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REVIEW TEST-3 Class : XIII (XYZ) PAPER CODE : A Time : 3 hour Max. Marks : 216 INSTRUCTIONS 1. The question paper contains 72 questions and 16 pages. Each question carry 3 marks and all of them are compulsory. There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Please ensure that the Question Paper you have received contains all the QUESTIONS and Pages. If you found some mistake like missing questions or pages then contact immediately to the Invigilator. 2. Indicate the correct answer for each question by filling appropriate bubble in your answer sheet. 3. Use only HB pencil for darkening the bubble. 4. Use of Calculator, Log Table, Slide Rule and Mobile is not allowed. 5. The answer of the questions must be marked byshading the circles against the question bydark HB pencil only. 6. The answer(s) of the questions must be marked by shading the circles against the question by dark HB pencil only. For example if only 'B' choice is correct then, the correct method for filling the bubble is A B C D the wrong method for filling the bubble are (i) A B C D (ii) A B C D (iii) A B C D The answer of the questions in wrong or any other manner will be treated as wrong. USEFUL DATA Atomic weights: Al = 27, Mg = 24, Cu = 63.5, Mn = 55, Cl = 35.5, O = 16, H = 1, P = 31, Ag = 108, N = 14, Li = 7, I = 127, Cr = 52, K=39, S = 32, Na = 23, C = 12, Br = 80, Fe = 56, Ca = 40, Zn = 65.4, Radius of nucleus =10–14 m; h = 6.626 ×10–34 Js; me = 9.1 ×10–31 kg, R = 109637 cm–1 XIII (XYZ) MATHS REVIEW TEST-3 PART-A Select the correct alternative. (Only one is correct) [24 × 3 = 72] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q.1cir The area of the region of the plane bounded above by the graph of x2 + y2 + 6x + 8 = 0 and below by the graph of y = | x + 3 |, is (A*) 4 2 (B) 4 (C) 2 (D) [Sol. Completing the square, the top curve is the circle (x + 3)2 + y2 = 1 and the lower curve which is a right angle "vee-shape", cuts the circle (of radius 1) into quarters. r2 The area of the region is 4 = 4 Ans.] Q.2st.line Consider straight line ax + by = c where a, b, c R+ and a, b, c are distinct. This line meets the coordinate axes at P and Q respectively. If area of OPQ, 'O' being origin does not depend upon a, b and c, then (A) a, b, c are in G.P. (B*) a, c, b are in G.P. (C) a, b, c are in A.P. (D) a, c, b are in A.P. 1 0 [Hint: A = 0 c a = 2 0 0 1 c b 1 0 1 If A is independent of a, b and c then c2 = ab a, c, b are in G.P. ] Q.3ph-1 If x and y are real numbers and x2 + y2 = 1, then the maximum value of (x + y)2 is (A) 3 (B*) 2 (C) 3/2 (D) [Sol. x2 + y2 = 1 let x = cos and y = sin so (x + y)2 can be written in the form of (cos + sin)2 = 2 sin 2 + 4 maximum value = 2 Ans. ] dx Q.4def The value of the definite integral (1+ xa )(1+ x2 ) (a > 0) is (A*) 4 [Hint: put x = tan (B) 2 (C) (D) some function of a. 2 d

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- 1. REVIEW TEST-3 PAPER CODE : A P A P E R - 1 Class : XIII (XYZ) Time : 3 hour Max. Marks : 216 INSTRUCTIONS 1. The question paper contains 72 questions and 16 pages. Each question carry 3 marks and all of them are compulsory.There is NEGATIVE marking. 1 mark will be deducted for eachwrong answer. Please ensure that the Question Paper you have received contains all the QUESTIONS and Pages. If you found some mistake like missing questions or pages then contact immediately to the Invigilator. 2. Indicate the correctanswer for each question byfillingappropriate bubble inyour answer sheet. 3. Use onlyHB pencil fordarkeningthe bubble. 4. Use ofCalculator, LogTable, SlideRule and Mobile is not allowed. 5. TheanswerofthequestionsmustbemarkedbyshadingthecirclesagainstthequestionbydarkHBpencilonly. 6. The answer(s) ofthe questions must bemarked byshading thecircles against the questionbydark HB pencilonly. For exampleifonly'B'choice is correct then, thecorrect method for fillingthe bubble is A B C D thewrongmethodfor fillingthebubbleare (i) A B C D (ii) A B C D (iii) A B C D The answer of the questions inwrong or anyothermanner will be treated as wrong. USEFUL DATA Atomic weights:Al = 27, Mg = 24, Cu = 63.5, Mn = 55, Cl = 35.5, O = 16, H = 1, P= 31,Ag = 108, N = 14, Li = 7, I = 127, Cr = 52, K=39, S = 32, Na = 23, C = 12, Br = 80, Fe = 56, Ca = 40, Zn = 65.4, Radius of nucleus =10–14 m; h = 6.626 ×10–34 Js; me = 9.1 ×10–31 kg, R = 109637 cm–1
- 2. XIII (XYZ) MATHS (24-9-2006) REVIEW TEST-3 PART-A Select the correct alternative. (Only one is correct) [24 × 3 = 72] There is NEGATIVE marking. 1 mark will be deducted for eachwrong answer. Q.1cir The area of the region of the plane bounded above bythe graph of x2 + y2 + 6x + 8 = 0 and below by the graph of y = | x + 3 |, is (A*) 4 (B) 4 2 (C) 2 (D) [Sol. Completing the square, the top curve is the circle (x + 3)2 + y2 = 1 and the lowercurvewhichis aright angle"vee-shape", cuts the circle (of radius 1)into quarters. The area of the region is 4 r2 = 4 Ans.] Q.2st.line Consider straight line ax + by = c where a, b, c R+ and a, b, c are distinct. This line meets the coordinate axes at Pand Q respectively. If area of OPQ, 'O' being origin does not depend upon a, b and c, then (A) a, b, c are in G.P. (B*) a, c, b are in G.P. (C) a, b, c are in A.P. (D) a, c, b are inA.P. [Hint: A = 2 1 1 0 a c 1 b c 0 1 0 0 = 2 1 ab c 0 2 IfAis independent of a, b and c then c2 = ab a, c, b are in G.P. ] Q.3ph-1 If x and y are real numbers and x2 + y2 = 1, then the maximum value of (x + y)2 is (A) 3 (B*) 2 (C) 3/2 (D) 5 [Sol. x2 + y2 = 1 let x = cos and y= sin so (x + y)2 can be written in the form of (cos + sin)2 = 2 4 sin2 maximum value=2 Ans. ] Q.4def The value of thedefinite integral 0 2 a ) x 1 )( x 1 ( dx (a >0) is (A*) 4 (B) 2 (C) (D) somefunction of a. [Hint: put x = tan I = 2 0 a ) (tan 1 d = 2 0 a a a d ) (cos ) (sin ) (cos I = 4 Ans. ]
- 3. Q.5mod Let a, b, c are non zero constant number then r c sin r b sin r c cos r b cos r a cos Lim r equals (A) bc 2 c b a 2 2 2 (B) bc 2 b a c 2 2 2 (C*) bc 2 a c b 2 2 2 (D) independent of a, b and c [Sol. Let r 1 = x so that as r , x 0 2 0 x x · bc · cx cx sin · bx bx sin cx cos · bx cos ax cos Lim = 2 0 x x cx cos · bx cos ax cos Lim bc 1 use L'Hospital's to get (C) ] Q.6de A curve y = f (x) such that f ''(x) = 4x at each point (x, y) on it and crosses the x-axis at (–2, 0) at an angle of 45°. The value of f (1), is (A) – 5 (B*) – 15 (C) – 3 55 (D) – 3 35 [Sol. f ' (x) = 2x2 + c f ' (–2) = 1 = 8 + c c = – 7 now f ' (x) = 2x2 – 7 f (x) = d x 7 x 3 2 f (–2) = 0 – 3 16 + 14 + d = 0 d = – 3 26 f (x) = 3 1 [2x3 – 21x – 26] f (1) = – 3 45 = – 15 Ans. ] Q.7ph-1 Theminimumvalueofthefunctionf(x)= x cos 1 x sin 2 + x sin 1 x cos 2 + 1 x sec x tan 2 + 1 x cosec x cot 2 as x varies over all numbers inthe largest possible domain of f (x) is (A) 4 (B*) – 2 (C) 0 (D) 2 [Hint: 2 1 1 1 1 y then quadrant 4 x if 0 1 1 1 1 y then quadrant 3 x if 2 1 1 1 1 y then quadrant 2 x if 4 1 1 1 1 y then quadrant 1 x if th rd nd st 2 ymin Q.8 A non zero polynomial with real coefficients has the property that f (x) = f ' (x) · f ''(x). The leading coefficient of f(x) is (A) 6 1 (B) 9 1 (C) 12 1 (D*) 18 1
- 4. [Sol. Degree of f (x) = n; degree of f ' (x) = n – 1 degree of f ''(x) = (n – 2) hence n = (n – 1) + (n – 2) = 2n – 3 n = 3 hence f (x) = ax3 + bx2 + cx + d, (a 0) f ' (x) = 3ax2 + 2bx + c f '' (x) = 6ax + 2b ax3 + bx2 + cx + d = (3ax2 + 2bx + c)(6ax + 2b) 18a2 = a a = 18 1 Ans. ] Q.9def Let Cn = n 1 1 n 1 1 1 dx ) nx ( sin ) nx ( tan then n 2 n C · n Lim equals (A) 1 (B) 0 (C) – 1 (D*) 2 1 [Sol. Cn = n 1 1 n 1 1 1 dx ) nx ( sin ) nx ( tan (put nx = t) Cn = 1 1 n n 1 1 dt ) t ( sin ) t ( tan n 1 L = 1 1 n n 1 1 n n 2 n dt t sin t tan · n Lim C · n Lim ( × 0); L = = n 1 dt t sin t tan 1 1 n n 1 1 applyingLeibnitzrule L = 2 2 1 1 n n 1 ) 1 n ( 1 1 n n sin 1 n n tan Lim = 2 · 4 = 2 1 Ans. ] Q.10complex Let z1, z2, z3 be complex numbers such that z1 + z2 + z3 = 0 and | z1 | = | z2 | = | z3 | = 1 then 2 3 2 2 2 1 z z z , is (A) greater than zero (B) equal to 3 (C*) equal to zero (D) equal to 1 [Sol. 1 1z z = 1 ; 2 2z z = 1 ; 3 3z z = 1 given, z1 + z2 + z3 = 0 3 2 1 z z z = 0 hence (z1 + z2 + z3)2 = 0 2 1 z + 2z1z2z3 3 2 1 z 1 z 1 z 1 = 0 ; hence 2 1 z + 2z1z2z3 [ 3 2 1 z z z ] = 0 2 1 z = 0 Ans. ]
- 5. Q.11p/c Numberof rectangleswith sides parallel to the coordinateaxes whose verticesare all of theform (a, b) with a and b integers such that 0 a, b n, is (n N) (A*) 4 ) 1 n ( n 2 2 (B) 4 n ) 1 n ( 2 2 (C) 4 ) 1 n ( 2 (D) n2 [Hint: number of rectangles = n+1C2 · n+1C2 = 4 ) 1 n ( n 2 2 ] Q.12aod Number of roots of the function f (x) = 3 ) 1 x ( 1 – 3x + sin x is (A) 0 (B) 1 (C*) 2 (D) more than 2 [Sol. f ' (x) = – 4 ) 1 x ( 3 – 3 + cos x < 0 hence f (x) is always decreasing,Also as x , f (x) – and as x – , f (x) + hence one positive and onenegative root Graph is as shown ] Q.13QE If p (x) = ax2 + bx + c leaves a remainder of 4 when divided by x, a remainder of 3 when divided by x + 1, and a remainder of 1 when divided byx – 1 then p(2) is (A) 3 (B) 6 (C) – 3 (D*) – 6 [Hint: p(x) = ax2 + bx + c p(x) = Q1x + 4 = Q2(x + 1) + 3 = Q3(x – 1) + 1 p(0) = c = 4 ....(1) p(–1) = a – b + 4 = 3 b – a = 1 p(1) = a + b + 4 = 1 b + a = – 3 b = – 1; a = – 2 p(x) = – 2x2 – x + 4 p(2) = – 8 – 2 + 4 = – 6 Ans. ] Q.14aod Let f (x) bea function that has a continuous derivative on [a, b], f (a) and f (b)have opposite signs, and f ' (x) 0 for all numbers x between a and b, (a < x < b). Number of solutions does the equation f (x) = 0 have (a < x < b). (A*) 1 (B) 0 (C) 2 (D) cannot be determined [Sol. As f (x) has contained derivative x [a, b] f (x) is continuous and as f (a) & f (b) have opposite signs at least one value a < x = c < b such that f (c) = 0 also since f '(x) 0 f ' (x) > 0 or f '(x) < 0 throughout function is increasing ordecreasing will have onlyonepossible root. ]
- 6. Q.15def Whichofthefollowingdefiniteintegralhasapositivevalue? (A) 3 2 0 dx ) x 3 sin( (B) 0 3 2 dx ) x 3 sin( (C*) 0 2 3 dx ) x 3 sin( (D) 2 3 0 dx ) x 3 sin( [Sol. put 3x + = t (I) 3 dt 3 t sin = 0; (II) 3 dt 3 t sin = 0; (III) 2 7 dt 3 t sin = + ve (IV) 2 7 dt 3 t sin = – ve (C) ] Q.16p&c Let setAconsists of 5 elements and set B consists of 3 elements. Number of functions that can be defined fromAtoBwhichareneitherinjectivenorsurjective,is (A) 99 (B*) 93 (C) 123 (D) none [Sol. numberofmapping = 3C1 (25 – 2) + 3C2 = 90 + 3 = 93 Ans. ] Q.17cir Acircle with centerAandradius 7 is tangent to thesides ofan angle of 60°.AlargercirclewithcenterBis tangent tothesides oftheangleand to the first circle. The radius of thelarger circle is (A) 3 30 (B*) 21 (C) 3 20 (D) 30 [Sol. r = 7 sin 30° = r R r R = 2 1 2R – 2r = R + r R = 3r = 21 Ans. ] Q.18vector The value of the scalar s r · q p can be expressed in the determinant form as (A) s · p r · p s · q r · q (B) r · q s · q s · p r · p (C) s · p r · q s · q r · p (D*) s · q r · q s · p r · p [Sol. s · r ) q p ( = s · p ) q · r ( q ) p · r ( = ) s · p )( q · r ( ) s · q )( p · r ( ] Q.19limit If x 1 0 1 x 1 0 1 x n · x Lim x l = – 5, where , , are finite real numbers then (A) = 2, = 1, R (B) = 2, = 2, = 5 (C) R, = 1, R (D*) R, = 1, = 5 [Hint: Let, L= x x n · x Lim 3 x l ; for limit to exist, 1 now L = x x 1 n x Lim 3 x l = x x x Lim 3 x = 2 x x Lim = – = – 5 5 and R ]
- 7. Q.20ITF If f (x, y) = sin–1( | x | + | y | ), then the area of the domain of f, is (A*) 2 (B) 2 2 (C) 4 (D) 1 [Sol. | x | + | y | 1 interior of a square with vertices (1, 0), (0, 1), (–1, 0), (0, – 1) area = 2Ans.] Q.21s&p A,B and C aredistinct positive integers, less than or equal to10.The arithmetic meanofAand Bis 9. The geometric mean ofAand C is 2 6 . The harmonic mean of Band C is (A*) 19 9 9 (B) 9 8 8 (C) 19 7 2 (D) 17 8 2 [Sol. A + B = 18 .....(1) AC = 72 .....(2) Thereareonlytwopossibilities A = 10 and B = 8 If A= 10 then from (2) C is not an integer. Hence A = 8 and B = 10; C = 9 H.M. between B and C = 9 10 9 · 10 · 2 = 19 180 = 19 9 9 Ans. ] Q.22 If x is real and 4y2 + 4xy+ x + 6 = 0, then the complete set of values of x for which yis real, is (A) x 2 or x 3 (B*) x – 2 or x 3 (C) – 3 x 2 (D) x – 3 or x 2 [Hint: D 0 16x2 – 16(x + 6) 0 (x – 3)(x + 2) 0] Q.23prob Ialternativelytoss afaircoinandthrowafairdieuntil I, eithertossahead orthrowa2. IfItoss thecoin first, the probabilitythat Ithrow a 2 before I toss a head, is (A*) 1/7 (B) 7/12 (C) 5/12 (D) 5/7 [Sol. H: tossing a Head, P(H) = 2 1 ; A : event of tossing a 2 with die, P(A) = 6 1 E: tossinga 2 beforetossinga head P(E) = P ....... or ) A H ( and ) A H ( or A H = 6 1 · 2 1 + 6 5 · 2 1 · 6 1 · 2 1 + ....... = 12 1 + 12 5 · 12 1 + ....... P(E) = 12 5 1 12 1 = 7 1 Ans. ] Q.24mat LetA, B, C, D be (not necessarilysquare) real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix S =ABCD, consider the two statements. I S3 = S II S2 = S4 (A) II is true but not I (B) I is true but not II (C*) both I and II are true (D) both I and II are false.
- 8. [Sol. S = ABCD = A(BCD) = AAT ....(1) S3 = (ABCD)(ABCD)(ABCD) = (ABC)(DAB)(CDA)(BCD) = DTCTBTAT = (BCD)TAT = AAT ....(2) from (1) and (2) 3 S S I is correct multiplybothsidesbyS 4 2 S S II is correct Both I and II are true Ans. ]