GOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD SCHEME OF VALUATION. Subject : MATHEMATICS Subject Code : 35

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1 GOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD II YEAR PUC EXAMINATION MARCH APRIL 0 SCHEME OF VALUATION Subject : MATHEMATICS Subject Code : 5 PART A Write the prime power factorization of the composite number Define a diagonal matri. Definition : A square matri in which all the elements ecept the principal diagonal elements are zero is called a diagonal matri. [Note : The diagonal elements may or may not be zero] Why the usual division is not a binary operation on the set of real numbers? Since a a,b, R, R when b 0 b OR Any counter eample like, 0 R but 0 is not defined. Simplify (i ˆ ˆ j) k ˆ (i ˆ ˆj) kˆ kˆ kˆ 0 r (Vector notation for zero vector is compulsory)

2 5 Find the centre of the circle + y 6 + y 0 Writing the equation as, + y + y 0 Centre is, 6 For the parabola y 5 what is the length of the latus rectum? Writing the equation as y 5 Length of LR a 5 7 Evaluate sin cos 5 Let cos θ 5 cos θ 5 θ cos θ 5 sin 0 8 Epress it in the polar form z + i() r, θ Polar form is z cis

3 9 Ans: If y sin + cos find d dy dy y 0 d OR dy + 0 d 0 Evaluate : ( + sin ) d Ans: Since the integrand is an odd function by a property ( + sin ) d 0 OR ( + sin ) d cos 0 PART -B If (a, b), (a, c) then prove that (a, bc) Ans: (a,b) a + by and (a,c) a + cy Where, y,, y I (If one of the above equations is correct give one mark) (a + by )(a + cy ) a(a + c y + b y ) + bc(y y ) a + bcy, y I (a, bc) where

4 Solve by Cramer s rule : y 5 + y 7, 7 7 y Prove that the identity element of a group (G, *) is unique. If e and e are two different identity elements Then e e e e e and e e e e e e e Identity element is unique Find the area of parallelogram whose diagonals are i ˆ + ˆj k ˆ and ˆi ˆj + kˆ r r Let d (,, ) and d (,, ) r r d ˆ ˆ ˆ d i 5j 7k r r 8 Area of the parallelogram d d sq. units 5 + y and y are equations of two diameters of a circle whose radius is 5 units. Find the equation of the circle. Finding the centre (, - ) Writing the equation ( ) + (y + ) 5

5 5 6 Find the centre of the ellipse 5 + 6y 00 + y 8 0 Reducing to the standard form ( ) (y + ) centre (, ) 7 If tan + tan y then show that + y + y + y tan y + y tan + y + y y 8 If + cos θ then show that one of the values of i is e θ Getting cos θ+ 0 cos θ ± cos θ cos θ ± isin θ, e iθ 9 If y sin + sin + sin upto then show that dy cos d y Ans: Getting y sin + y Squaring and differentiating and getting dy cos d (y )

6 6 0 Find the y-intercept of the tangent drawn to the curve y at (,-) on it Ans: Getting dy m d (, ) Equation of the tangent is y the y-intercept of the tangent is Evaluate : cos d 8 + cos cos cos d d 8 + cos 9 sin Put sin t cos d dt dt sin + 9 t G.I sin C Find the order and degree of the differential equation d y dy a + d d Ans: Order of D.E I Degree of D.E PART - C Find the G.C.D of and 09 and epress it in two ways in the form m + 09 n where m, n z Ans: Division Writing (, 09) 09 (9) + ( 70) OR ( 70) + 09(9)

7 7 m 70, n 9 ( 70) + 09 (9) getting m 9, n 9 OR m 79, n 0. Solve by matri method +y+z 0, y+z and +y z 9 Ans: Writing AdjA X A B and A A A Cofactor matri (any four correct co-factors give one mark) 5 7 Adjoint of A Getting, y and z 5 Show that the set G { n } n z w.r.t usual multiplication. forms an abelian group Ans: Note : Particular eamples do not carry any mark Closure law Associative law

8 8 0, is the identity element n n n G, ( ) G Commutative law r r r r r r r r r 6 (a) Prove that a + b b + c c + a a b c Ans: r r r r r r r r r r r r r r r r (a + b). b c + b a + c c + c a r r r r r r r r r a. (b c) + a.(b a) + a.(c a) r r r r r r r r r + b.(b c) + b.(b a) + b.(c a) Writing L.H.S (a + b). ( b + c) ( c + a) r r r r r r a. (b c) + b. (c a) r r r a b c [ Deduct one mark if and are not mentioned ] 6 (b) r r θ being angle between a ˆi + ˆj k ˆ and b i ˆ + ˆj kˆ find the value of cos θ r r a. b cos θ r r a b II 7(a) (Without the formula if the substitution is correct give one mark) Find the equation of the tangent to the circle + y + g + fy + c 0 at a point P(, y ) on it. Correct figure [ No figure or wrong figure carries zero mark] + g Slope of the tangent at (, y ) y + f

9 9 Getting the equation, + yy + g( + ) + f (y + y ) + c 0 7 (b) What is the value of k if the circles and + y y + k 0 + y + k + y + 0 cut each other orthogonally? Ans: Writing the condition g g + f f c + c Getting k 0 8 (a) y Find the equation of the ellipse in the form +, if the a b distance between foci is 8 and the distance between the directricies is. Writing ae 8 and Getting a 8, e and Writing the equation a e OR b be 8 & e b 8 y b 8, e and a 8 y (b) Find a point on the parabola y whose focal distance is 5 Focal distance + a 5 Getting a and, y ± The point is (, ) OR (, ) (any one point is sufficient)

10 0 9 (a) Solve sin + sin Let sin A sin A cos A sin B sin B cos B and A + B Taking cos and getting Simplifying and getting ( ) (b) Find the general solution of tan θ+ tan θ+ tan θ tan θ Getting tan θ+ tan θ tan θ tan θ Writing n tan θ and θ +, n I 9 III 0 (a) Differentiate tan (a ) w.r.t from first principles. Writing dy tan (a + a δ) tan(a ) lim d δ 0 δ sin(a + a δ) cos cos(a + a δ) sin lim cos(a + a δ)cos δ 0 sin aδ lim δ 0 (cos a ) δ. cos(a + a δ ) dy d a sec a OR

11 dy d lim δ 0 [ ] a tan (a δ ) + tan (a + a δ) tan a a δ a sec a 0 (b) Differentiate w.r.t : (cos ) Let y (cos ) log y log (cos ) dy d + Differentiating and getting (cos ) [ tan log cos ] (a) If y e mtan then show that ( + ) y + ( + ) y m y 0 m tan y me. + Again differentiating ( + ) y + y m y Substituting for y in the R.H.S and getting ( + ) y + ( + ) y m y 0 (b) Differentiate cos( + ) w.r.t cos Let y cos ( + ) and z cos Getting dy d dz sin( + ). and sin d dy sin( + ) dz sin

12 (a) If + y + y + 0 and y then show that dy d ( + ) + y y + ( + y) y ( + ) Getting [ ] ( y) + y + y 0 Writing y + dy d ( + ) [for correct direct differentiation give one mark] (b) Evaluate ( ) d + 5 Writing I d d ( 5) C (a) Evaluate e ( + )log ( e ) d e Put e t I log t dt t log t t. dt t t log t t + C

13 (b) Evaluate (tan + cot ) d I (tan + cot + ) d (sec cos ec + ) d tan cot + C OR sin cos I + d d cos sin sin cos ec d cot + C Find the area of the circle + y a by the method of integration Correct figure [ no figure, deduct one mark ] Area of the circle a a y d a d 0 0 a a + sin a a 0 a a + + (0) sin () [0 0] a sq. units PART D 5 (a) Define hyperbola as locus of a point and derive its equation in y the standard form a b

14 Definition as locus of a point only correct figure getting cs ae and a cz e Writing ps e p m (e > ) Substituting for ps and pm and simplifying to get y where b a (e ) a b (No figure or wrong figure carries zero mark) 5 (b) Prove that b + c c + a a + b a b c q + r r + p p + q p q r y + z z + + y y z By c c + c c L.H.S C c + a a + b r r + p p + q z z + + y c c + a a + b r r + p p + q z z + + y by c c c, c a a + b r p p + q z + y by c c c, c a b r p q z y

15 5 Interchanging c, c and c, c a b c p q r y z 6 (a) State and prove the De Moivre s theorem for all rational indices Statement : If n is an integer and n (cos θ+ isin θ ) cos nθ+ isin nθ if n is a fraction then one of the values of n (cos θ+ isin θ) is cos nθ+ isin nθ Proving for positive integers Proving for negative integers Proving for fractions 6 (b) Prove by vector method : cos( α +β ) cosα cosβ sin α sin β Let X O X and YOY be the co-ordinate aes uuur uuuur Let OP, OQ be two unit vectors with XOP ˆ α and XOQ ˆ β X Y O Y β P M Q X Correct figure with eplanation uuur OP cos α ˆi + sin α ˆj uuur and OQ cosβˆi sinβˆj uuur uuur uuur uuur OP. OQ OP OQ cos ( α +β) uuur uuur Writing OP. OQ cos α cosβ sin αsin β

16 6 and concluding cos( α +β ) cosα cosβ sin α sin β 7 (a) A man 6 ft tall walks at a rate of ft / sec away from the source of light which is hung 5 ft above the horizontal ground. i) How fast is the length of his shadow increasing? ii) How fast is the tip of his shadow moving? Figure A [ no figure carries zero mark] 5' 6' P B Q y O From similar triangles, AB PQ BO QO 5 + y y 6 y d dy dy. feet / sec dt dt dt Let z + y dz d dy 0 + feet / sec dt dt dt 7 (b) Find the general solution of cos θ + cos θ+ cos 5θ + cos 7θ 0 Writing cosθ cos θ + cos5θcos θ 0 cos θ [cosθ+ cos5 θ ] 0 cos θ. [cos θ.cos θ ] 0 θ n ±

17 7 n OR θ ± θ (n + ) 8 θ (n + ) θ n ± where n is an integer θ (n + ) 8 8 (a) Prove that log e ( + ) d log Put tan θ and writing I log ( + tan θ) dθ 0 Also I log + tan θ dθ 0 0 log d θ + tan θ log dθ log ( + tan θ)dθ 0 0 I log.dθ log [ θ] I log (b) Find the general solution of dy ( ) ( y ) d dy y d dy y y d +

18 8 dy d C + y sin - y + sin - +C (without C deduct one mark) PART E 9 (a) Find the cube roots of + i and represent them in Argand diagram. Ans: r θ + i cis cis n + 6n + cis Writing 7 z0 cis, z cis and 9 9 z cis 9 Y Argand diagram : Z Z0 X O X Z Y

19 9 9 (b) Find the length of the common chord of the intersecting circles + y + 6y 0 and + y + y 0 Figure A 5 5 (,-) (-,) C P C B Getting radical ais 6 8y 0 C P AP 5 length of the chord AB 5 unit 9 (c) Find the number of incongruent solutions and the incongruent solutions of linear congruence 5 (mod 6) (5, 6) no. of incongruent solution (a) r If a. b r r + r are two unit vectors such that ( a b) r r vector then show that angle between a and b is r r show that a b is also a unit and also r r r r Given a + b a + b r r r r a + b. a + b Getting ( ) ( ) r r a. b and θ

20 0 r r r r r r r r a b. a b a + b a.b Writing ( ) ( ) Getting r r a b and a b r r 0 (b) Evaluate tan ( )d I tan. tan d (sec ) tan d sec tan d (sec ) d tan tan + + C 9 [ If two integrals are correct give one mark] + 0 (c) Differentiate w.r.t : log5 ( log e ) Writing y log (log ) log5 e dy.. d log 5 log *****

PRACTICE PAPER 6 SOLUTIONS

PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6