JEE MAIN - Mathematics (2003)

• 1

  The value of $$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {{{\sec }^2}tdt} } \over xsinx}$$ is

  回答

  (D)

  1
• 2

  The function $$f\left( x \right)$$ $$ = \log \left( {x + \sqrt {{x^2} + 1} } \right)$$, is

  回答

  (C)

  an odd function
• 3

  A function $$f$$ from the set of natural numbers to integers defined by $$$f\left( n \right) = \left\{ {\matrix{ {{{n - 1} \over 2},\,when\,n\,is\,odd} \cr { - {n \over 2},\,when\,n\,is\,even} \cr } } \right.$$$ is

  回答

  (D)

  one-one and onto both
• 4

  If $$f:R \to R$$ satisfies $$f$$(x + y) = $$f$$(x) + $$f$$(y), for all x, y $$ \in $$ R and $$f$$(1) = 7, then $$\sum\limits_{r = 1}^n {f\left( r \right)} $$ is

  回答

  (A)

  $${{7n\left( {n + 1} \right)} \over 2}$$
• 5

  Domain of definition of the function f(x) = $${3 \over {4 - {x^2}}}$$ + $${\log _{10}}\left( {{x^3} - x} \right)$$, is

  回答

  (A)

  (-1, 0)$$ \cup $$(1, 2)$$ \cup $$(2, $$\infty $$)
• 6

  If $$\mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x}$$ = k, the value of k is

  回答

  (D)

  $${2 \over 3}$$
• 7

  Let $$f(a) = g(a) = k$$ and their n^th derivatives
  $${f^n}(a)$$, $${g^n}(a)$$ exist and are not equal for some n. Further if
  
  $$\mathop {\lim }\limits_{x \to a} {{f(a)g(x) - f(a) - g(a)f(x) + f(a)} \over {g(x) - f(x)}} = 4$$
  
  then the value of k is

  回答

  (B)

  4
• 8

  $$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\left[ {1 - \tan \left( {{x \over 2}} \right)} \right]\left[ {1 - \sin x} \right]} \over {\left[ {1 + \tan \left( {{x \over 2}} \right)} \right]{{\left[ {\pi - 2x} \right]}^3}}}$$ is

  回答

  (D)

  $${1 \over 32}$$
• 9

  If $$f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0} \cr 0 & {,x = 0} \cr } } \right.$$
  
  then $$f(x)$$ is

  回答

  (C)

  continuous for all x but not differentiable at x = 0
• 10

  In an experiment with 15 observations on $$x$$, then following results were available:
  $$\sum {{x^2}} = 2830$$, $$\sum x = 170$$
  One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is :

  回答

  (D)

  78.00
• 11

  The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set :

  回答

  (D)

  remains the same as that of the original set
• 12

  Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is :

  回答

  (A)

  $${{2 \over 5}}$$
• 13

  The trigonometric equation $${\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a$$ has a solution for :

  回答

  (D)

  $$\left| a \right| \le {1 \over {\sqrt 2 }}$$
• 14

  If the function $$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$$ where $$a>0,$$ attains its maximum and minimum at $$p$$ and $$q$$ respectively such that $${p^2} = q$$ , then $$a$$ equals

  回答

  (D)

  $$2$$
• 15

  If $$A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]$$ and $${A^2} = \left[ {\matrix{ \alpha & \beta \cr \beta & \alpha \cr } } \right]$$, then

  回答

  (C)

  $$\alpha = {a^2} + {b^2},\,\beta = 2ab$$
• 16

  $$\Delta = \left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr } } \right|$$ is equal to

  回答

  (B)

  $$0$$
• 17

  If the system of linear equations
  $$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$
  has a non - zero solution, then $$a, b, c$$.

  回答

  (D)

  are in H.P.
• 18

  The area of the region bounded by the curves $$y = \left| {x - 1} \right|$$ and $$y = 3 - \left| x \right|$$ is :

  回答

  (D)

  $$4$$ sq. units
• 19

  Let $$f(x)$$ be a function satisfying $$f'(x)=f(x)$$ with $$f(0)=1$$ and $$g(x)$$ be a function that satisfies $$f\left( x \right) + g\left( x \right) = {x^2}$$. Then the value of the integral $$\int\limits_0^1 {f\left( x \right)g\left( x \right)dx,} $$ is

  回答

  (D)

  $$e - {{{e^2}} \over 2} - {3 \over 2}$$
• 20

  If $$f\left( {a + b - x} \right) = f\left( x \right)$$ then $$\int\limits_a^b {xf\left( x \right)dx} $$ is equal to

  回答

  (C)

  $${{a + b} \over 2}\int\limits_a^b {f\left( x \right)dx} $$
• 21

  The value of the integral $$I = \int\limits_0^1 {x{{\left( {1 - x} \right)}^n}dx} $$ is

  回答

  (D)

  $${1 \over {n + 1}} - {1 \over {n + 2}}$$
• 22

  The solution of the differential equation
  
  $$\left( {1 + {y^2}} \right) + \left( {x - {e^{{{\tan }^{ - 1}}y}}} \right){{dy} \over {dx}} = 0,$$ is :

  回答

  (C)

  $$2x{e^{{{\tan }^{ - 1}}y}} = {e^{2{{\tan }^{ - 1}}y}} + k$$
• 23

  The degree and order of the differential equation of the family of all parabolas whose axis is $$x$$-axis, are respectively.

  回答

  (C)

  $$1,2$$
• 24

  Events $$A, B, C$$ are mutually exclusive events such that $$P\left( A \right) = {{3x + 1} \over 3},$$ $$P\left( B \right) = {{1 - x} \over 4}$$ and $$P\left( C \right) = {{1 - 2x} \over 2}$$ The set of possible values of $$x$$ are in the interval.

  回答

  (B)

  $$\left[ {{1 \over 3},{1 \over 2}} \right]$$
• 25

  If $$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$ then $$\overrightarrow a + \overrightarrow b + \overrightarrow c = $$

  回答

  (C)

  $$0$$
• 26

  Let $$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$ and $$\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.$$ If $$\widehat n$$ is a unit vector such that $$\overrightarrow u .\widehat n = 0$$ and $$\overrightarrow v .\widehat n = 0\,\,,$$ then $$\left| {\overrightarrow w .\widehat n} \right|$$ is equal to :

  回答

  (A)

  $$3$$
• 27

  The vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k$$ are the sides of triangle $$ABC.$$ The length of the median through $$A$$ is :

  回答

  (D)

  $$\sqrt {33} $$
• 28

  The two lines $$x=ay+b,z=cy+d$$ and $$x = a'y + b',z = c'y + d'$$ will be perpendicular, if and only if :

  回答

  (A)

  $$aa' + cc' + 1 = 0$$
• 29

  The lines $${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$$ and $${{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}$$ are coplanar if :

  回答

  (D)

  $$k=0$$ or $$-3$$
• 30

  $$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$ are $$3$$ vectors, such that
  
  $$\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$$ , $$\left| {\overrightarrow a } \right| = 1\,\,\,\left| {\overrightarrow b } \right| = 2,\,\,\,\left| {\overrightarrow c } \right| = 3,$$,
  
  then $${\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a }$$ is equal to :

  回答

  (C)

  $$-7$$
• 31

  A tetrahedron has vertices at $$O(0,0,0), A(1,2,1) B(2,1,3)$$ and $$C(-1,1,2).$$ Then the angle between the faces $$OAB$$ and $$ABC$$ will be :

  回答

  (B)

  $${\cos ^{ - 1}}\left( {{{19} \over {35}}} \right)$$
• 32

  If $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0$$ and vectors $$\left( {1,a,{a^2}} \right),\,\,$$
  
  $$\left( {1,b,{b^2}} \right)$$ and $$\left( {1,c,{c^2}} \right)\,$$ are non-coplanar, then the product $$abc$$ equals :

  回答

  (C)

  $$-1$$
• 33

  Consider points $$A, B, C$$ and $$D$$ with position
  
  vectors $$7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \widehat i - 3\widehat j + 4\widehat k$$ and $$5\widehat i - \widehat j + 5\widehat k$$ respectively. Then $$ABCD$$ is a :

  回答

  (D)

  None
• 34

  Let $${Z_1}$$ and $${Z_2}$$ be two roots of the equation $${Z^2} + aZ + b = 0$$, Z being complex. Further , assume that the origin, $${Z_1}$$ and $${Z_2}$$ form an equilateral triangle. Then :

  回答

  (D)

  $${a^2} = 3b$$
• 35

  If $${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$ then :

  回答

  (B)

  x = 4n , where n is any positive integer
• 36

  If the sum of the roots of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the sum of the squares of their reciprocals, then $${a \over c},\,{b \over a}$$ and $${c \over b}$$ are in

  回答

  (D)

  Harmonic Progression
• 37

  The value of '$$a$$' for which one root of the quadratic equation $$$\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0$$$
  is twice as large as the other is

  回答

  (B)

  $$ {2 \over 3}$$
• 38

  The number of real solutions of the equation $${x^2} - 3\left| x \right| + 2 = 0$$ is

  回答

  (C)

  4
• 39

  The real number $$x$$ when added to its inverse gives the minimum sum at $$x$$ equal :

  回答

  (C)

  1
• 40

  If $$x$$ is positive, the first negative term in the expansion of $${\left( {1 + x} \right)^{27/5}}$$ is

  回答

  (D)

  8th term.
• 41

  The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by

  回答

  (B)

  $$6!\, \times 5!$$
• 42

  The number of integral terms in the expansion of $${\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}$$ is

  回答

  (C)

  33
• 43

  A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

  回答

  (C)

  196
• 44

  If $${}^n{C_r}$$ denotes the number of combination of n things taken r at a time, then the expression $$\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$$ equals

  回答

  (C)

  $${}^{n + 2}{C_{r + 1}}$$
• 45

  A square of side a lies above the $$x$$-axis and has one vertex at the origin. The side passing through the origin makes an angle $$\alpha \left( {0 < \alpha < {\pi \over 4}} \right)$$ with the positive direction of x-axis. The equation of its diagonal not passing through the origin is :

  回答

  (A)

  $$y\left( {\cos \alpha + \sin \alpha } \right) + x\left( {\cos \alpha - \sin \alpha } \right) = a$$
• 46

  If $${x_1},{x_2},{x_3}$$ and $${y_1},{y_2},{y_3}$$ are both in G.P. with the same common ratio, then the points $$\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)$$ and $$\left( {{x_3},{y_3}} \right)$$ :

  回答

  (B)

  lie on a straight line
• 47

  Locus of centroid of the triangle whose vertices are $$\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \right)$$ and $$\left( {1,0} \right),$$ where $$t$$ is a parameter, is :

  回答

  (C)

  $${\left( {3x - 1} \right)^2} + {\left( {3y} \right)^2} = {a^2} + {b^2}$$
• 48

  If the equation of the locus of a point equidistant from the point $$\left( {{a_{1,}}{b_1}} \right)$$ and $$\left( {{a_{2,}}{b_2}} \right)$$ is
  $$\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0$$ , then the value of $$'c'$$ is :

  回答

  (B)

  $${1 \over 2}\left( {{a_2}^2 + {b_2}^2 - {a_1}^2 - {b_1}^2} \right)$$
• 49

  The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :

  回答

  (D)

  $${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,47$$
• 50

  The foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$ and the hyperbola $${{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \over {25}}$$ coincide. Then the value of $${b^2}$$ is :

  回答

  (D)

  $$7$$
• 51

  If $$f\left( y \right) = {e^y},$$ $$g\left( y \right) = y;y > 0$$ and
  
  $$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,} $$ then :

  回答

  (C)

  $$F\left( t \right) = {e^t} - \left( {1 + t} \right)$$
• 52

  $$f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\left( 1 \right)} \over {2!}} - {{f'''\left( 1 \right)} \over {3!}} + ..........{{{{\left( { - 1} \right)}^n}{f^n}\left( 1 \right)} \over {n!}}$$ is

  回答

  (D)

  $$0$$
• 53

  Let $$f\left( x \right)$$ be a polynomial function of second degree. If $$f\left( 1 \right) = f\left( { - 1} \right)$$ and $$a,b,c$$ are in $$A.P, $$ then $$f'\left( a \right),f'\left( b \right),f'\left( c \right)$$ are in

  回答

  (B)

  $$A.P$$
• 54

  If $$z$$ and $$\omega $$ are two non-zero complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$Arg(z) - Arg(\omega ) = {\pi \over 2},$$ then $$\,\overline {z\,} \omega $$ is equal to

  回答

  (A)

  $$- i$$