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# Dpp 13th Maths WA.pdf

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### Dpp 13th Maths WA.pdf

1. MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) DPP. NO.-29 This is the test paper of Class-XI (J-Batch) held on 24-09-2006.Take exactly 75 minutes. Q.1 If tan , tan  are the roots of x2 – px + q = 0 and cot , cot  are the roots of x2 – rx + s = 0 then find the value of rs in terms of p and q. [4] Q.2 Let P(x)=ax2 +bx + 8 isaquadratic polynomial. Iftheminimum valueofP(x)is 6when x =2,find the values of a and b. [4] Q.3 Let P =             1 n 2 1 1 n 10 then find log0.01(P). [4] Q.4 Prove the identity 1 A 4 sec 1 A 8 sec   = A 2 tan A 8 tan . [4] Q.5 Find the general solution set of the equation logtan x(2 + 4 cos2x) = 2. [4] Q.6 Find the value of                 17 cos ........ 5 cos 3 cos cos 17 sin ......... 5 sin 3 sin sin when  = 24  . [4] Q.7(a) Sumthefollowingseriestoinfinity 7 · 4 · 1 1 + 10 · 7 · 4 1 + 13 · 10 · 7 1 + ........... (b) Sum thefollowingseries upton-terms. 1 · 2 · 3 · 4 + 2 · 3 · 4 · 5 + 3 · 4 · 5 · 6 + ............. [3 + 3] Q.8 The equation cos2x – sin x + a = 0 has roots when x  (0, /2) find 'a'. [6] Q.9 A, B and C are distinct positiveintegers, less than or equal to 10.The arithmeticmean ofAand B is 9. The geometric mean ofAand C is 2 6 . Findthe harmonic mean of B and C. [6] Q.10 Express cos 5x in terms ofcos x andhence find general solution ofthe equation cos 5x = 16 cos5x. [6] Q.11 If x is real and 4y2 + 4xy+ x + 6 = 0, then find the complete set of values of x for which yis real. [6] Q.12 Findthesumofalltheintegralsolutionsoftheinequality [6] 2 log3x – 4 logx27  5. Q.13 If + +  =  2 , show that                                                 2 tan 1 2 tan 1 2 tan 1 2 tan 1 2 tan 1 2 tan 1 =            cos cos cos 1 sin sin sin . [7] Q.14(a) In any ABC prove that c2 = (a – b)2cos2 2 C + (a + b)2sin2 2 C . (b) In any  ABC prove that a3cos(B – C) + b3cos(C – A) + c3cos(A – B) = 3 abc. [4 + 4]
2. MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) Time: 40 Min for each Dpp DPP. NO.-30,31,32 DPP-30 Q.1 8 claytargets have been arranged in vertical column, 3 being in the first column, 2 in the second, and 3 in the third. In how manyways can theybe shot (one at a time) if no target below it has been shot. Q.2 Evaluate:      0 2 2 dx ) x (cos cos ) x (sin sin x Q.3 Evaluate:     0 2 2 dx ) x cos(sin ) x sin(cos x Q.4          2 0 2 dx x cos x sin x x Q.5 Prove that             0 n 2 n 3 1 1 n 3 1 = 3 3  DPP-31 Q.1 If cosA, cos B and cos C are the roots of the cubic x3 + ax2 + bx + c = 0 whereA, B, C arethe angles of a triangle then find the value of a2 – 2b – 2c. Q.2 Find all functions, f : R  R satisfying    2 x ) x ( F ) x ( F 2 ) x ( x   f = 0  x  R where f (x) = F'(x). Q.3          2 2 3 2 1 dx x 3 1 x Q.4 For a > 0, b > 0 verify that     0 2 dx a bx ax x n l reduces to zero by a substitution x = 1/t. Using this or otherwiseevaluate:     0 2 dx 4 x 2 x x n l Q.5            0 3 1 dx x x tan DPP-32 Q.1 Evaluate:   dx x cos x sin x cos x sin 3 4 4 , x         2 , 0 Q.2 Evaluate:   dx x cos x sin x cos x sin 3 4 4 Q.3 Integrate:   x sin x cos dx 3 3 . Q.4 1 n n 2 2 n 2 n 1 n n Lim               Q.5 Let I =    2 0 dx x sin b x cos a x cos and J =    2 0 dx x sin b x cos a x sin , where a > 0 and b > 0. Compute the values of Iand J.
3. MATHEMATICS Daily Practice Problems Select the correct alternative. (Only one is correct) There is NEGATIVE marking and 1 mark will be deducted for each wrong answer. Q.1 Findthesum oftheinfiniteseries ..... 63 1 45 1 30 1 18 1 9 1      (A) 3 1 (B) 4 1 (C) 5 1 (D) 3 2 Q.2 Numberofdegreesin thesmallest positiveanglex such that 8 sin x cos5x – 8 sin5x cos x = 1, is (A) 5° (B) 7.5° (C) 10° (D) 15° Q.3 There exist positive integersA, Band C with no common factors greater than 1, such that Alog2005 + B log2002 = C. The sumA+ B + C equals (A) 5 (B) 6 (C) 7 (D) 8 Q.4 A trianglewith sides 5, 12 and 13 has both inscribed and circumscribed circles. The distance between the centres of these circles is (A) 2 (B) 2 5 (C) 65 (D) 2 65 Q.5 Thegraphofacertaincubic polynomialis as shown. Ifthepolynomial can be written in the form f (x) = x3 + ax2 + bx + c, then (A) c = 0 (B) c < 0 (C) c > 0 (D) c = – 1 Q.6 Thesidesofa triangleare6 and8andtheangle betweenthesesidesvariessuch that0° <  < 90°.The length of 3rd side x is (A) 2 < x < 14 (B) 0 < x < 10 (C) 2 < x < 10 (D) 0 < x < 14 Q.7 The sequence a1, a2, a3, .... satisfies a1 = 19, a9 = 99, and for all n  3, an is the arithmetic mean of the first n – 1 terms. Then a2 is equal to (A) 179 (B) 99 (C) 79 (D) 59 Q.8 If b is the arithmetic mean between a and x; b is the geometric mean between 'a' and y; 'b' is the harmonic mean between a and z, (a, b, x, y, z > 0) then the value of xyz is (A) a3 (B) b3 (C) a b 2 ) b a 2 ( b3   (D) b a 2 ) a b 2 ( b3   Q.9 GivenA(0, 0),ABCD is a rhombus of side 5 units where the slope ofAB is 2 and the slope ofAD is 1/2. Thesum of abscissa and ordinate of the point C is (A) 5 4 (B) 5 5 (C) 5 6 (D) 5 8 CLASS : XIII (ALL) TIME: 60 Min. DPP. NO.-33
4. Q.10 A circle of finite radius with points (–2, –2), (1, 4) and (k, 2006)can exist for (A) no value of k (B) exactlyone value of k (C) exactlytwovalues of k (D)infinitevaluesofk Q.11 If a ABC is formed by3 staright lines u = 2x + y – 3 = 0; v = x – y = 0 and w = x – 2 = 0 then for k = – 1 the line u + kv = 0 passes throughits (A)incentre (B)centroid (C) orthocentre (D)circumcentre Q.12 If a,bandc arenumbers for whichtheequation 2 2 ) 3 x ( x 36 x 10 x    = x a + 3 x b  + 2 ) 3 x ( c  is an identity, , then a + b + c equals (A) 2 (B) 3 (C) 10 (D) 8 Q.13 If a, b, c are in G.P. then a b 1  , b 2 1 , c b 1  are in (A)A.P. (B) G.P. (C) H.P. (D) none Q.14 How manyterms are there in the G.P. 5, 20, 80, .........20480. (A) 6 (B) 5 (C) 7 (D) 8 Q.15 The sum of the first 14 terms of the sequence x 1 1  + x 1 1  + x 1 1  + ....... is (A) x 1 ) x 11 2 ( 7   (B) x 1 ) x 7 1 ( 7   (C) ) x 1 )( x 1 )( x 1 ( 14    (D) none Q.16 If x, y> 0, logyx + logxy= 3 10 and xy = 144, then arithmetic mean of x and y is (A) 24 (B) 36 (C) 2 12 (D) 3 13 Q.17 Acircleofradius R is circumscribedabout aright triangleABC. If ris theradius ofincircleinscribed in triangle thenthearea ofthetriangle is (A) r(2r + R) (B) r(r + 2R) (C) R(r + 2R) (D) R(2r + R) Q.18 The simplest form of a 1 1 1 a 1    is (A) a for a  1 (B) a for a  0 and a  1 (C) – a for a  0 and a  1 (D) 1 for a  1 Select the correct alternatives. (More than one are correct) Q.19 If the quadratic equation ax2 + bx + c = 0 (a > 0) has sec2 and cosec2 as its roots then which of the followingmustholdgood? (A) b + c = 0 (B) b2 – 4ac  0 (C) c  4a (D) 4a + b  0
5. Q.20 Which of the followingequations can have sec2 and cosec2 as its roots ( R)? (A) x2 – 3x + 3 = 0 (B) x2 – 6x + 6 = 0 (C) x2 – 9x + 9 = 0 (D) x2 – 2x + 2 = 0 Q.21 The equation 1 x 10 2 | 2 x |   = x 3 | 2 x |  has (A)3integralsolutions (B)4 real solutions (C)1primesolution (D)noirrational solution Q.22 Whichofthefollowingstatementsholdgood? (A)If M is the maximum and m is the minimum valueof y= 3 sin2x + 3 sinx · cosx + 7 cos2x then the mean of M and m is 5. (B)The value of cosec  18 – 3 sec  18 is a rational which is not integral. (C) If x lies in the third quadrant, then the expression x 2 sin x sin 4 2 4  + 4 cos2         2 x 4 is independent ofx. (D) There are exactly 2 values of  in [0, 2] which satisfy4 cos2 2 2 cos 1 = 0. PART-B MATCH THE COLUMN INSTRUCTIONS: Column-Iand column-IIcontainsfour entries each. Entries ofcolumn-Iareto bematchedwith some entriesofcolumn-II.Oneormorethanoneentriesofcolumn-Imayhavethematchingwiththesameentries ofcolumn-IIandoneentryofcolumn-Imayhaveoneormorethanonematchingwithentriesofcolumn-II. Column-I Column-II Q.1 (A) Area ofthetriangle formedbythe straight lines (P) 1 x + 2y – 5 = 0, 2x + y – 7 = 0 and x – y + 1 = 0 insquareunits is equal to (Q) 3/4 (B) Abscissaofthe orthocentre of thetriangle whose vertices are the points (–2, –1); (6, – 1) and (2, 5) (R) 2 (C) Variable line 3x( + 1) + 4y( – 1) – 3( – 1) = 0 for different values of  areconcurrent at the (S) 3/2 point (a, b). The sum (a + b) is (D) The equation ax2 + 3xy – 2y2 – 5x + 5y + c = 0 represents twostraight lines perpendiculartoeach other, then | a + c | equals Column-I Column-II Q.2 (A) In a triangleABC,AB = 3 2 , BC = 6 2 ,AC > 6, (P) 60° and area of the triangleABC is 6 3 .  B equals (Q) 90° (B) In a triangleABC is b = 3 , c = 1 andA= 30° (R) 120° thenangleBequals (C) In a  ABC if (a + b + c)(b + c – a) = 3bc (S) 75° then Aequals (D) Area of a triangleABC is 6sq. units. If the radii of its excircles are2, 3 and 6then largest angle of the triangle is
6. Column-I Column-II Q.3 (A) The sequence a, b, 10, c, d is an arithmetic progression. (P) 10 The value of a + b + c + d (B) Thesidesofright triangleform athreeterm geometric (Q) 20 sequence. Theshortest side has length2. Thelength of the hypotenuse is of the form b a  where a  N (R) 26 and b is a surd, then a2 + b2 equals (C) Thesumoffirst threeconsecutivenumbers ofan (S) 40 infinite G..P.is70, ifthetwoextremes bemultipled each by4, and the mean by5, the products are inA.P. The first term of the G.P. is (D) The diagonals ofaparallelogram haveameasure of 4 and6 metres. Theycut off formingan angleof 60°. If theperimeterofthe parallelogram is   b a 2  where a, b  N then (a + b) equals
7. MATHEMATICS Daily Practice Problems This is the test paper of Class-XI (PQRS & J) held on 19-11-2006. Take exactly 75 minutes. Q.1 Consider the quadratic polynomial f (x) = x2 – 4ax + 5a2 – 6a. (a) Findthesmallest positiveintegral value of'a'for which f(x) is positive foreveryreal x. (b) Find the largest distance between the roots of the equation f (x) = 0. [2.5 + 2.5] Q.2(a) Find the greatest value of c such that system of equations x2 + y2 = 25 x + y = c has areal solution. (b) The equations to a pair of opposite sides of a parallelogram are x2 – 7x + 6 = 0 and y2 – 14y + 40 = 0 findtheequationstoits diagonals. [2.5+2.5] Q.3 Find the equationof the straight linewith gradient 2 ifit intercepts a chord of length 5 4 on the circle x2 + y2 – 6x – 10y + 9 = 0. [5] Q.4 Thevalueoftheexpression, x sin x cos x 2 cos 3 x 2 cos 6 6 3   whereverdefinedisindependentof x.Withoutallotting aparticularvalueofx, find thevalueofthis constant. [5] Q.5 Findthegeneralsolutionoftheequation sin3x(1 + cot x) + cos3x(1 + tan x) = cos 2x. [5] Q.6 Ifthethirdandfourthtermsofanarithmeticsequenceare increasedby3and8respectively,thenthe first fourterms form ageometric sequence. Find (i) thesum of the first fourterms ofA.P. (ii) second term of the G.P. [2.5+2.5] Q.7(a) Let x = 3 1 or x = – 15 satisfies the equation, log8(kx2 + wx + f ) = 2. If k, w and f are relativelyprime positive integers then find the value of k + w + f. (b) The quadratic equation x2 + mx + n = 0 has roots which are twice those of x2 + px + m = 0 and m, n and p  0. Find the value of p n . [2.5+2.5] Q.8 Line 1 8 y 6 x   intersects the x and y axes at M and N respectively. If the coordinates of the point P lying inside the triangle OMN (where 'O' is origin) are (a, b) such that the areas of the triangle POM, PON and PMN are equal. Find (a) the coordinates ofthe point P and (b) the radius of the circle escribed opposite to the angle N. [2.5+2.5] Q.9 Startingattheorigin,abeamoflighthitsamirror(intheformofaline)atthepointA(4,8)andisreflected at the point B(8, 12). Compute the slope of the mirror. [5] Q.10 Find the solution set of inequality, ) x x ( log 2 3 x   < 1. [5] CLASS : XIII (XYZ) TIME: 75 Min. DPP. NO.-34
8. Q.11 If the first 3 consecutive terms of a geometrical progression are the roots of the equation 2x3 – 19x2 + 57x – 54 = 0 find the sum to infinite number of terms of G.P. [5] Q.12 Find the equationto the straight lines joining the origin to thepoints of intersection ofthe straight line 1 b y a x   andthe circle 5(x2 +y2 +bx +ay)=9ab.Also find thelinearrelation betweena and bso that thesestraightlinesmaybeatright angle. [3+2] Q.13 Let f (x) = | x – 2 | + | x – 4 | – | 2x – 6 |. Find the sum of the largest and smallest values of f (x) if x  [2, 8]. [5] Q.14 If c x 4 x 3 x b x 3 x 2 x a x 2 x 1 x          = 0 then all lines represented by ax + by + c = 0 pass through a fixed point. Findthe coordinates ofthat fixedpoint. [5] Q.15 If S1, S2, S3, ... Sn,.... are the sums of infinite geometric series whose first terms are1, 2, 3, ... n, ... and whose common ratios are 2 1 , 3 1 , 4 1 , ...., 1 n 1  , ... respectively, then find the value of    1 n 2 1 r 2 r S . [5] Q.16 In anytriangle if tan 2 A = 6 5 and tan 2 B = 37 20 then find the value of tan C. [5] Q.17 The radii r1, r2, r3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimeter is 24 cm, find the lengths of its sides. [5] Q.18 Findtheequationofacircle passingthroughtheorigin ifthelinepair, xy– 3x +2y– 6 =0is orthogonal to it. If this circle is orthogonal to the circle x2 + y2 – kx + 2ky – 8 = 0 then find the value of k. [5] Q.19 Findthelocus ofthecentres of thecircles whichbisects the circumferenceofthe circles x2 + y2 =4 and x2 + y2 – 2x + 6y + 1 = 0. [5] Q.20 Find the equation of the circle whose radius is 3 and which touches the circle x2 + y2 – 4x – 6y– 12=0 internallyat thepoint (–1, – 1). [5] Q.21 Find the equation of the line such that its distance from the lines 3x – 2y– 6 = 0 and 6x – 4y– 3 = 0 is equal. [5] Q.22 Findtherangeofthevariablex satisfyingthequadraticequation, x2 + (2 cos )x – sin2 = 0    R. [5] Q.23 If tan         2 y 4 = tan3         2 x 4 then prove that sin y = x sin 3 1 ) x sin 3 ( x sin 2 2   . [5]
9. MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) TIME: 40 Min. DPP. NO.-35 Select the correct alternative. (Only one is correct) Comprehension (4 questions together) Consider the circle S: x2 + y2 – 4x – 1 = 0 and the line L: y= 3x – 1. If the line Lcuts the circle at A and B then Q.1 Length of the chord AB equal (A) 5 2 (B) 5 (C) 2 5 (D) 10 Q.2 The angle subtended by the chord AB in the minor arc of S is (A) 4 3 (B) 6 5 (C) 3 2 (D) 4  Q.3 Acute angle between the line L and the circle S is (A) 2  (B) 3  (C) 4  (D) 6  Q.4 If the equation of the circle on AB as diameter is of the form x2 + y2 + ax + by + c = 0 then the magnitude of the vector k̂ c j ˆ b î a V     has the value equal to (A) 8 (B) 6 (C) 9 (D) 10 Q.5 How manybaseball nines can be chosen from 13 candidates ifA, B, C, D are the onlycandidates for two positions and can playat no otherposition? Q.6 Thevalues of a, for which oneof the roots oftheequation 2x2 – 2(2a +1)x +a(a+1) = 0is greater than a andtheother is smaller, is (A)     , 2 1 (B) (0, 1) (C) (– , –3)(0, ) (D) (– , –1)(0, ) Q.7 If and  are the roots of a(x2 – 1)+ 2bx = 0then, which one of thefollowing are the roots of the same equation? (A)  + ,  –  (B)    1 2 ,    1 2 (C)    1 ,    1 (D)    2 1 ,    2 1 Q.8 The solutions of the equation, (1+ cos x) 2 x tan – 2 + sin x = 2 cos x are identical to the solutions of theequation (A) sin x = 1 (B) cos x = 0 (C) sin 2x = 0 (D) sec (x/2) = 2 Q.9 The solution of the equation ) x 2 3 ( log 2 x cos  < ) 1 x 2 ( log 2 x cos  is (A) (1/2, 1) (B) (– , 1) (C) (1/2, 3) (D) (1, ) –  n 2 , n  N Q.10 In ABC if B = 2  , s – a = 3; s – c = 2, then (A) r = 5/2 (B)  = 12 (C) r1 = 2 (D) R = 3 SUBJECTIVE: Q.11 Find the least value of a, for which (5x + 1 + 51 – x); 2 a 3 1 and (25x + 25–x) and the successive terms of anA.P. for every x  R. Q.12 Consider the quadratic polynomial f (x) = x2 – px + q where f (x) = 0 has prime roots. If p + q = 11 and a = p2 + q2 then find the value of f (a) where a is an odd positive integer.
10. MATHEMATICS Daily Practice Problems This is the test paper of Class-XI (PQRS & J) held on 24-12-2006. Take exactly 90 minutes. PART-A Select the correct alternative. (Only one is correct) [15 × 3 = 45] There is NEGATIVE marking and 1 mark will be deducted for each wrong answer. Q.1 A triangle with sides a = 15, b = 28 andc = 41. The length of the altitude from the vertex Bon the side AC is (A) 6 (B) 7 (C) 9 (D) 16 Q.2 If sides a, b and c of triangleABC satisfy c b a c b a 3 3 3     = c2 then tan       4 C has the value equal to (A) 2 – 1 (B) 2 – 3 (C) 1/ 3 (D) 2 + 3 Q.3 In a triangleABC, ABC = 45°, point D is onBC so that2BD =CD and DAB= 75°. ACBequals (A) 15° (B) 60° (C) 30° (D) 75° Q.4 The first term of an infinite geometricseries is 2 andits sum be denoted byS. If |S – 2 |<1/10 then the true set of the range of common ratio of theseries is (A)       5 1 , 10 1 (B)        2 1 , 2 1 – {0} (C)        20 1 , 19 1 – {0} (D)        21 1 , 19 1 – {0} Q.5 Number of solution satisfyingthe equation, tan22x = 2 tan 2x · tan 3x + 1 in [0, 2] is (A) 0 (B) 1 (C) 2 (D) 4 Q.6 Numerical value of 12 cos           4 cos 12 5 sin + 12 sin           4 sin 12 5 cos , is (A) 2 1 (B) 2 3 (C) 2 + 3 (D) 2 3 1 Q.7 Two circles both touching the coordinate axes and pass through the point (6, 3). The radii of the two circles are the roots of the equation (A) t2 – 12t + 20 = 0 (B) t2 – 15t + 36 = 0 (C) t2 – 18t + 45 = 0 (D) t2 – 14t + 48 = 0 Q.8 Let 'a' and 'b' are the roots of the equation x2 – mx + 2 = 0. Suppose that        b 1 a and        a 1 b are the roots of the equation x2 – px + q = 0. If p = 2q then the value of m is equal to (A) 4 (B) 6 (C) 8 (D) 9 Q.9 The value of the determinant y x 1 x y x 1 1 x 1 0 1     depends on (A)onlyx (B)onlyy (C) both x and y (D)neitherx nor y CLASS : XIII (XYZ) TIME: 90 Min. DPP. NO.-36
11. Q.10 The sum of all the positive integers greater than 1 and less than 1000, which leave a remainderof one when divided by2, 3, 4, 5 and 6, is (A) 8176 (B) 7936 (C) 8167 (D) none Direction forQ.11 and Q.12 (2 questions together) Consider the digits 1, 2, 2, 3, 3, 3 and answer the following Q.11 If all the 6 digit numbers using these digits only are formed and arranged in ascending order of their magnitudethen29th numberwillbe (A) 213332 (B) 233321 (C) 233312 (D) none Q.12 LetMdenotesthenumberofsixdigitnumbersusingonlythegivendigitsifnotallthe2'saretogetherand N denotes the corresponding figure if no 3's are together then M – N equals (A) 16 (B) 28 (C) 54 (D) 36 Q.13 Number of selections that can be madeof 6 letters from the word "COMMITTEE" is (A) 20 (B) 17 (C) 34 (D) 35 Q.14 Acircle of radius r touches the lines given bythe equation 4x2 – 4xy+ y2 – 18x + 9y– 36 = 0.Area of thecircleinsquareunitsis (A) 45  (B) 75  (C) 45/2 (D) 45/4 Q.15 Ifthemaximum andminimumvalueoftheexpression 6 x 3 x 2 2 x 2    (x R)are Mand mrespectively then the value of m 1 M 1  equals to (A) – 13 (B) – 10 (C) 10 (D) 16 Select the correct alternatives. (more than one are correct) [5 × 5 = 25] ThereisNO NEGATIVE marking. Q.16 If sin (x + 20°) = 2 sin x cos 40° where x         2 , 0 then which of the following hold good (A) sec 2 x = 2 6  (B) cot 2 x =(2 + 3 ) (C) tan 4x = 3 (D) cosec 4x = 2 Q.17 If the vertices of an equilateral triangleABC are (1, 1); (–1, –1) and (a, b) then (A) a2 + b2 must be equals to 6 (B) a + b must be equals to zero (C) a + b can be equal to 3 2 (D)lengthofits median is 6 Q.18 The sides ofa right triangleT1 are 20, xand hypotenuse y.Thesides ofanotherright triangleT2 are30, x – 5 and hypotenuse y+ 5. If P1 and P2 are the radii of the circles inscribed and 1 and 2 arethe areas ofthetriangles T1 and T2 respectivelythenwhichofthefollowinghold good? (A) 61 = 52 (B) 81 = 72 (C) P1 = P2 (D) 2P1 = P2 Q.19 ABCD is aquadrilateral co-ordinates of whosevertices areA(1, 0),B(–1, 0), C(3,4) andD(–3, 4) then (A)Thediagonals of the quadrilateral are equal but not at right angle (B)Areaofthe quadrilateral is 16 (C) Circlepassingthrough anythreepoints of this quadrilateralalso passes through thefourthpoint (D)ThequadrilateralABCDisanequilateral trapezium Q.20 LetA (1, 2); B  (3, 4) and C  (x, y) be any point satisfying (x – 1)(x – 3) + (y– 2)(y – 4) = 0 then whichofthefollowingholdgood? (A) Maximum possible area of thetriangleABC is 2 squareunits (B) MaximumnumberofpositionsofCintheXYplanefortheareaofthetriangleABCtobeunity,is4 (C) Least radius of the circle passing throughAand B is 2 (D) If'O'is theoriginthen the orthocentre aswell as circumcentre ofthe triangle OABlies outside this triangle
12. PART-B MATCH THE COLUMN [3 × 8 = 24] There is NEGATIVE marking. 0.5 Marks will be deducted for each wrong match. INSTRUCTIONS: Column-Iand column-IIcontainsfour entries each. Entries ofcolumn-Iareto bematchedwith some entriesofcolumn-II.Oneormorethanoneentriesofcolumn-Imayhavethematchingwiththesameentries ofcolumn-IIandoneentryofcolumn-Imayhaveoneormorethanonematchingwithentriesofcolumn-II. Q.1 ColumnI ColumnII (A) Numberofincreasingpermutationsof msymbols (P) nm are there from the n set numbers {a1, a2, , an} where theorder amongthenumbersis given by a1 < a2 < a3 <  an–1 < an is (B) There are m men and n monkeys. Number of ways (Q) mCn in whicheverymonkeyhas a master, ifa man can haveanynumberofmonkeys (C) Number of ways in which n red balls and (m – 1) green (R) nCm balls canbe arranged in a line, so that no two red balls are together, is (balls of the same colour are alike) (D) Number of ways in which 'm'different toys can be (S) mn distributed in 'n'childrenifeverychildmayreceive anynumberof toys, is Q.2 ColumnI ColumnII (A) If the lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 (P) ArithmeticProgression and cx + 4y+ 1 = 0 passes through the same point, then a, b, c are in (B) Let a, b, c be distinct non-negative numbers. (Q) GeometricProgression If the lines ax + ay + c = 0, x + 1 = 0 and cx + cy + b = 0 passes through the same point, then a, b, c are in (C) If the lines ax + amy + 1 = 0, bx + (m + 1)by + 1 = 0 (R) HarmonicProgression and cx + (m + 2)cy + 1 = 0, where m  0 are concurrent then a, b, c are in (D) If the roots of the equation (S) None x2 – 2(a + b)x + a(a + 2b + c) = 0 be equal then a, b, c are in Q.3 ColumnI ColumnII (A)     n 1 n n 2 n n 2 C Lim equals (P) 0 (B) Let the roots of f (x) = 0 are 2, 3, 5, 7 and 9 (Q) 1 and the roots of g (x) = 0 are – 1, 3, 5, 7 and 8. Number of solutions of the equation ) x ( ) x ( g f =0 is (C) Let y = x cos x sin3 + x sin x cos3 where 0 < x < 2  , (R) 3/2 thentheminimumvalueofyis
13. (D) Acircle passesthrough vertex D of the squareABCD, (S) 2 and is tangent to the sidesAB and BC. IfAB = 1, the radius of the circle can be expressed as p + 2 q , then p + q has the value equal to PART-C SUBJECTIVE: [4 × 9 = 36] ThereisNO NEGATIVE marking. Q.1 If x1 and x2 arethe twosolutions oftheequation 3 16 2 3 3 9 log x log 3 1 log x 12 3         , then find thevalue of 2 2 2 1 x x  . Q.2 A circle with center in thefirst quadrant is tangent to y = x + 10, y= x – 6, and the y-axis. Let (h, k) be the center of the circle. If the value of (h + k) = a + a b where a is a surd, find the value of a + b. Q.3 Suppose that there are 5 red points and 4 blue points on a circle. Find the number of convex polygons whose vertices areamongthe 9 points and having at least one blue vertex. Q.4 TriangleABC lies intheCartesianplaneand has an areaof 70sq. units.The coordinates of Band C are (12, 19)and(23, 20) respectivelyand the coordinates ofAare (p, q).The line containing themedian to the side BC has slope –5. Find the largest possible value of (p + q).
14. MATHEMATICS Daily Practice Problems This is the test paper of Class-XI (PQRS & J) held on 21-01-2007. Take exactly 2 hrs. P A P E R - 1 PART-A Select the correct alternative. (Only one is correct) [8 × 3 = 24] There is NEGATIVE marking and 1 mark will be deducted for each wrong answer. Q.1 Let x, y, z be real numbers such that 3x, 4yand 5z form a geometric progression while x, y, z form an H.P. Then the value of z x + x z = n m where m and n are relativelyprime then, (m + n) is equal to (A) 29 (B) 39 (C) 49 (D) 59 Q.2 Number of different colour arrangements are possiblebyplacing 2 orangeballs, 3 blue balls and 4 red balls in arow if the terminal balls are ofthesame colour, is (A) 250 (B) 350 (C) 810 (D) none Q.3    100 1 r 1 r 100 r 100 C C · r equals (A) 100 (B) 4950 (C) 5050 (D) 5151 Q.4 A triangle has base 10 cm long and the base angles of 50° and 70°. If the perimeter of the triangle is x + y cos z° where z  (0, 90) then the value of x + y + z equals (A) 60 (B) 55 (C) 50 (D) 40 Q.5 A triangle has two of its vertices at (0, 1) and (2, 2) in the Cartesian plane. Its third vertex lies on the x-axis. If the area of the triangle is 2 square units then the sum of the possible abscissae of the third vertex,is (A) – 4 (B) 0 (C) 5 (D) 6 Q.6 AnequilateraltriangleABC is inscribedinacircleofradius unitycenteredatorigin.IfPbeanypointon the circle then the value of (PA)2 + (PB)2 + (PC)2 equals (A) 3 (B) 6 (C) 8 (D) 12 Q.7 Number of real numbers x with 0  x  , satisfyingthe equation sin 3x(sin 3x – cos x) = sin x(sin x – cos 3x), is (A) 2 (B) 3 (C) 4 (D) 5 Q.8 Onesideofa rectangularpieceofpaperis 6cm, theadjacent sidesbeinglonger than 6 cms. One corner of the paper is folded so that it sets on the opposite longer side.Ifthelengthof thecrease is l cms and itmakes an angle with the long side as shown, then l is (A)   2 cos sin 3 (B)  cos sin 6 2 (C)  cos sin 3 (D)  3 sin 3 CLASS : XIII (XYZ) TIME: 120 Min. DPP. NO.-37
15. Select the correct alternatives. (more than one are correct) [4 × 5 = 20] ThereisNO NEGATIVE marking. Q.9 Let x be an irrational number.Whichofthe followingstatement(s)need not bealways correct? (A)tanxis irrational. (B)sinx isirrational. (C) x 1 1   is irrational. (D) | x | isirrational. Q.10 Afamilyoflinearfunctions isgivenby f (x)=1 +c(x +3)wherecR.Ifamemberofthis familymeets a unit circle centred at origin in two coincident points then 'c'can beequal to (A) – 3/4 (B) 0 (C) 3/4 (D) 1 Q.11 Consider the circles S1 : x2 + y2 = 4 and S2 : x2 + y2 – 2x – 4y+ 4 = 0 which ofthe following statements are correct? (A) Number of common tangents to these circles is 2. (B) If the power of a variable point P w.r.t. these two circles is same then P moves on the line x + 2y – 4 = 0. (C) Sum of the y-intercepts of both the circles is 6. (D) The circles S1 and S2 are orthogonal. Q.12 Number of ways in which the letters of the word 'B U L B U L' can be arranged in a line in a definite order is also equal to the (A)numberofways in which2alikeApplesand4 alikeMangoes canbedistributedin 3children so that each child receives anynumber of fruits. (B) Number ofways in which 6 different books can betied up into 3 bundles, if each bundleis to have equal number of books. (C) coefficient of x2y2z2 inthe expansion of (x + y+ z)6. (D)numberofways in which6different prizes canbedistributed equallyinthreechildren. PART-B MATCH THE COLUMN [2 × 8 = 16] There is NEGATIVE marking. 0.5 Marks will be deducted for each wrong match. INSTRUCTIONS: Column-Iand column-IIcontainsfour entries each. Entries ofcolumn-Iareto bematchedwith some entriesofcolumn-II.Oneormorethanoneentriesofcolumn-Imayhavethematchingwiththesameentries ofcolumn-IIandoneentryofcolumn-Imayhaveoneormorethanonematchingwithentriesofcolumn-II. Q.1 Column-I Column-II (A) The equation  x x x x x x  hastwo solutions inpositivereal (P) 8/3 numbers x.One obvious solution is x = 1. The other oneis x = (B) Suppose atriangleABC is inscribedin a circle ofradius 10 cm. (Q) 9/4 If theperimeter of the triangleis 32 cm thenthe value of sinA+ sin B + sin C equals (R) 5/4 (C) Sum ofinfinteterms oftheseries 1 + 4 3 + 16 7 + 64 15 + 256 31 + .... equals (S) 8/5 (D) The sum of               1 r ) 2 r )( 1 r ( r 3 r equals
16. Q.2 Column-I Column-II (A) Sum ofthevalues ofnsatisfyingtheequation (P) 11 2 ! 4 ! n       + ! 4 · 4 ) ! 5 )( ! 7 ( =         ! 4 ! n 240 (Q) 12 (B) If the points with coordinates (a, 0) and (0, b) are equidistant (R) 13 from the point (1, 4) and (9, 0) then (a – b) equals (C) The valueof the expression                  11 1 k 2 k 12 2 1 1 log (S) 14 (D) If logb3 = 4 and 27 log 2 b = 2 a 3 then the value of a2 – b4 is PAPER-2 Q.1 Consider the word F OU ND AT IO N. Find the (i) numberofwaysinwhichthelettersofthis word canbearranged without changingtheorderofvowels. (ii) Number of wordsin which vowels andconsonants are alternate. [2.5 + 2.5] Q.2 In a kite ABCD,AB = AD and CB = CD. If A = 108° and C = 36° then the ratio of the area of ABD to the area of CBD can be written in the form c 36 tan b a 2   where a, b and c are relatively prime positive integers. Determine the ordered triple(a, b, c). [5] Q.3 Circles C1 andC2 areexternallytangentandtheyarebothinternallytangent tothe circleC3. Theradiiof C1 and C2 are 4 and 10, respectivelyand the centres of the three circles are collinear.Achord of C3 is also a common internal tangent of C1 and C2. Given that the length of the chord is p n m where m, n and p are positive integers, m and p are relatively prime and n is not divisible by the square of any prime, find the value of (m + n + p). [5] Q.4 Let ,  and  are the roots of the cubic x3 – 3x2 + 1 = 0. Find a cubic whose roots are 2    , 2    and 2    . Hence or otherwise find the value of ( – 2)( – 2)( – 2). [5] Q.5 The two line pairs y2 – 4y + 3 = 0 and x2 + 4xy + 4y2 – 5x – 10y + 4 = 0 enclose a 4 sided convex polygonfind (i) area ofthe polygon (ii) lengthofitsdiagonals. [5] Q.6 If Pn denotes the product of all the coefficients in the expansion of (1 + x)n, n  N, show that ! n ) 1 n ( P P n n 1 n    . [5] Q.7 With usual notations, prove that in a triangleABC, cot 2 A + cot 2 B + cot 2 C = r s . [5]
17. Q.8 (a) Findthedomainofdefinitionofthefunction f (x) = 5 x 2 x 3 20 1 x 2 x 16 2 x 1 x P C 6 x x log                where [ * ] denotes the greatest integer function. (b) Suppose setAconsists of 4 distinct elements and set B consists of 6 distinct elements. Find (i) numberofinjectivemappingdefined fromAB. (ii) number ofmappingfromA B whichare not surjective. [2 + 1.5 + 1.5] OR Q.8 (a) If the coefficient of xp and xp+1 in (3x + 2)19 areequal then find the value of p. (b) If rth term in the expansion of 12 2 x 3 1 x 9        is independent of x, then find the valueof 'r'and alsothe valueofthis term. [2.5+2.5]
18. MATHEMATICS Daily Practice Problems SPECIAL DPP ON PROBABILITY Q.1 Three events A, B, C are defined with the following probabilities P(A  B  C) = 0.1; P(A  B) = 0.13; P(A  C) = 0.12; P(B  C) = 0.2; P(A) = 0.25; P(B) = 0.43; P(Ac  Bc  Cc) = 0.3 Find (i) P(B/Ac) (ii) P(A  Cc) (iii) P(Bc  Cc) Q.2 Five persons entered the lift cabin on the ground floor of an eight floor house . Suppose that each of them , independently & with equal probability can leave the cabin at any floor beginning with the first, find out the probability of all 5 persons leaving at different floors. Q.3 Three political parties namely the Congress, the B.J.P. and the Janta Dal are contesting for a state legislative assembly elections. The state does not have a common entrance test after the 12th standard, for the admissions to the medical or engineering colleges. The probabilities of these parties winning the elections are 1 3 4 9 2 9 , & respectively. If the Congress comes to the power, the probability of its introducing the common entrance test for the state is 0.6 and the corresponding probabilities for the B.J.P. and Janta Dal are 0.7 and 0.5 respectively. Find the probability that the common entrance test is introduced. Q.4 Three persons A, B, C independently fire at a target. Suppose P (A) = 6 1 ; P (B) = 4 1 ; P (C) = 3 1 denote their probabilities of hitting the target. (a) Find the probability that at least one of them hits the target. (b) Find the probability that exactly one of them hits the target. (c) If the target is hit only once, find the probability that it was the man A. Q.5 Each of the two rifles are fired independently at a target. The probability of the first rifle destroying the target is 0.8 and that of the second is 0.4. The target is destroyed by a single hit. Determine the probability that it was destroyed by the first rifle. Q.6 If n different biscuits are randomly distributed among N beggars. Find the chance that a particular beggar receives r biscuits (r < n). Q.7 Urn-I contains 3 red balls and 9 black balls. Urn-II contains 8 red balls and 4 black balls. Urn-III contains 10 red balls and 2 black balls.Acard is drawn from a well shuffled pack of 52 playing cards. If a face card is drawn, a ball is selected from Urn-I. If an ace is drawn, a ball is selected from Urn-II. If any other card is drawn, a ball is selected from Urn-III. Find (a) the probability that a red ball is selected. (b) the conditional probability that Urn-I was one from which a ball was selected, given that the ball selected was red. Q.8 In a college, four percent of the men and one percent of the women are taller than 6 feet. Further 60 percent of the students are women. If a randomly selected student is taller than 6 feet, find the probability that the student is a women. Q.9 A hunter knows that a deer is hidden in one of the two near by bushes, the probability of its being hidden in bushI being 4/5. The hunter having a rifle containing 10 bullets decides to fire them all at bush–I or II . It is known that each shot may hit one of the two bushes , independently of the other with probability 1/2. How many bullets must he fire on each of the two bushes to hit the animal with maximum probability. (Assume that the bullet hitting the bush also hits the animal). Q.10 The probability that an ancher hits the target when it is windy is 0.4; when it is not windy, her probability of hitting the target is 0.7. On any shot, the probability of a gust of wind is 0.3. Find the probability that (a) She hit the target on first shot (b) Hits the target exactly once in two shots CLASS : XIII (XYZ) DPP. NO.-38
19. MATHEMATICS Daily Practice Problems HOME TEST Q.1 A = x 1 x cos x 2 sin Lim 0 x    ; B = x x x Lim 2 x     ; C = x 1 n 0 x x Lim l  ; D = 2 x 3 x 3 x Lim 2 2 x     FindAB + C + D 1 . Q.2 A box contains 2 fifty paise coins, 5 twenty five paise coins & a certain fixed number N( 2)often&five paisecoins. Fivecoinsaretakenout ofthebox atrandom. Find theprobabilitythat the total value of these five coins is less than Re.1 & 50 paise. Q.3 Findtheparametricequationforthelinewhichpasses throughthepoint(0, 1,2)and isperpendicularto the line x = 1 + t, y = 1 – t and z = 2t and also intersects this line. Q.4 If thenormals to the curvey= x2 atthe points P, Qand R pass throughthe point       2 3 , 0 , findtheradius ofthecirclecircumscribingthetrianglePQR. Q.5 Let A = {a  R | the equation (1 + 2i)x3 – 2(3 + i)x2 + (5 – 4i)x + 2a2 = 0} has at least one real root. Find the value of  A a 2 a . Q.6 Let f (x)be a differentiable function suchthat f '(x) + f (x) = 4xe–x · sin 2x and f (0)= 0.Find the value of      n 1 k n ) k ( f Lim . Q.7 A factoryA produces 10% defective valves and another factory B produces 20% defective. A bag contains 4valves offactoryAand 5 valves offactoryB.Two valves are drawn at random from the bag. If the probabilitythat atleastone valveis defectiveis q p where pand q arerelativelyprimefind (p+ q). Q.8 Let S be the set of all x such that x4 – 10x2 + 9  0. Find the maximum value of f (x) = x3 – 3x on S. Q.9 Usingthe fact that sin x = i 2 e e ix ix   orotherwise prove that if   0 n n 3 ) nx sin( = c b b a  andsin x = 3 1 where 0 < x < /2, find the value of (a + b + c), where a, b, c are positive integers. Q.10 A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any combination of1's and 2's. Find the total number ofways in which the person can go up the stairs. Q.11 Let a and b be two positive real numbers. Evaluate   b a x b a x dx x e e . CLASS : XIII (XYZ) TIME : 60 Min. DPP. NO.- 39
20. MATHEMATICS Daily Practice Problems This is the partial test paper of Class-XI (PQRS & J) held on 25-02-2007. Take exactly 45 minutes. Q.1 If  and  are the roots of the equation x2 + 5x – 49 = 0 then find the value of cot(cot–1 + cot–1). [3] [Ans:10] Q.2 Findthegeneralsolutionoftheequation               ... x sin ) 1 ..( x sin x sin x sin 1 .... x sin ...... x sin x sin x sin 1 n n 3 2 n 3 2 = x tan 1 4 2  where x  k + 2  , k  I. [Ans. n + (–1)n 6  , n  I] [4] Q.3 In a triangleABC if 2 cos 2 B cos 2 C = 2 1 + 2 A sin a c b        then find the measure of angleA. A. [4] [Ans:A= 60° ] Q.4 Givenbelow is apartial graph of an even periodic function f whose period is 8. If [*] denotes greatest integerfunctionthenfindthevalueoftheexpression. [Ans: sum= 5] f (–3) + 2 | f (–1) | +             8 7 f + f (0) + arc cos  ) 2 ( f + f (–7) + f (20) [4] Q.5 Tendogs encounter8 biscuits. Dogs donot sharebiscuits. In howmanydifferent ways can the biscuits be consumed (a) if weassume that thedogsare distinguishable,but thebiscuits arenot. (b) if we assume that both dogs andbiscuits are different and anydog can receive anynumber of biscuits. (c) if dogs andbiscuits are different andeverydog can get atmost one biscuit. [1+1+2] [Ans. (a) 17C8; (b) 108; (c) 10C8 · 8! ] Q.6 Given   35 1 k k 5 sin = tan       n m , where angles are measured indegrees, and m and n arerelativelyprime positive integers that satisfy n m <90, find thevalue of (m + n). [5] [Ans. 177] Q.7 Let f (n) =           n 0 r n r k r k . Find the total number of divisors of f (9). [Ans: 8] [5] Q.8 AsquareABCDlyinginI-quadrant hasarea36sq. units andis such thatits sideABis parallelto x-axis. VerticesA, B and C are on the graph of y = logax, y= 2 logax and y = 3 logax respectivelythen find the value of 'a'. [Ans: a = 6 3 ] [6] CLASS : XIII (XYZ) TIME : 45 Min. DPP. NO.-40
21. MATHEMATICS Daily Practice Problems PART-A Select the correct alternative. (Only one is correct) [8 × 3 = 24] There is NEGATIVE marking and 1 mark will be deducted for each wrong answer. Q.1 Thefunction f (x)= 1 x 1 x 3   canbe writtenas thesum ofanevenfunctionand an oddfunction. Theeven functionis (A) 1 x 1 x 6 4   (B) 1 x 1 x 6 4   (D) 1 x 1 x 6 4   (D) 1 x 1 x 6 4   Q.2 For a positive integer n, let fn () = (2 cos  + 1) (2 cos   1) (2 cos 2 1) (2 cos 22  1) ...... (2 cos 2n  1  1). Which one ofthefollowingdoes not holdgood? (A) f2 (/6) = 0 (B) f3 (/8) =  1 (C) f4 (/32) = 1 (D) f5 (/128) = 2 Q.3 For a > 0 and 0 < b < , if sin x + 3 cos x = a sin(x + b) then the value of arc sin(sin ab) equals (A) 3 2 (B) 3  (C) 6  (D) – 6  Q.4 Let a,b,c, dand ebe five consecutivetermsofan arithmetic sequence such that a + b + c + d + e = 30. Thenwhichofthefollowingcanbe determined? (A) a (B) b (C) c (D) d Q.5 The sum of the zeroes, product of the zeroes and the sum of the coefficients of the quadratic function f (x) = ax2 + bx + c are equal. Their common value must be equal to (A)thecoefficient of x2 (B)thecoefficient ofx (C) the y-intercept of the graph of y= f (x) (D) the meanof the x-intercept of the graphof y= f(x) Q.6 If sin a + sin b = 3 5 and cos a + cos b = 1 then the value of cos(a – b) equals (A) 3 1 (B) 2 1 (C) 3 2 (D) 1 Q.7 The line y= 1 – x cuts the curve ky= x2 + x in points P and Q. If POQ is a right angle (where 'O' is the origin)then 'k'equals (A) 2 (B) 1 (C) 2 (D) 1/2 Q.8 Number of 4 letter collections that can be had usingthe letters of the word ASSISTING, equals (A) 10 (B) 41 (C) 60 (D) none Select the correct alternatives. (One or more than one is/are correct) [4 × 5 = 20] ThereisNO NEGATIVE marking. Q.9 The equation 8 8 x log ) x 4 (log 2 2 2 2 1           has (A)oneintegralsolution (B)tworational solutions (C)noprimesolution (D)onereal solution Q.10 Whichofthecubicpolynomialsgiven belowdonot havetheirrootsin arithmeticprogression? CLASS : XIII (XYZ) TIME : 45 Min. DPP. NO.-41
22. (A) x3 + 3x2 – 2x – 1 (B) x3 + 3x2 – 2x – 2 (C) x3 + 3x2 – 2x – 3 (D) x3 + 3x2 – 2x – 4 Q.11 A function f : R  R is suchthat         x 1 x 1 f =x for allx  –1. Then which ofthefollowingstatements are true? (A)   ) x ( f f = x (B)         x 1 x 1 f = f (– x), x  1 (C)       x 1 f = – f (x), x  0 (D) f (– x – 2) = – f (x) – 2 Q.12 Whichofthefollowingfunction(s)is/areodd? (A) f (x) = cos(tan–1x) (B) g (x) = tan(cot1x) (C) h (x) = sin(tan–1x) (D) k (x) = tan(cos1x) PART-B MATCH THE COLUMN [2 × 8 = 16] There is NEGATIVE marking. 0.5 Marks will be deducted for each wrong match. Q.1 Column-I Column-II (A) In the given figureAC = 2x, BC = 2x + 1 (P) 0 and ACB = 30°. If the area of ABC is 18 then the value of x, is equal to (Q) 1 (B) Number of real solution(s) oftheequation log2x – 1(x3 + 3x2 – 13x + 10) = 2 is (R) 2 (C) Number of intervals givenbelow arenot thesubsets of the (S) 4 the domain ofthe function f (x)= 4 x 6 x 7 x 3 2    (a) (– , – 1) (b) (– 7, 0) (c) [, 4) (d) [3, 10] (e) (8, ) (D) If f (t) = sin          2 t thenthe smallest positivevalue of 't'for whichf(t)attainsitsminimumvalueis Q.2 Column-I Column-II (A) The complex number 1 – 4i is a zero ofthe function (P) 2 f (x) = x4 – 4x3 + 18x2 – 28x – 51. If the product of the other three zeroes is written as + i, (Q) 3 where ,   R then the ratio   equals (R) 4 (B) If A =          0 k k 3 2 and B =           0 k k ) k cos( 2 1 (S) None of P, Q and R then the valueofABequals (C) x-coordinate ofthecentreofthe circletouchingthe line 3x + y+ 2 = 0 at the point (–1, 1) and passing through the point (3, 5) is (D) Giventhat thesum ofthesolutionsoftheequation sin x · tan x – sin x + tan x – 1 = 0 over [0, 2] = k, where k  N then the value of k equals Answer Key: Q1. A, Q2. D, Q3. B, Q4. C, Q5. A, Q6. A, Q7. C, Q8. B Q9. AB, Q10. ABC, Q11. ACD, Q12. BCD. Match the Column: Q1. A-S, B-Q, C-R, D- R; Q2. A-R, B-P, C-P, D-S
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