 = 2)Show that tan 4 5 19 4 2 3)Solve for x: cos-1x -sin-1 x =cos-1x 3 4)Solve for x: sin-1 x +sin-12x = 3 -1 -1 4)Find sin(cos 1/3 . 12 6 5)Find the equation to the parabola with vertex (-3. (Each carrying 5 Marks) 2)Find the equation of the parabola whose vertices's is on the line y=x and axis parallel to x axis and passing through (6. 5)If y=cos(atan-1x) show that (1+x2)y2 +2x(1+x2)y1 +a2y=0. 6)Find the General solution tan5x tan2x=1 7) Find the general solution of cos 2x +cos3x=0 CALCULUS ANSWER ANY THREE QUESTIONS: 6. sin ax. a)Some standard problems on successive differentiation. .2). logx. cos2x etc iv)sin-1x. sec x. etc iii)sin2x. 3)Find the equation of the parabola whose vertex is (-2. and the distance between the directrices is 9/2. Standard problems using chain rule.Page 9 8)Find the condition for the line y=mx+c to be tangent to the a)paraboal b)Ellipse c)Hyperbola in standard form.-2) and (3. cos-1x. Derivatives of inverse trignometric functions by substitution. cosax etc ii)ex. 5.(Each carrying 3Marks) b) Questions carrying 2M each 1)Find the equations of the asymptotes of hyperbola 9x2 -4y2 =36.1) and directrix y=6. Prove that (x2 -1)y1 +xy1=0 2)If y= e mcos x Prove that (1-x2)y2 -xy1 -m2y=0 3)If y=sin(m sin-1x) Prove that (1-x2)y2 -xy1 +m2y=0 4)If y=acos(logx)+bsin(logx). Prove that x2y2 +xy1 +y=0. r.r. Find the position of P such that AP2 +BP2 is minimum.(2M each) 1).r.t x 2 4x −4x2 .Page 10 b) Standard problems on Differentiation or Application of Differentiation(2M each) 1)P is a point on the line AB=8cms. 9. 5)Show that sinx(1+cosx) is maximum when x= 3 8. a)QUESTIONS CARRYING 3M EACH: Evaluating the integrals of particular types a 2xinxcosx a−x dx dx 1)Evaluate ∫ 2)Evaluate ∫ 3sinx−2cosx ax −a 3)Differentiate tan-1 4)Differentiate tan-1 1x 2−1 x w. a)Some standard problems on Application of Differentiation or Integration(3M each) x1 1).3). 4)When the breakes are applied to a moving car .t tan-1 2x 1x 2 dy 1− y 2 5)If 1−x 2 1− y 2=a x− y Prove that = dx 1−x 2 x 2 2 a2 x 6)If y= x −a − 2 cosh−1 a then prove that dy = x 2−a 2 dx 2 1−x 1x w. 2)Find the minimum value of xex 3)Find the angle between the curves x2 +y2 +3x-8=0 and x2 +y2 =5 at (1. When and where does the car stops. x 2 dx 2)Differentiate (sin-1x)x w.t sin-1 2x 1x 2 b)QUESTIONS CARRYING 2M EACH: 1)Evaluate ∫ 4 x . 5)Prove that x 21 dx 2)Evaluate ∫ 4 ∫ 2 dx 2 3/ 2 = 2 x2 2 C x 1 a x a a x dx dx dx 3)Evaluate ∫ 6)Evaluate ∫ 54sinx 54 cosx b)Some standard problems on application of Differentiation. then find the initial velocity.t x 6 4x 1 1 w. Show that the velocity is inversly proportional to displacement.Find the range in which the funtion x2 -6x+3 is a)increasing b)decreasing 3)If the displacement s metres of a particle at time 't' seconds is given by s=2t3 -5t2 +4t-3. If the displacement 's' at time't' is given by s= 1−t .r.t x 1 dx 3)Evaluate ∫ 54x−4x 2 3 4)Integrate 5)Integrate 6)Integrate sinx w.t x 2 13−9sin x x2 w. Show that the curves 2y=x3 +5x and dx 4)Evaluate ∫ 2 2 y=x +2x+1=0 touch each other at (1. 2).r. the car travels a distance of 's' metres in time 't' seconds given by s=20t-40t2 .r. x 4x5 Find the equation to common tangnet.2) 4)Find the equation to tangnet to the curve y=6x-x2 where the slope of the tangent is -4. a−v 2 5)If the law of motion is s2 =at2 +2bt+c then show that acceleration is where v is s velocity. a)Total 6Marks Question ONE OF THE QUESTION IS asked on CONIC SECTION (All Derivations of Parabola . What is meant by rectangular hyperbola. find its centre. 3)Find the area of the ellipse 2 a b 4)Find the area enclosed between the parabola y2 =4x and the line y=2x-4.. 12. 36. Prove that . x2 y2 − =1 . ON COMPLEX NUMBERS: (6M each) 1)If cos +cos +cos =0=sin +sin +sin . Hyperbola and others) x2 y2 =1 1)Define Ellipse and Derive standard equation to the ellipse a2 b2 2)Define Hyperbola as a locus of a point and Derive the equation of the same in the standrad x2 y2 − =1 form a2 b2 3)Find the condition for the line y=mx+c to be tangent to a)parabola b)Ellipse c)Hyperbola in standard form also find the point of contact. 37. 11. 5)Find the area of the circle x2 +y2 =a2 using integration. a2 b2 8)Show that an equation 9x2 +5y2 -36x-50y-164=0 represents an ellipse. Find the equations of the asymptotes of the Hyperbola x2 y2 − =1 . 6)Find the area between the curves x2 =y and y=x+2 7)Find the area enclosed between the parabola y2 =4ax and x2 =4by PART D ANSWER ANY TWO: (Includes Question Numbers 35. length of latus rectum and foci. eccentricity. Hence deduce the condition for the line x cosα+ysinα=p to be a tangent to the a) ellipse b)Hyperbola (Each carrying 6 Marks) 4)Define a parabola and obtain its equation in the standard form x2 y2 − =1 5) a)Obtain the equation of asymptotes of the hyperbola a2 b2 b)Prove that the locus of point of intersection of perpendicular tangnets to the parabola y2 =4ax is the directrix x+a=0 (Each carrying 3M) 6)Show that locus of point of intersection of perpendicular tangents to a)Ellipse b)Hyperbola is the director circle a)x2 +y2 =a2 +b2 b)x2 +y2 =a2 -b2 7)Define asymptotes of Hyperbola. Question is exclusively asked in AREA UNDER A CURVE : (5M EACH) 1)Find the area bounded by parabola y=11x-24-x2 and the line y=x. 2)Find the area enclosed between the parabola y2 =4ax and x2 =4ay x2 y2 2 by integration. Ellipse.Page 11 10. 38. Write equation to the 9)Derive the equation of the hyperbola in the standard form a2 b2 x2 y2 − 2 =1 and locus of the point of intersection of perpendicular tangents to the hyperbola 2 a b write the name of the equation. Problems asked on topics a)Conic section (6M) b)Complex numbers (6M) c)Application of Differentiation or Integration (6M) d)Vectors (6m or 4m) e)Matrices and Determinants(4M) f)Differential equation(4M) g)General solution of trigonometric equation (4m) h)Matrices and Determinants. Prove that z1 |z1z2|=|z1||z2| and arg(z1 z2)=arg z1 +arg z2 and arg =argz1 -argz2 . 2)A metal cube expands on heating such that its side is increasing uniformly at 2 mm/sec.Page 12 i) cos 3 +cos 3 + cos 3 =3 cos( + + ) sin 3 + sin 3 + sin 3 =3 sin( + + ) 3 2 2)Find the fourth roots of the complex number -1+ 3 i and represent them in the Argand diagram. Also. problems on maxima and minima) 1)Water is being poured into a right circular cone of base radius 15 cms and height 40 cms at the rate of 12π cc per minute. Prove that |z1z2|=|z1||z2| and arg(z1 z2)=arg z1 +arg z2 1sin icos ii)Prove that =i(tanθ+secθ) 1−sin −icos 6) If z1 and z2 are any two non zero complex numbers. (6M each)(Problems related to Derivative as a rate measure . Find the rates at which the depth of water and radius of the water cone increase when the depth of water is 16 cms. x2 y2 x2 y2 2 =1 cut each other orhoganally if 8)Show that the curve =1 and 2 A B a b . 3 )3. also find their 3/4 8)Show that the continued product of four values of 1 3 i 2 2 is 1 and reprsent all values in the argand diagram 13. represent them on an Argand plane. find their continued product. 3)State and Prove Demoivres theorem 4)Find all the fourth roots of the complex number (i. is walking away from a source of light. 3i 1 3 and represent then on an Argand diagram. Find the rate of rise of water level and the rate of increase of the 2 surface area when the depth of water is 4 cm. the power of subtangent varies as the nth power of subnormal. Water is poured into it 1 at the rate of 1 cc/sec. Find the rate at which i) his shadow lengthens b)tip of the shadow moves 4)Prove that the greates size rectangle than can be inscribed in a circle of radius 'a' is a square. angle of intersection between two curves. 1−i 7)Find all values of continued product. ii) cos2 +cos2 +cos2 =sin2 +sin2 +sin2 = 5)i) If z1 and z2 are any two non zero complex numbers. 5)Prove that the rectangle of least perimeter for a given area is a square. Find the rate at which its i)volume ii)Surface area and iii)diagonals are increasing when the side is 10 mm 3)A man 160 cm tall. Use these results z2 1i1− 3 i to find that modulus and amplitude of . 6)An inverted circular cone has depth 12 cms and base radius 9 cms. which is 480 cm above the ground at 5 kms /hr. ON CALCULUS: ON DIFFERENTIATION OR APPLICATION OF DERIVATIVES OR INTEGRATION. Also find the continued product of the roots. 7)Define subtangent and subnormal to a curve and Prove that in the curve xm+n =am-n y2n . Vector product. Vector triple product. 1.p. 11)Show that right circular cone of greatest volume which can be inscribed in a given sphere is such that three times its height is twice the diameter of the sphere. Problems using properties of Determinants.General solution of Trignometric equation: 4. ∫ sinx cosx dx sinx 2 . variable seperable forms. 12)A man 6 feet in height moves away at a uniform rate of 4m.Page 13 A-B =a-b 9)Prove that the portion of tangent to the curve x2/3 +y2/3 =a2/3 intercepted between the coordinate axes is of constant length. Hence evaluate ∫ x 3x −7x dx 2a 0 a 2a 0 ∫ f x dx=∫ f x dx∫ f 2a−x dx 0 a −99 a and evaluate 3)Prove that b a xsinx ∫ 1sinx dx 0 8)Prove that ∫ f x dx=∫ f a−x dx 0 0 4 and b hence show that and 9)Evaluate 3 ∫ f x dx=∫ f ab−x dx a ∫ log 1tanx dx 0 4−x dx hence evaluate ∫ 1 4−x x 4)Evaluate 5)Prove that 1 tanx ∫ secxxtan x dx 0 x dx ∫ a 2 cos2 xb2 sin2 x 0 b a 10)Show that log sinx dx = − 2 log2 ∫ 0 a 0 2 ∫ a f x dx=∫ f t dt and 0 log 1 x dx evaluate ∫ 1 x 2 0 6)Prove that ∫ log sinx dx = 2 log(1/2) 0 a a 2 11)Prove that a ∫ f x dx = ∫ f a− xdx 0 and hence evaluate 7)Prove that ∫ −a f x dx=2∫ f x dx 0 0 if f(x) is even In the above questions section (b) contains 4 marks questions on following topics merged with any questions given above. verifying cayley Hamilton theorem. 3. Vectors: Application of vectors. 10)Show that a rectangle of maximum area that can be inscribed in a circle is a square. Find the rate at which the shadow lengthens and the rate at which the tip of his shadow is moving. ON INTEGRATION: ON DEFINITE INTGRALS: (6M each) 1 function log 1 x dx= log2 1)Prove that ∫ 2 =0 if f(x) is odd function 8 1 x 0 99 2 3 2)Prove that . From a source of light which is 20 feet above the ground. Reducing to variable seperable form. Scalar triple product. 2. 14.Differential equations: Finding the perticular solution. finding Inverse using caley Hamilton theorem. Matrices and Determinants: Calyey Hamilton theorem.h. Page 14 The question paper pattern in second PUC mathematics has changed from 2007-2008 for 100 marks instead of earlier 90 marks.Find the incongruent solution of 3x ≡ 9(mod 6) (3 incongruent solutions x=3.Let . a c a b c a c a b=0 and ∣∣ b | =4.Find the digit in the unit place of a) 237 b)753 (2M each) 2. c b a b a b c c a (2M) x 6. Here is some likely questions on specific units for practice which may be asked in PART E .If ∣ b a b a a b are two vectors such that 3. c b. 1. If .7) 4. and the digit in the unit place of a given number using congruence. iii)Students will have to answer only one of the two questions ALGEBRA: (a) Problems on scalar product of type a i)Show that ∣ ∣=∣ −∣ is perpendicular to a b a b b c ii) Given c =0 | | | | | | to find the angle between any two vectors etc a a b b Practice questions: and be three vectors such that c =0 and and ∣∣ =3. . . Prove by vector method cos(A+B) =cosA cosB – sin A sin B (4M) ∣ −∣ a b 2 ∣ ∣ a b 2 2 a . confined to II PU syllabus . . a b b a b a b 2 2 2 2 2 = | | + | | -2. b. and are vectors of equal magnitudes prove that is orthogonal to − (2M) b a b a b 10. prove that a) ∣ ∣ = | | 2 + | |2 +2. | |=5 and 1.e. If . a =2. which are included in assignments/projects. For any two vectors b) d) 11.Find all incongruent solutions of 2x ≡ 4(mod10) (2M) 6.Find the sum of all positive divisors of 360 5.Find the reminder in the following divisions: a)350 by 7 b)5225 by 3 c)2125 by 11 d)2575 by 13 e)2100 by 19 (Each question carries 2M) 10.If . and to find the incongruent solutions of a linear congruence. PART E .(x+ a )=8 . (Each carries 2M) PUC PCMB a b a b (b) To find the least +ve remainder. and b c a ∣=∣ −∣ show that and are orthogonal. If .5 and . c) ∣ ∣ + ∣ −∣ =2( | | + | |2 ) a a b a b b a b a b 2 . Each question will have three sub divisions. =3 and . c a b perpendicular to c .4.Prove that 310 ≡ 1(mod 31). a 5. If . 4. a b a b a b and be three vectors whose magnitudes are respectively 3. | b c a b and be three vectors such that c =0 . find the angle between . (4M) and are unit vectors find ∣ −∣ (2M) 7. BEST OF LUCK. | | =8 then find c 2 (2M) =1. a c a a b b b . (4M) (Ans:600) | |=7.(2M) 7.5. .(4M) 2. . ii)There will two questions of 10 marks each. Solve 4x-3 ≡ -2x+6(mod 11) .∣ −∣ =4. should be selected from the following topics. If | a | is a unit vector and (x.Find the least non negetive remainder when 250 is divided by 7. Department has given some specific topics for giving questions in the last part of question paper i. is perpendicular to and is perpendicular to the c a c b b a b c |= 5 2 show that | b a 9.solve the following congruences: a)2x ≡ 3(mod 5) b)4x ≡ 7(mod 12) 3. 9. find the value of . then find | |. If a .Find the reminder when 768x217x87 is divided by 11 (2M) 8. =4 find ∣ −∣ (2M) a a b a b b b and be three vectors such that a.a ). is a 8. The first and second questions carry 4 marks each and the third question carries 2 marks. PART E: i)Questions in Part E . 4. Ans: 4(2M) TRIGNOMETRY: a)To find the cube roots of a complex number and their representation in argand plane and to find their continued product. tan(x/2)0 etc(Here degree must be converted into radians) PRACTICE QUESTIONS: Find dy/dx if a) y=log sinx b)y=log e1+sinx c)y=log tanx when x is measured in degrees.If 1. ..3). Ans: 4 3 5...Find the length of the chord of the circle x2 +y2 -6x-15y-16=0 intercepted by x axis .25) 2. then find AM using pythogorean formula and length of chord = 2AM RA r C1 A p M B PRACTICE QUESTIONS: 1. 3 I and represent them in the argand diagram.ω10)(2.Page 15 ANALYTICAL GEOMETRY: a) To find the length of the common chord of two intersectiong circles Method: Find the RA.A wire of length 8cms is cut into two pieces.ω11)=49 d)(a+b)(a+b ω)(a+b ω2)=a3 +b3 e)(1+ ω)(1+ ω2)(1+ ω4)(1+ ω8). ii)sin(3x)0 . (4M)(Ans: 25. 3. w2 are the cube roots of unity then a)show that (1+w-2w2)(1+w2 -2w)(w+w2 -2)+27=0 (2M) b)Show that (1+ω+5 ω2 )(1+5 ω+ ω2)(5+ ω+ ω2)=64 c)(2. Ans: 2 21 (EACH ABOVE QUESTIONS CARRIES 4M) 6. One piece is bent into the form of a square and the other piece is bent into the shape of circle.Find the length of the chord of the cirlc x2 +y2 -4x-8y+12=0 intercepted by the y axis .ω2)(2. Practice questions: (Each question carries 4M) 1.Prove that among all the right angled traingles of given hypotenus. d)y=log a−x ax e)y=log10(logx) f)y=logxe g)y=cos(log(sinx)) f) y=sin(2x)0 h) y=sin35x0 (Each carries 2M) (b)Applications of derivative in finding the maxima and minima of functions involving two dimensions only.. PRACTICE QUESTIONS: (EACH CARRIES 4M): 1.ω)(2..Find the length of common chord of intersecting circles x2 +y2 -4x-5=0 and x2 +y2 -2x+8y+9=0 4.Find the length of the chord of the circle x2 +y2 -4x-2y-20=0 which is bisected at (2..Ans :10(2M) 7.to 100 factors =1 (Each carries 2M) CALCULUS: (a)Finding the derivative of functions of the following type only Logaf(x).Find the length of the common chord of intersecting circles (x-h)2 +(y-k)2 =a2 and (x-k)2 +(y-h)2 =a2 ans: length= 4a 2 −2 h−k 2 2. Find the length of the perpendicular(p) from centre of one of the circle to RA. w. Show that the sum of the areas of the square and circle will be minimum when the side of the square is equal to the diameter of the circle.Show that largest rectangle of given perimeter is a square.Find the fourth rootsof 3-3i b)Problems related to the cube roots of unity 1. the isosceles traingle has the maximum area. 5. 1. 4.. using the properties of .The perimeter of a rectangle is 100 meteres .Find continued product of cube roots of 1+i 3 3. Find centre(C1) and radius(r) of one of the either circles.Find the length of the common chord of two intersecting circles x2 +y2 +2gx+2fy+c=0 and x2 +y2 +2fy+2gy +c=0 length= 2 g f 2−4c 3.Find all the fourth roots of 3 -i PUC PCMB 2.Find the fourth roots of complex number 1.Show that rectangle of maximum area that can be inscribed in a circle of given radius is a square.Find the length of the chord of the circle x2 +y2 -x+3y-10=0 intercepted by the line x+y+2=0. Find the sides when the area is maximum. cos3x etc problems confined to power 3 and 4 only of any trigonometric functions) PRATICE QUESTIONS: Inegrate the following w. 3. Answer any one question: 39.E= ∫ 2− x5 and 2. 1. ∫ 0 0 2 ∫ 25−5c 1 23 x 2 −5c3 22 x 35c 4 2 x 4− x5 dx (Hint: G. a)If is a unit vector and − . a)Find all the fourth roots of 1 i 3 b)Find the length of the common chord of intersecting circles x2 +y2 -4x-5=0 and x2 +y2 -2x+8y+9=0 c)Find the remainder when 520 is divided by 7 40. a)Show that maximum rectangle that can be inscribed in a circle is a square tan 4 x dx b)Evaluate ∫ 4 4 2 4 4 . [ ] 2 dy =ytan2x when x=0. ∫ 4018−x x 3. d y dy = 1 2 dx d x 2 [ ] 3 2 2. 7.Evaluate 4.r.Show that triangle maximum area that can be inscribed in a given circle is an equilateral triangle. d2 y dy = 1 dx d2 x f)Finding the particular solution of a differential equations of Find the particular solution of following differential equation first order and first degree only. cosec3x.t x a) sin3x. d) sec3 x e)sec3 2x f)cosec3 2x g)sin5 2x (Each carries 4M) e) cosec 2x f)sec2x g)tan2x h)cot2x i)tanax j)sec ax (Each carries 2M) For question c and d use integration by parts (d)Integrals of the type PRACTICE QUESTIONS: 3 ∫ a−xx x 0 a etc 4018 PUC PCMB 1. =8 then find ∣∣ x 2 2 b)Find the equation of chord of circle x +y -2x+4y-17=0 bisected at (-1. y=0 dx 4. Evaluate 1 0 ∫ 4−x x x x 4018 2x dx 2 x 2 4018− x use the properties( of definite integral)(2M) (e)Finding the order and degree of a differential equations having with fractional powers. d y dy a = b 2 dx dx 2 2 [ 2 3 4 ] 3. y=2 dx dy 2. =2ex y3 when c=0.Page 16 6.a)Show that maximum rectangle that can be inscribed in a circle is a square b)Evaluate ∫ cosec 3 x d x c)Find the order and degree of Differential equation 4 4 2 4 4 1 2 d2 y dy = 1 2 dx d x [ ] Example -2: PART E: Answer any ONE question x a x a a 1. tan(ax) etc (sin3x. Find the order ad degree of differential equation.2) c) Find the derivative of sin(3x)0 2. (c)Indefinite Integrals of the type sec(ax).xy dy =y+2 when x=2.Prove that maximum rectangle inscribed in a circle of radius r is square of side r 2 8. y=1/2 dx 1. sec3x .What is the largest size rectangle that can be inscribed in a semi circle of radius 1 so that two vertices lie on the diameter. b) cos3x c) cosec3x.(y2 +y)dx +(x2 +x)dy=0 given that x=1 when y=2 Example 1: PART -E: I. Page 17 d2 y dy 2 a = b 2 dx dx 2 c)Find the order and degree of the differential equation [ ] 3 4 2 Example-3: PART E:Answer any ONE question 1.wordpress. |b|=5 |c|=7 find the angle between a and b c) Find the derivative of tan(x/2)0 2.com . a)Find all values of (1+i 3 )2/3 b)If a+b+c=0 and |a|=3. 4 4 2 4 4 http://pucpcmb. a)Of all the rectangles of constant area show that the square has the least perimeter 3 b)Evaluate ∫ sec x dx c)Find the unit digit of 7129 Example-4: 2 Part E: Answer any ONE question 1.
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