Maths Bansal Dpp

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008

Daily Practice Problems

CLASS : XI (J1 & J2) DATE : 21/06/2006 TIME : 40 to 50 Min. Q.1 State whether the following statements are True or False.

DPP. NO.-1

(i)

 1 1 1 1 1 1    If x, y, z are all different real numbers, then =    2 2 2 (x  y) (y  z) (z  x)  x  y y  z z  x

(ii)

tan2 · sin2 = tan2 – sin2.

(v)

(iii)

1  102 1  99·101 = 101. There exist natural numbers, m & n such that m2 = n2 + 2002.

(iv)

2

sin(1) – cos(1) < 0

Fill in the blanks.

tan180    cos180    tan90    wherever it is defined, is equal to ______. sin90    cot90    tan90   

Q.2

The expression

Q.3

If 2 cos2 ( + x) + 3 sin ( + x) vanishes then the values of x lying in the interval from 0 to 2 are _____.

Q.4

If tan 25º = a then the value of

Q.5

Select the correct alternative : (Only one is correct) The number of real solution(s) of the equation, sin (2x) = x + x is : (A) 0 (B) 1 (C) 2 (D) none of these

Q.6

   3   7  tan  x   . cos  x   sin 3   x 2 2 2       simplifies to The expression  3      cos x   . tan   x  2   2 

(A) (1 + cos2x)

tan 205  tan115 in terms of ‘a’ is _____. tan 245  tan 335

(B) sin2x

(C) – (1 + cos2x) 2x

(D) cos2x 2

Q.7

Number of values of ‘x’  (–2,2) satisfying the equation 2sin (A) 8 (B) 6 (C) 4

Q.8

Let m = tan 3 and n = sec 6 , then which one of the following statement holds good? (A) m & n both are positive (B) m & n both are negative (C) m is positive and n is negative (D) m is negative and n is positive.

Q.9

Let y =

1

, then the value of y is

1

2

 4.2cos x  6 is (D) 2

1

3 2

1 3  .....

15  3 15  3 13  3 13  3 (B) (C) (D) 2 2 2 2 a Q.10 If tan = where a, b are positive reals and   1st quadrant then the value of b sin sec7 + cos cosec7 is (a  b ) 3 (a 4  b 4 ) (a  b ) 3 (a 4  b 4 ) (a  b ) 3 ( b 4  a 4 ) (a  b ) 3 ( a 4  b 4 )  (A) (B) (C) (D) (ab) 7 / 2 (ab) 7 / 2 (ab) 7 / 2 (ab) 7 / 2 (A)

DPP

on

the

path

of

success

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (J1 & J2)

Daily Practice Problems

DATE : 23/06/2006

TIME : 45 to 55 Min.

DPP. NO.-2

Fill in the blanks. Q.1

Q.2

cos   3  If tan = 2 and   ,  then the value of the expression is equal to _______.  2 sin 3   cos3  Select the correct alternative : (Only one is correct) The number of values of k for which the system of equations (k + 1)x + 8y = 4k ; kx + (k + 3)y = 3k – 1 has infinitely many solutions is (A) 0 (B) 1 (C) 2 (D) infinite

Q.3

116 people participated in a knockout tennis tournament. The players are paired up in the first round, the winners of the first round are paired up in the second round, and so on till the final is played between two players. If after any round, there is odd number of players, one player is given a bye, i.e. he skips that round and plays the next round with the winners. The total number of matches played in the tournament is (A) 115 (B) 53 (C) 232 (D) 116

Q.4

PQRS is a square. SR is a tangent (at point S) to the circle with centre O and TR = OS. Then, the ratio of area of the circle to the area of the square is 11 3 7  (A) (B) (C) (D)  11 3 7

Q.5

 3  The two legs of a right triangle are sin + sin  3   and cos – cos   . The length of its  2   2  hypotenuse is (A)1

Q.6

(B) 2

(C)

2

(D) some function of 

 4  3   6     4 6 If f (x) = 3sin   x   sin (3  x ) – 2 sin   x   sin (5  x ) then, for all permissible  2  2      values of x, f (x) is (A) – 1 (B) 0 (C) 1 (D) not a constant function

Q.7

Subjective : If an equilateral triangle and a regular hexagon have the same perimeter then find the ratio of their areas.

Q.8

Prove the identity,

Q.9

A polynomial in x of degree three which vanishes when x = 1 & x = – 2, and has the values 4 & 28 when x = – 1 and x = 2 respectively is ______.

tan 3  cot 3  1  2 sin 2  cos 2    . sin  cos  1  tan 2  1  cot 2 

Q.10 The length of a common internal tangent to two circles is 7 and a common external tangent is 11. Compute the product of the radii of the two circles. Q.11

Prove that x4 + 4 is prime only for one value of x  N.

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (J1 & J2)

DATE : 23/06/2006

Daily Practice Problems TIME : 45 to 55 Min.

DPP. NO.-3

Fill in the blanks. Q.1

The smallest natural number of the form 1 2 3 X 4 3 Y, which is exactly divisible by 6 where 0  X, Y 9, is ______ .

Q.2

The line AB is 6 meters in length and is tangent to the inner one of the two concentric circles at point C. It is known that the radii of the two circles are integers. The radius of the outer circle is _______

Q.3

The positive integers p, q & r are all primes. If p2  q2 = r then the set of all possible values of r is ______ . Select the correct alternative : (Only one is correct)

Q.4

Q.5

Q.6

2 2 Solution set of the equation 32x  2.3x  x  6  32(x  6)  0 is (A) {–3, 2} (B) {6, –1} (C) {–2, 3}

(D) {1, – 6}

Exact value of cos2 73º + cos2 47º  sin2 43º + sin2 107º is equal to : (A) 1/2 (B) 3/4 (C) 1 (D) none sin2 sin3 sin4 If cos2cos3 cos4 = tan k is an identity then the value k is equal to :

(A) 2

(B) 3

(C) 4

(D) 6

Select the correct alternative : (More than one are correct)

Q.7

sin22 cos8 cos158 cos98 The expression when simplified reduces to : sin23 cos7 cos157 cos97 (A) sec(–100)

Q.8

If

 3  (B) cosec     2 

 7  (C) sin    2 

 5  (D) cot    4 

1  sin A sin A 1   , for all permissible values of A, then A belongs to 1  sin A cos A cos A

(A) First Quadrant

(B) Second Quadrant (C) Third Quadrant

(D) Fourth Quadrant

ACME

Q.9

The sines of two angles of a triangle are equal to (A)

245 1313

Q.10 If secA = (A)

85 36

(B)

255 1313

5 99 & . The cosine of the third angle is : 13 101

(C)

735 1313

(D)

765 1313

17 5 and cosecB = then sec(A + B) can have the value equal to 8 4

(B) –

85 36

(C) 

85 84

(D)

85 84

Match the Column. This question contains two columns. Column-I contains four questions and column-II contains their answers written in random order. Each entry in column-I is associated with some or the other entry of column-II. Some entries in column-II may not be the answers of any entry of column-I. Credit will be given only when all the matching are correct. Column-I Column-II (i) Number of right triangle on a given hypotenuse, is (A) 2 (ii) (iii)

(iv)

In a scalene triangle, centroid divides the line joining orthocentre and circumcentre in a ratio K where K equals

(B)

Three sides of a regular hexagon, no two of which share a vertex of the hexagon are exteded to form an equilateral triangle. The perimeter of the triangle thus formed is p times the perimeter of the original hexagon where p equals

(C) 3

In the figure shown BC is tangent to the circle with centre D and diameter 12. Length of FB is

(D) infinite

3 2

(E) 5

1 5

1 3

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (J1 & J2)

Daily Practice Problems

DATE : 23/06/2006

TIME : 45 to 55 Min.

DPP. NO.-4

Fill in the blanks : Q.1

sin 4 t cos4 t 1 The expression when simplified reduces to ______ . sin 6 t cos6 t 1

Q.2

sin24cos6sin6sin66 The exact value of sin21cos39cos51sin69 is ______.

Q.3

A rail road curve is to be laid out on a circle. If the track is to change direction by 280 in a distance of 44 meters then the radius of the curve is ________. [use  = 22/7] Select the correct alternative : (Only one is correct)

Q.4

If cos ( + ) = m cos (), then tan  is equal to :  1  m  tan   1  m

(A)  Q.5

 1  m  tan   1  m

(B) 

 1  m  cot   1  m

(C) 

 1  m  cot   1  m

(D) 

The side of a regular dodecagon is 2 cm. The radius of the circumscribed circle in cms. is : (A) 4( 6  2 )

(B)

6 2

(C)

2 2 3 1

(D)

6 2

Q.6

Which of the following conditions imply that the real number x is rational? I x1/2 is rational II x2 and x5 are rational III x2 and x4 are rational (A) I and II only (B) I and III only (C) II and III only (D) I, II and III

Q.7

The number of all possible triplets (a1, a2, a3) such that a1 + a2cos2x + a3sin²x = 0 for all x is : (A) 0 (B) 1 (C) 3 (D) infinite Select the correct alternative : (More than one are correct)

Q.8

Which of the following when simplified reduces to unity? (A)

1  2 sin2      2 cot   cos2    4  4 

(B)

1 (1  tan2 )2  (C) 4 sin2  cos2  4 tan2 

(D)

sin      sin   cos  tan 2

+ cos ( – )

1  sin 2 (sin   cos )2

Subjective : 2

Q.9

  3   3     cos     = a + b sin 2 then find the If [1  sin (+) + cos ( + )] + 1  sin   2   2   2

value of a and b. Q.10 If secA – tanA = p, p  0, find the value of sinA. ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (J1 & J2)

Daily Practice Problems

DATE : 23/06/2006

TIME : 45 to 55 Min.

DPP. NO.-5

Fill in the blanks : Q.1

Exact value of tan 200º (cot 10º  tan 10º) is ______ .

Q.2

The greatest value of the expression   15    17    4x  sin2   4x for 0  x   8   8  8

sin2 

is ___________.

Select the correct alternative : (Only one is correct) Q.3

The expression

1  sin 2 cos 2  2  . tan  



reduces to : (A) 1 Q.4

Q.6



1    3    sin 2  cot  cot     when simplified  2 4 2 2 



(C) sin2 (/2)

(B) 0

(D) sin2 

Exact value of cos 20º + 2 sin2 55º  2 sin 65º is : (A) 1

Q.5

3 4

(B)

1

F G H

sin 3 18 23 = p where   , cos 2 48 48 (A) p > 0 and q > 0 (C) p < 0 and q < 0

Let

1 + cos 290

(A)

1 3 sin 250

2 3 3

(C)

2

IJ & K

(D) zero

2

sin 3 cos 2 = q

where  

13 14  I F G H48 , 48 J KThen

(B) p > 0 and q < 0 (D) p < 0 and q > 0

=

(B)

4 3 3

(C) 3

(D) none

Subjective : 3   4  + sin (3 8)  sin(412) = 4 cos 2 cos 4 sin 6.  2 

Q.7

Prove the identity, cos 

Q.8

Prove that:

Q.9

Prove the identity, sin 2 (1 + tan 2 . tan ) +

Q.10 Prove that

cos 5x  cos 4 x = cos x + cos 2x. 2 cos 3x  1 1  sin     = tan 2 + tan2    .  4 2 1  sin 

tan 8 = (1 + sec2) (1 + sec4) (1 + sec8) tan 

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL) Q.1

Daily Practice Problems

DATE : 30/06/2006

TIME : 45 Min.

DPP. NO.-6

If sin2 = 4 sin2, show that 5 tan( – ) = 3 tan( + ).

Q.2

Find the degree measure of all angles ‘x’ such that 0  x  180° and

Q.3

If 0 < x <

Q.4

Three real numbers a, b, c satisfy 2b = a + c, show that

cos6x – sin6x +

sin 2 2 x ·cos 2 x =0 4

 5 and cos x + sin x = , find the numerical values of cos x – sin x. 4 4

sin a  sin b  sin c = tan b. cos a  cos b  cos c

2

Q.5

2

 sin 3   cos 3     = 8 cos2, wherever it is defined. Prove the identity   sin    cos  

Q.6

Find the value of  lying in the interval [0, 2] and satisfying the cubic, 2sin3 – 5sin2 + 2 sin = 0.

Q.7

Find the exact value of cos236° + sin218°.

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL)

Daily Practice Problems

DATE : 30/06/2006

TIME : 45 Min.

DPP. NO.-7

Fill in the blanks : 64 is equal to _________. 27

Q.1

The value of log

Q.2

The solution set of the system of equations , x + y =

11 cos 6

3 2 , cos x + cos y = , 3 2

where x & y are real , is _______.      x . sin   x  b then the ordered pair (a, b) is ______. 3  3 

Q.3

If a  sin x . sin 

Q.4

The value of b satisfying log

Q.5

The number of integral pair(s) (x , y) whose sum is equal to their product is ______.

Q.6

A mixture of wine and water is made in the ratio of wine : total = k : m. Adding x units of water or removing x units of wine (x  0), each produces the same new ratio of wine : total. The numerical value of the new ratio is ______.

Q.7

If x2  5x + 6 = 0 and log2 (x + y) = log4 25, then the set of ordered pair(s) of (x, y) is ______.

8

b3

1 is _______. 3

Select the correct alternative : (Only one is correct) Q.8

 

If A + B + C =  & sin  A  (A)

Q.9

k 1 k1

(B)

C A B C tan =  = k sin , then tan 2 2 2 2

k1 k 1

(C)

k k1

(D)

k1 k

2

The equation 7 log7 ( x  4 x  2) = x – 2 has (A) two natural solution (C) no composite solution

(B) one prime solution (D) one integral solution

Q.10 The number of real solution of the equation log10 (7x  9)2 + log10 (3x  4)2 = 2 is (A) 1 (B) 2 (C) 3 (D) 4

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL) Fill in the blanks : Q.1

If x = 3 7  5 2 

Daily Practice Problems

DATE : 30/06/2006 1

3

TIME : 45 Min.

DPP. NO.-8

, then the value of x3 + 3x  14 is equal to ______.

75 2

Select the correct alternatives : (More than one are correct) x

 x

x

Q.2

If p, q  N satisfy the equation x (A) relatively prime (C) coprime

Q.3

...... p where p  2, p  N, when simplified is : The expression, logp logp   

p p p



then p & q are : (B) twin prime (D) if logqp is defined then logpq is not & vice versa

p

n radical sign

(A) independent of p, but dependent on n (C) dependent on both p & n Q.4

Q.5

(B) independent of n, but dependent on p (D) negative .

Which of the following when simplified, reduces to unity ? 2 (A) log105 . log1020 + log10 2

(B)

2 log 2  log 3 log 48  log 4

(C)  log5 log3

(D)

1  64  log 3   6 2  27 

The number N =

5

9

1  2 log3 2

1  log3 2

2

 log26 2 when simplified reduces to :

(A) a prime number (C) a real which is less than log3 Q.6

(B) an irrational number (D) a real which is greater than log76

Subjective : If tan A & tan B are the roots of the quadratic equation, a x2 + b x + c = 0 then evaluate a sin2 (A + B) + b sin (A + B). cos (A + B) + c cos2 (A + B).

a Q.7

If cos + cos = a and sin + sin= b then prove that, cos2 + cos2 =

Q.8

Establish tricotomy in each of this following pairs of numbers

(i) (iii)

Q.9

log 27 3

3

and2





 b2 a 2  b2  2

a

2

 b2



(ii) log 4 5 and log1/16 (1 / 25)

log 4 2

4 and log 3 10  log 10 81

Compute the value of

2

81

(iv) log 1/ 5 (1 / 7) and log 1/ 7 (1 / 5) 1 log 5 3



27

log 9 36



Q.10 Given, log712 = a & log1224 = b . Show that, log54168 =

3

4 log 7 9

1  ab . a (8  5 b )

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL)

Daily Practice Problems

DATE : 30/06/2006

TIME : 45 Min.

DPP. NO.-9

Fill in the blanks :





2 8 =

1 . Then the value of 1000 x is equal to _____. 3

Q.1

If logx log18

Q.2

The expression log0.52 8 has the value equal to ______.

Q.3

Solution set of the equation 1  log 1 x + 2 = 3  log 1 x is _______.

Q.4

log x log 2 log 27 The solution set of the equation 4 9  6.x 9  2 3 = 0 is ______.

6

6

Select the correct alternative : (Only one is correct) Q.5

Which one of the following when simplified does not reduce to an integer? log 2 32

2 log 6

(A)

(B) log 243 3

log12  log 3

log5 16  log5 4 (C) log5 128

 1 (D) log1/4    16 

2

Q.6

Let u = (log2x)2 – 6 log2x + 12 where x is a real number. Then the equation xu = 256 has (A) no solution for x (B) exactly one solution for x (C) exactly two distinct solutions for x (D) exactly three distinct solutions for x

Q.7

The equation, log2 (2x2) + log2 x . x log x log2 x 1 + (A) exactly one real solution (C) 3 real solutions

1 log42 (x4) + 2 3 log1 / 2 log2 x  = 1 has : 2

(B) two real solutions (D) no solution .

Select the correct alternative : (More than one are correct) Q.8

The equation

log8

  8 x2

log8 x 2

= 3 has :

(A) no integral solution (C) two real solutions

(B) one natural solution (D) one irrational solution

Subjective Q.9

Find the exact value of tan2

 3 5 7 + tan2 + tan2 + tan2 16 16 16 16

Q.10 In any triangle, if (sin A + sin B + sin C) (sin A + sin B  sin C) = 3 sin A sin B, find the angle C. Q.11

Which is smaller? log 1 3

1 80

or

 1  log 1   15  2  2

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL)

Daily Practice Problems

DATE : 07/07/2006

TIME : 50 Min.

DPP. NO.-10,11

NOTE: Dpp-10 & 11 can be simultaneously done for a better test practice. DPP-10 Q.1 Simplify whenever defined

Q.2

cot( 270  ) sin( 270   ) cos 3 (720   )  sin( 270   ) sin 3 (540   ) + cosec 2 ( 450  ) sin( 90  ) sin(  )  cos 2 (180   )) where  is taken such that the denominator appearing in any fraction in the expression does not vanish. 1 Given x2 + 4y2 =12xy, where x>0, y>0 then prove that, log(x + 2y) – 2log2 = (log x + log y). 2

log( x )  log x 2 .

Q.3

Solve the equation,

Q.4

Let fn(x) = sinnx + cosnx. Find the number of values of x in [0, ] for which the relation 6f4(x) – 4f6(x) = 2f2(x) holds valid.

Q.5

If 2cos = x +

Q.6

b a If a  b > 1, then find the largest possible value of the expression log a   + log b   . a b

Q.7

  1 x    1 x    Prove that solution of the equation, 2 log 9 2   1  log 27     4  is an irrational number..  2   4      

1 , find the values of the following in terms of cosine of the multiple angle of . x 1 1 1 (i) x2 + 2 ; (ii) x3 + 3 and (ii) x4 + 4 x x x 1 Hence deduce the value of xn + n , n  N. x

3

DPP-11 sin   cos   tan  4 if tan = – . sec   cosec   cot  3

Q.1

Find the possible value(s) of

Q.2

If log a  log b  log c , show that aa . bb . cc = 1. bc ca a b

Q.3

Find the value of sin

Q.5

Prove that the expression, cos2

Q.5

        2 cos   1 Show that, tan    tan    =  6 2   6 2  2 cos   1

Q.6

Let y =

Q.7

Solve the equation log x 1 ( x  0.5)  log x 0.5 ( x  1) .

 3    323  and cos . If sin = – and    , 2  2 2 325 

 3 5 7 + cos2 + cos2 + cos2 is not irrational. 8 8 8 8

sin x  sin 2 x  sin 4 x  sin 5x  . Find the value of y where x = . cos x  cos 2 x  cos 4 x  cos 5x 36

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL)

Daily Practice Problems

DATE : 07/07/2006

TIME : 40 Min.

Q.1

Find the number of degree in the acute angle  satisfying cos  =

Q.2

If x satisfies log2x + logx2 = 4, then log2x can be (A) tan(/12) (B) tan(/8) (C) tan (5/12)

1 2

DPP. NO.-12

2 2 ?

(D) tan(3/8)

Q.3(a) Solve (x + 2)(x – 2)(x – 13) = (x + 2)(x – 7)(x – 11) for x. (b) Solve (x – 3)(x – 2)(x – 13) = (x – 3)(x – 4)(x – 11) for x.

Q.4

Find all real numbers such that

x 5 –

x  7 = 2.

Q.5

Let D be any point on the base of an isosceles triangle ABC. AC is extended to E so that CE = CD. ED is extended to meet AB at F. If angle CED = 10°, find the cosine of the angle BFD.

Q.6

In the figure, E is the midpoint of AB and F is the midpoint of AD. If the area of FAEC is 13 sq. units, find the area of the quadrilateral ABCD.

Q.7

In the figure, 'O' is the centre of the circle and A, B and C are three points on the circle. Suppose that OA = AB = 2 units and angle OAC = 10°. Find the length of the arc BC.

Q.8

Find all values of a such that the three equations ax + y = 1 x+y=2 x–y=a are simultaneously satisfied by same ordered pair (x, y).

Q.9

In a triangle ABC, BC = 8, CA = 6 and AB = 10. A line dividing the triangle ABC into two regions of equal area is perpendicular to AB at the point X. Find the length BX.

Q.10 If m, n > 1 and for all x > 0 and x  1 lognx = 3 logmx. Write an equation expressing m explicitly in terms of n.

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL)

Daily Practice Problems

DATE : 07/07/2006

TIME : 45 Min.

DPP. NO.-13

Q.1

If logab + logbc + logca vanishes where a, b and c are positive reals different than unity then the value of (logab)3 + (logbc)3 + (logca)3 is (A) an odd prime (B) an even prime (C) an odd composite (D) an irrational number

Q.2

Each of the four statements given below are either True or False. sin765° = –

I.

1 2

II.

cosec(–1410°) = 2

1  15  13  =–1 = IV. cot   3 3  4  Indicate the correct order of sequence, where 'T' stands for true and 'F' stands for false. (A) F T F T (B) F F T T (C) T F F F (D) F T F F tan

III.

Q.3

The value of p which satisfies the equation 122p–1 = 5(3p ·7p) is ln 5  ln 12 (A) ln 21  ln 12

Q.4

ln 12  ln 5 (B) ln 12  ln 21

If tan (A)

Q.6

ln 12 (D) ln 12  5ln 21

tan 2 20  sin 2 20 The expression simplifies to tan 2 20 ·sin 2 20 (A) a rational which is not integral (C) a natural which is prime

Q.5

ln 5  ln 12 (C) ln 144  ln 21

(B) a surd (D) a natural which is not composite

 1  2 sin 2 ( / 2) = m, then the value of is 2 1  sin 

2m 1 m

The value of

(B)

1 m 1 m

(C)

1 m 1 m

(D)

1 m 2m

3  cot 76 cot 16 is : cot 76  cot 16

(A) tan 46º

(B) tan 44º

(C) cot 46º

(D) cot 2º

Q.7

An unknown polynomial yields a remainder of 2 upon division by x – 1, and a remainder of 1 upon division by x – 2. If this polynomial is divided by (x – 1)(x – 2), then the remainder is (A) 2 (B) 3 (C) – x + 3 (D) x + 1

Q.8

If sec x + tan x =

22 x , find the value of tan . Use it to deduce the value of cosec x + cot x. 7 2

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL) Q.1

Daily Practice Problems

DATE : 14/07/2006

TIME : 45 Min.

DPP. NO.-14

d  4 3   sin x  sin x  when x = 12° is dx  3 

(A) 0

(B)

6 2 4

5 1 4

(C)

(D)

5 1 4

Q.2

Number of real x satisfying the equation | x – 1 | = | x – 2 | + | x – 3 | is (A) 1 (B) 2 (C) 3 (D) more than 3

Q.3

A rectangle has its sides of length sin x and cos x for some x. Largest possible area which it can have, is (A)

Q.4

1 4

(B) 1

(C)

1 2

(D) can not be determined

If logAB + logBA 2 = 4 and B < A then the value of logAB equals (A)

2 1

(B) 2 2  2

(C) 2  3

(D) 2  2

Q.5

The sum of 3 real numbers is zero. If the sum of their cubes is C then their product is (A) a rational greater than 1 (B) a rational less than 1 (C) an irrational greater than 1 (D) an irrational less than 1

Q.6

The sides of a triangle ABC are as shown in the given figure. Let D be any internal point of this triangle and let e, f, and g denote the distance between the point D and the sides of the triangle. The sum (5e + 12f + 13g) is equal to (A) 120 (B) 90 (C) 60 (D) 30

Q.7

The value of tan27° + tan18° + tan27° tan18°, is (A) an irrational number (B) rational which is not integer (C) integer which is prime (D) integer which is not a prime.

Q.8

If cos( + ) + sin( – ) = 0 and tan  =

1 . Find tan . 2006

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL) Q.1

Q.2



 2

(B)

(B)

2 3



(C)

5 6

(D) none

3 3 8

(C)

3 3 16

(D)

7 3 16

There is an equilateral triangle with side 4 and a circle with the centre on one of the vertex of that triangle. The arc of that circle divides the triangle into two parts of equal area. How long is the radius of the circle? 12 3 

(B)

24 3 

30 3 

(C)

(D)

6 3 

If log3(x) = p and log7(x) = q, which of the following yields log21(x)? (A) pq

Q.6

DPP. NO.-15

The difference (sin8 75° – cos8 75°) is equal to

(A) Q.5

TIME : 45 Min.

 sin 2 x  Smallest positive solution of the equation, 4 16 = 6 sin x , is   2

(A) 1 Q.4

DATE : 14/07/2006

A diameter and a chord of a circle intersect at a point inside the circle. The two parts of the chord are length 3 and 5 and one part of the diameter is length unity. The radius of the circle is (A) 8 (B) 9 (C) 12 (D) 16

(A) Q.3

Daily Practice Problems

(B)

1 pq

(C)

1 pq 1 (D) 1 p q p  q 1 1

The value of the expression 2(sin 1  sin 2  sin 3  .......  sin 89) equals 2(cos 1  cos 2  .................  cos 44)  1 (A)

2

(B)

1 2

(C)

1 2

(D) 1

cos A cos B cos C + + sin B sin C sin C sin A sin A sin B (A) is prime (B) is composite (C) is rational which is not an integer (D) an integer

Q.7

In a triangle ABC, the value of

Q.8

ABC is a right angled triangle. Show that sinA·sinB·sin(A–B)+sinB·sinC·sin(B–C)+sinC·sinA·sin(C–A)+sin(A–B)·sin(B–C)·sin(C–A)=0.

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL)

Daily Practice Problems

DATE : 14/07/2006

TIME : 50 Min.

DPP. NO.-16

Q.1

Given that log (2) = 0.3010....., number of digits in the number 20002000 is (A) 6601 (B) 6602 (C) 6603 (D) 6604

Q.2

If logx(logyz) = 0 and logy(logzx) = 0, where x, y, z > 1, then 2z – x – y equals (A) 0 (B) 1 (C) xy (D) yz

Q.3

Which of the following is the largest? (A) 2

Q.4

Q.5

Q.6

log5 6

(B) 3log6 5

(C) 3log5 6

(D) 3

If sin  and cos  are the roots of the equation ax2 – bx + c = 0, then (A) a2 – b2 = 2ac (B) a2 + b2 = 2ac (C) a2 + b2 + 2ac = 0 (D) b2 – a2 = 2ac 2  4  2  4       + cos  x   and b = sin x + sin  x   + sin  x   then which Let a = cos x + cos  x  3  3  3  3      one of the following does not hold good? (A) a = 2b (B) b = 2a (C) a + b = 0 (D) a  b

Suppose that log10(x – 2) + log10y = 0

and

x  y2  x y

Then the value of (x + y), is (A) 2 Q.7

(D) 4 + 2 2

(C) 2 + 2 2

(B) 2 2

If x, y, z are real numbers greater than 1 and 'w' is a positive real number. If logxw = 24, logyw = 40 and logxyzw = 12 then logwz has the value equal to (A)

1 120

(B)

2 120

(C)

3 120

(D)

5 120

Q.8

If  and  are the roots of the quadratic equation (sin 2a)x2 – 2(sin a + cos a)x + 2 = 0, find them and hence prove that 2 + 2 = 2 · 2.

Q.9

Find all integral solution of the equation, 4 log x

 x  2 log x   3 log x . 2

4x

3

2x

2

ACME

M A T H E M A T I C S

BANSAL CLASSES Target IIT JEE 2008 CLASS : XI (ALL)

Daily Practice Problems

DATE : 14/07/2006

TIME : 50 Min.

DPP. NO.-17

2

Q.1

Q.2

Q.3

Q.4

Let N =  22  12 2  22  12 2  then log2N equals   (A) 2 (B) 3 (C) 4 The sum of all values of x so that 16( x (A) 0 (B) 3

Q.6

(B) 12

The reals x and y satisfy log8x + log4(y2) = 5 and then the value of xy is (A) 1024 (B) 512

If sin 2x = 1 45

(C) 27

(D) – 5

(D) more than two roots

(D) 3 3

log8y + log4(x2) = 7 (C) 256

(D) 81

2024 5 9 , where <x< , the value of the sin x – cos x is equal to 2025 4 4

(B)

1 45

  k1 k   The equation ln  1 ( k 1)  = F(k) · ( k  1 )   F(100) has the value equal to

(A) 100

Q.8

2

 8( x 3x  2) , is (C) – 3

Given log2(log8x) = log8(log2x) then (log2x)2 has the value equal to

(A) –

Q.7

3x 1)

The equation, | sin x | = sin x + 3 in [0, 2] has (A) no root (B) only one root (C) two roots

(A) 9 Q.5

2

(D) none

(B)

1 101

(C) ±

1 2025

(D) none

   1  1 ln 1  k  1   k ln k  is true for all k wherever defined.    

(C) 5050

Let a and b are two real numbers such that, sin a + sin b =

(D)

1 100

2 6 and cos a + cos b = . Find the value of 2 2

(i) cos(a – b) and (ii) sin(a + b). Q.9

For any 3 angle ,  and , prove that         ·sin   ·sin  . sin  + sin  + sin  = sin( +  + ) + 4 sin   2   2   2 

ACME