37965451 bansal classes 12th standard maths dpps

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5/23/2018 37965451BansalClasses12thStandardMathsDPPs-slidepdf.com http://slidepdf.com/reader/full/37965451-bansal-classes-12th-standard-maths-dpps 1/24 | JgBANSAL CLASSES IpTargef CLASS: XII SIT J E E 2 0 0 7 MATHEMATICS Daily Practice Problems TIME: 50 Min. DPP. NO.-53 (ABCdT DATE: 11-12/12/2006 Q. 1 Q.2 Revision Dpp on Permutation & combination Select the correct alternative. (Only one is correct) Number of natural numbers between 100 and 1000 such that at leas t one of their digits is 7, is (A) 225 (B) 243 (C) 252 (D)none The number of way s in which 100 persons may be seated at 2 round tables T, and T 2 , 5 0 persons being seated at each is : ( A ) f m M ! m l ! Q. 3 There are six periods in each working day of a school. Number of ways in which 5  subjects can be arranged if each subject is allotted at least one period and no  period remains vacant is (A)210 (B)1800 (C)360 (D)120 The number of ways in whi ch 4 boys & 4 girls can stand in a circle so that each boy and each girl is one after the other is: (A) 4 ! . 4 ! (B) 8 ! (C) 7 ! (D) 3 ! . 4 ! If letters ofthe  word "PARKAR" are written down in all possible manner as they are in a dictiona ry, then the rank of the word "PARKAR" is: (A) 98 (B) 99 (C) 100 (D) 101 The num ber of different words of three letters which can be formed from the word "PROPO SAL", if a vowel is always in the middle are: (A) 53 (B) 52 (C) 63 (D) 32 Consid er 8 vertices of aregular octagon and its centre. If T denotes the number of tri angles and S denotes the number of straight lines that can be formed with these 9 points then T - S has the value equal to (A) 44 (B)48 (C) 52 (D)56 A polygon h as 170 diagonals. How many sides it will have ? (A) 12 (B) 17 (C) 20 (D) 25 Q. 4 Q.5 Q. 6 Q.7 Q. 8 Q. 9 The number of ways in which a mixed double tennis game can be arranged from amon gst 9 married couple if no husband & wife plays in the same game is; (A) 756 (B)  1512 (C) 3024 (D) 4536 4 normal distinguishable dice are rolled once. The numbe r of possible outcomes in which atleast one die shows up 2 is: (A) 216 (B) 648 ( C) 625 (D) 671 Il-l OQ X nr x . p is equal to : Q. 10

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    | JgBANSAL CLASSESIpTargef CLASS: XII SIT J E E 2 0 0 7

    MATHEMATICSDaily Practice ProblemsTIME: 50 Min. DPP. NO.-53

    (ABCdT

    DATE: 11-12/12/2006

    Q. 1 Q.2

    Revision Dpp on Permutation & combination Select the correct alternative. (Onlyone is correct) Number of natural numbers between 100 and 1000 such that at least one of their digits is 7, is (A) 225 (B) 243 (C) 252 (D)none The number of ways in which 100 persons may be seated at 2 round tables T, and T 2 , 5 0 personsbeing seated at each is :( A ) f m

    M !

    m

    l !

    Q. 3

    There are six periods in each working day of a school. Number of ways in which 5subjects can be arranged if each subject is allotted at least one period and noperiod remains vacant is (A)210 (B)1800 (C)360 (D)120 The number of ways in which 4 boys & 4 girls can stand in a circle so that each boy and each girl is oneafter the other is: (A) 4 ! . 4 ! (B) 8 ! (C) 7 ! (D) 3 ! . 4 ! If letters oftheword "PARKAR" are written down in all possible manner as they are in a dictionary, then the rank of the word "PARKAR" is: (A) 98 (B) 99 (C) 100 (D) 101 The number of different words of three letters which can be formed from the word "PROPOSAL", if a vowel is always in the middle are: (A) 53 (B) 52 (C) 63 (D) 32 Consid

    er 8 vertices of aregular octagon and its centre. If T denotes the number of triangles and S denotes the number of straight lines that can be formed with these9 points then T - S has the value equal to (A) 44 (B)48 (C) 52 (D)56 A polygon has 170 diagonals. How many sides it will have ? (A) 12 (B) 17 (C) 20 (D) 25

    Q. 4

    Q.5

    Q. 6

    Q.7

    Q. 8 Q. 9

    The number of ways in which a mixed double tennis game can be arranged from amongst 9 married couple if no husband & wife plays in the same game is; (A) 756 (B)1512 (C) 3024 (D) 4536 4 normal distinguishable dice are rolled once. The number of possible outcomes in which atleast one die shows up 2 is: (A) 216 (B) 648 (C) 625 (D) 671 Il-l OQ X nr x . p is equal to :

    Q. 10

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    Q-l l

    fQ. 12

    ( B ) f ^

    ( Q ^

    There are counters available in x different colours, The counters are all alikeexcept for the colour. The total number of arrangements consisting of y counters

    , assuming sufficient number of counters of each colour, if no arrangement consists of all counters of the same colour is: (A) x y - x (B) x y - y (C) y x - x (D)yx-y In a plane a set of 8 parallel lines intersects a set of n parallel lines, that goes in another direction, forming a total of 1260 parallelograms. The value of n is: (A) 6 (B) 8 (C) 10 (D) 12

    Q. 13

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    Q. 14

    A team of 8 students goes on an excursion, in two cars, of which one can seat 5and the other only 4. If internal arrangement inside the car does not matter then the number of ways in which they can travel, is (A) 91 (B) 126 (C) 182 (D)3920In a conference 10 speakers are present, If S5 wants to speak before S 2 & S 2wants to speak after S 3 , then the number of ways all the 10 speakers can givetheir speeches with the above restriction if the remaining seven speakers have no obj ection to speak at any number is (A)10

    Q. 15

    C3

    (B)

    10

    Pg

    (C)

    I0

    P3

    (D) i i l

    Q. 16

    There are 8 different consonants and 6 different vowels. Number of different words of 7 letters which can be formed, ifthey are to contain 4 consonants and 3 vowels if the three vowels are to occupy even places is (A) 8 P 4 . 6 P 3 (B) 8 P4 . 6 C 3 (C) s P 4 . 7 P 3 (D) 6 P 3 . 7 C 3 . 8 P 4 Number of ways in which 5different books can be tied up in three bundles is (A) 5 (B) 10 (C) 25 (D) 50 How many words can be made with the letters of the words "GENIUS" if each word nei

    ther begins with G nor ends in S is : (A) 24 (B) 240 (C) 480 (D) 504 Number of numbers greater than 1000 which can be formed using only the digits 1,1,2,3,4,0 taken four at a time is (A) 332 (B) 159 (C) 123 (D) 112 Select the correct alternative. (.More than one are correct)

    Q.17 Q. 18

    Q. 19

    Q.20

    Identify the correct statement(s). (A) Number of naughts standing at the end of1125 is 30. (B) Atelegraph has 10 arms and each aim is capable of 9 distinct pos

    itions excluding the position of rest. The number of signals that can be transmitted is 1010 - 1 . (C) In a table tennis tournament, every player plays with every other player. If the number of games played is 5050 then the number of players in the tournament is 100. (D) Number of numbers greater than 4 lacs which canbe formed by using only the digits 0,2,2,4, 4 and 5 is 90.n+

    Q.21 Q.22

    '-Cg + C4 >

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    n+2

    C 5 - n C 5 for all ' n ' greater than : (B) 9 (C) 10 (D) 11 (200^

    (A) 8

    The number of ways in which 200 different things can be divided into groups of 100 pairs is: (10fl (102^1 (103^1 (A) 2 ( 1 . 3 . s..199) (C) -,100 /lnn\ i 200!00

    I t J r r J _ (D) 200! ->100 to n factors is equal to : (B) 2Cn ( D ) 2 n (1 -3 - 5

    B

    I T

    Q.23

    2' (100)! The continued product, 2 . 6 . 1 0 . 1 4 (A) 2n P n (C) ( n + 1)(n + 2) ( n + 3) (n + n)

    2n-l)

    Q.24

    The Number of ways in which five different books to be distributed among 3 persons so that each person gets at least one book, is equal to the number of ways inwhich (A) 5 persons are allotted 3 different residential flats so that and eachperson is alloted at most one flat and no two persons are alloted the same flat. (B) number of parallelograms (some of which may be overlapping) formed by oneset of 6 parallel lines and other set of 5 parallel lines that goes in other direction. (C) 5 different toys are to be distributed among 3 children, so that each child gets at least one toy. (D) 3 mathematics professors are assigned five different lecturers to be delivered, so that each professor gets at least one lecturer.

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    4Q.l (a) (b)

    J BANSAL CLASSES{Target BIT JEE 2007DATE: 22-23/11/2006

    MATHEMATICSDaily Practice ProblemsTIME: 75 Min. DPR NO.-S2

    CLASS: XII (ABCD)

    This is the test paper ofClass-XI

    (PQRS & J) held on 19-11-2006. Take exactly 75 minutes.

    Consider the quadratic polynomial f (x) = x 2 - 4ax + 5 a 2 - 6a. Find the smallest positive integral value of'a' for which f (x) is positive for every real x.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 x 2 + y 2 = 25x+y= c has a real solution. (b) The equations to a pair of opposite sides of a

    parallelogram are x 2 - 7x + 6 = 0 and y 2 - 1 4 y + 40 = 0 find the equations to its diagonals. Q. 3

    [2.5+2.5]

    Find the equation of the straight line with gradient 2 if it intercepts a chordof length 4^/5 on the circle x 2 + y 2 - 6x - 1 Oy + 9 = 0. [5]

    Q.4

    The value ofthe expression,

    cos^ 2x + 3 cos 2x 7 7 wherever defined is independent of x. Without allotting c

    os x - s i n x a particular value of x, find the value of this constant. [5] Find the general solution of the equation sin 3 x(l + cot x) + cos 3 x(l + tan x) =cos 2x.

    Q. 5

    [5]

    Q. 6

    If the third and fourth terms of an arithmetic sequence are increased by 3 and 8respectively, then the first four terms form a geometric sequence. Find (i) thesum of the first four terms ofA.P. (ii) second term of the G.P. [2.5+2.5]

    Q.7(a) Let x = or x = - 15 satisfies the equation, log 8 (&x 2 +wx + / ) = 2 . If k, w and/are relatively prime positive integers then find the value of k+w +f.(b) The quadratic equation x 2 + mx + n - 0 has roots which are twice those ofx 2 + px + m = 0 and n m, n and p* 0. Find the value of ~ . [2.5+2.5] x y Lme + = 1 intersects the x and y axes at M and N respectively. If the coordinates of the point P 6 8 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 of the point P and (b) the radius of the circle escribed opposite to theangle N. [2.5+2.5] Starting at the origin, a beam oflight hits a mirror (in the

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    fomi of a line) at the point A(4,8) and is reflected at the point B(8,12). Compute the slope of the mirror. [5]

    Q. 8

    Q. 9

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    Q. 10 Q.ll

    Find the solution set of inequality,

    log x + 3 (x 2 - x) < 1.

    [5]

    If the first 3 consecutive terms of a geometrical progression are the roots of the equation 2x 3 - 1 9 x 2 + 57x - 5 4 = 0 find the sum to infinite number of terms of G.P. [5] Find the equation to the straight lines joining 1 lie o- "m to t

    he points of intersection of the straight line 2L + L = i and the circle 5(x 2 +y 2 + bx+ay) = 9ab. Also find the linear relation between a and b so that a b these straight lines may be at right angle. [3+2]

    Q. 12

    Q. 13

    L e t / ( x ) = | x - 2 | + | x - 4 | | 2 x - 6 j . Find the sum of the largestand smallest values of f (x) if x e [2, 8], [5] x+1 x+2 If x+3 x+2 x+3 x+4 x+a x+ b = 0 then all lines represented by ax + by + c = 0 pass through a fixed point. x+c [5]

    Q.14

    Find the coordinates of that fixed point. Q. 15

    If Sj, S 7 , S 3 ,... S ,.... are the sums of infinite geometric series whose first terms are 1,2,3,... n,... and 1 1 1 1 whose common ratios are , - , ,...., ,... respectively, then find the value of 2 J nr * O ** 1 T A 5 B 20 In any triangle if tan = 7 and tan = then find the value of tan C. 2 6 2 3/2(1-1

    r=l

    -

    . [5]

    Q. 16

    [5]

    Q.17

    The radii r p r 2 , r 3 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 lengthsof its sides. [5] Find the equation of a circle passing through the origin if the line pair, xy - 3x + 2y - 6 = 0 is orthogonal to it. If this circle is orthog

    onal to the circle x 2 + y 2 - kx + 2ky - 8 = 0 then find the value of k. [5] Find the locus of the centres of the circles which bisects the circumference of the circles x 2 + y 2 - 4 and x 2 + y 2 2x + 6y + 1 = 0. [5] Find the equation ofthe circle whose radius is 3 and which touches the circle x 2 + y 2 - 4x 6y - 12=0 internally at the point ( - 1 , - 1 ) . [5] Find the equation of the line such that its distance fiom the lines 3x - 2y - 6 = 0 and 6x - 4 y - 3 = 0 is equal. [5] Find the range of the variable x satisfying the quadratic equation, x 2 +(2 cos (j))x - sin2c|> = 0 V

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    Q. 18

    Q. 19

    Q.20

    Q.21

    Q. 22

    [5]

    Q.23

    [5]

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    1Q.l Q. 2

    i BANSAL CLASSESTarget IIT JEE 2007DATE: 10-11/11/2006

    MATHEMATICSDaily Practice ProblemsTIME: 60 Min. DPP. NO.-51

    CLASS : XII (ABCD)

    Select the correct alternative. (Only one is correct) There is NEGATIVE markingand 1 mark will be deducted for each wrong answer. 1 1 1 1 1 Find the sum of theinfinite series 7 + 7T: + T r + 7 7 + 7 r + T 9 18 30 45 63

    (A) }

    (B) i

    (C) |

    (D) f

    Number of degrees in the smallest positive angle x such that 8 sin x cos 5 x - 8sin5x cos x = 1, is (A) 5 (B) 7.5 (C)10

    (D) 15

    Q. 3

    There exist positive integers A, B and C with no common factors greater than 1,such that Alog 200 5 + B log 200 2 = C. The sumA + B + C equals (A) 5 ~ (B) 6 (C) 7 (D) 8 A triangle with sides 5,12 and 13 has both inscribed and circumscribedcircles. The distance between the centres of these circles is (A) 2 (B)| (C) V65 (D)^f y

    Q. 4

    Q. 5

    The graph of a certain cubic polynomial is as shown. If the polynomial can be written in the form / ( x ) = x 3 + ax2 + bx + c, then (A) c = 0 (B) c < 0 (C) c >0 (D) c = - 1

    Q. 6

    The sides of a triangle are 6 and 8 and the angle 0 between these sides varies such that 0 < 0 < 90. The length of 3rd side x is (A) 2 < x < 14 (B) 0 < x < 10 (C)

    2 < x < 10 (D)0 0) then the valu

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    e of xyz is (A) a 3 (B,b3 ( C ) ' t a 2b-a 2a-b

    Q.9

    Given A(0,0), ABCD is a rhombus of side 5 units where the slope of AB is 2 and the slope of AD is 112. The sum of abscissa and ordinate of the point C is (A) 4V5 (B)5V5 (C)6V5 (D) 8V5

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    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) exactly one value of k (C) exactly two values of k (D) infinite values of k If a A ABC is formed by 3 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 through its (A) incentre (B) centroid (C) orthocentre (D) circumcentre x2 + 1 0 x -3 6 a b c = + ~ + If a, b and c are numbers for which the equation - + + x x-3 (x-3) then a + b + c equals (A) 2 (B) 3 (C)10 (D)8

    Q. 11

    Q. 12

    is an identity,

    Q. 13

    1 1 1 If a, b, c are in G.P. then ~ , , b - a 2 b b - c (A) A. P. (B) G.P.

    are in (C)H.P. (D) none

    Q. 14

    How many terms are there in the G.P. 5,20, 80, 20480. (A) 6 (B)5 (C) 7 1

    (D)8

    Q. 15

    The sum of the first 14 terms of the sequence

    1 1 j= + h t= + 1 + Vx 1-X 1 v x7

    is

    (

    A

    ) 14

    (

    B

    )

    ^ f >

    (C)

    (l + V x ) ( l - x ) ( l - V x ) 10

    (D)none

    Q. 16

    If x, y > 0, logyx + logxy = and xy = 144, then arithmetic mean of x and y is (A

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    ) 24 (B) 36 (C)12V2 (D)13V3

    Q. 17

    A circle of radius R is circumscribed about a right triangle ABC. If r is the radius of incircle inscribed in triangle then the area of the triangle is (A)r(2r+ R) (B)r(r + 2R) (C)R(r + 2R) (D)R(2r + R) The simplest form of 1 + 1 (A) a fora * 1 (C) - a for a * 0 and a * 1 is 1-a (B) a for a * 0 and a * 1 (D)lfora*l

    Q. 18

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    Select the correct alternatives. (More than one are correct) Q. 19 If the quadratic equation ax2 + bx + c = 0 (a > 0) has sec29 and cosec 2 0 as its roots thenwhich of the following must hold good? (A) b + c = 0 (B) b 2 - 4ac > 0 (C) c > 4a (D) 4a + b > 0 Which of the following equations can have sec29 and cosec29 asits roots (9 e R)? (A) x 2 - 3x + 3 = 0 (B) x 2 - 6x + 6 = 0 (C) x 2 - 9x + 9 =0 (D) x 2 - 2x + 2 = 0 The equation | x - 2 | 10x2_1 = | x - 2 | 3x has (A) 3 integral solutions (C) 1 prime solution Q. 22 (B) 4 real solutions (D) no irrational solution

    Q.20

    Q.21

    Which of the following statements hold good? (A) If Mis the maximum and m is theminimum value of y = 3 sin2x + 3 sin x cos x + 7 cos2x then the mean of M and mis 5, 71 .71 (B) The value of cosec sec is a rational which is not integral.18 ^ 18

    (C) If x lies in the third quadrant, then the expression 1/4 s i n 4 independentofx.

    x

    + sin 2 2x

    +

    4 cos 2

    4

    2

    is

    (D) There are exactly 2 values of 9 in [0, 2tt] which satisfy 4 cos 2 9 - 2 -Jlcos 9 - 1 = 0 .

    MATCH THE COLUMN INSTRUCTIONS: Column-I and column-II contains four entries each. Entries of column-I are to be matched with some entries of column-El. One or more than one entries of column-I may have the matching with the same entries ofcolumn-H and one entry of column-I may have one or more than one matching with entries of column-II. Column-I Column-II (A) Area of the triangle formed by the straight lines (P) 1 x + 2y - 5 = 0, 2x + y - 7 = 0 and x - y + 1 = 0 in square units is equal to (Q) 3/4 (B) (C) Abscissa of the orthocentre of the triangle whose vertices are the points (-2, -1); (6, - 1) and (2, 5) Variable line 3x(A. + 1) + 4y(A. - 1) - 3 ( 1 - 1) = 0 for different values of A, are concurrent at thepoint (a, b). The sum (a + b) is The equation ax2 + 3xy - 2y2 - 5x + 5y + c = 0represents two straight lines perpendicular to each other, then | a + c | equals (R) (S) 2 3/2

    Q.l

    (D)

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    Column-I Q.2 (A) In a triangle ABC, AB = 2^3 , BC = 2-J6 , AC > 6, and area of the triangle ABC is 3 V

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    J g BANSAL CLASSESI B Target I1T JEE 2007CLASS: XII (ABCD) Q. 1 DATE: 04-07/10/2006

    Pa/7/ Practice ProblemsTIME: 40 Min.for each

    MATHEMATICSDPP. NO.-49, 50

    -49

    8 clay targets have been arranged in vertical column, 3 being in the first column, 2 in the second, and 3 in the third. In how many ways can they be shot (one at a time) if no target below it has been shot. [4]

    Q.2

    Evaluate: /x(sin 2 (sinx) + cos 2 (cosx))dx o Evaluate: jx(sin(cos 2 x)cos(sin 2x ) ) d x

    [4]

    Q.3

    [4]

    Q.4

    J - . x dx * V xYQ111 YJ-fAQY . sin x + cos x / 0 R satisfying ( x / ( x ) - 2F ( x ) ) ( F ( x ) - X 2 ) = 0 V x e R where f (x) = F'(x). [4]

    Q.2

    0 '3 Q 3 Q.4

    J j f ^ 2l3-xJ

    f *

    HI 0 0 J J^ dx reduces to zero by a substitution x = 1 /t. Using this or ax" + bx + a oaAx

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    For a > 0, b > 0 verify that

    f fax otherwise evaluate: i 2 0 tan - 1 xx"\3

    d

    [7]

    Q.5

    1 v

    dx

    y

    [81

    A

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    JABANSAL CLASS ESl ^ P T a r g e t HT J E E CLASS: XII (ABCD) 2007 DATE: 29-30/9/2006

    MATHEMATICSDaily Practice ProblemsDPP. NO.-47

    This is the test paper-1 of Class-XIII (XYZ) held on 24-09-2006. Take exactly 60minutes. P A R X - A Select the correct alternative. (Only one is correct) [24x 3 = 72] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q. 1 The area of the region of the plane bounded above by the graph of x 2 +

    y2 + 6x + 8 = 0 and below by the graph of y = | x + 3 is (A) jc/4 (B) ti2/4 (C)7c/2 (D) it Q.27' Consider straight line ax + by = c where a , b , c e R+ and a, b, c are distinct. This line meets the coordinate axes at P and Q respectively. If area of AOPQ, '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. If x and y are real numbers and x2 + y2 = 1, then the maximum value of (x + y)2 is (A) 3 Q.4 (B) 2 (C) 3/2 (D) J 5

    Q. y

    dx The value of the definite integral j n (a > 0) is q (1 + x )(1 + x ) (A) ti/4(B) nil (C) tc (D) some function of a. a b e cos coscos Let a, b, c are non zeroconstant number then Lim - equals

    r-co

    sinsin r r . . . _ _ _ J (D) independent of a, band c

    . b

    . C

    ... a 2 + b 2 - c 2 (A) 2bc Q.6 ^

    (B)

    c2 + a 2 - b 2 2bc

    ^xb2+c2-a2 (C) 2bc

    A curve y =/(x) such that/"(x) = 4x at each point (x, y) on it and crosses the x-axis at (-2, 0) at an angle of 450. The value of / (1), is (A) - 5 (B) - 15 (C)- f (D) y

    Q.7/ v

    sinx cosx tanx cotx = The minimum value of the function/(x) = 1 + / + 7 + ~T as2 9 Vl-cos x vl-sin x vsec x - 1 Vcosec x - 1 x varies over all numbers in the largest possible domain of / ( x ) is (A) 4 (B) - 2 (C) 0 (D) 2 A non zero polynomial with real coefficients has the property that f (x) = / ' (x) f"(x). The lea

    ding coefficient of / (x) is (A) 1/6 ' (B) 1/9 (C) 1/12 (D) 1/18l_

    Q.8

    Q-9

    r tan -1 (nx) ^ 2 X 2 Let Cn = J s i n - V ) then Lim n -C f l"n+l

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    equais

    (A) 1 / Q. 10

    (B) 0

    (C) - 1

    (D) 1/2 2 2 2 | = | z31 = 1 then z, + z 2 + z 3 , (D) equal to 1

    Let Zj, z2, z3 be complex numbers suchthat zx + z2 + z3 = 0 and | zx \ - \ is (A

    ) greater than zero (B) equal to 3 (C) equal to zero

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    Q.ll

    Number of rectangles with sides parallel to the coordinate axes whose vertices are all of the form (a, b) with a and b integers such that 0 < a, b < n, is (n eN)

    (A)Q.12 ^.13

    n 2 (n + l)2

    (B)

    (n - l ) 2 n 2

    (C)+

    (n + 1)2

    (D) n2

    Number of roots of the function/(x) ~ (A) 02

    1 ^ 3 - 3x + sin x is (C)2 (D) more than 2

    (B) 1

    If p (x) = ax + bx + c leaves a remainder of 4 when divided by x, a remainder of3 when divided by x + 1, and a remainder of 1 when divided by x - 1 then p(2) is (A) 3 (B) 6 (C) - 3 (D) - 6 Let/(x) be a function that has a continuous derivative on [a, b],/(a) and/(b) have opposite signs, and / ' (x) * 0 for all numbersx between a and b, (a < x < b). Number of solutions does the equation / ( x ) =0 have (a < x < b). (A) 1 (B) 0 (C) 2 (D) cannot be determined -3tt/2 j Sin(3x+ 7t)dx

    V^l

    5

    Which of the following definite integral has a positive value? 2it/3 0 0 Jsin(3x+ 7i)dx (g) Jsin(3x + 7t)dx ^ q Jsin(3x + Jt)dx 2tc/3 -3it/2

    ^

    Q. 16

    Let set A consists of 5 elements and set B consists of 3 elements. Number of functions that can be defined from A to B which are neither injective nor surjectiv

    e, is (A) 99 (B) 93 (C) 123 (D) none A circle with center A and radius 7 is tangent to the sides of an angle of 60. A larger circle with center B is tangent to the sides of the angle and to the first circle. The radius of the larger circle is (A) 30V3 (B) 21 (C) 20V3 (D) 30 The value of the scalar (p x q)-(r x s) can beexpressed in the determinant form as q-r (A) p-r qs p-s 1 1/x 0 p-r (B) q-s y p1/x p-s q-r

    vXl7

    \J2[- 18

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    (C) q-r p-s

    p-r

    q-s

    p-r (D) q-r

    p-s q-s

    Q.19

    a/x jf Lim x In 0 x->00 1_1

    5, where a, p, y are finite real numbers then (C) a e R, p=l, yeR (C) 4 (D) a eR, p = 1, y = 5 (D) 1

    (A) a = 2, p=l, yeR Q.20 Q.21 (A) 2

    (B) a =2, p=2, y = 5 (B) 2 / 2

    If / (x. y) = sin ( | x [ + | y |), then the area of the domain of / is A, B and

    C are distinct positive integers, less than or equal to 10. The arithmetic meanof A and B is 9. The geometric mean of A and C is 5 / 2 The harmonic mean of Band C is 9 (A) 9 (B) (C) (D) 2-^r v v_/ 2 ~ ' 19 9 19 17 If x is real and 4y2 + 4xy + x + 6 = 0, then the complete set of values of x for which y is real, is (A)x < 2 or x > 3 ( B ) x < - 2 or x > 3 ( C ) - 3 < x < 2 ( D ) x < - 3 or x > 2 Ialternatively toss a fair coin and throw a fair die until I, either toss a heador throw a 2. If I toss the coin first, the probability that I throw a 2 beforeI toss a head, is (A) 1/7 " (B) 7/12 (C) 5/12 (D) 5/7 Let A, B. C, D be (not necessarily square) 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.

    Q.22 Q.23

    Q.24

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    J s B A N S A L CLASS ESV S Target NT JEE 2 0 0 7CLASS: XII (ABCD) DATE: 02-03/10/2006

    MATHEMATICSDaily Practice ProblemsDPP. NO.-48

    This is the test paper-2 of Class-XIII (XYZ) held on 24-09-2006. Take exactly 60minutes. Select the correct alternative. (More than one is/are correct) There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q. 1 [ 3 x 6

    = 18]

    The function/(x) is defined for x > 0 and has its inverse g (x) which is differentiable. I f / (x) satisfies g(x) J f (t) dt = X2 and g (0) = 0 then (A)/(x) isan odd linear polynomial (C)/(2) = 1 (B)/(x) is some quadratic polynomial (D)g(2) = 4

    Q. 2

    Consider a triangle ABC in xy plane with D, E and F as the middle points of thesides BC, CA and AB respectively. If the coordinates of the points D, E and F are (3/2, 3/2); (7/2,0) and (0, -1/2) then which of the following are correct? (A)circumcentre of the triangle ABC does not lie inside the triangle. (B) orthocen

    tre, centroid, circumcentre and incentre of triangle DEF are collinear but of triangle ABC are non collinear. (C) Equation of a line passes through the orthocentre of triangle ABC and perpendicular to its plane is r = 2(i - j) + A.k 5V2 (D)distance between centroid and orthocentre of the triangle ABC is .

    Q. 3

    X X If a continuous function/ ( x ) satisfies the relation, j t / ( x - t ) dt =j / ( t ) dt + s j n X-+ cos x - x - 1 for 0 0 . all real numbers x, then whichof the following does not hold good? it (A)/(0) = 1 ( B ) / ' (0) = 0 (C)f" (0)= 2 (D) J / ( x ) d x = e * 0 MATCH THE COLUMN [ 3 x 8 = 24] There is NEGATIVEmarking. 0.5 mark will be deducted for each wrong match within a question.

    Q.l ,.. (A)T.

    Column IY VY tl X-*co

    Column II (P) 0

    Lim

    In x r dt IS V X JJ3 /n tt Ine

    (B) (C)

    ' vx +1 xz+l /~T7 2 -e Lim e

    is is where n e N\

    (Q) 4n

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    : 1

    Lim (-1)" s i n f W n 2 + 0.5n + l l sin J tan-1/

    (R)

    (D)

    The value of the integral j 0

    VX

    + ly9A A

    f

    tan"

    1

    l + 2x-2x

    dx is

    (S)

    non existent

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    Q.2

    Consider the matrices A=

    andB :

    a 0

    b T 1 and let P be any orthogonal matrix and Q = PAP Column II (P) G.P. with common ratio a (Q) A. P. with common difference 2 (R) GP. with common ratio b (S) A. P. with common difference - 2.

    and R = P T Q K P also S = PBP T and T = P T S K P Column I (A) If we vary K from 1 to n then the first row first column elements at Rwill form (B) If we vary Kfrom 1 to n then the 2 nd row 2nd column elements at Rwill form (C) If we varyK from 1 to n then the first row first column elements of T will form (D) If wevary K from 3 to n then the first row 2nd column elements of T will represent the sum of Q.3 (A) (B) (C) (D) Column I

    Column II (P) (Q) (R) (S) 30 4560

    Given two vectors a and b such that | a | = | b | = |a + b | = 1 The angle between the vectors 2a + b and a is In a scalene triangle ABC, if a c o s A = b c o s

    B then Z C equals In a triangle ABC, BC = 1 and AC = 2. The maximum possible value which the Z A can have is In a A ABC Z B = 75 and BC = 2AD where AD is the altitude from A, then Z C equals

    90

    SUBJECTIVE: Q.l 96V 2 1 SupposeV= J x sin x dx, find the value of 71 2tc/2

    [ 5 x 1 0 = 50]

    Q. 2

    " , where m r is non zero integer and n and r are relatively prime natural numbers. Find the value of m + n + r. A circle C is tangent to the x and y axis in the first quadrant at the points P and Q respectively. BC and AD are parallel tangents to the circle with slope - 1 . If the points A and B are on the y-axis while C and D are on the x-axis and the area of thefigureABCD is 900 V2 sq. units then find the radius of the circle.

    One of the roots of the equation 2000x6 + 100x5 + 1 Ox3 + x - 2 = 0 is of the form

    m +

    Q.3

    Q. 4

    Let/(x) = ax2 - 4ax+b (a > 0) be defined in 1 < x < 5. Suppose the average of the maximum value and the minimum value of the function is 14, and the differencebetween the maximum value and minimum value is 18. Find the value of a 2 + b2. 1x

    Q.5

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    If the Limx-*0

    1 + ax Vl + x 1 + bx

    1 2 3 exists and has the value equal to I, then find the value of - y + .

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    JGBANSAL CLASS>ESTarget I I T JEE 2 0 0 7CLASS: XII (ABCD) Q.l Q. 2 DATE: 27-28/9/2006

    MATHEMATICSDaily Practice ProblemsDPR NO.-46

    This is the test paper of Class-XI (J-Batch) held on 24-09-2006. Take exactly 75minutes. If tan a . tan P are the roots of x 2 - px + q = 0 and cot a,cot p arethe roots of x 2 - rx + s = 0 then find the value of rs in terms ofp and q. [4]

    Let P(x) = ax2 + bx + 8 is a quadratic polynomial. If the minimum value of P(x)is 6 when x = 2 , find the values of a and b. 14]( \_\

    Q.3

    LetP= f j 102" n=l

    .n-i

    then find log 001 (P). sec 8A - 1 sec 4A - 1 tan 8 A tan 2 A

    [4]

    Q. 4 Q.5 Q.6

    Prove the identity

    [4] [4] [4]

    Find the general solution set of the equation loglan x (2 + 4 cos hi) - 2. Findthe value of sin a + sin 3a + sin 5a + cos a + cos 3a + cos 5a + + sinl7a n - when a = . + cosl7a 24

    Q.7(a) Sum the following series to infinity 1 1-4-7+

    1 4-7-10

    +

    1 7-10-13

    +

    (b) Sum the following series upton-terms. 1 -2-3-4 + 2-3-4-5 + 3-4-5-6 + Q.8 Q.9 The equation cos 2 x - sin x + a = 0 has roots when x e (0, rc/2) find 'a'. The geometric mean ofA and C is 5 / 2 Fi n d the harmonic mean of B and C. Q. 10 Q. 11 Q. 12 Express cos 5x in terms of cos x and hence find general solution ofth

    e equation cos 5x = 16 cos 5 x.

    [3 + 3]

    [6]{6]

    A, B and C are distinct positive integers, less than or equal to 10. The arithmetic mean ofA and B is 9.

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    [6]

    If x is real and 4y 2 + 4xy + x + 6 = 0, then find the complete set of values ofx for which y is real.[6]

    Find the sum of all the integral solutions of the inequality 21og3x-41ogx27

    sin a + sin P + sin y - 1 cos a + cos p +cosy [7]

    j y i 4(a) In any A ABC prove that C C c 2 = (a - b) 2 cos 2 + (a + b) 2 sin 2 .(b) In any A ABC prove that a 3 cos(B - C) + b 3 cos(C - A) + c 3 cos(A - B) = 3abc.

    [4 + 4]

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    dQ. 1 Q.2 Q.3

    l BANSAL CLASSES5Targe* liT JEE 2007DATE: 20-21/9/2006

    MATHEMATICSDaily Practice ProblemsDPP. NO.-44

    CLASS: XII (ABCD)

    This is the test paper of Class-XI (PQRS) held on 17-09-2006. Take exactly 75 minutes. Evaluate 8 n -.2 r 5 s where 5 r s = r=l s=l Will the sum hold i f n - >oo?n n r O i f r ^ S

    1

    if r = s [4] |4J

    x x Find the general solution of the equation, 2 + tan x cot + cot x tan = 0ven that 3 sin x + 4 cos x = 5 where x e (0, n/2). Find the value of 2 sin x + c

    os x + 4 tan x.

    14] Q.4 Find the integral solution of the inequality log 0 3 ( x - 1 ) ' 1 + cos x true for every x e R Solveforx, s ig 2 * 2 ^ l o g J x V s ) = ^ l o g ^ x 2 _ 5 i o g 2 * 1 , In a tringle ABC if a 2 + b 2 = 101 c 2 then find the value of & cot C . cot A + cot B [5] [5] [5] 1 1

    Q.10 Q. 11

    Q.12

    Solve the equation for x, 5 2

    +52

    +!0g5(smx)

    = 152

    +l08l5(C0Sx)

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    [5]

    Z~n .n=l6

    00

    [5]

    Q. 14 Q . 15

    Suppose that P(x) is a quadratic polynomial such that P(0) = cos 3 40, P( 1) = (cos 40)(sm240) and P(2) t 0 . Find the value of P(3). [8] If /, m, n are 3 numbersin G.P. prove that the first term of an A.P. whose 7th, mth, nth terms are in H.P. is to the common difference as (m + 1) to 1. [8]

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    BAN SAL CLASSESy g Target I IT JEE 2007CLASS : XII (ABCD) Q. 1 DATE: 22-23/9/2006

    MATHEMATICSDaily Practice ProblemsTIME: 55 to 60 Min. DPP. NO.-45

    Let a, b, c, d, e, f e R such that ad + be + cf = ^ ( a 2 + b 2 + c 2 ) ( d 2 +e 2 + f 2 ) use vectors or otherwise to prove that, a+b+c Va2+b2+c2 d+e+f Vd2+e2+f2 '

    Q.2

    Let the equation x 3 - 4x 2 + 5x - 1.9 = 0 has real roots r, s, t. Find the areaof the triangle with sides r, s, and t. 50 2

    Q. 3

    Suppose x + ax + bx + c satisfies f (-2) = - 1 0 and takes the extreme value where x = . Find the value of a, b and c.

    J

    2

    Q- 4

    f i-y Hv L e t I x l d X ^/nx +xy-

    and

    r / n x x + xy _ I 1y

    dy

    x d dy where ~ = x y . Show that I J = (x + d)(y + c) where c, d e R. Hence show

    that (I J) = I + J y dx dx

    Q.5

    Let a;, i = 1, 2, 3, 4, be real numbers such that aj + % + % + a 4 = 0. Show that for arbitrary real numbers bi5 i = 1,2, 3 the equation a, + bjx + 3a 2 x 2 + b2 x 3 + Sa^x4 + b 3 x 5 + 7a 4 x 6 = 0 has at least one real root which lies onthe interval - 1 < x < 1.

    V3

    Q.6

    Evaluate:

    x 2l l x t = x J I x + x +3x" + X r

    dx+ 1

    Q. 7

    Let x, y e R in the interval (0, 1) and x + y = 1. Find the minimum value of the

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    expression x x + yy

    r | (1 - sin x)(2 - sin x) ^ y (1 + sin x)(2 + sin x)

    ^

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    i l l SBANSALCLASS: XII (ABCD)

    C L A S SE S l U l a r g e t NT JEE 2 0 0 7DATE: 08-12/9/2006

    M A T H E M A T I C S

    Daily Practice ProblemsDPP. NO.-42, 43 TIME : 60 Min.

    DATE : 08-09/09/2006 O P P - 4 2 This is the test paper of Class-XIII (XYZ) heldon 27-08-2006. Take exactly 60 minutes. S^'S-yV Select the correct alternative,(Only one is correct) There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. sin 2 (x 3 + x 2 + x - 3 ) Li rn ~ ~ ~ ~~ has the value equal to M x->i 1 cos(x 4x + 3) (A) 18 (B) 9/2 (C) 9 (D) none

    [16 x 3 = 48]

    Q. I

    dt Q.2 / Let/(x)= r . . If g'(x) is the inverse of / ( x ) then g'(0) has the value equal to 4 2 * 3-v t +3t +13 (A) 1/11 (B) 11 (C)Vl3 (D) l / V n Q.3 The function/(x) has the property that for each real number x in its domain, 1/x is also

    in its domain and /(x) + /(l/x) = x. The largest set of real numbers that can be in the domain of /(x), is (A){x|x*0) (B) { x | x > 0) (C) { x | x * - l a n dx * 0 a n d x * 1) Q.4 j 6/ Let w =Z2

    (D) {-1, 1}

    z +1

    37 + 6

    ,

    and z = 1 + i. then | w | and amp w respectively are (B) , - 71/4 (C) 2, 3TC/4 (D) ^ , 3n/4

    (A) 2, - n /4

    1 - cos a - tan 2 (a/2) k cos a Q.5 A If . j/ " ~= where k, w and p have no common factor other than 1, then the F ./! sin (a/2) w + pcosa 2 2 2 value of k + w+ p is equal to (A) 3 (B)4 (C)5 (D)64 Q.6 In a birthday .party, each man shook hands with eveiyone except his spouse, and no handshakes took place between women. If 13 married couples attended, how many handshakes were there among these 26people? (A) 185 (B)234 (C)312 (D)325 If x and y are real numbers such that x2 +y2 = 8, the maximum possible value of x - y, is (A) 2 (B) (C) V2/2 (D) 4 Let w(x

    ) and v(x) are differentiable functions such that u(x) = 7. If U^x) ~ P and ' u(x) v(x) = q, then

    Q.7

    Q.8/

    p+q

    M p - q has the value equal to

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    (A) 1 Q.9 Q. 10

    (B)0

    (C)7

    (D)-7

    The coefficient of x9 when (x + (2/Vx j)30 is expanded and simplified is (A) 30C|4 29 (B) 30C]6 214 (C) 30 C 9 -2 21 (D) 10C9 Let C be the circle described by (

    x - a)2 + y2 = r2 where 0 < r < a. Let m be the slope of the line through the origin that is tangent to C at a point in the first quadrant. Thenr Va2-r2 r (A) m = r ^ 7 (B) m = (C) m = (D) m = Va - r r a What can one say about the local extrema of the function/(x) = x + (1/x)? (A) The local maximum off(x) is greater than the local minimum of/(x). (B) The local minimum off (x) greater than the local maximum off (x). (C) The function/(x) does not have any localextrema. (D)/(x)

    a r

    Q. 11

    has one asymptote.

    Q.l 2

    / r_2^ + arctan(5) equals tan arc tan I 3 v (A) - / 3 (B)-l

    (C)l

    (D)V3

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    / y/Q- ip

    A line passes through (2, 2) arid cuts a triangle of area 9 square units from the first quadrant. The sum of all possible values for the slope of such a line, is (A) - 2.5 (B) - 2 (C) - 1.5 (D) - 1 Which of the following statement is/are true concerning the general cubic / ( x ) = ax3 + bx2 + cx + d (a * 0 & a, b, c, de R) I The cubic always has at least one real root II The cubic always has exactly one point of inflection (A) Only I (B) Only II (C) Both I and II are true IfS = 1 + 3 + 5 + (A) S + 25502 2 2 2 2 2 2

    ^gf. 14

    (D) Neither 1 nor II is true

    Q. 15 Q. 16

    + (99) then the value of the sum 2 + 4 + 6 + + (100)2 is (B)2S (C) 4S (D) S + 5050

    Through the focus of the parabola y2 = 2px (p > 0) a line is drawn which intersects the curve at A(x,, y,) y\y 2 and B(x,, v.). The ratio x x equals l 2 (A) 2 (B) - 1 (C) - 4 (D) some function of p Select the correct alternative. (Only oneis correct) There is NEGATIVE marking. 1 mark will-be deducted for each wrong an

    swer, i n-3n ^ i If 6 N) ^n(x-9)+n-3D+1-3n = 3 ^ ^ ^ f X iS (A) [2,5)' ' (B) (1) (C) (-1,5) (D)(-co,oo) The area of the region(s) enclosed by the curves v = x2and y = ^ | x | is (A) 1/3 (B) 2/3 (C) 1/6 (D) 1 Suppose that the domain of thefunction/(x) is set D and the range is the set R, where D and R are the subsetsof real numbers. Consider the functions:/(2x),/(x + 2), 2/(x), /(x/2), / ( x )/ 2 - 2 . If m is the number of functions listed above that must have the same domain as/and n is the number of functions that must have the same range as f (x), then the ordered pair (m, n) is (A) (1,5) (B) (2, 3) (C)(3,2) (D) (3, 3) r x 2+ 2mx - 1 for x < 0 / : R - R is defined as / ( x ) = - mx - 3 for x > 0 If / (x ) is one-one then m must lies in the interval(A) ( oo, 0) (B) ( oo, 0] (C)(0,oo) (D) [0, co)B = { x | (m - 1)X2 + m x + 1 = 0, X e R } Let A = { x | x 2 + (M - l ) x - 2(m+ 1 ) = 0 , X G R } ;

    [ 9 x 4 = 36 j

    '! 7 18 Q.l 9

    Q.20

    Q.21

    . Number of values of m such that A u B has exactly 3 distinct elements, is (A)4 (B) 5 (C) 6 (D) 7 ^Q.22 If the function/(x) = 4x2 - 4x - tarra has the minimumvalue equal to - 4 then the most general values of 'a' are given by (A) 2n7t +ti/3 (B) 2nn - rc/3 (C) im n/3 (D) 2nn/3 where n e I Direction for Q.23 to Q.25.

    ^/Q.23 sinx-xcosx x Consider the function defined on [0, i] -> R, / ( x ) = 5 *0 anc f (0) = 0 The function/(x) (A) has a removable discontinuity at x = 0 (B)has a non removable finite discontinuity at x=0 (C) has a non removable infinitediscontinuity at x = 0 (D) is continuous at x = 01

    ^jQ.24

    J / ( x ) d x equals (A) 1 - sin (1)t

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    (B) sin (1) - 1

    (C) sin (1)

    (D)-sin(l)

    ^.25

    1 L i m z j / ( x ) d x equals t->o t 7 1 0 (A) 1/3 (B) 1/6

    (C) 1/12

    (D) 1/24

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    DATE : 11-12/09/2006

    i>B>S>-43

    TIME : 60 Min. [ 7 x 4 = 28]

    Select the correct alternative. (More than one are correct) xe x Q.26 Let / (x)=L

    There is NO NEGATIVE marking. Marks will be awarded only if all the correct alte

    rnatives are selected. x 0 ( A )/ is continuous and differentiate for all x. ( B ) / is continuous but not differentiate at x = 0. ( C ) / ' is continuous and differentiate for all x. ( D ) /' is continuous but not differentiate at x = 0. x2-l Suppose/ is defined from R [1, 1] as / ( x ) = z > where R is the set of real number. Then the x" + 1 statent which does not hold is ( A ) / is many one onto ( B ) / increases for x > 0and decrease for x < 0 (C) minimum value is not attained even though f is bounded (D) the area included by the curve y = f (x) and the line y = 1 is n sq. units.2

    Q.27

    Q.28

    The value of the definite integral

    r , (3 + cosx V J x ' n i 3 _ c o s x J > is v 0 ]dx J ^3-cosx J ) ( CV z e r o' (D) V* 0 V3 + c o s x ;

    (A) n ] l n ( Jdx J V3 cosx J

    (B)

    0-29

    r x 3 (l-x)sin(l/x 2 J if 0 < x < l f : [0. 1] -> R is defined as / ( x ) = j __, then 0 if x = 0 (A)/ is continuous but not derivable in [0, 1 ] ( C ) / is bounded in [0, 1 ] ( B ) / is differentiate in [0, 1 ] ( D ) / ' is bounded in [0,1]

    Q.30

    Let 2 sin x + 3 cos v = 3 and 3 sin y + 2 cos x = 4 then (A) x + y = (4n + 1)TE/2, n e l (B) x + y = (2n + l)rc/2, n E I (C) x and y can be the two non right angles of a 3-4-5 triangle with x > y. (D) x and v can be the two non right anglesof a 3-4-5 triangle with y > x. The equation cosec x + sec x = 2V2 has (A) no solution in (0, n/4) (C)no solution in (n/2, 3n/4) (B) a solution in [tc/4 , n/2)(D) a solution in [37r/4, tc)

    2

    Q.31

    Q.32

    For the quadratic polynomial / ( x ) = 4x - 8kx + k, the statements which hold good are (A) there is only one integral k for which/(x) is non negative V x e R (B) for k < 0 the number zero lies between the zeros of the polynomial. (C)/(x) =0 has two distinct solutions in (0, 1) for k e (1/4, 4/7) (D) Minimum value of

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    y V k e R is k(l + 12k) I^A. l^TI-S^ MATCH THE COLUMN [ 3 x 8 = 24]

    Q. i

    Column-I contain four functions and column-II contain their properties. Match every entry of column-1 with one or more entries of column-II. Column-I Column-II(A) / ( x ) = sin"](in x) + cos""1 (cos x) (P) range is [0,71] (B) g (x) = sin-'j-x | + 2 tair'j x | (Q) is increasing V x e (0, 1) (C) (D) ( 2x 1 h (x) = 2sirr>! j j , x 6 [0, 1] k (x) = cot(cor'x) (R) (S) period is 2% is decreasing V x e (0, 1)

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    Q.2 (A) (B)

    Column-I

    Column-II +c a+b+c 3

    Centre of the parallelopipeci whose 3 coterminous edges OA, OB and (P) OC have position vectors a, b and c respectively where O is the origin, is OABC is a tetrahedron where O is the origin. Positions vectors of its angular points A, B andC are a, b and c respectively. Segments joining each vertex with the centroid ofthe opposite face are concurrent at a point P whose p. v.'s are Let ABC be a tr

    iangle the position vectors of its angular points are a, b and c respectively. If\a-b\ = \b-c\=\c-a\then the p.v.of the orthocentre of the triangle is Let a, b,c be 3 mutually perpendicular vectors of the same magnitude. If an unbiown vector x satisfies the equation a x[fx -b)xaj+b x[(x-c)xbj+c x({x -a)xc) = G. Then xis given by Column-I

    (Q)

    (C)

    (R) a+b+c 2

    (D)

    (S)

    Q.3 (A)

    Column-II (a - b)(b - c)(c - a)(a + b + c) then the solution 1 (x-b)2 (x-c)(x-a)1 (x-c)2 =0, is (x-a)(x-b) (P) a +b+c

    If

    1 a a~

    1 b

    1 (x-a)2 of the equation (x-b)(x-c) (B) The value of the limit,f , a X + b X + cX

    (Q)

    L X

    (^/(x + a)(x + b)(x + c) - x), iis

    (R)(S)

    a +b+c

    (C) (D)

    Lim x->0

    equals

    Let a, b, c are distinct reals satisfying a3 + b3 + c3 = 3abc. If the quadraticequation (a + b - c)x2 + (b + c - a)x + (c + a - b) = 0 has equal roots then a r

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    oot of the quadratic equation is

    SUBJECTIVE: Q.l

    [ 4 X 6 = 24]

    Let / ( x ) = (x + l)(x + 2)(x + 3)(x + 4) + 5 where x e [-6, 6], If the range of the function is [a, b] where a, b e N then find the value of (a + b).tu/4

    Q.2 Q.3

    Let I

    o

    j

    (TCX - 4x ) /n(l + tan x)dx. If the value of 1

    2

    7i "7n 2 k

    where k e N, find k.

    Suppose/and g are two functions such that f g : R -> R,2 / ( x ) ^/n^l + V l ^ ]

    and ( fiW

    g(x) = /n! x + \ / l T x 2

    then find the value of x egW

    + g'(x) at x = 1. 120ti is equal to -, find the value of k.K

    Q.4

    If the value of limit

    L,m

    -1 l + 7 ( k - l ) k ( k + lXk + 2) Z cos k(k + l) k=2

    /

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    JHBANSAL CLASSIES^ T a r g e t 1ST JEE 2 0 0 7CLASS: XII (ABCD) DATE: 04-07/9/2006

    MATHEMATICSDaily Practice ProblemsDPR N0.-40, 41

    DATE: 04-05/09/2006 TIME: 50 Min. Q. 1 Let/(x) = 1 - x - x 3 . Find all real values of x satisfying the inequality, 1 - / ( x ) - / 3 ( x ) > / ( 1 - 5x) g2x _gX j Integrate: j dx 3 (e x sin x + cos x)(e x cos x - sin x) The circle C : x 2

    + y 2 + kx + (1 + k)y - (k + 1) = 0 passes through the same two points for every real number k. Find the coordinates of these two points. the minimum value ofthe radius of a circle C. i Comment upon the nature of roots of the quadratic equation x + 2 x = k + J| t + k | dt depending on the 0 value of k e R.2

    Q.2

    Q.3 (i) (ii) Q. 4

    Q.5

    1/n a C Given Lim = where a and b are relatively prime, find the value of (a + b)

    . 2n f\ b n->oo \ ny3n

    DFP-41DATE: 06-07/09/2006 Q. 1 TIME: 50 Min.

    Let a, b, c be three sides of a triangle. Suppose a and b are the roots of the equation x 2 - (c + 4)x + 4(c + 2) = 0 and the largest angle of the triangle is 9degrees. Find 0. 7 1 Find the value of the definite integral j|V2sinx + 2 c o sx jdx. o 1 Let tan a tan (3 = 7 ^ 5 . Find the value of (1003 - 1002 cos 2a)(1003 - 1002 cos 2(3)1+V5

    Q.2

    Q.3

    0 * 4 Q.5

    2

    r

    /

    X2 + l /. j( .l n + X X +1 V

    x

    n dx XJ and e 2 is 60. The angle

    Two vectors Sj and e 2 with | e ( | = 2 and \ e 2 | = 1 and angle between

    between 2t e, + 7 e 2 and ej +1 e 2 belongs to the interval (90, 180). Find the range of t. Q.6 Afimction fix) continuous on Rand periodic with period 2% satisfies f (x) + sin x - / ( x + n) = sin 2 x. Find/(x) and evaluate f / ( x ) d x .

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    CLASS: XII (ABCD) DATE; 30-31/8/2006 TIME: 60 Min. DPP. NO.-39 This is the testpaper of Class~XI (J-Batch) held on 27-08-2006. Take exactly 60 minutes. Q. 1 Find the set of values of'a' for which the quadratic polynomial (a + 4)x 2 - 2ax +2a - 6 < 0 V x e R . x+1 x+5 Solve the inequality by using method of interval, -^ [3] I31 [3] [3]

    4

    | BAN SAL CLASSESglTarget SIT JEE 2007

    MATHEMATICS^Daily Practice Problems

    Q. 2 Q.3 Q.4 Q.5

    Find the minimum vertical distance between the graphs of y = 2 + sin x and y = cos x. d (3 ^ cos x - c o s J x Solve: dx 4 whenx = 18. If p, q are the roots of the quadratic equation x + 2bx + c = 0, prove that 2 l o g [ j y - p + y f y - q} = log2 + log(y + b + j,2

    [4] [4]

    Q. 6 Q.7 Q. 8 Q.9 Q.10 Q.ll Q. 12 Q. 13

    x 2 +14x + 9 Find the maximum and minimum value of y = , VxeR. x +2x + 3

    Suppose that a and b are positive real numbers such that log 2 7 a + log 9 b = 7/2 and log 27 b + log 9 a=2/3. Find the value of the ab. [4] Given sin 2 y=sin x sin z where x, y, z are in an A.P. Find all possible values of the common difference of the A.R and evaluate the sum of all the common differences which lie inthe interval (0,315). [4] tan 86 Prove that = (1 + sec29) (1 + sec40) (1 + sec86). [4] jl 371 571 In Find the exact value of tan 2 : + tan 2 + tan2~ + tan 2 16 16 1689 i

    [4] 151

    Evaluate Y ^ l + (tann) 2

    Find the value of k for which one root of the equation of x 2 - (k + 1 )x + k 2+ k-8=0 exceed 2 and other is smaller than 2. [5] Let an be the 0 th term of anarithmetic progression. Let Sn be the sum of the first n terms of the arithmeticprogression with aj = 1 and a 3 = 3a g . Find the largest possible value of S n. [5]V Z. J

    ( C^ C A B Q. 14(a) IfA+B+C = n & sin A + = k sin , then find the value of tan an in terms of k. ( \ X +x (b) Solve the inequality, log. log 6 -

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    4Q. 1 Q.2

    | BANSAL CLASSESj Target III JEE 2 0 0 7DATE: 23-24/8/2006

    MATHEMATICSDaily Practice ProblemsTIME: 60 Min. DPP. NO.-38

    CLASS: XII (ABCD)

    r 2 1/2 Find the value of a and b where a < b, for which the integral j (24 - 2x- x ) dx } i a s the largesta

    value.

    Solve the differential eqaution: y' +

    sin x - cos x Ve -cosx y= e x - c o s x

    Q.3

    Integrate: J. -dx (x cos x - sin x)(x sin x + cos x)

    Q.4

    In a A ABC, given sin A: sin B : sin C = 4 : 5 : 6 and cos A: cos B : cos C = x: y : z. Find the ordered pair that (x, y) that satisfies this extended proportion.

    QQ.6

    5

    V sin 1 V x

    FCNdxX X

    Find the general solution of the equation, 2 + tan x cot + cot x tan = 0

    Q.7

    Let a , (3 be the distinct positive roots of the equation tan x = 2x then evaluate J(sinax-sin[3x)dx , o independent of a and {3.

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    J | BANSAL CLASSESI g g T a r g e f HT JEE 2 0 0 7

    MATHEMATICSDaily Practice ProblemsDPR NO.^37

    CLASS: XII (ABCD) " DATE: 18-19/8/2006 TIME: 75 Min. This is the test paper of Class-XI (PQRS) held on 13-07-2006. Take exactly 75 minutes. Q. 1 Q.2 Q.3

    The sum of the first five terms of a geometric series is 189, the sum of the fir

    st six terms is 381, and the sum of the first seven terms is 765. What is the common ratio in this series. [4] Form a quadratic equation with rational coefficients if one of its root is cot 2 l 8. 1( a + 1)(p + 1 }

    [4]

    Let a and (3 be the roots ofthe quadratic equation ( x - 2 ) ( x - 3)+(x-3)(x +l ) + ( x + l)(x-2)=0.Find the value of +(a

    1 _ 2)(p _2) +

    (a

    _m

    1

    _ 3)

    W [4]

    Q.4 Q.5 Q. 6 Q.7

    If a sin2x +Mies in the interval [-2,8] foreveryx < R then find the value of ( a

    - b ) . =

    For x > 0, what is the smallest possible value of the expression log(x 3 - 4x 2+ x + 26) - log(x + 2)? [4] The coefficients of the equation ax 2 + bx + c = 0 where a * 0, satisfy the inequality (a + b + c)(4a - 2b + c) < 0. Prove that thisequation has 2 distinct real solutions. [4]

    In an arithmetic progression, the third term is 15 and the eleventh term is 55.An infinite geometric progression can be formed beginning with the eighth term of this A.P. and followed by the fourth and second term. Find the sum of this geometric progression upto n terms. Also compute Srjo if it exists. [5] Find the solution set of this equation log)sin X|(x2 - 8x + 23) > l o g ( s i n x j ( 8 ) in x e [0,2n). Find the positive integers p, q, r, s satisfying tan = ( j p - Jq)

    (yfr - s)Find the sum to n terms of the series. 1+

    Q.8 Q.9 Q. 10

    [5] [5]

    2 + -

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    3+

    4+

    5+

    2 4 8 16 32 Also find the sum if it exist if n -> oo. Q. 11 Q. 12 Q.13 (a) (b) (c) Q. 14

    2

    [5]

    If sin x, sin 2x and cos x sin 4x form an increasing geometric sequence, find the numerial value of cos 2x. Also find the common ratio of geometric sequence. [5] Find all possible parameters 'a' for which, f (x) = (a 2 + a - 2)x 2 - (a + 5)x - 2 is non positive for every x e [0,1 ].st nd rd 2 st nd

    [5 j

    The 1 , 2 and 3 terms of an arithmetic series are a, band a where 'a' is negativ

    e. The 1 , 2 and 3rd terms of a geometric series are a, a 2 and b find the val ue of a and b sum of infinite geometric series if it exists. If no then find thesum to n terms of the G P sum ofthe 40 term ofthe arithmetic series. [5] j) Then th term, a n of a sequence of numbers is given by the formula a n = a n _ } +2n for n > 2 and aj = 1. Find an equation expressing an as a polynomial in n. Also find the sum to n terms ofthe sequence. [8] 2x x Let/(x) denote the sum of the infinite trigonometric series, / ( x ) = ^ sin sin . 3 n=J 3 Find/ ( x ) (independent of n) also evaluate the sum ofthe solutions ofthe equation f (x) = 0 lying in the interval (0,629). [8]00

    Q. 15

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    . k B A N S A L CLASSESI B Target I I T JE 2 0 0 7CLASS: XII (ABCD) I > I* I " - 3 DATE: 16-17/08/2006 Q.l Q.2 Q.3x 3

    MATHEMATICSDaily Practice ProblemsDPP. NO.-35, 36 5 TIME: 45 Min.

    d If y = Jx 2 V^nt dt, find at x = e . l Find the equation of the normal to thecurve y = (l +x) y + sin -1 (sin2 x) at x = 0.

    x

    Find the real number 'a' such that 6 + Ja2

    f f(t)dt

    -j = 2 v x

    Q.4 Q.5

    7 The tangent to y = ax + bx + - at (1,2) is parallel to the normal at the point

    (-2, 2) on the curve y = x 2 + 6x + 10. Find the value of a and b. Let f be a real valued function satisfying f(x) + f(x+4) = f(x + 2) + f(x + 6) then prove that the functionx+8

    g(x) = | f(t) dt is a constant function.X

    Q. 6

    A tangent drawn to the curve C l = y = x 2 + 4 x + 8 at its point P touches thecurve C 2 = y = x 2 + 8x + 4 at its point Q. Find the coordinates of the point Pand Q, on the curves C j and C 2 .

    3S DATE: 16-17/08/2006 TIME: 45 Min. 2 3 4 Q. 1 Given real numbers a and r, consider the following 20 numbers: ar, ar , ar , ar , , ar20. If the sum of the 20 numbers is 2006 and the sum of the reciprocal of the 20 number is 1003, find the product of the 20 numbers. Q.2 Let f(x) and g(x) are differentiable functions satisfyingthe conditions; (i)f(0) = 2 ; g ( 0 ) = l (ii)f'(x) = g(x) & Find the functions f(x) and g(x).3

    (iii)g'(x) = f(x).

    Q.3

    Let f(x) = L 2x-3

    (b3-b2+b-l) Y (b 2 + 3b + 2 j

    _ ,0

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    nstant. Q.5 Q. 6 r Given Jf(tx) dt = nf( x ) then find f(x) where x > 0. o Tangent at a point P j [other than (0,0)] on the curve y = x 3 meets the curve againat P 2 . The tangent at P 2 meets the curve at P 3 & so on. Show that the abscissae of P,, P 2 , P 3 , P n , form a GP. Also find the ratio ^(P^P,) area (P 2 P3 P 4 )Xf

    V 2

    X16

    X18

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    ft

    4Q.l Q.2

    | BANSAL CLASSES|Target 8iT JEE 2007DATE: 11-12/8/2006

    MATHEMATICSDaily Practice Problems

    TIME: 60 Min. DPP. NO.-34

    CLASS 7 XII (ABCD)

    Let F (x) = jV4 +1 2 dt and G (x) = JV4 +1 2 dt then compute the value of (FG)'(0) where dash-1 X

    denotes the derivative.

    10 identical balls are to be distributed in 5 different boxes kept in a row andlabelled A, B, C, D and E. Find the number of ways in which the balls can be distributed in the boxes if no two adjacent boxes remain empty.

    Q. 3

    Iff (x) = 4x 2 + ax + (a - 3) is negative for atleast one negative x, find all possible values of a.

    Q.4 (a) (b) (c)

    Let/(x) = sin 6 x + cos 6 x + k(sin 4 x + cos 4 x) for some real number k. Determine all real numbers k for which/(x) is constant for all values of x. all realnumbers k for which there exists a real number 'c' such that f (c) = 0. I f k =- 0 . 7 , determine all solutions to the equation/(x) = 0.

    7 T Q.5

    ,

    Letx 0 = 2cos a n d x n = ^ 2 + x ^ , n = 1 , 2 , 3 , n-*>o find Lim 2< n+1) -V2^T n ~.

    Q.6

    f Let/(x)= - then find the value of the sumy j 20C>6 / + ^

    1 1 f 2 1 +f U 0 0 6 j ^ ^2006 J

    f 3 ^ [2006J

    (2005^ 2006 J

    Q.7

    V j ^ d * 8 + sin x

    x

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    .

    Q.8

    Va For a > 0, fmdthe minimum value ofthe integral J(a 3 + 4 x - a 5 x 2 ) e a xdx.0

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    I BANSAL CLASSESTarget liT JEE 2007CLASS: XII (ABCD) O P P1

    MATHEMATICSDaily Practice ProblemsDPP. NO.-33 E K X H E W E

    DATE: 31/7/2006 to 5/08/2006 O F

    This is the test paper of Class-XIII (XYZ) held on 30-07-2006. Take exactly 2 Hours. N O T E : Leave Star ( *) marked problems. " P A R T ' - A . Select the correct alternative. (Only one is correct) Q.l Number of zeros of the cubic f (x) =x3 + 2x + k V k e R, is (A) 0 (B) 1 (C) 2 /x Q.2 The value of Lim dr, is x-> dx yL(r + l ) ( r - l ) (A) 0 Q.3 (B) 1 (C) 1/2 -2 x 4 (D) non existent 5 - 1 equalto 86. The sum of 2x (D)9 (D)3 [26 x 3 = 78]

    t

    There are two numbers x making the value of the determinant these two numbers, is (A)-4 (B)5 (C)-3

    Q.4

    A function / (x) takes a domain D onto a range R if for each y e R , there is some x e D for which / (x) = y. Number of function that can be defined from the domain D = {1,2,3} onto the range R = {4, 5} is (A) 5 (B)6 (C)7 (D)8 Suppose/,/' and/" are continuous on [0, e] and that/' (e) = / ( e ) = / ( l ) = 1 and je 1

    Q.5

    f/(x),2X

    = Z, then1

    1

    the value of f / " ( x ) / n x d x equals I 5 1 3 1 (B) j Q.6

    (C)

    1

    1

    (D) 1 -

    1

    A circle with centre C (1, 1) passes through the origin and intersect the x-axisat A and y-axis at B. The area of the part of the circle that lies in the firstquadrant is (A) n + 2 (B) 2n - 1 (C) 2n - 2 (D) n + 1 The planes 2x - 3y + z =4 and x + 2y - 5z = 11 intersect in a line L. Then a vector parallel to L, is (A) 13i + l l j + 7 k (B) 1 3 i + l l j - 7 k (C) 1 3 i - l l j + 7 k (D) i + 2 j- 5 k

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    &Q.8 A fair dice is thrown 3 times. The probability that the product of the three outcomes is a prime number, is (A) 1/24 (B) 1/36 (C) 1/32 (D) 1/8 Q.9 Period of the function, / ( x ) = [x] + [2x] + [3xj + + [nx] n(n +1)n J.

    where n e N and [ J denotes the greatest integer function, is (A) 1 (B) n (C) 1/n Q. 10

    (D) non periodic

    2i - i 1 Let Z be a complex number given by, Z = 3 i - 1 the statement which does not hold good, is (A) Z is purely real 10 1 1 (B) Z is purely imaginary (C) Zis not imaginary (D) Z is complex with sum of its real and imaginary part equalsto 10 Let/(x, y) = xy2 if x and y satisfy x2 + y2 = 9 then the minimum value of f (x, y) is (A) 0 (B) - 3-^3 (Q-6V3 (D)-3V6

    Q. 11

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    Q. 12

    Vl + 3 x - l - x Eim ^ has the value equal to x^o (1 + x) -l-101x (A)3 5050 (B)5050 (C) 5051 (D) 4950

    Q. 13

    Number of positive solution which satisfy the equationlog 2 x log 4 x log 6 x = log 7 x log 4 x + log 2 x log 6 x + log 4 x loggX

    (A) 0 Q.14 Q. 15

    (B) 1

    (C) 2_1 3

    (D) infinite

    Number of real solution of equation 16 sin"'x tan x cosec"'x = n is/are (A) 0 (B) 1 (C) 2 (D) infinite Length of the perpendicular from the centre of the ellipse 27x2 + 9y2 = 243 on a tangent drawn to it which makes equal intercepts on thecoordinates axes is (A) 3/2f,

    (B) 3/V2 1x2n2

    (C) 3V2 2x 1-x2

    (D) 6

    Q.l 6

    Let/(x) = cos"1 (A) 0

    1+ x

    + tan (B) ti/4

    where x e (-1, 0) then/simplifies to (C) n/2 (D) 7t

    Q. 17A person throws four standard six sided distinguishable dice. Number of ways in which he can throw if the product of the four number shown on the upper faces is 144, is (A) 24 (B) 36 (C) 42 (D)48 Q.18 a Let A = p x b q y 4x c r and suppose that det.(A) = 2 then the det.(B) equals, where B = 4y 4z z (B) det(B) = -8 (C) det(B) = - 162003

    2a 2b 2c

    -p -q -r

    (A) det(B) = - 2 Q. 19

    (D) det(B) = 8 (D)9

    The digit at the unit place ofthe number (2003) is (A) 1 (B) 3 (C) 7

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    Q.20

    AB AF Let ABCDEFGHIJKL be a regular dodecagon, then the value of + is Ar AB (A)4 (B)2-s/3 (C) 2V2 (D)2

    &Q.21 Urn A contains 9 red balls and 11 white balls. Urn B contains 12 red ballsand 3 white balls. One is to roll a single fair die. If the result is a one ora two, then one is to randomly select a ball from urn A. Otherwise one is to randomly select a ball form urn B. The probability of obtaining a red bail, is (A)41/60 (B) 19/60 (C) 21/35 (D)35/60 Q.22 L e t / be a real valued function of real and positive argument such that / ( x ) + 3x / (l/x) = 2(x + 1) for all real x

    > 0. The value of /(10099) is (A) 550 Q.23 (B) 5052

    (C)5050

    (D) 10010 a\2 /

    If a and P be the roots of the equation x + 3x + 1 = 0 then the value of

    1+P

    +

    P a +1

    is equal to

    Q.24

    (A) 15 (B) 18 (C) 21 (D) none The equation (x - l)(x - 2)(x - 3) = 24 has the real root equal to 'a' and the complex roots b and c. Then the value of b c / a ,is (A) 1/5 (B) - 1/5 (C) 6/5 (D) - 6/5 cos m0 sin n0 then m + n is equal to sin0(D) 12 If m and n are positive integers satisfying 1 + cos 20 + cos 40 + cos 60

    + cOs 80 + cos 100 = (A) 9 (B) 10 (C) 11

    Q.25

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    Q.26

    A circle of radius 320 units is tangent to the inside ofa circle ofradius 1000.The smaller circle is tangent to a diameter of the larger circle at the point P.Least distance of the point P from the circumference of the laiger circle is (A)300 (B)360 (C)400 (D) 420 Select the correct alternative. (More than one are correct)

    [8x4 = 32]x e 1/x

    Q.27

    In which of the following cases limit exists at the indicated points. at x = 0 xl + e 1/x where [x] denotes the greatest integer functions. tan-11 x | (C)/(x)= (x - 3)1/5 Sgn(x - 3) at x = 3, (D)/(x) = at x = 0. x where Sgn stands for Signum function.

    (A) /(x)

    [x+|x|]

    at x = 0

    (B)/(x) =

    &Q.28 Let A and B are two independent events. If P(A) = 0.3 and P(B) = 0.6, then(A) P(A and B) = 0.18 (B) P(A) is equal to P(A/B) (C) P(A or B) = 0 (D) P(A orB) = 0.72 Q.29 Let T be the triangle with vertices (0, 0), (0, c2) and (c, c2) and let R be the region between y = cx and y = x2 where c > 0 then (A) Area (R)=c3 (B) Area of R= 3 In Area (T) Area(T) _ 3 (C) Lim =3 (D) Lim + + c-o Area (R) cArea(R) 2( x+3

    Q.30

    Q.31

    Consider the graph of the function f (x) = e U+i . Then which of the following is correct. (B) / (x) has no zeroes. (A) range of the function is (1, oo) (D) domain of f is ( - oo, - 3) u (-1, oo) (C) graph lies completely above the x-axis.1 1 x x-1 Let /,(x) = x, / 2 (x) = 1 - x; / 3 (x) = - ,/ 4 (x) = ; / 5 (x) = ; /6(x) = X I X x-1 Suppose that (A) m = 5 / 6 ( / m ( x ) ) =/ 4 (x) and / n ( / 4( x ) ) =/ 3 (x) then (B) n = 5 (C) m = 6 (D) n = 6

    Q.32

    The graph of the parabolas y = - (x - 2)2 - 1 and y = (x - 2)2 - 1 are shown. Use these graphs to decide which of the statements below are true. (A) Both function have the same domain. (B) Both functions have the same range. (C) Both graphs

    have the same vertex. (D) Both graphs have the same y-intercepts. Consider thefunction / ( x ) = f a x + l"\ vbx + 2y where a2 + b2 * 0 then Lim / ( x )X-CO

    Q.33

    (A) exists for all values of a and b

    (B) is zero for a < b

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    (D) is e~ (5/a) or e~ (l/b) if a = b (C) is non existent for a > b Q.34 Which ofthe following fiinction(s) would represent a non singular mapping. (A) / : R - Rf (x) = | x | Sgn x (B) g : R -> R g(x) = v 3/5 where Sgn denotes Signum function 3x 2 - 7 x + 6 (C) h : R R h (x) = x4 + 3x2 + 1 (D) k : R R k (x) : x -x 2 -2 MATCH THE COLUMN ^^^E^TT-S [4x4 = 16] INSTR UCTIONS: Column-I and column-II contains four entries each. Entries of column-I are to be matched with some entries of column-II. One or more than one entries of column-I may have the matching with the same entries of colurnn-II and one entry of column-I may have one or more than one matching with entries of column-II. Q.l (A) (B) Column I Constant function/(x) = c, c e R The function g (x) = P (x > 0), is Column II (P) Bound (Q)periodic Monotonic neither odd nor even

    Ji t

    (C)(D)

    The function h (x) = arc tan x is The function k (x) = arc cot x is

    (S)

    (R)

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    Q.2 (A) (B) (C) (D) Q.3

    Column I cor (tan(-37 )) cos" (cos(-233)) A 1 -cos T sin v9, cos - arc cos ColumnI (A) Number of integral values of x satisfying the inequality x-11 1 0

    Column II (P) (Q) (R) 143 127 3 4 2 3 Column II 1 - 2 - 1 0

    (S)

    2 (P) 4 x-3 (B) The quadratic equations 2006 x2 + 2007 x + 1 = 0 and x2 + 2007x

    + 2006 = 0 have a root in common. Then the product of the uncommon roots is (Q)3 3 (C) Suppose sin 9 - cos 9 = 1 then the value of sin 9 - cos 9 is (9 e R) (R)sin2x-2tanx (D) The value ofthe limit, L l ~ ; 3 ; i s (S) /n(i + x ) Q.4 A qudratic polynomial / ( x ) = x2 + ax + b is formed with one of its zeros being 4+ 3^3

    where a and b 2 + V3 are integers. Also g (x) = x 4 + 2x 3 - 10x2 + 4x - 10 is abiquadratic polynomial such that

    8(A) (B) (C) (D)

    4 + 3y3 2 + V3

    =

    +

    d where c and d are also integers. Column II (P) 4 (Q) 2 (R) -1 (S) -11 13 x 8 =24]

    _1 _i

    Column I a is equal to b is equal to c is equal to d is equal to

    SUBJECTIVE: Q.l Q.2

    Let y = sin"'(sin 8) - tan (tan 10) + cos (cos 12) - sec"'(sec 9) + cor '(cot 6)- cosec "'(cosec 7). If y simplifies to an + b then find (a - b). Suppose a cubic polynomial / (x) = x3 + px2 + qx + 72 is divisible by both x2 + ax + b and x2+ bx + a (where a, b, p, q are constants and a ^ b). Find the sum of the squares of the roots ofthe cubic polynomial. ~3~

    r44

    Q.3

    The set of real values of'x' satisfying the equality

    V

    X

    = 5 (where [ ] denotes the greatest integer

    ( b function) belongs to the interval a , - where a, b, c e N and ~ is in its lowest form. Find the value of c I c. a + b + c + abc.

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    4Q. 1 Q.2

    | BANSAL CLASSES| Target IIT JEE 2007DATE: 26-27//07/2006

    MATHEMATICSDaily Practice ProblemsTIME: 45 Min. DPP. NO.-32

    CLASS: XII (ABCD)

    This is the test paper of Class-XI (J-Batch) held on 23-07-2007. Take exactly 45minutes. If (sin x + cos x) 2 + k sin x cos x = 1 holds V x e R then find the value of k. If the expression r cosX

    [3]

    371

    v2 , V 2y is expressed in the form of a sin x + b cos x find the value of a + b.Q.3

    r>. 371 + x + sin (327t + x) - 18 cos(19rt - x) + + sin

    cos(56tc

    + x) - 9 sin(x + 17tc) [3]

    3 statements are given below each of which is either True or False. State whether True or False with appropriate reasoning. Marks will be allotted only if appropriate reasoning is given. I (log 3 169)(log 13 243) = 10 II cos(cos 7t) = cos (cos 0) III cos x + 3 =T cosx 2 1 S3]

    Q.4 Q. 5

    Prove the identity cos 4 t = ~ + - cos 2t + r cos 4t. o 2 o Suppose that for some angles x and y the equations i 3a 0 sin^x + cos^y = and a2 cos x + sin y = J 2 2

    3

    1

    1

    [3]

    hold simultaneously. Determine the possible values of a. Q. 6 Find the sum of all the solutions of the equation (log 27 x 3 ) 2 = log 27 x 6 .

    [3] [3]

    7i % 10y-10~y If - < x < and y = log 10 (tan x + sec x). Then the expression E = simplifies to one ** JL the six trigonometric functions,findthe trigonometric function. 13] Q.8 If log 2 (log 2 (log 2 x))= 2 then find the number of digits in

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    x. You may use log ?0 2 = 0,3010. [3] Q. 9 Assuming that x and y are both + vesatisfying the equation log ( x + y ) = l o g x + l o g y find y in terms of x.Base of the logarithm is 10 everywhere. [3] If x = 7.5 then find the value of cosx ~ cos 3x ~ : . sin 3x - sin x [3]

    Q.10

    Q. 11

    Find the solutions of the equation, log ^ sm x (1 + cos x) = 2 in the interval xe [0,2n].

    [4]

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    Q. 12 Q, 13

    Given that log a2 (a 2 +1) = 16 find the value of log a32 (a + - )a

    [4]

    If cos e = - find the values of (i) cos 36 (ii)tam [4] [5]

    Q. 14

    If log 12 27 = a find the value of log 6 16 in term of a.

    Q . 15

    sin x - c o s x + 1 1 + sinx Prove the identity, r = =tan + , wherever it is defed. Starting with left 4 2 sin x + c o s x - 1 cosx hand side only. [5] [5] [6]

    Q. 16 Q. 17

    Find the exact value of cos 24 - cos 12 + cos 48 - cos 84. S olve the system of tions 5 (logxy + log y x) =26 and xy = 64.r=4

    Q.18

    Prove that

    sinr=l V

    (2r-l)7c'

    8

    -2 r=l

    r=4

    cos

    (2r-l)7t -\

    4

    8[6]

    Also find their exact numerical value. r i 1a Solve for x: log 2 (4 - x ) + log(4 -x). log f x + - 1 - 2 log 2 x + 2, V 2J

    0,19

    = 0.

    [6]

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    4

    | BANSAL CLASSESSTarget iiT JEE 2007DATE: 05-06/06/2006 The value of Lim / n x - / n Vx +1 + xX-00

    MATHEMATICSDaily Practice ProblemsTIME: 50 Min. DPR NO.-28

    CLASS : XII (ABCD)

    (A)

    1

    (B)/n7t/4

    vw

    (C) does not exist

    (D) 0

    Evaluate J(tanx-sec 4 x ) d x . (A) 1/4 (B) 1/2 (C) 3/4 (D)l

    The product of two positive numbers is 12. The smallest possible value of the sum of their squares is (A) 25 Q4 (B) 24 (C) 18 V2 (D) 18

    Given that log (2) = 0.3010 number of digits in the number 2000 2000 is (A) 6601(B) 6602 (C) 6603 (D)6604 , , 1 1 1 Given that a, b and c are the roots of theequation x" - 2x 2 - 1 1 x + 12 = 0, then the value of + + ~ (A) (B) n 12 (C) 1312 (D) 7

    If Jtan x dx = 2, then b is equal to (A) arc cos(2e) t. Q/7 (B) arc sec(2) (C) a

    rc sec 2 (e) = 8 ( x 2 + 3 x + 2 ) , is (C)-3 (D)-5 (D)none

    The sum of all values of x so that 16 ( " 2+3x (B)3 (A) 0

    A certain function/(x) satisfies f (x) + 2 / ( 6 - x) = x for all real numbers x. The value o f / ( l ) , is (A) 3 (B)2 (C)l (D) not possible to determine Q.9 Number of ways in which the letters A, B, C and D be arranged in a sequence so that A is not in position 3, B is not in position 1, C is not in position 2 and Dis not in position 4, is (A) 8 (B) 15 (C) 9 (D) 6 Using only the letter from theword WILDCATS with no repetitions allowed in a codeword, number of 4 letter codewords are possible that both start and end with a consonant, are (A)360 (B)900(C) 1680 (D)2204

    Q.10

    {

    Q:ll

    Find j(x/nx)dx

    (A)-

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    (B)-

    (C)-l

    (D)l

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    Q.12

    IfP(x) is a polynomial with rational coefficients and roots at 0,1, -Jl P(x) isat least (A) 4 (B) 5 V + (C) 49 (C)6+ 00

    anci

    1 - \/3 , then the degree of

    (D)8 >

    cc ua t0

    7 Sum of the infinite series, 4 - ^ + (A) ^ Q.14 (B) 24

    l l

    (D)

    A florist has in stock several dozens of each of the following: roses, carnations, and lilies. How many different bouquets of half dozen flowers can be made? 8!( ) 2!-6!A

    9!(B)

    12!

    3!-6!

    (D) 56

    ^ 1 5

    Let/(x) = (A)-9

    e3x - 1 . if x * 0 x then/'(0), is 3 if x = 0 (B)9 (C) 9/2 (D) nonexistent

    n6

    I f f "(x) = 10 and f ' (1) = 6 and f ( l ) = 4 then f (-1) is equals (A)-4 (B)2 ' (C)8 The coefficient of x in the expansion of (A) 97 (B)983

    (D)12

    Q.17

    x"

    v

    2+

    \12

    4

    xy

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    ,is (D)100

    (C) 99

    Q.18

    In how many ways can six boys and five girls stand in a row if all the girls areto stand together but the boys cannot all stand together? (A) 172,800 (B) 432,000 (C) 86,400 (D)none The composite of two functions f and g is denoted by fog and defined by (fog)(x) = f (g(x)). When f(x) 5x and g (x) = which one of the fol

    lowing is equal to (fog)(x)? x 1 x-2 4-x x-2 The equation Ink iA (k + 1)i/(k+D

    6x

    3 Ox >

    (C) In 1 -

    x-2 4x + 2 1 +Ink k

    (D)

    15x 2x + l

    = F(k)

    k+1

    is true for all k wherever defined.

    F(100) has the value equal to (A) 100 (B) 1 101 (C)5050 (D) 1 100

    Q.21

    Compute f ,_ ^Vx+K/x

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    ill BANSAL CLASSESH Target I I I JEE 2007CLASS: XII (ABCD) DATE: 28-29/06/2006 Q.l tan 9 =2+ -

    MATHEMATICSDaily Practice ProblemsTIME: 50 Min each DPR DPR NO.-25 TIME: 50 Min

    DATE: 28-29/06/2006

    I > P P - 2 5

    1~

    where 9 e (0,2n), find the possible value of 6.

    {2]

    2 +

    '--co+

    Q. 2 Q.3

    Find the sum of the solutions of the equation2e2x

    _

    5e

    x

    4

    =

    o_

    [2]

    Suppose that x and y are positive numbers for which log 9 x = log 12 y = log 16(x + y). If the value of --2 cos 9, where 9 e (o, rc/2) find 9. [3]

    Q. 4

    Using L Hospitals rule or otherwise, evaluate the following limit: Limit

    X->0+

    Limitn->eo

    [l2 (sinx)" j + 22 (sinx)x + n3

    +

    n2 (sinx)x " where [. ] denotes the [4]

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    greatest integer function. Q.5 1 Consider f ( x ) = - ^ = , ~sin2

    x

    I . V a + htan'x , f o r b > a > 9 & the functions g(x)&h(x)

    |1 + I 5

    (V

    sinx

    are defined, such that g(x) = [f(x>] - j - ^ J & h(x) = sgn (f(x)) for x e domain of f, otherwise g(x) = 9 = h(x) for x domain o f ' f , where [x] is the greatest integer function of x & {x} is the fractional 7t part of x. Then discuss the continuity of'g' & *h' at x = and x = 9 respectively.~ ^ f x 2 tan

    _1

    [5]

    x

    ,

    Q. 7

    Using substitution only, evaluate: jcosec 3 x dx.

    [5j

    DATE: 30-01/06-07/2006 Q.l 12 A If sin A = . Find the value of tan ,

    JIME: 50 Min. [2]

    Q.2

    x v The straight line - + ^ = 1 cuts the x-axis & the y-axis in A& B respectively& a straight line perpendicular to AB cuts them in P & Q respectively. Find thelocus of the point of intersection ofAQ & BP.

    [2]

    Q.J

    tan 9 1 cot 9 If - - - = , find the value of - . tan 9 - tan 39 3 cot9-cot3

    HI

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    Q.4

    If a A ABC is formed by the lines 2x + y - 3 = 0; x - y + 5 = 0 and 3x - y + 1 =0, then obtain a cubic equation whose roots are the tangent of the interior angles of the triangle. [4] Integrate

    Q.5

    fQ.6

    J a 2 - tan2 x)Vb2 - tan2 x

    dx

    (a>b)

    15]

    xsmxcosx I ((a 2 cos 2 x +, b sin x) dx T,2 ;2 \2

    [5]

    Q.7

    d dy Let ~ (x 2 y) = x - 1 where x ^ 0 and y = 0 when x = 1. Find the set of values of x for which dx [5] is positive.

    DATE: Q. 1

    03-04/07/2006

    TIME;

    50Min.

    Let x = (0.15) 20 . Find the characteristic and mantissa in the logarithm of x,

    to the base 10. Assume log 10 2 = 0.301 and log 10 3 = 0.477. [2] Two circles ofradii R & r are externally tangent. Find the radius ofthe third circle which isbetween them and touches those circles and their external common tangent in terms of R & r. [2]

    Q. 2

    Q. 3

    Let a matrix A be denoted as A = diag. 5 x , 5 5 \ 5 5 S

    then compute the value ofthe integral j( det A)dx.

    Q. 4

    P] Using algebraic geometry prove that in an isosceles triangle the sum ofthe distances from any point of the base to the lateral sides is constant. (You may assume origin to be the middle point of the base of the isosceles triangle) [4] Evaluate: f1 +-x + x

    Q.5

    J

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    dx Vx +X2

    +x 3fav 3

    [5] a2-3];?

    Q.6

    If the three distinct points,

    fb3 [b-r

    b 2 -3^1 b-ij?

    r c3 [c-l '

    c 2 -3^1 c-lj

    a-l ' a-1

    are collinear then [5] [5]

    show that abc + 3 (a + b + c) = ab + be + ca. Q.7 Integrate: j^/tanx dx

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    ill BANSAL CLASSESI g l T a r g e t HT JEE 2007CLASS: XII (ALL) DATE: 23-24/06/2006 Select the correct alternative: (Only one is correct) Q. 1

    MATHEMATICSDaily Practice ProblemsTIME: 50 Min. DPP. NO.-24

    [16 x 3 = 48]

    A circle of radius 2 has center at (2,0). Acircle of radius 1 has center at (5,0). Aline is tangent to the two circles at points in the first quadrant. Which ofthe following is the y-intercept ofthe line? (A) 3 (B)

    V2

    (Q3

    8

    (D) 2a/2

    Q.2

    In a triangle ABC, the length ofAB is 6, the length of BC is 5, and the length of CAis 4. If K lies on BC BK 3 such that the ratio of length r is , then the length ofAK isKC 2

    (A) 2V3 Q. 3

    (B)4

    (C) 3V2

    (D) 2,

    Which one of the following quadrants has the most solutions to the inequality, x- y < 2? (A) I quadrant (B) II quadrant (C) ID quadrant (D) I and III quadranthave same The range of the function / ( x ) = sin _1 x + tan~'x + cos _1 x, is (A) (0,71) (B)7t 371

    Q. 4

    4'T

    (C) [0,71]

    (D)R

    Q.5

    The area of the region of the plane consisting of all points whose coordinates (x, y) satisfy the conditions 4 < x 2 + y 2 < 36 and y < | x is (A) 24n (B) 27TI(C) 20TT (D) 32tc A straight wire 60 cm long is bent into the shape of an L. Theshortest possible distance between the two ends of the bent wire, is (A) 30 cm(B) 3 0 V 2 c m (C)10V26N

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    Q. 6

    (D) 20^571

    Q.7

    '7t X holds, is Sum of values of x, in (0, n/2) for which tan + X = 9 tan 4' 4 v(A) 0 (B) 71 - tan _1 (2) (C) cor'(O) (D) tan -1 (2)Given/"(x) = cos x, / ' ^ y J = e a n d / ( 0 ) = 1, then/(x) equals. (A)sinx-(e

    +l)x (B) sinx + (e+ l)x ( C ) ( e + l)x + c o s x (D) ( e + l ) x - c o s x + 2

    Q. 8

    Q.9

    Evaluate the integral: j x e c o s x 2 sin x 2 dx (A) | e c o s x 2 + C (B)- -esmx +C

    1

    (C)

    1 _sin x 2

    + C

    (D)- iecosx2 + C

    Q.10

    The value of Lim x->n (A)0

    e~n -e" x

    is sin x (B)-e-

    (D)e-

    (D)nonexitent

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    Q.LL

    Let/(x) = x e R, is (A) ( - , , - 2 ]

    ^" k x + x2-k

    1

    . The interval(s) of all possible values of k for which/is continuous for every(B)[-2,0) (C) R - ( - 2,2) (D)(-2,2)

    Q.12 Q, 13

    Suppose F (x) = / (g(x)) and g(3) = 5, g'(3) = 3,/'(3) - 1 , / ' ( 5 ) = 4. Thenthe value of F'(3), is (A) 15 (B) 12 (C) 9 (D)7 From a point P outside of a circle with centre at O, tangent segments PA and PB arc drawn. If

    1( A O )2

    1"+

    ~

    Ye ' t

    1

    b e n l e n t b

    chord AB is (C) 8 (D) 9

    (A) 6a

    (B)4 na

    Let

    l2

    al3 a23 a33

    a21 a31

    a22 a32

    , Aj * 0

    b

    i,

    b 12b22 b32

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    b13 b23 b33

    b21

    where b- is cofactor of a^ V i, j = 1,2, 3

    b3]c

    and

    n

    C12 C22 C32

    C13 C 23 C 33

    C 21 C31

    where c^ is cofactor of

    V i, j = 1,2, 3.

    then which one of the following is always correct. (A) Aj, A2, A3 are in A.P. (B

    ) Aj, A2, A3 are in G.P. (C) A2 Q. 15A

    3

    (D) A,

    A0

    The first three terms of an arithmetic sequence, in order, are 2x + 4,5x - 4 and3x + 4. The sum of the first 10 terms of this sequence, is (A) 176 (B) 202.4 (C) 352 (D) 396 The value of V3 i (A) + w 2 2/ n

    Q. 16

    r

    4 71 \ wI 71 . . 7n \ 5 T . 7t / 7i cos+ zsin cos +1 sin is equal to 15 15, 8

    7 L 7t

    . .

    K J

    2

    2

    (C)

    S

    i

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    (D)-^--i w 2 2

    Subjective: Evaluate: Q.l

    J"

    dx

    xV a x - :x 5

    q.2

    ff s5* s* j 5 x5 rsin j , sm1

    dx

    Q.3

    V x - cos 1 yfx r , r* r dx + COS

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    BANSAL CLASSES8 T a r g e * I I T JEE 2 0 0 7CLASS: XII(ALL) DATE: 16-22/06/2006 2 DATE: 16-17/06/2006 Q. 1 For x > 0 and ^ 1and n e N, evaluate,n-co

    MATHEMATICSDaily 1 TIME: 45 Min. 1 logx 2 ~o n_1

    Practice Problems

    DPR NO.-21, 22, 23

    Lim

    1 1 + + logX 2 . log 4' logx 4 . log 8 V =>x

    +

    .Iog_ -2 n ~ o x

    y

    Q. 2

    Show that (a + b + c), (a 2 + b 2 + c 2 ) are the factors of the determinant a2b2 c2 (b + c) 2 (c + a) 2 (a + b) 2 be ca . Also find the remaining factors. ab

    Q. 3 Q. 4 Q.5

    Prove that a non singular idempotent matrix is always an involutaiy matrix. Findan upper triangular matrix A such that A 3 = 8 0 -57 27d2

    ^ y dy I f ' y' is a twice differentiable function of x, transform the equation,(1 - x 2 ) -7 - x - + y = 0 by dx dx means of the transformation, x = sin t, in terms of the independent variable' t'.

    Atangent line is drawn to a circle of radius unity at the point A and a segmentAB is laid offwhose length is equal to that of the arc AC. A straight line BC isdrawn to intersect the extension of the diameter AO at the point P. Prove that:9 (1 - cos 0) (ii)L^tpA=3. (i) PA = e - sin e Use of series expansion or L1 Hospital's rule prohibited.

    Q. 6

    DATE: Q. 1

    19-20/06/2006

    TIME:

    \ l-x\

    45Min.

    Without using any series expansion or L' Hospital's rule, Evaluate: Lim x la e|1 + x/ VT3+V3 Find the value of the determinant V15+V26 3 + V65 2V5 5 VPS V5 V105

    /

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    Q. 2

    Q. 3

    / ( x ) is a diffrentiable function satisfy the relationship f2 (x) + f 2(y) + 2(xy - 1 ) = f 2 (x + y) V x, y e R. Also f (x) > 0 V x e R , and f (V2 )= 2. Determine f (x).

    Q.4

    Let,y = t a n - | j

    5

    + tan"

    x 2.3 + x jz

    + tan - 1

    j 3.4 + x^

    +

    upto n terms.

    dy Find - expressing your answer in two terms, dx

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    Q. 5

    0 x+a Without expanding the determinant show that the equation x+b root.

    x-a 0 x+c

    x-b x-c 0

    :

    0 has zero as a

    Q.6

    Let a j , a 2 & p j, (3, be the roots of ax 2 +bx + c = 0 & px 2 + qx + r = 0, respectively. If the system b ac of equations a , y + a 2 z = 0 & p t y + p 2 z =0 has a non-trivial solution, then prove that = . D r P - 2 3 X + ^X 0 ~1 TIME45 Min. Asi ~ Zs>-J j-2..

    DATE: Q. 1

    21-22/06/2006

    Compute x in terms x 0 , x,, and n. Also evaluate Lim x n =

    Q.2

    A 2 vb

    a

    5 c 8 2

    d is Symmetric and B = b - a -2

    3 e 6

    a - 2b - c is Skew Symmetric, then find AB. -f

    Is AB a symmetric, Skew Symmetric or neither of them. Justify your answer.x +1

    Q.3

    Let f ( x ) = e x , x 0 Discuss continuity and differentiability of f (x) at x = 0 . 1 0 Show that the matrix A = 2 1 can be decomposedas a sum of a unit and a nilpotent marix. Hence evaluate the matrix 1 0 2 12007

    Q. 4

    Q. 5

    dv Find , if (tan"1 xV + y cotx = 1. dx f w -)_ bY^) f If / is differentiate andLim ^ h L'Hospital's rule. 1 + e" x

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    Q.7

    Consider the function / ( x ) =

    x + 2 , 0 < x 3 x Find all points where f (x) is discontinuous. Find all points when f (x)is not differentiable. Draw the graph, showing clearly the points of discontinui

    ty or non derivability.

    6

    3

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    | h B A N S A L C L A S S IESV S T a r g e t I I T JEE 2 0 0 7CLASS: XII (EXCEPT A-2) Q. 1 TIME: 60 Min.

    MATHEMATICSDaily Practice ProblemsDPR NQ.-20

    3 10 The set of all x for which > 27 consists of the union of a finite and an innite interval. The length x x +1 ofthe finite interval is (A) 3 (B)2

    1

    C I O

    (D)2 T

    Q.2

    Five persons put their hats in a pile. When they pick up hats later, each one gets some one else's hat. Number of ways this can happen, is (A) 40 (B)44 (C) 96 (D) 120 Suppose the origin and the point (0,5) are on a circle whose diameter isalong the y-axis and (a, b) lies on the circle. Let L be the line that passes through the origin and (a, b). If a 2 + b 2 = 16 and a > 0 then the equation of th

    e line L is (A) 3 x - 4 y = 0 (B) 2 0 x - 3y = 0 (C)2x-y = 0 (D)4x-3y = 0 If 1 lies between the roots of the equation y 2 - my + 1 = 0 then the value of has thevalue equal to (Here [x] denotes gratest integer function) (A) 0 (B) 1 (C) 2 4[x] VxeR

    Q.3

    Q.4

    IxI +16

    (D) none3 2

    Q.5 Q.6

    The sum of the squares of the three solutions to the equation x + x + x + 1 = 0,is (A)~ 1 (B)0 (C)l (D)2 Let / ( x ) = 1 + x 3 . If g (x) = / _ 1 ( x ) , i.e.if g is the inverse / , then g'(9) equal to (A) 1/12 (B) 1/243 (C) 1/8 (D) 1/24.Lim j V x - V x - Vx + Vx x-oo v (A) equal to 0 is (C) equal t o - 1 (D) equal to - 1/2

    Q.7

    (B) equal to 1

    Q. 8

    Suppose f is a differentiable function such that / ( x + y ) = / ( x ) + / ( y )+ 5xy for all x, y and f'(0) = 3. The minimum value of f (x) is (A) - 1 (B) 9/10(C) - 9/25 (D)none x-1

    Q-9 f i i n Jfg . x + l = 3x then the value of g (3), is v y (A)Q. 10 Q. 11 15

    V2 (B)-

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    (C)9

    (D)

    V3

    9 sin( A + B) For acute angles A and B if (tan A)(cot B) = - then the value of equal to 5 sin(A d) (A) 7/4 (B) 2/7 (C) 4/7 (D) 7/2 The value of this product of98 numbers

    !

    (A)

    3y 1

    1 - -

    1-2

    5y

    1 - -

    98

    1-

    99

    1 - -

    100

    ,is

    Q. 12

    10 5050 2 If T = 3 /n(x + x) with > 0 and x > 0, then 2x + is equal to

    98 (B) 100

    (C)

    (D)

    14950

    (A) V-f'2 + 4eT/3

    (B) V^2 + 4e-T/3

    (C)

    (D) V^2-4eT/3

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    Q.13

    Evaluate:

    -2 - , ^ U -12X + 35 (A)-1.25 (B)-1.5

    (C)-1.75

    (D)-2

    Q.14

    Q.15 Q.16

    Let/be a polynomial function such that for all real x f(x 2 + 1) = x 4 + 5x 2 +3 then the premitive o f / ( x ) w.r.t. x, is 3 2 x 3 3x 2 x 3 3x 2 x 3 3x 2 x +C vD)+ ^ + x + C ( ( AJ ) + x + C w (B)- + x + C (C) K 3 2 3 2 3 2 3 Nuular polygons that have integral interior angle measure, is (A) 20 (B)21 (C) 22(D)23 Suppose/ is a differentiable function such that for every real number x, /( x ) + 2 / ( - x ) = sin x, then f'(n/4) has the value equal to

    (A)l/V2Q.17

    (B)-l/V2

    (B) -1/2V2

    (D)V2

    The number of permutation of the letters A A A A B B B C i n which the A's appear together in a block of four letters or the B's appear in a block of 3 letters,is (A) 44 (B) 50 (C) 60 (D)none If {x} denotes the fractional part function then the number x = TT^a (A) 1/2 (B)0 (C) - 1 / 2

    Q.18

    {sf-iyif

    f/)2 simplifies to (D)none

    Q.19

    Which one of the following is wrong? 2

    (A) JtanOsec2 0dO = tankx x

    +C

    (B) JtanOsec2OdO =

    (D)none

    +C

    (C) Jxsinxdx = s i n x - x c o s x + Cfor x < 0 Q.20 Let/(x) = (A) none Q.21 Find L y->2 (A)0 Q.22 1

    . I f / ( x ) is continuous at x = 0 then the number of values of k is (B) 1 1 (B)/nx3 2

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    3x + 2k for x >