Ma Marks : Topics : Ch to Ch 8 HALF SYLLABUS TEST Time : Minutes General instructions : (i) All questions are compulsory (ii) Please check that this question paper contains 9 questions (iii) Questions - in Section - A are Very Short Answer type questions and carry mark each (iv) Questions 5 - in Section - B are Short Answer type question and carry marks each (v) Question - in Section - C are Long Answer type questions and carry marks each (vi) Question - 9 are in Section - D are Long Answer type questions and carry 6 marks each (vii) Please write down the serial numer of the question efore attempting it SECTION A (Each question carries One mark) Q If A is an order n matri such that adj( A) kadja, then find the value of k Q What is the principal domain of cos ec? Q Is a solution of the equation sin ( ) sin? Justify your answer Q Draw a free-hand graph of sin SECTION B (Each question carries Two marks) Q5 If A and B are two matrices such that AB B and BA A then, show that A B A B Q6 To raise money for an orphanage, students of three schools P, Q and R organized an ehiition in their area, where they sold paper ags, used ooks and files made y them using recycled paper at the rate of `5, ` and ` per unit School P sold paper ags, 5 ooks and files School Q sold 6 paper ags, 5 ooks and 5 files and, school R sold 5 paper ags, ooks and 5 files Using matrices, find the total amount raised y each school Q7 Evaluate : cos sin 5 Q8 If tan (cos ) tan (cos ec ), ( ), then find the value of Q9 If the derivative of tan (a ) takes the value at, prove that a Q Side of an equilateral triangle is increasing at the rate of cm/s At what rate is its area increasing when the side of the triangle is cm? Q For what values of, the function f (), is inverse of itself? Q Let * e a inary operation defined on Q as a * a 5 a for all a, Q Find 6* 5 SECTION C (Each question carries Four marks) a Q If A and B and (A B) A B, then find the values of a and
Q Using properties of determinants, prove that a a (a )( c)(c a)(a c) c c OR Using properties of determinants, prove the following : a a a a ( a ) a a y Q5 Prove that : cot cot y cot y OR If sin[cot ( )] cos(tan ), then find Q6 If sin sin y sin z, then prove that y y z z yz d y dy Q7 If a cosθ sin θ and, y a sin θ cosθ, show that y y Q8 Show that the function f (), for all R, is not differentiale at the point Q9 If f () ; g() and h(), then find f [h {g ()}] Q Determine the local maima and local minima, of the function f () sin cos, Also find the local maimum and local minimum values OR Find the minimum value of (a y), where y c Q Evaluate : ( )( ) / cos Q Evaluate : Q Evaluate : sin sin cos sin SECTION D (Each question carries Si marks) Q Find the area enclosed etween the curves y and y, OR Sketch the graph of f () Evaluate, f () What does the value of this integral represent on the graph? Q5 Find the value of p for which 9p(9 y) and p(y ) cut each other at right angles log Q6 Integrate wrt OR Find : ( ) sin sin k Q7 Find the anti-derivative of sin sin k cos cos k Q8 Find the interval in which the function f () log, is increasing Q9 Determine whether the relation R defined on the set R of all real numers as R {(a, ):a, R and a S, where S is the set of all irrational numers}, is refleive, symmetric and transitive /
Half Syllaus Test - SECTION A Q n k, where n is the order of matri Q R (, ) Q Consider LHS and replace : sin sin sin 6 So isn t a solution of the given equation Also, note that sin can t e possile as domain of sin is [, ] Q See NCERT Page 5 Fig (ii) SECTION B Q5 Given AB B ABB BB AI I ie, A I Similarly we have BA A B I Consider LHS: A B I I II II I I A B RHS Q6 Let the amounts raised y schools P, Q and R e, y and z (in `) respectively Paper ags Used Books Files Cost Amounts y school P 5 5 So, Amounts y school Q y 6 5 5 Amounts y school R z 5 5 8 6 y 8 5 57 z 75 6 5 789 By equality of matrices, we get : 6, y 57, z 789 So the amounts raised y school P is `6, y school Q is `57 and y school R is `789 Q7 Consider y cos sin Let sin sin cos 5 5 5 5 cos y cos y 5 y cos Q8 We ve tan cos tan cosec tan tan cosec cos cos cos tan tan tan tan cos sin sin sin cossin sin sin (cos sin ) sin or cos sin or But, so is the only required solution dy Q9 Let y tan (a ) (a ) dy As the derivative of y takes the value at = so, at (a ) a a da Q Let a denote the side length of the triangle so, cms dt of RHS
Since area of the equilateral triangle is A a da cm s dt at acm y Q Let y f (), f () Since f () f () On comparing the coefficients of like terms on oth sides, we get :, Q Given that a * a 5 a for all a, Q da a da dt dt 6 9 So, 6* 6 5 6 5 5 5 5 5 5 SECTION -- C a Q If A and B and (A B) A B, then find the values of a and We ve (A B) A B (A B)(A B) A B AA AB BA BB AA BB a a AB BA So, a a a a By equality of matrices, we get : a a, a, a, On solving these equations, we get : a, Q LHS : Let a a By R R R, R R R c c a a c c Taking (a ) & ( c) common from c c a a (a )( c) c c By R R R c c (a c)(a c) (a c) R and R respectively (c a) common from R (a )( c) c c Taking c c a c (a )( c)(c a) c c Epanding along c c (a )( c)(c a) ( a c)[ ] (a )( c)(c a)(a c) RHS OR LHS : Let a a a a By C C ac a a R
Q5 Let a a a a Taking a common from C a a a a a By R R ar a a a a a Taking a a a a a Epanding along C a a a cot, cot y common from R RHS (i) cot, y cot cot cot y y cot( ) cot( ) cot cot cot y y y By using (i), we get : cot cot y cot y OR Q6 Let We ve sin[cot ( )] cos(tan ) ( ) sin, sin y, sin z Consider LHS : ( ) sin sin cos cos (i) y y z z sin sin sin sin sin sin sin cos sin cos sin cos [sin cos sin cos sin cos ] [sin sin sin ] [sin( ) cos( ) sin cos ] sin( )cos( ) sin cos sin cos( ) sin cos sin [cos( ) cos ] sin cos cos sin cos cos sin cos cos sin sin sin yz RHS [By using (i) Q7 Given a cosθ sin θ and, y a sin θ cosθ a sin θ cosθ and, dy a cos sin θ d d dy dy d a cos sin θ (i) d cosθ a sin θ y dy y By (i), we get : d y dy dy d y dy y y y d y dy y y
( ) ( ), if Q8 We ve f () ( ) ( ), if, if f ( ), f () Differentiaility at = : ( ) LHD (at = ) : lim lim lim ( ) ( ) RHD (at = ) : lim lim lim LHD (at ) ( ) f () is not differentiale at Differentiaility at = : LHD (at = ) : lim lim lim ( ) RHD (at = ) : lim lim LHD (at ) f () is not differentiale at Q9 We have f () ; g() and h() ( ) ( ) f () ; g () and h () ( ) f () ; g () and h () ( ) Now f [h {g ()}] f h f [] ( ) 5 Q We have f () sin cos, f () cos sin, f () sin cos For local points of maima and minima f () cos sin 7 tan,, 7 7 7 f cos sin and, f cos sin 7 f () is maimum at and, minimum at So, Local maimum value f sin cos 7 7 7 And, Local minimum value f sin cos OR Given y c (i) c ds c Let S (a y) S a a and, d S c ds c For local points of maima and/or minima, a c a d S at c a c c a / S is minimum at c a
Also, minimum value of S a y a That is, S ac S c a a c c a a Replacing value of c in (i), we get a y Q Let I I ( )( ) ( )( ) ( )( ) I I log ( )( ) (i) ( )( ) Consider A B C A( ) B( ) C( ) ( )( ) On equating the coefficients of like terms, we get : A, B,C 5 5 5 I log 5 5 5 I log log log tan C 5 5 5 Therefore, I log log tan C 5 5 5 / / cos sec Q Let I I sin On dividing Nr & Dr y cos sec sec tan I Put / So, I sec sec ( tan tan ) I / sec ( tan )( tan ) dt I dt ( t )( t ) t t tan t sec dt When t & when t tan (t) I tan t I 6 / sin cos Q Let I sin Put sin cos t (sin cos ) dt And, (sin cos ) t dt t I I log t () t log I log I SECTION D Q Given curves are y (i) and y (ii) On solving (i) and (ii), we get :, Required area, ar(oabco) [ ] t sin Also when t and when t [y y ] ii i I log log
/ OR Let y f () Here / [ ], f (), When, y y ( ) ( ) verte of the paraola is (, ) When, y Required area f () ( ) ( ) squnits 6 squnits This value represent the area of the curve y = f () enclosed etween = and = Q5 Given 9p(9 y) (i) and, p(y ) (ii) By (i) & (ii), p(y ) 9p(9 y) y 8, 9p (A) On diff (i) & (ii) wrt oth sides, dy dy and 9p p Since the curves (i) & (ii) cut each other at right angles so, p 9p By (A), 9p 9p p(p ) p, We shall reject p since it doesn t fit into the given conditions so, p 9p Q6 Let I I I I I sin sin I sin C ie, I sin C OR log d Let I I log log ( ) ( ) ( ) log log I ( ) I ( ) ( ) log log I log log C I log C ( ) ( ) Q7 sin sin k ( sin ) ( sin k) sin sin k cos cos k sin sin k cos cos k (cos sin ) (cos k sin k) sin sin k cos cosk (cos sin cos k sin k)(cos sin cos k sin k) sin sin k cos cos k (cos sin cos k sin k) sin cos (cosk sin k) C
Q8 We have f () log, f () log( ) log f () f () f () (, ) f () is increasing for all, ie (, ) Q9 We have R {(a, ):a, R and a S, where S is the set of all irrational numers} Refleivity : Let a e any real numer a a S (a, a) R So, R is refleive Symmetry : Let (, ) R S But if (, ) R S That is, (a, ) R doesn t necessarily imply (,a) R a, R So, R isn t symmetric Transitivity : Let (, 5), ( 5, ) R 5 S and 5 5 S But if (, ) R S That is, (a, ) and (,c) R doesn t necessarily imply (a,c) R a,,c R So, R isn t transitive as well y y OR We ve y y [Dividing y oth sides y log log ( ) Clearly y is defined iff y log log ( ) log So, domain is (, ) Now as so, log ( ) log So, the range is y (, ) log ( ) log log (i) y log ( ) log ( ) y