POSSIBLE AREAS OR UNITS OF SELECTION OF QUESTIONS IN II P.U. ANNUAL EXAMINATION- OF KARNATAKA, INDIA

ALGEBRAELEMENTS 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 Ruleb). 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 productTIPS: 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 (x1,y1).

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 conicCONCENTRATE MORE ON: CONIC SECTION

TRIGNOMETRY:INVERSE TRIGNOMETRIC FUNCTIONS: Problems using the concept of tan-x ± tan-1y, sin-1x ± sin-1y etcFinding 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. Sin2x, cos2x, log ax, etc by first principles method. (Definite)b). Problems on second order Differentiation.

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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 minimaINTEGRATION:

c). One question on Problems on Integrals of the form(Particular forms)

1abcosx ,

1absinx ,

1acosxbsinxc

1acos2 xbcos2 xc

1a2±x2

1 x2−a2

1x x2±a2

1a2±x2

1x2−a2 a2± x2 , x2−a2 ,

pxqax2bxc

, pxq

ax2bxc,

pcosxqsinxacosxb sinx , ex [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 formSTRESS 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 MATHEMATICSPART -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/c2.Find the number of incongruent solutions of 6x≡ 9(mod15)

3.If a/b and b/a then prove that a= ± b4.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/b7.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−30 2 ] is a scalar matrix ,

find x and y

• Evaluate ∣4996 49974998 4999∣

• Find x such that [ x−1 20 3] has no

multiplicative inverse.

• Find the value of ∣1 3 32

3 32 33

32 33 34∣• Solve for x in ∣x x

3 x∣ =4

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• If A= 2 34 5 B= 6 7

8 9 find AB'

• Find x such that [ 2 4x5x2 1 ] is

symmetric matrix..

• If 0 2x−63x−4 0 is a skew

symmetric matrix, find x.

• If the matrix [a 2 −3b c 5d e f ] is skew

symmetric matrix , find the value of a+b+c+d+e+f.

• Find the characteristic roots of the matrix

[1 42 3 ] (2M or 1M)

• Evaluate ∣4996 49974998 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 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=ab prove that * is not a binary operation.•On Q, a*b= ab/4 find the identity•On the set Q1 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 2a3b•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× 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 x2 +y2 -2x-7y-8=0 on the x axis.•Find the equation of circle with centre (3,-2) and touching x axis.•If the equation 3x2 -py2 +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 x2 +y2 +4x+6y-7=0.

•Show that the circle x2 +y2 -4x-2y+1=0 touches y axis.•Find the length of tangent from (1,2) to the cricle x2

+y2 -x+2y+1=0• Find the power of the point (-2,1) w.r.t. the circle x2 +y2 -3x +5y -7 = 0• Find the equation of the tangent to the circle x2+y2=30 at ( 5, 5 ) on it.

6.Conic section: Level: skill•Find the angle between asymptotes of

hyperbola x2

a2−y2

b2 =1

•Find the eccentricity of the ellipse x2

a2y2

b2=1

•Write the equation of the director circle of the

hyperbola x2

a2−y2

b2 =1 having 10 and 8 as the

lengths of axes.•Find the vertex of the parabola 20x2 +20x-32y+53=0

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•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) x2

25

y2

1=1

x2

9 y2

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 y2 =8x

7. Inverse trigonometric function (Level: U): •Show that cos-1(-x)=π-cos-1x for x>0.•Show that tan-1x +cot-1x =π/2•Evaluate cos(cosec-15/4)

•Find the value of cos-1(cos(-5850))

•Find 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]

8. Complex number: Level : knowledge● Express the complex number 3 -i in

polar form

● Find the imaginary part of 1

74i● Find the multiplicative inverse of (5+6i)2 ● Find the complex conjugate of (3-7i)2 ● Find the modulus of 1+ 3 I

● Express 32i4−3i in the form a+ib.

● Write the modulus of the function 1+cosθ+i sinθ, 0<θ<π

● Find the amplitude of

1+cos16 +i sin

16

9. Differentiation and application of derivatives :Level: skill•Define derivative of y=f(x) at a point x=a.•If y=cosec3 (1/x) find dy/dx•If x=ct y=c/t find dy/dx•Differentiate x2 log2x w.r.t x•Find the derivative of log(secx+tanx)

•Differentiate sin(1/x) w.r.t x

•Differentiate 5x2

6x2 w.r.t x

•If y=log e1+sinhx then find dy/dx•Differenatiate elog sinx w.r.t x

10.Integration or definite integrals: Level: knowledge

Evaluate ∫1

e 1xdx Evaluate ∫

1

3dx

1x2

Evaluate ∫1

2

x2dx Evaluate ∫1

2

x15 /3dx

Evaluate ∫0

1 1−x1x

dx Evaluate ∫2

3 dxx−1

Evaluate ∫0

3

tanx dx

Evaluate ∫ x 3 x21dx Evaluate ∫1

∞ 1x2 dx

Evaluate ∫−

2

2

sin3 x dx

PART -CANSWER 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 5700 ≡ 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 2301 is divided by 7?

b)If (a, b)=1 and (a, c)=1, Prove that (a, bc)=16) 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.

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7)a) Prove that the number of primes is infinite. b)Find the digit in the unit place of 7123 .

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 31 x2 31 2 x3∣ =0

2)State cayley Hamilton theorem and verify

it for A= 1 −34 −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 a2

ca b b2

ab c c2∣ =(a-b)(b-c)(c-

a)

5)a)Prove that ∣bc ca abca ab bcab bc ca∣

=2 ∣a b cb c ac a b∣

b)Find x, y and z if

[ x 2 −35 y 21 −1 z ] [3 −1 2

4 2 52 0 3]

= [ 5 3 319 −5 161 −3 0 ]

6)Prove that ∣x x2 y zy y 2 z xz z 2 x y∣

=(x-y)(y-z)(z-x)(x+y+z)7)Find the inverse of the matrix :

[−3 5 −14 −1 20 8 −2]

8)Prove that ∣a21 ab acab b21 bcac bc c21∣

=1+a2 +b2 +c2

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 bc b−x ab a c−x∣

=0

11)Prove that ∣ a2 bc acc2

a2ab b2 acab b2bc c2 ∣

=4a2b2c2

12)Show that ∣ −bc b2bc c2bca2ac −ac c2aca2ab b2ab −ab ∣

=(ab+bc+ca)3

13) a) Show that ∣x y yy x yy y x∣ =(x+2y) (x- y )2

b)If the matrix [ 2 x 34 1 6−1 2 7] has

no inverse, find x14)State caley-Hamilton theorm and find A3 if

A= [1 22 1] using Caley-Hamilton theorem

15)State Cayley-Hamilton theorem and find the

inverse of the matrix [1 32 −1] using the

theorem.

16) a)Find the inverse of [a 0 00 b 00 0 c] where a,

b,c are real numbers b)If matrix A=[aij] = i+2j, i=1,2,3 and j=1,2 . Find the matrix A

see other standard problems using properties of determinants.

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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 xx 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 = ab2 iii) a*b =

ab5 a, b ЄQ+ (each carries 5 M)

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, Show that (Q1,*) is an abelian group and solve 5*x=3 in Q1.6)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 x12 x=7

12)Show that set of all matrices of the form Aα = [cos −sinsin cos ] 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 ×cb)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 ABCb)If a =i+j+2k and b =3i+j-k, find the cosine of the angle between a and b3)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 2a=i×a×i j×a× j k×a×k 5)a) Prove that [a×b , b×c c×a]=[a bc ]2 b)Find the sine of the angle between the vectors i-2j+3k and 2i+j+k=06)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

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the direction of a×b×c b)If cosα, cosβ and cosγ are the direction ratios of the vector 2i+j-2k, Show that cos2α +cos2 β+cos2γ=1.8)a)Prove that [ab ,bc ,ca]=2 [abc ] 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

sinC b)For any three vectors a, b, c Prove that a×b×c b×c×ac×a×b =010) 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

AB . BC BC . CA CA. AB=−32

a2

11)a)Prove that sin(A+B) =sinAcosB +cosA.sinB by vector method b)If | ab |=5 and a is perpendicular to b . Find | a−b |

12)a)Prove that [ab bc ca] =2 [a b c ]

b)Find the projection of a =i+2j+3k on b =2i+j+2k

13. a) If a=a1 ia2 ja3 k show that i×a×i j×a×j k×a×k =2a 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 Marks1)Define orhoganality of two circles. Find the condition for the circles x2 +y2 +2g1x+2f1y+c1

=0 and x2 +y2 +2g2x +2f2y +c2 =0 to cut orthoganally.2)Find the condition that the line y=mx+c may be tangent to the circle x2 +y2 =a2 . Also find the point of contact3)Derive the equation to the tangent to the circle x2 +y2 +2gx+2fy+c=0 at point (x1 ,y1 ) on it.

4)Define Radical axis, Show that radical axis is perpendicular to line joining the centres of two circles5)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=06)Find the equation of the circle which passes through the point (2,3), cuts orthogonally the circle x2 +y2 -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 x2 +y2 +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 (x1 , y1 ) to the circle x2 +y2 +2gx+2fy+x=0.11)Find the equation of the circle which passes through the point (2,3), cuts orthoganally the circle x2 +y2 -2x-4y-5=0

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Questions carrying 5 Marks12)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 3 313)Find the equation of the circle, cutting the three circles x2 +y2 +4x+2y+1=0, 2x2 +2y2 +8x+6y-3=0 and x2 +y2 +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 x2 +y2 -4x+2y-3=0 orthogonally.b) Questions carrying 2 Marks1)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 (4,0) and (0, -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−c4)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 x2 +y2 -4x-10y+25=0 and x2 +y2 +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 3b)Find the points on the circle x2 +y2 =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 x2 +y2 -2x+3y+k=0 and x2 +y2 +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

x2 +y2 +2g1x +2f1y +c1 =0 and x2 +y2 +2g2x +2f2y+c2 =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 x2+y2-4x-6y-3=0;

4. a)QUESTION ON CONIC SECTION: ( Questions carrying 3 M)1)Find the centre and foci of the hyperbola 9x2 -4y2 +18x-8y-31=02)Find the equation of parabola having vertex (3,5) and focus (3,2)3)Find the centre and foci of the ellipse 3x2 +y2 -6x-2y-5=0

4)If the normal at one end of latus rectum of the ellipse x2

a2y2

b2=1 passes through other

end of the minor axis the prove that e4 +e2 =1, where 'e' is the eccentricity of the ellipse.5)Find the eccentricity and equations to directrices of the ellipse 4x2 +9y2 -8x+36y+4=06)Find the equaton of tangent and normal to a)Parabola b)Ellipse c)Hyperbola at (x1 , y1 )[Each carrying 3 marks].7)Find the equation of tangent and normal to a)parabola b)Ellipse c)Hyperbola at t (Each carrying 3M)

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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 each1)Find the equations of the asymptotes of hyperbola 9x2 -4y2 =36. Also find the angle between them.2)Find the equation of the tangent at any point (t) on the hyperbola y2 =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 x2

12

y2

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-1x +cos-1y +cos-1 z=π, then prove that x2 +y2 +z2 +2xyz=1(other problems of this type)

2)Show that tan−1 34tan−1 3

5tan−1 8

19=4

3)Solve for x: cos-1x -sin-1 x =cos-1x 3 4)Solve for x: sin-1 x +sin-12x =23

4)Find sin(cos-1 1/3 - sin-1 2/3)

5)If sin−1 2x1x2 cos−1 1− y2

1 y2 tan−1 2z1−z2=2 Prove that x+y+z=xyz

b)QUESTION ON GENERAL SOLUTION OF TRIGNOMETRIC EQUATIONS

1)Find the General solution of sin2 θ = 2−34

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. 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)ex, logx, , ax, xn , etc iii)sin2x, cos2x etc iv)sin-1x, cos-1x, tan-1x 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+ x2−1 ), Prove that (x2 -1)y1 +xy1=02)If y= emcos

−1 x Prove that (1-x2)y2 -xy1 -m2y=03)If y=sin(m sin-1x) Prove that (1-x2)y2 -xy1 +m2y=04)If y=acos(logx)+bsin(logx), Prove that x2y2 +xy1 +y=0.5)If y=cos(atan-1x) show that (1+x2)y2 +2x(1+x2)y1 +a2y=0.

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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 AP2 +BP2 is minimum.2)Find the minimum value of xex 3)Find the angle between the curves x2 +y2 +3x-8=0 and x2 +y2 =5 at (1,2)4)Find the equation to tangnet to the curve y=6x-x2 where the slope of the tangent is -4.

5)Show that sinx(1+cosx) is maximum when x= 3

8. a)Some standard problems on Application of Differentiation or Integration(3M each)1). Show that the curves 2y=x3 +5x and y=x2 +2x+1=0 touch each other at (1,3). Find the equation to common tangnet.

2)Evaluate ∫ x21x41

dx

3)Evaluate ∫ dx54sinx

dx

4)Evaluate ∫ x1x24x5

dx

5)Prove that

∫ dxa2x23/2

= xa2a2 x2

C

6)Evaluate ∫ dx54cosx

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 x2 -6x+3 is a)increasing b)decreasing3)If the displacement s metres of a particle at time 't' seconds is given by s=2t3 -5t2 +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-40t2 . When and where does the car stops.

5)If the law of motion is s2 =at2 +2bt+c then show that acceleration is a−v2

swhere v is

velocity.9. a)QUESTIONS CARRYING 3M EACH: Evaluating the integrals of particular types

1)Evaluate ∫−a

a

a−xax

dx 2)Evaluate ∫ 2xinxcosx3sinx−2cosx

dx

3)Differentiate tan-1 1x2−1x w.r.t tan-1 2x

1x2 4)Differentiate tan-1 1−x

1x w.r.t sin-1 2x1x2

5)If 1−x21− y2=a x− y Prove that dydx = 1− y2

1−x2

6)If y= x2 x2−a2−

a2

2cosh−1 xa then prove that

dydx = x2−a2

b)QUESTIONS CARRYING 2M EACH: 1)Evaluate ∫ 4x3

. x2dx 2)Differentiate (sin-1x)x w.r.t x

3)Evaluate ∫ 154x−4x2

dx

4)Integrate sinx

13−9sin2 xw.r.t x

5)Integrate x2

4x61w.r.t x

6)Integrate 1

4x2−4x2 w.r.t x

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10. 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

3)Find the area of the ellipse x2

a2y2

b2 by integration.

4)Find the area enclosed between the parabola y2 =4x and the line y=2x-4.5)Find the area of the circle x2 +y2 =a2 using integration.6)Find the area between the curves x2 =y and y=x+27)Find the area enclosed between the parabola y2 =4ax and x2 =4by

PART DANSWER 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 x2

a2y2

b2=1

2)Define Hyperbola as a locus of a point and Derive the equation of the same in the standrad

form x2

a2−y2

b2=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 x2

a2−y2

b2=1

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. Find the equations of the asymptotes of the Hyperbola x2

a2−y2

b2=1 . What is meant by rectangular hyperbola.

8)Show that an equation 9x2 +5y2 -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 x2

a2−y2

b2=1 , Write equation to the

locus of the point of intersection of perpendicular tangents to the hyperbola x2

a2−y2

b2=1 and

write the name of the equation..12. ON COMPLEX NUMBERS: (6M each)

1)If cos +cos +cos =0=sin +sin +sin , Prove that

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i) cos 3 +cos 3 + cos 3 =3 cos( + + ) sin 3 + sin 3 + sin 3 =3 sin( + + )

ii) cos2 +cos2 +cos2 =sin2 +sin2 +sin2 =32

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 theorem4)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 z1 and z2 are any two non zero complex numbers, Prove that |z1z2|=|z1||z2| and arg(z1 z2)=arg z1 +arg z2

ii)Prove that 1sinicos1−sin−icos =i(tanθ+secθ)

6) If z1 and z2 are any two non zero complex numbers, Prove that

|z1z2|=|z1||z2| and arg(z1 z2)=arg z1 +arg z2 and arg z1

z2 =argz1 -argz2 . Use these results

to find that modulus and amplitude of 1i1−3 i1−i .

7)Find all values of 3i 13 and represent then on an Argand diagram, also find their

continued product.

8)Show that the continued product of four values of 12i 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 mm3)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 moves4)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 112 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 xm+n =am-n y2n , the power of subtangent varies as the nth power of subnormal.

8)Show that the curve x2

Ay2

B=1 and

x2

a2y2

b2 =1 cut each other orhoganally if

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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.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 x1 x2 dx=

8log2

2)Prove that

∫0

2a

f x dx=∫0

a

f x dx∫0

2a

f 2a−x dx

and evaluate ∫0

xsinx

1sinxdx

3)Prove that

∫a

b

f x dx=∫a

b

f ab−x dx and

hence evaluate ∫1

3 4−x4−x x

dx

4)Evaluate ∫0

x dx

a2 cos2 xb2 sin2 x

5)Prove that ∫a

b

f x dx=∫0

a

f t dt and

evaluate ∫0

1 log 1 x1 x2 dx

6)Prove that ∫0

2

log sinx dx =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

x23x3−7x dx

8)Prove that ∫0

a

f x dx=∫0

a

f a−x dx and

hence show that ∫0

4

log 1tanxdx

9)Evaluate ∫0

x tanx

sec xtan xdx

10)Show that ∫0

2

log sinx dx = −2log2

11)Prove that ∫0

a

f xdx =

∫0

a

f a−xdx and hence evaluate

∫0

2 sinx sinxcosx

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.

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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 ∣ab∣=∣a−b∣ a is perpendicular to bii) Given abc=0 | a | | b | | c | to find the angle between any two vectors etcPractice questions: 1. If a , b and c be three vectors such that abc=0 and and ∣a∣ =3, | b |=5 and| c |=7. find the angle between a , and b . (4M) (Ans:600)2.If ∣ab∣=∣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. bc b. ca c.ab=0 and ∣a∣=1, |b | =4, |c | =8 then find abc 2 (2M)5. If a , b and c be three vectors such that abc=0 , find the value of a .bb.cc.a(2M)6. If |a | is a unit vector and (x- a ).(x+ a )=8 , then find | x |. (4M)7. If a , b and ab 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 bc , b is perpendicular to ca and c is perpendicular to ab the show that | abc |= 529. If a , and b are vectors of equal magnitudes prove that ab is orthogonal to a−b (2M)10. For any two vectors a , b prove that a) ∣ab∣2 = |a |2 + |b |2 +2. a .bb) ∣a−b∣2 = |a |2 + |b |2 -2. a .b c) ∣ab∣2 + ∣a−b∣2 =2( |a |2 + |b |2 )

d) ∣ab∣2 - ∣a−b∣2 =4. a .b (Each carries 2M) PUC PCMB11. 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) 237 b)753 (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 3605.Find all incongruent solutions of 2x ≡ 4(mod10) (2M)6.Find the least non negetive remainder when 250 is divided by 7.(2M)7.Find the reminder when 768x217x87 is divided by 11 (2M)8.Prove that 310 ≡ 1(mod 31).9.Find the reminder in the following divisions: a)350 by 7 b)5225 by 3 c)2125 by 11d)2575 by 13 e)2100 by 19 (Each question carries 2M)10. Solve 4x-3 ≡ -2x+6(mod 11)

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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 =a2 and (x-k)2 +(y-h)2 =a2 ans: length= 4a2−2h−k 22.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 2−4c3.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 -x+3y-10=0 intercepted by the line x+y+2=0. Ans: 4 35.Find the length of the chord of the circle x2 +y2 -4x-2y-20=0 which is bisected at (2,3). Ans: 221 (EACH ABOVE QUESTIONS CARRIES 4M)6.Find the length of the chord of the circle x2 +y2 -6x-15y-16=0 intercepted by x axis ,Ans :10(2M)7.Find the length of the chord of the cirlc x2 +y2 -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):

32.Find continued product of cube roots of 1+i 33.Find the fourth roots of complex number 1- 3 I and represent them in the argand diagram.4.Find the fourth rootsof 3-3ib)Problems related to the cube roots of unity 1, using the properties of .1.If 1, w, 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)=64c)(2- ω)(2- ω2)(2- ω10)(2- ω11)=49d)(a+b)(a+b ω)(a+b ω2)=a3 +b3

e)(1+ ω)(1+ ω2)(1+ ω4)(1+ ω8)........to 100 factors =1 (Each carries 2M)CALCULUS:

(a)Finding the derivative of functions of the following type onlyLogaf(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 e1+sinx c)y=log tanx when x is measured in degrees.

d)y=log a−xax 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 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.

rp

RA

A

BMC1

1.Find all the fourth roots of -i PUC PCMB

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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 28.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 (sin3x, cosec3x, sec3x ,cos3x etc problems confined to power 3 and 4 only of any trigonometric functions)PRATICE QUESTIONS: Inegrate the following w.r.t x a) sin3x, b) cos3x c) cosec3x, 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 ∫0

a xa−x x

etc PUC PCMB

PRACTICE QUESTIONS:

1.Evaluate ∫1

3 x4−x x

2. Evaluate ∫0

4018 x 4018−x x

3. ∫0

4018 2x

2x24018− x dx

4. ∫0

2

25−5c1 23 x2−5c3 22 x35c4 2 x4− x5dx (Hint: G.E= ∫ 2− x5 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 yd 2 x

=[1 dydx ]32

2. a2 d 2 ydx2 =[b dy

dx

2]34 3. d 2 y

d 2 x=[1 dydx

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.dydx =ytan2x when x=0, y=2

2.dydx =2ex y3 when c=0, y=1/2

3.xy dydx =y+2 when x=2, y=0

4.(y2 +y)dx +(x2 +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 1i 3 4 b)Find the length of the common chord of intersecting circles x2 +y2 -4x-5=0 and x2 +y2 -2x+8y+9=0 4 c)Find the remainder when 520 is divided by 7 240.a)Show that maximum rectangle that can be inscribed in a circle is a square 4 b)Evaluate ∫ cosec3 x dx 4

c)Find the order and degree of Differential equation d 2 yd 2 x

=[1dydx ]12

Example -2: PART E: Answer any ONE question

1. a)If a is a unit vector and x−a . xa =8 then find ∣x∣ 4b)Find the equation of chord of circle x2 +y2 -2x+4y-17=0 bisected at (-1,2) 4c) Find the derivative of sin(3x)0 2

2. a)Show that maximum rectangle that can be inscribed in a circle is a square 4b)Evaluate ∫ tan4 x dx 4

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c)Find the order and degree of the differential equation a2 d 2 ydx2 =[b dy

dx

2]34 2

Example-3: PART E:Answer any ONE question

1. a)Find all values of (1+i 3 )2/3 4b)If a+b+c=0 and |a|=3, |b|=5 |c|=7 find the angle between a and b 4c) 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 4b)Evaluate ∫ sec3 x dx 4c)Find the unit digit of 7129

Example-4: 2Part E: Answer any ONE question1.

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