Namma Kalvi Mathematics (Sample Question Papers only for Practice)

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1 th STD. Time : ½ hours Namma Kalvi Sample Question Paper Mathematics (Sample Question Papers only for Practice) Kind Attention to the Students From this year onwards, blue print system has been abolished. Please note that questions will be framed from IN-TEXT portions ALSO. Written Eam Marks : 9 Marks Approimately % of the questions will be asked from IN-TEXT portions. These questions will be based on Reasoning and Understanding of the lessons. Further, Creative and Higher Order Thinking Skills questions will also be asked. It requires the students to clearly understand the lessons. So the students have to think and answer such questions. It is instructed that henceforth if any questions are asked from out of syllabus, grace marks will not be given. Term Test, Revision Test and Model Eam will be conducted based on the above pattern only. Concentrating only on the book-back questions and/or previous year questions, henceforth, may not ensure to score % marks. Also note that the answers must be written either in blue ink or in black ink. Avoid using both the colour inks to answer the questions. For MCQs, the answers should be written in full. Simply writing (a) or (b) etc. will not get full marks. You have to write (a) or (b) etc., along with the answer given in the options. 7

2 th STD. Sura s Model Question Paper Time :. Hours Mathematics Marks : 9 Section - I Note : (i) Answer all the questions. [ ]. If A (ii) Choose the correct or most suitable answer from the given four alternatives. Write the option code and the corresponding answer. and B and (A B) A B, then the values of a and b are () a, b () a, b () a, b () a, b. If A to a y and if y, then det (A A T ) is equal a () (a ) () (a ) () a () (a ). If A is skew-symmetric of order n and C is a column matri of order n, then C T AC is () an identity matri of order n () an identity matri of order () a zero matri of order () an identity matri of order. The vectors a b, b c, c a are () parallel to each other () unit vectors () mutually perpendicular vectors () coplanar vectors. 5. Two vertices of a triangle have position vectors i j k and i j k. If the position vector of the centroid is i j k, then the position vector of the third verte is () i j 9k () i j 6k () i j 6k () i j 6k between them is 6 and their scalar product is then a is () () () 7 () 7. The derivative of f () at is () 6 () 6 () does not eist () 8. If the derivative of (a 5)e at is, then the value of a is () 8 () () 5 () 9. If f (), then f (f()) at is () 8 () () () 5. If f : R Ris defined by f() for R, then the lim f is equal to () () () () e sin. lim is () () () (). If lim sin p, then the value of p is tan () 6 () 9 () (). If f ()d g () c,then f () g ()d () f d ò ò () f gd () f ( ) gd () ( g ) d. If d k c, then the value of k is () log () log () - () log log 6. If a and b having same magnitude and angle

3 Std - Mathematics Sura s Model Question Paper If f e d e ( ) c, then f () is. Let a and b be the position vectors of the points () c () () 6 c () c c 6. The gradient (slope) of a curve at any point (, y) is. If the curve passes through the point (, 7), then the equation of the curve is, () y () y () y () y 6 cos 7. tan d is cos () c () c () c () c 8. There are three events A, B and C of which one and only one can happen. If the odds are 7 to against A and 5 to against B, then odds against C is () : 65 () 65: () : 88 () 88: 9. If A and B are two events such that P(A)., P(B).8 and P(B/A).6, then P( A B) is ().96 (). ().56 ().66. A man has fifty rupee notes, hundred rupees notes and 6 five hundred rupees notes in his pocket. If notes are taken at random, what are the odds in favour of both notes being of hundred rupee denomination? () : () : () : () : (i) (ii) Section - II Answer any SEVEN questions. Question number is compulsory. 7. Determine the value of y if y y y 6 a. If A, then compute A. A and B. Prove that the position vectors of the points which trisects the line segment AB are a b b a and.. If PO OQ QO OR, prove that the points P, Q, R are collinear. 5. Evaluate : lim sin. 6. If the limit does not eist eplain why? lim( ) 5 y 7. Find the derivatives from the left and from the right at (if they eist) of the following functions. Are the functions differentiable at? f (). 8. Show that the following functions are not differentiable at the indicated value of, <. f ;, 6 9. cos( 5 ) 9e. 6. An eperiment has the four possible mutually eclusive and ehaustive outcomes A, B, C, and D. Check whether the following assignments of probability are permissible. (i) P(A).5, P(B)., P(C)., P(D). (ii) P(A)., P(B).8, P(C). 6, P (D). (iii) P(A) 5, P(B) 5, P(C) 5, P(D) 5

4 5 Std - Mathematics Sura s Model Question Paper (i) (ii) Section - III Answer any SEVEN questions. Question number is compulsory. 7. If A, show that A is a unit matri. a b. Give your own eamples of matrices satisfying the following conditions in each case: (i) A and B such that AB BA. (ii) and B such that AB BA, A and B. (iii) A and B such that AB and BA.. If D and E are the midpoints of the sides AB and AC of a triangle ABC, prove that BE DC BC.. Verify whether the following ratios are direction cosines of some vector or not. (i),, (ii),, (iii), 5 5 5, 5. lim y tan sec 7. y tan q (sin q cos q) 8. If f 5 and f (), find f (). 9. If f 9 6 and f (), find f ().. An integer is chosen at random from the first positive integers. What is the probability that the integer chosen is a prime or multiple of 8? Section - IV Answer all questions (a) If A and such that (A I)(A I), find the value of. (b) Give your own eamples of matrices satisfying the following conditions in each case: (i) A and B such that AB BA. (ii) A and B such that AB BA, A and B. (iii) A and B such that AB and BA.. (a) Find the direction cosines and direction ratios for the following vectors. (i) i j 8k (ii) i j k (iii) j (iv) 5i j 8k (v) i k j (vi) i k (b) Show that the points whose position vectors i 5 j k, j k, i 9 j k and i j k are coplanar.. (a) Eamine the continuity of the following: (i) sin (ii) cos (iii) e tan (iv) e sin (v). ln (vi) 6 (vii) (viii) (i) () cot tan (b) Find the points of discontinuity of the function f, where 5, if (i) f() 5, if >, if (ii) f (), if <, if (iii) f(), if > π sin, (iv) f () π π cos, < <. (a) If f(), test whether f ( ) eists. (b) Eamine the differentiability of functions in by drawing the diagrams. (i) sin (ii) cos.

5 Std - Mathematics Sura s Model Question Paper (a) A ball is thrown vertically upward from the ground with an initial velocity of 9. m/sec. If the only force considered is that attributed to the acceleration due to gravity, find (i) how long will it take for the ball to strike the ground? (ii) the speed with which will it strike the ground? and (iii) how high the ball will rise? (b) A wound is healing in such a way that t days since Sunday the area of the wound has been decreasing at a rate of cm per day. If on Monday ( t ) the area of the wound was cm (i) What was the area of the wound on Sunday? (ii) What is the anticipated area of the wound on Thursday if it continues to heal at the same rate? 6. (a) (i) The odds that the event A occurs is 5 to 7, find P(A). (ii) Suppose P(B). Epress the odds that the 5 event B occurs (b) The probability that a new railway bridge will get an award for its design is.8, the probability that it will get an award for the efficient use of materials is.6, and that it will get both awards is.. What is the probability, that (i) it will get at least one of the two awards ( ii) it will get only one of the awards. 7. (a) The probability that a car being filled with petrol will also need an oil change is.; the probability that it needs a new oil filter is.; and the probability that both the oil and filter need changing is.5. (i) If the oil had to be changed, what is the probability that a new oil filter is needed? (ii) If a new oil filter is needed, what is the probability that the oil has to be changed? (b) An advertising eecutive is studying television viewing habits of married men and women during prime time hours. Based on the past viewing records he has determined that during prime time wives are watching television 6% of the time. It has also been determined that when the wife is watching television, % of the time the husband is also watching. When the wife is not watching the television, % of the time the husband is watching the television. Find the probability that (i) the husband is watching the television during the prime time of television (ii) if the husband is watching the television, the wife is also watching the television.. () a, b SECTION - I. () (a ). () a zero matri of order. () coplanar vectors. 5. () i j 9k 6. () 7. () 8. () 9. (). (). (). (). () f d. () - log 5. () c 6. () y 7. () c 8. () 65: 9. ().56. () : SECTION - II y 7 7y. Solution : Given 5 7 y 6 Equating the corresponding entries on both sides we get, y 7 [Equating a ]... () 6 [Equating a ]... () From (), Substituting in () we get, y 7 y 7 y \ y 5

6 5 Std - Mathematics Sura s Model Question Paper a. Solution : Given A A a a a a a A A.A a a A a a A a. Solution : a b A B P Q Let a and b be the position vectors of the points A and B. OA a and OB b. Let P divides the line segment AB in the ratio : and Q divides the line segment AB in the ratio : 5. Solution : f () Let f() sin \ OP.( OB) ( OA) ( b) ( a) b a and OQ ( OB) ( OA) b a a b Hence, the required position vectors are b a and a b.. Solution : Given PO OQ QO OR PQ QR [By triangle law of addition] PQ QR and Q is a common point. Hence, the points P, Q, R are collinear f () sin.. sin (. )..998 lim sin sin.. sin (. ).999. sin.. sin (. ) sin sin sin...998

7 Std - Mathematics Sura s Model Question Paper Solution : lim 5 ( ) y At, the value of the curve on y-ais is. \ lim ( ) 7. Solution : Given f() f ( ) f f () lim ( ) lim \f ( ) \f ( ) f f () lim \f ( ) Since the one sided derivatives f ( ) and f ( ) are not equal, f () does not eist. \ f is not differentiable at., < 8. Solution : f ;, \f ( ) f f lim f f ( ) \f ( ) [f() ]... () From () and (), f ( ) f ( ) Hence f() is not differentiable at. 9. Solution : cos( 5 ) 9e cos( 5 d ) 9 e d 5. sin... () f f ( ) ( ) 6- d d 6 d 6 9. e - 6. log c 6 sin 5 e - -6 log 6 c. Solution : (i) Given P(A).5, P(B)., P(C)., P(D). P (A), P (B), P(C) and P(D) Also, P(S) P(A) P(B) P(C) P(D).5... The assignment of probability is permissible. (ii) Given P(A)., P(B).8, P(C).6 and P(D). Also, P(S) The assignment of probability is not permissible. (iiii) Since P (C) is negative, the assignment 5 of probability is not permissible

8 5 Std - Mathematics Sura s Model Question Paper Solution : SECTION - III Given A a b A A.A a b a b a a b b \ A is a unit matri.. Solution : (i) Let A AB BA 5 and B \ AB ¹ BA (ii) Let A and B AB BA Hence AB BA and A ¹, B ¹. (iii) Let A AB BA and B \ AB and BA ¹. Solution : Let the position vectors of the vertices of the DABC be a, b and c respectively. OA a, OB b and OC c D A a B C b c Since D is the mid-point of the side AB, E

9 Std - Mathematics Sura s Model Question Paper OD a b and E is the mid-point of the AC... () OE a c... () BE OE OB a c b a c b DC OC OD a b c c a b a c b c a b \ BE DC a c b c a b c b ( c b) ( OC OB ) \ BE DC BC Hence proved.. Solution : (i) Given ratios are 5, 5 and 5. Let the ratios are l \ l m n 5, m 5, n [From ()] Hence, the given ratios are not the direction cosines of any vector. (ii) Let l \ l m n, m and n Hence, the given ratios are direction cosines of some vector. (iii) Let l, m, n \ l m n ¹ Hence, the given ratios are not the direction cosines of any vector. 5. Solution : lim 5 5 Multiplying and dividing by we get, lim 5 5 ( ) 5 ( 5) ( ) 9 5 ( 5) 5 5 ( 5) Solution : tan Given y sec sec d d. tan tan sec dy ( ). d d d sec sec ( sec ) tan sec tan sec sec sec tan sec tan sec

10 56 Std - Mathematics Sura s Model Question Paper sec ( sec tan ) sec tan () sec sec sec tan sec tan sec sec ( tan ) tan sec sec tan sin cos sec sec cos dy cos sin. cos d 7. Solution :Given y tan q (sin q cos q) dy d tan θ d d. ( sin θ cos θ) sin θ cos θ. tan θ dθ dθ tan θ( cos θ sin θ) ( sin θ cos θ) sec θ sin θ ( cos θ sin θ) sin θ cos θ cos θ cos θ sin θ sin θ cos θ sin θ cos θ cos θ cos θ sin θ sin θ sin θ. cos θ cos θ cos θ cos θ sin θ sin θ tan θ. sec θ cos θ cos θ sin θ tan θ sec θ. sin θ cos θ cos θ dy sin q cos q tan q sec q. d 8. Solution :Given f () 5and f() Integrating on both sides with respect to we get, f d ( 5) d f() d 5 d 5 c f() 5 c... () Given f() f() () 5() c 8 c c c Substituting C in () we get, f() 5 9. Solution : Given f () 9 6 and f() Integrating on both sides we get, f d 9 6 d f () 9 d6 d 9 f () - 6 c c... () Since f() f() () () c c Substituting c. Solution : - in () we get, f () f () ( Let S {,,... } n (S) Let A be the event of getting a prime number and B be the event of getting multiple of 8. A {,, 5, 7,,, 7,9,, 9,, 7,,, 7, 5, 59, 6, 67, 7, 7, 79, 8, 89, 97} n (A) 5 n (B) B {8, 6,,,, 8, 56, 6, 7, 8, 88, 96} n (A or B) n (A) n (B) 5 7 P(A or B) P( A) P( B) 7 n( S) [ A and B are mutually eclusive events, A B ϕ] SECTION - IV.(a) Solution : Given A Also, (A I) (A I) \ A I

11 Std - Mathematics Sura s Model Question Paper A I 9 9 \(A I) (A I) 9 9 ( ) ( ) ( 9 ) Equating the corresponding entries we get, or or 7 or (b) Solution : (i) Let A Since alone satisfies the equation (A I) (A I), we get. AB BA 5 and B \ AB ¹ BA (ii) Let A AB BA and B Hence AB BA and A ¹, B ¹. (iii) Let A AB and B BA \ AB and BA ¹.(a) Solution : (i) Given vector is i j 8k The direction ratios of i j 8k are,, 8. y z 8 Hence, its direction cosines are ,,

12 58 Std - Mathematics Sura s Model Question Paper (ii) Give vector is i j k (iii) The direction ratios of i j k are,,. r y z Hence, its direction cosines are,, Given vector is j The direction ratios of j are,,. y z Hence, its direction cosines are,,,,. (iv) The given vector is 5i j8k The direction ratios are 5,, 8. ( ) r y z Hence, the direction cosines are 5 8,, (v) The given vector is i k j i jk The direction ratios are,,. r y z Hence, the direction cosines are (vi) The given vector is i k The direction ratios are,,.,, y z Hence, the direction cosines are,,,, (b) Solution : Let the position vectors of the given vector be OA i 5 j k OB jk OC i 9 j k and OD i j k Let a b and c Also, let a AB OB OA ( jk) ( i 5 j k ) i6 j k AC OC OA ( i 9 j k ) ( i 5 j k ) i j k AD OD OA ( i j k ) ( i 5 j k ) 8i j k s b t c i6 j k s( i j k ) t( 8i j k ) i6 j k ( s 8t) i (s t) j (s t) k Equating the like components on both sides, we get s 8t... () 6 s t... () s t... () () 6 s t () 6 s t Adding, t t Substituting t in () we get, s 8 s 6 s 6 6 Substituting t, and s in () we get, which satisfies equation ().

13 Std - Mathematics Sura s Model Question Paper Thus, one vector is the linear combination of other two vectors. Hence, the given points are co-planar..(a) Solution : (i) Let f () sin (ii) (iii) Since the algebraic function is continuous for all R and the circular function sin is continuous for all Î R. f () sin is continuous for all R. Let f() cos The algebraic function and the circular function cos is continuous for all R. f() cos is continuous for all Î R. Let f() e tan We know the eponential function e is continuous for all R. But tan is not continuous at odd multiples of π. \ f() e tan is continuous for all R (n ) π, n Z. (iv) Let f() e (v) (vi) Since the eponential function e and algebraic function ( ) is continuous for all R. f() e is continuous for all R. Let f(). In() The algebraic function is continuous for all R, but the logarithmic curve eists only for positive values of f() is continuous in (, ). Let f() sin f() does not eist for. But sin is continuous for all R. f() is continuous only in R {} (vii) Let f() 6 The algebraic function is continuous for all R. Since the curve f() does not eist for, the given function is continuous only in R { }. (viii) Let f() The modulus function is continuous for all R. \ f() is continuous for all R. (i) Let f() () The modules function is continuous for all R. But the given curve does not eist for. \ f() is continuous in R { }. Let f() cot tan cot is not continuous in multiples of p and tan is not continuous in (n ) π. π π \ f() cot tan is not continuous in ( n ), f() is continuous in R nπ, n Z. (b) Solution : 5, if (i) Given f() 5, if > lim f lim ( 5 ) () 5 7 lim f lim ( 5 ) 5 7 Since the lim f f ¹ lim f() is not continuous at. (ii) Given f() lim f lim f, if, if < Also f() \ lim f f f () \ f() is continuous in R., if (iii) Let f (), if > lim f 8 5 lim f 5 Also, f() 8 5 \ lim f f \ f () is continuous in R. f() 5

14 6 Std - Mathematics Sura s Model Question Paper π sin, (iv) f() π π cos, < < lim f cos cos π π π lim f sin sin π π π π Also, f sin π π \ lim f f f π π \ f() is continuous in, π.(a) Solution :Given f() \ f ( ) f f f f ( ) ( ) ( ) lim ( ) ( ) ( ) lim \ f ( ) ( ) ( )... () f f ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ]99... () From () and (), f () is not differentiable at f () does not eist. (b) Solution : (i) Let f() sin when, f()! sin! π π π π π π 5π 5 π π π f() 5π π π π π π π π π y y π The curve f() sin has got vertical tangents at π, π, π, π etc. \ f() sin is not differentiable at nπ, n Z. (ii)let f() cos when, f() cos π π π π π π 5π 5π f() 5π π π π π π π π π y f() cos has got vertical tangents at π, π, π, π, 5π, 5π etc. π \ f() cos is not differentiable at (n ), n Z. 5.(a) Solution : Given initial velocity u 9. m/sec. y Distance ut gt when g 9.8 m/sec 9.t (9.8) t \ 9.t.9 t... () (i) For the ball to strike the ground, 9.t.9t 9.t.9t 9..9t [ t ¹ ] 5π π

15 Std - Mathematics Sura s Model Question Paper t \ t 8 sec. (ii) To find the speed with which will strike the ground to find velocity at t 8 sec. we know 9.t.9t v when t d 9..9 (t) dt 8 sec. v (8) Since velocity cannot be negative, m/sec. v (iii) At maimum height, velocity t t t 9. m/sec. 9. sec. 98. \ Maimum height 9.().9 (6) [From ()] m (b) Solution :Let Sunday be the initial period, da Given dt ( t ) cm/day da ( t ) dt Integrating both sides we get, ò da ( t ) dt A t ( ) c A (t ) c A c t... () When t, A cm c c c () becomes, A... () t (i) (ii) On Sunday (t ), initial period A 5.5 sq. cm From Sunday to Thursday, there are days When t, () becomes A.5 sq. cm. 6 (b) Solution : < as is large 7 ( ) 7...! ( ) < ( )...! Since is large, is very small and hence higher powers of are negligible. Thus 7 7 and.therefore (a) Solution :The odds of an event A are a : b in favour of an event and a P(A) a b (i) Given that odds of an event A occurs is 5 : 7 P(A) (ii) Given P(B) [ a and b ] 5 The odds that the event B occurs is to.

16 6 Std - Mathematics Sura s Model Question Paper (b) Solution :. Let the events be as follows : A : getting an award for its design B : getting an award for efficient use of materials. Given P (A).8, P(B).6 and P (A B). (i) (ii) P (getting atleast one of the two awards) P (A B) P(A) P (B) P (A B) P (getting only one of the awards) P(A B ) P( A B) P (A) (A B) P(B) P (A B) P(A) P(B) P (A B).8.6 (.) (a) Solution : Let the events be defined as follows : B : Car being filled with petrol will also need an oil change. P (B). E : Car needs a new oil filter P (E ). P (B E ).5 (i) If the oil had to be changed, the probability that a new oil filter is needed P(E /B) P E B 5. P(B)..5 (ii) If a new filter is needed, the probability that the oil has to be changed. (b) Solution : ( ) P(B/ E ) P B E P E Let the events be defined as follows : A : Event of husband and watching the television B : Husband is watching the television. Given P(A ). P(B/A ).6 P (A ). P(B/A ).6. (i) P (Husband watching the television) P(B) P(A ). P(B/A ) P(A ). P(B/A ) P(B) (.) (.6) (.) (.) P(B).. P(B) P(B) 5 (ii) P (if the husband is watching, the wife is also watching the television) P (A / B) P (A / B) P (A/ B) P(A ). P(B /A ) P(A ). P(B /A ) P(A ). P(B /A ) (. )( 6. ) P(B)