JEE MAIN - Mathematics (2006)

• 1

  Suppose a population A has 100 observations 101, 102,........, 200, and another population B has 100 observations 151, 152,......., 250. If V_A and V_B represent the variances of the two populations, respectively, then $${{{V_A}} \over {{V_B}}}$$ is

  回答

  (A)

  1
• 2

  The value of $$\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a > 1$$ where $${\left[ x \right]}$$ denotes the greatest integer not exceeding $$x$$ is

  回答

  (B)

  $$\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( {\left[ a \right]} \right)} \right\}$$
• 3

  The set of points where $$f\left( x \right) = {x \over {1 + \left| x \right|}}$$ is differentiable is

  回答

  (C)

  $$\left( { - \infty ,\infty } \right)$$
• 4

  If $${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$$ then $${{{dy} \over {dx}}}$$ is

  回答

  (A)

  $${y \over x}$$
• 5

  The function $$f\left( x \right) = {x \over 2} + {2 \over x}$$ has a local minimum at

  回答

  (A)

  $$x=2$$
• 6

  A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $$x$$. The maximum area enclosed by the park is

  回答

  (C)

  $${1 \over 2}{x^2}$$
• 7

  If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that
  $${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$$ then which of the following will be always true?

  回答

  (B)

  $$AB=BA$$
• 8

  Let $$A = \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)$$ and $$B = \left( {\matrix{ a & 0 \cr 0 & b \cr } } \right),a,b \in N.$$ Then

  回答

  (D)

  there exist infinitely many $$B'$$s such that $$AB=BA$$
• 9

  $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} $$ is equal to

  回答

  (D)

  $$\pi \int\limits_0^{\pi /2} {f\left( {\cos x} \right)dx} $$
• 10

  $$\int\limits_{ - {{3\pi } \over 2}}^{ - {\pi \over 2}} {\left[ {{{\left( {x + \pi } \right)}^3} + {{\cos }^2}\left( {x + 3\pi } \right)} \right]} dx$$ is equal to

  回答

  (C)

  $${\pi \over 2}$$
• 11

  The differential equation whose solution is $$A{x^2} + B{y^2} = 1$$
  where $$A$$ and $$B$$ are arbitrary constants is of

  回答

  (D)

  second order and first degree
• 12

  At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of $$5$$ phone calls during $$10$$ minute time intervals. The probability that there is at the most one phone call during a $$10$$-minute time period is :

  回答

  (D)

  $${6 \over {{e^5}}}$$
• 13

  The values of a, for which the points $$A, B, C$$ with position vectors $$2\widehat i - \widehat j + \widehat k,\,\,\widehat i - 3\widehat j - 5\widehat k$$ and $$a\widehat i - 3\widehat j + \widehat k$$ respectively are the vertices of a right angled triangle with $$C = {\pi \over 2}$$ are :

  回答

  (A)

  $$2$$ and $$1$$
• 14

  The two lines $$x=ay+b, z=cy+d;$$ and $$x=a'y+b' ,$$ $$z=c'y+d'$$ are perpendicular to each other if :

  回答

  (A)

  $$aa'+cc'=-1$$
• 15

  If $$0 < x < \pi $$ and $$\cos x + \sin x = {1 \over 2},$$ then $$\tan x$$ is :

  回答

  (C)

  $$ - {{\left( {4 + \sqrt 7 } \right)} \over 3}$$
• 16

  If $${z^2} + z + 1 = 0$$, where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \over {{z^6}}}} \right)^2}$$ is :

  回答

  (D)

  12
• 17

  The value of $$\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)} $$ is :

  回答

  (D)

  - i
• 18

  If the roots of the quadratic equation $${x^2} + px + q = 0$$ are $$\tan {30^ \circ }$$ and $$\tan {15^ \circ }$$, respectively, then the value of $$2 + q - p$$ is

  回答

  (B)

  3
• 19

  All the values of $$m$$ for which both roots of the equation $${x^2} - 2mx + {m^2} - 1 = 0$$ are greater than $$ - 2$$ but less then 4, lie in the interval

  回答

  (C)

  $$ - 1 < m < 3$$
• 20

  If $$x$$ is real, the maximum value of $${{3{x^2} + 9x + 17} \over {3{x^2} + 9x + 7}}$$ is

  回答

  (B)

  $$41$$
• 21

  At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is

  回答

  (C)

  385
• 22

  If the expansion in powers of $$x$$ of the function $${1 \over {\left( {1 - ax} \right)\left( {1 - bx} \right)}}$$ is $${a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}.....$$ then $${a_n}$$ is

  回答

  (D)

  $${{{b^{n + 1}} - {a^{n + 1}}} \over {b - a}}$$
• 23

  If $${{a_1},{a_2},....{a_n}}$$ are in H.P., then the expression $${{a_1}\,{a_2} + \,{a_2}\,{a_3}\, + .... + {a_{n - 1}}\,{a_n}}$$ is equal to

  回答

  (D)

  $$(n - 1)\,\,{a_1}{a_n}$$
• 24

  Let $${a_1}$$, $${a_2}$$, $${a_3}$$.....be terms on A.P. If $${{{a_1} + {a_2} + .....{a_p}} \over {{a_1} + {a_2} + .....{a_q}}} = {{{p^2}} \over {{q^2}}},\,p \ne q,\,then\,{{{a_6}} \over {{a_{21}}}}\,$$ equals

  回答

  (D)

  $${{11} \over {41}}$$
• 25

  A straight line through the point $$A (3, 4)$$ is such that its intercept between the axes is bisected at $$A$$. Its equation is :

  回答

  (C)

  $$4x + 3y = 24$$
• 26

  If $$\left( {a,{a^2}} \right)$$ falls inside the angle made by the lines $$y = {x \over 2},$$ $$x > 0$$ and $$y = 3x,$$ $$x > 0,$$ then a belong to :

  回答

  (C)

  $$\left( {{1 \over 2},3} \right)$$
• 27

  If the lines $$3x - 4y - 7 = 0$$ and $$2x - 3y - 5 = 0$$ are two diameters of a circle of area $$49\pi $$ square units, the equation of the circle is :

  回答

  (D)

  $${x^2} + {y^2} - 2x\, + 2y - 47 = 0$$
• 28

  Let $$C$$ be the circle with centre $$(0, 0)$$ and radius $$3$$ units. The equation of the locus of the mid points of the chords of the circle $$C$$ that subtend an angle of $${{2\pi } \over 3}$$ at its center is :

  回答

  (D)

  $${x^2} + {y^2} = {{9} \over 4}$$
• 29

  In the ellipse, the distance between its foci is $$6$$ and minor axis is $$8$$. Then its eccentricity is :

  回答

  (A)

  $${3 \over 5}$$
• 30

  Let $W$ denote the words in the English dictionary. Define the relation $R$ by
  
  $R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is

  回答

  (A)

  reflexive, symmetric and not transitive
• 31

  The $$rms$$ value of the electric field of the light coming from the Sun is $$720$$ $$N/C.$$ The average total energy density of the electromagnetic wave is

  回答

  (A)

  $$4.58 \times {10^{ - 6}}\,J/{m^3}$$