JEE MAIN - Mathematics (2005)

• 1

  Let $$f:( - 1,1) \to B$$, be a function defined by
  $$f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}$$,
  then $$f$$ is both one-one and onto when B is the interval

  Jibu

  (D)

  $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$
• 2

  A real valued function f(x) satisfies the functional equation
  
  f(x - y) = f(x)f(y) - f(a - x)f(a + y)
  
  where a is given constant and f(0) = 1, f(2a - x) is equal to

  Jibu

  (A)

  - f(x)
• 3

  A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

  Jibu

  (C)

  Interval	Function
  $$\left( { - \infty ,{1 \over 3}} \right]$$	3x2 - 2x + 1
• 4

  Suppose $$f(x)$$ is differentiable at x = 1 and
  
  $$\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5$$, then $$f'\left( 1 \right)$$ equals

  Jibu

  (C)

  5
• 5

  Let $$\alpha$$ and $$\beta$$ be the distinct roots of $$a{x^2} + bx + c = 0$$, then
  
  $$\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}}$$ is equal to

  Jibu

  (A)

  $${{{a^2}{{\left( {\alpha - \beta } \right)}^2}} \over 2}$$
• 6

  If $$f$$ is a real valued differentiable function satisfying
  
  $$\left| {f\left( x \right) - f\left( y \right)} \right|$$ $$ \le {\left( {x - y} \right)^2}$$, $$x, y$$ $$ \in R$$
  and $$f(0)$$ = 0, then $$f(1)$$ equals

  Jibu

  (B)

  0
• 7

  If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately :

  Jibu

  (C)

  24.0
• 8

  Let x₁, x₂,...........,x_n be n observations such that
  
  $$\sum {x_i^2} = 400$$ and $$\sum {{x_i}} = 80$$. Then a possible value of n among the following is

  Jibu

  (A)

  18
• 9

  The value of $$a$$ for which the sum of the squares of the roots of the equation $${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is :

  Jibu

  (A)

  $$1$$
• 10

  If the cube roots of unity are 1, $$\omega \,,\,{\omega ^2}$$ then the roots of the equation $${(x - 1)^3}$$ + 8 = 0, are :

  Jibu

  (C)

  $$ - 1,1 - 2\omega ,1 - 2{\omega ^2}$$
• 11

  The value of $$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a > 0,} $$ is

  Jibu

  (B)

  $${\pi \over 2}$$
• 12

  Area of the greatest rectangle that can be inscribed in the
  ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

  Jibu

  (A)

  $$2ab$$
• 13

  Area of the greatest rectangle that can be inscribed in the
  ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

  Jibu

  (A)

  $$2ab$$
• 14

  A spherical iron ball $$10$$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $$50$$ cm$$^3$$ /min. When the thickness of ice is $$5$$ cm, then the rate at which the thickness of ice decreases is

  Jibu

  (B)

  $${1 \over {18\pi }}$$ cm/min
• 15

  If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is :

  Jibu

  (D)

  $$I-A$$
• 16

  The system of equations
  

  $$\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr {x + y + \alpha \,z = \alpha - 1} \cr } $$

  has no solutions, if $$\alpha $$ is :

  Jibu

  (A)

  $$-2$$
• 17

  If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant $$$\Delta = \left| {\matrix{ {\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr {\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr {\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {a_{n + 8}}} \cr } } \right|$$$
  is equal to :

  Jibu

  (B)

  $$0$$
• 18

  If $${a^2} + {b^2} + {c^2} = - 2$$ and

  f$$\left( x \right) = \left| {\matrix{ {1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr {\left( {1 + {a^2}} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \right)x} \cr {\left( {1 + {a^2}} \right)x} & {\left( {1 + {b^2}} \right)x} & {1 + {c^2}x} \cr } } \right|,$$

  then f$$(x)$$ is a polynomial of degree :

  Jibu

  (D)

  $$2$$
• 19

  $$\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx} $$ is equal to

  Jibu

  (D)

  $${x \over {{{\left( {\log x} \right)}^2} + 1}} + C$$
• 20

  If $${I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}}dx} } } $$ and $${I_4} = \int\limits_1^2 {{2^{{x^3}}}dx} $$ then

  Jibu

  (B)

  $${I_1} > {I_2}$$
• 21

  The area enclosed between the curve $$y = {\log _e}\left( {x + e} \right)$$ and the coordinate axes is :

  Jibu

  (A)

  $$1$$
• 22

  The parabolas $${y^2} = 4x$$ and $${x^2} = 4y$$ divide the square region bounded by the lines $$x=4,$$ $$y=4$$ and the coordinate axes. If $${S_1},{S_2},{S_3}$$ are respectively the areas of these parts numbered from top to bottom ; then $${S_1},{S_2},{S_3}$$ is :

  Jibu

  (D)

  $$1:1:1$$
• 23

  Let $$f(x)$$ be a non - negative continuous function such that the area bounded by the curve $$y=f(x),$$ $$x$$-axis and the ordinates $$x = {\pi \over 4}$$ and $$x = \beta > {\pi \over 4}$$ is $$\left( {\beta \sin \beta + {\pi \over 4}\cos \beta + \sqrt 2 \beta } \right).$$ Then $$f\left( {{\pi \over 2}} \right)$$ is

  Jibu

  (D)

  $$\left( {1 - {\pi \over 4} + \sqrt 2 } \right)$$
• 24

  If $${\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha ,$$ then $$4{x^2} - 4xy\cos \alpha + {y^2}$$ is equal to :

  Jibu

  (C)

  $$4{\sin ^2}\alpha $$
• 25

  The value of integral, $$\int\limits_3^6 {{{\sqrt x } \over {\sqrt {9 - x} + \sqrt x }}} dx $$ is

  Jibu

  (B)

  $${3 \over 2}$$
• 26

  The differential equation representing the family of curves $${y^2} = 2c\left( {x + \sqrt c } \right),$$ where $$c>0,$$ is a parameter, is of order and degree as follows:

  Jibu

  (C)

  order $$1,$$ degree $$3$$
• 27

  If $$x{{dy} \over {dx}} = y\left( {\log y - \log x + 1} \right),$$ then the solution of the equation is :

  Jibu

  (C)

  $$\log \left( {{y \over x}} \right) = cx$$
• 28

  Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is :

  Jibu

  (B)

  $${1 \over 9}$$
• 29

  Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is :

  Jibu

  (B)

  $${1 \over 9}$$
• 30

  A random variable $$X$$ has Poisson distribution with mean $$2$$.
  Then $$P\left( {X > 1.5} \right)$$ equals :

  Jibu

  (C)

  $$1 - {3 \over {{e^2}}}$$
• 31

  Let $$A$$ and $$B$$ two events such that $$P\left( {\overline {A \cup B} } \right) = {1 \over 6},$$ $$P\left( {A \cap B} \right) = {1 \over 4}$$ and $$P\left( {\overline A } \right) = {1 \over 4},$$ where $${\overline A }$$ stands for complement of event $$A$$. Then events $$A$$ and $$B$$ are :

  Jibu

  (C)

  independent but not equally likely
• 32

  If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB,$$ then :

  Jibu

  (A)

  $$\overrightarrow {PA} + \overrightarrow {PB} = 2\overrightarrow {PC} $$
• 33

  The angle between the lines $$2x=3y=-z$$ and $$6x=-y=-4z$$ is :

  Jibu

  (B)

  $${90^ \circ }$$
• 34

  If a vertex of a triangle is $$(1, 1)$$ and the mid points of two sides through this vertex are $$(-1, 2)$$ and $$(3, 2)$$ then the centroid of the triangle is :

  Jibu

  (C)

  $$\left( { 1,{7 \over 3}} \right)$$
• 35

  If $${z_1}$$ and $${z_2}$$ are two non-zero complex numbers such that $$\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$$, then arg $${z_1}$$ - arg $${z_2}$$ is equal to :

  Jibu

  (C)

  0
• 36

  If $$\,\omega = {z \over {z - {1 \over 3}i}}\,$$ and $$\left| \omega \right| = 1$$, then $$z$$ lies on :

  Jibu

  (C)

  a straight line
• 37

  If $${z_1}$$ and $${z_2}$$ are two non-zero complex numbers such that $$\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$$, then arg $${z_1}$$ - arg $${z_2}$$ is equal to :

  Jibu

  (C)

  0
• 38

  If $$\,\omega = {z \over {z - {1 \over 3}i}}\,$$ and $$\left| \omega \right| = 1$$, then $$z$$ lies on :

  Jibu

  (C)

  a straight line
• 39

  In a triangle $$PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$$ and $$ \tan \left( {{Q \over 2}} \right)$$ are the roots of $$a{x^2} + bx + c = 0,\,\,a \ne 0$$ then

  Jibu

  (B)

  $$c = a + b$$
• 40

  If both the roots of the quadratic equation $${x^2} - 2kx + {k^2} + k - 5 = 0$$ are less than 5, then $$k$$ lies in the interval

  Jibu

  (C)

  $$\left( { - \infty ,\,4} \right)$$
• 41

  If the coefficient of $${x^7}$$ in $${\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}$$ equals the coefficient of $${x^{ - 7}}$$ in $${\left[ {ax - \left( {{1 \over {b{x^2}}}} \right)} \right]^{11}}$$, then $$a$$ and $$b$$ satisfy the relation

  Jibu

  (D)

  $$ab = 1$$
• 42

  If $$x$$ is so small that $${x^3}$$ and higher powers of $$x$$ may be neglected, then $${{{{\left( {1 + x} \right)}^{{3 \over 2}}} - {{\left( {1 + {1 \over 2}x} \right)}^3}} \over {{{\left( {1 - x} \right)}^{{1 \over 2}}}}}$$ may be approximated as

  Jibu

  (C)

  $$ - {3 \over 8}{x^2}$$
• 43

  If the coefficients of r^th, (r+1)^th, and (r + 2)^th terms in the binomial expansion of $${{\rm{(1 + y )}}^m}$$ are in A.P., then m and r satisfy the equation

  Jibu

  (C)

  $${m^2} - m(4r + 1) + 4\,{r^2} - 2 = 0$$
• 44

  If $$x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,$$ where a, b, c are in A.P and $$\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| < 1$$ then x, y, z are in

  Jibu

  (D)

  H.P.
• 45

  The line parallel to the $$x$$ - axis and passing through the intersection of the lines $$ax + 2by + 3b = 0$$ and $$bx - 2ay - 3a = 0,$$ where $$(a, b)$$ $$ \ne $$ $$(0, 0)$$ is :

  Jibu

  (A)

  below the $$x$$ - axis at a distance of $${3 \over 2}$$ from it
• 46

  If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2} = {p^2}$$ orthogonally, then the equation of the locus of its centre is :

  Jibu

  (D)

  $$2ax\, + \,\,2\,by\,\, - \,({a^2}\, + \,{b^2} + {p^2}) = 0$$
• 47

  If the circles $${x^2}\, + \,{y^2} + \,2ax\, + \,cy\, + a\,\, = 0$$ and $${x^2}\, + \,{y^2} - \,3ax\, + \,dy\, - 1\,\, = 0$$ intersect in two ditinct points P and Q then the line 5x + by - a = 0 passes through P and Q for :

  Jibu

  (B)

  no value of a
• 48

  If the pair of lines $$a{x^2} + 2\left( {a + b} \right)xy + b{y^2} = 0$$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then :

  Jibu

  (D)

  $$3{a^2} + 2ab + 3{b^2} = 0$$
• 49

  An ellipse has $$OB$$ as semi minor axis, $$F$$ and $$F$$' its focii and theangle $$FBF$$' is a right angle. Then the eccentricity of the ellipse is :

  Jibu

  (A)

  $${1 \over {\sqrt 2 }}$$
• 50

  Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the parabola $${y^2} = 8x$$. The locus of mid point of $$PQ$$ is :

  Jibu

  (A)

  $${y^2} - 4x + 2 = 0$$
• 51

  The value of $$a$$ for which the sum of the squares of the roots of the equation
  $${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is

  Jibu

  (A)

  $$1$$
• 52

  Let $$f:R \to R$$ be a differentiable function having $$f\left( 2 \right) = 6$$,
  $$f'\left( 2 \right) = \left( {{1 \over {48}}} \right)$$. Then $$\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {{{4{t^3}} \over {x - 2}}dt} $$ equals :

  Jibu

  (D)

  $$18$$
• 53

  If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals

  Jibu

  (D)

  $$1$$
• 54

  For any vector $${\overrightarrow a }$$ , the value of $${\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \times \widehat j} \right)^2} + {\left( {\overrightarrow a \times \widehat k} \right)^2}$$ is equal to :

  Jibu

  (C)

  $$2{\overrightarrow a ^2}$$
• 55

  If non zero numbers $$a, b, c$$ are in $$H.P.,$$ then the straight line $${x \over a} + {y \over b} + {1 \over c} = 0$$ always passes through a fixed point. That point is :

  Jibu

  (C)

  $$(1, -2)$$
• 56

  Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :

  Jibu

  (D)

  reflexive and transitive only