JEE Advance - Mathematics (2002)

• 1

  The point(s) in the curve $${y^3} + 3{x^2} = 12y$$ where the tangent is vertical, is (are)

  إجابة

  (D)

  $$\left( { \pm {4 \over {\sqrt 3 }}, 2} \right)$$
• 1

  Let $$\omega $$ $$ = - {1 \over 2} + i{{\sqrt 3 } \over 2},$$ then the value of the det.
  $$\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - 1 - {\omega ^2}} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^4}} \cr } } \right|$$ is

  إجابة

  (B)

  $$3\omega \left( {\omega - 1} \right)$$
• 2

  Let $$\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $$ and $$\overrightarrow W = \overrightarrow i + 3\overrightarrow k .$$ If $$\overrightarrow U $$ is a unit vector, then the maximum value of the scalar triple product $$\left| {\overrightarrow U \overrightarrow V \overrightarrow W } \right|$$ is

  إجابة

  (C)

  $$\sqrt {59} $$
• 2

  Let a complex number $$\alpha ,\,\alpha \ne 1$$, be a root of the equation $${z^{p + q}} - {z^p} - {z^q} + 1 = 0$$, where p, q are distinct primes. Show that either $$1 + \alpha + {\alpha ^2} + .... + {\alpha ^{p - 1}} = 0\,or\,1 + \alpha + {\alpha ^2} + .... + {\alpha ^{q - 1}} = 0$$, but not both together.

  إجابة

  (A)

  The given equation can be factored, and the roots must satisfy either the p-th or q-th cyclotomic polynomial.
• 3

  If $${\overrightarrow a }$$ and $${\overrightarrow b }$$ are two unit vectors such that $${\overrightarrow a + 2\overrightarrow b }$$ and $${5\overrightarrow a - 4\overrightarrow b }$$ are perpendicular to each other then the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is

  إجابة

  (B)

  $${60^ \circ }$$
• 3

  Use mathematical induction to show that
  $${\left( {25} \right)^{n + 1}} - 24n + 5735$$ is divisible by $${\left( {24} \right)^2}$$ for all $$ = n = 1,2,...$$

  إجابة

  (D)

  The inductive hypothesis is used to prove the statement for n = k+1.
• 4

  Let $$T>0$$ be a fixed real number . Suppose $$f$$ is a continuous
  function such that for all $$x \in R$$, $$f\left( {x + T} \right) = f\left( x \right)$$.

  If $$I = \int\limits_0^T {f\left( x \right)dx} $$ then the value of $$\int\limits_3^{3 + 3T} {f\left( {2x} \right)dx} $$ is

  إجابة

  (C)

  $$3I$$
• 4

  Let a, b be positive real numbers. If a, $${{A_1},{A_2}}$$, b are in arithmetic progression, a, $${{G_1},{G_2}}$$, b are in geometric progression and a, $${{H_1},{H_2}}$$, b are in harmonic progression, show that $$\,{{{G_1},{G_2}} \over {{H_1},{H_2}}} = {{{A_1} + {A_2}} \over {{H_1} + {H_2}}} = {{(2a + b)\,(a + 2b)} \over {9ab}}$$.

  إجابة

  (A)

  The question asks to prove an equality related to arithmetic, geometric, and harmonic progressions.
• 5

  Let $$T>0$$ be a fixed real number . Suppose $$f$$ is a continuous
  function such that for all $$x \in R$$, $$f\left( {x + T} \right) = f\left( x \right)$$.

  If $$I = \int\limits_0^T {f\left( x \right)dx} $$ then the value of $$\int\limits_3^{3 + 3T} {f\left( {2x} \right)dx} $$ is

  إجابة

  (C)

  $$3I$$
• 5

  A straight line $$L$$ through the origin meets the lines $$x + y = 1$$ and $$x + y = 3$$ at $$P $$ and $$Q$$ respectively. Through $$P$$ and $$Q$$ two straight lines $${L_1}$$ and $${L_2}$$ are drawn, parallel to $$2x - y = 5$$ and $$3x + y = 5$$ respectively. Lines $${L_1}$$ and $${L_2}$$ intersect at $$R$$. Show that the locus of $$R$$, as $$L$$ varies is a straight line.

  إجابة

  (A)

  The locus of R is a straight line.
• 6

  Let $$f\left( x \right) = \int\limits_1^x {\sqrt {2 - {t^2}} \,dt.} $$ Then the real roots of the equation
  $${x^2} - f'\left( x \right) = 0$$ are

  إجابة

  (A)

  $$ \pm 1$$
• 6

  A straight line $$L$$ with negative slope passes through the point $$(8, 2)$$ and cuts the positive coordinate axes at points $$P$$ and $$Q$$. Find the absolute minimum value of $$OP + OQ,$$ as $$L$$ varies, where $$O$$ is the origin.

  إجابة

  (C)

  18
• 7

  The integral $$\int\limits_{ - 1/2}^{1/2} {\left( {\left[ x \right] + \ell n\left( {{{1 + x} \over {1 - x}}} \right)} \right)dx} $$ equal to

  إجابة

  (A)

  $$ - {1 \over 2}$$
• 7

  A triangle with vertices $$(4, 0), (-1, -1), (3, 5)$$is

  إجابة

  (A)

  isosceles and right angled
• 8

  The area bounded by the curves $$y = \left| x \right| - 1$$ and $$y = - \left| x \right| + 1$$ is

  إجابة

  (B)

  $$2$$
• 8

  Locus of mid point of the portion between the axes of $$x$$ $$\cos \alpha + y\sin \alpha = p$$ where $$p$$ is constant is

  إجابة

  (D)

  $${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}$$
• 9

  For all complex numbers $${z_1},\,{z_2}$$ satisfying $$\left| {{z_1}} \right| = 12$$ and $$\left| {{z_2} - 3 - 4i} \right| = 5,$$
  the minimum value of $$\left| {{z_1} - {z_2}} \right|$$ is
  

  إجابة

  (B)

  2
• 9

  If the pair of lines $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ intersect on the $$y$$ axis then

  إجابة

  (A)

  $$2fgh = b{g^2} + c{h^2}$$
• 10

  The length of a longest interval in which the function $$3\,\sin x - 4{\sin ^3}x$$ is increasing, is

  إجابة

  (A)

  $${\pi \over 3}$$
• 10

  The pair of lines represented by
  $$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$ are perpendicular to each other for

  إجابة

  (A)

  two values of $$a$$
• 11

  Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $$ABC$$ ($$R$$ being the radius of the circumcircle)?

  إجابة

  (D)

  $$a,\,\sin \,A,R$$
• 11

  Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

  إجابة

  A

  B

  C

  D
• 12

  The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix

  إجابة

  (C)

  $$x = 0$$
• 12

  Prove that $$\cos \,ta{n^{ - 1}}\sin \,{\cot ^{ - 1}}x = \sqrt {{{{x^2} + 1} \over {{x^2} + 2}}} $$.

  إجابة

  (C)

  The equation can be proven with trigonometric identities and inverse trigonometric functions.
• 13

  The equation of the common tangent to the curves $${y^2} = 8x$$ and $$xy = - 1$$ is

  إجابة

  (D)

  $$y= x + 2$$
• 13

  For any natural number $$m$$, evaluate
  $$\int {\left( {{x^{3m}} + {x^{2m}} + {x^m}} \right){{\left( {2{x^{2m}} + 3{x^m} + 6} \right)}^{l/m}}dx,x > 0.} $$

  إجابة

  (B)

  ${1 \over 6}{{{{\left( {2{x^{3m}} + 3{x^{2m}} + 6{x^m}} \right)}^{{{m + 1} \over m}}}} \over {m + 1}} + C
• 14

  If $$a > 2b > 0$$ then the positive value of $$m$$ for which $$y = mx - b\sqrt {1 + {m^2}} $$ is a common tangent to $${x^2} + {y^2} = {b^2}$$ and $${\left( {x - a} \right)^2} + {y^2} = {b^2}$$ is

  إجابة

  (A)

  $${{2b} \over {\sqrt {{a^2} - 4{b^2}} }}$$
• 14

  Find the area of the region bounded by the curves $$y = {x^2},y = \left| {2 - {x^2}} \right|$$ and $$y=2,$$ which lies to the right of the line $$x=1.$$

  إجابة

  (A)

  $$\frac{20}{3} - 4\sqrt{2}$$
• 15

  If the tangent at the point P on the circle $${x^2} + {y^2} + 6x + 6y = 2$$ meets a straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the lenght of PQ is

  إجابة

  (C)

  5
• 15

  A box contains $$N$$ coins, $$m$$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $$1/2$$, while it is $$2/3$$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. what is the probability that the coin drawn is fair?

  إجابة

  (A)

  $$rac{9m}{m + 8N}$$
• 16

  A straight line through the origin $$O$$ meets the parallel lines $$4x+2y=9$$ and $$2x+y+6=0$$ at points $$P$$ and $$Q$$ respectively. Then the point $$O$$ divides the segemnt $$PQ$$ in the ratio

  إجابة

  (B)

  $$3 : 4$$
• 16

  Let $$V$$ be the volume of the parallelopiped formed by the vectors $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k,$$ $$\,\,\,\,\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + {b_3}\widehat k,$$ $$\,\,\,\,\,\overrightarrow c = {c_1}\widehat i + {c_2}\widehat j + {c_3}\widehat k.$$ where $$r=1, 2, 3,$$ are non-negative real numbers and $$\sum\limits_{r = 1}^3 {\left( {{a_r} + {b_r} + {c_r}} \right) = 3L,} $$ show that $$V \le {L^3}\,\,.$$

  إجابة

  (B)

  The maximum possible volume V is L^3.
• 17

  Let $$P = \left( { - 1,\,0} \right),\,Q = \left( {0,\,0} \right)$$ and $$R = \left( {3,\,3\sqrt 3 } \right)$$ be three points.
  Then the equation of the bisector of the angle $$PQR$$ is

  إجابة

  (C)

  $$\sqrt 3 x + y = 0$$
• 18

  Let $$0 < \alpha < {\pi \over 2}$$ be fixed angle. If $$P = \left( {\cos \theta ,\,\sin \theta } \right)$$ and $$Q = \left( {\cos \left( {\alpha - \theta } \right),\,\sin \left( {\alpha - \theta } \right)} \right),$$ then $$Q$$ is obtained from $$P$$ by

  إجابة

  (D)

  reflection in the line through origin with slope tan $$\left( {\alpha /2} \right)$$
• 19

  Suppose $$a, b, c$$ are in A.P. and $${a^2},{b^2},{c^2}$$ are in G.P. If $$a < b < c$$ and $$a + b + c = {3 \over 2},$$ then the value of $$a$$ is

  إجابة

  (D)

  $${1 \over 2} - {1 \over {\sqrt 2 }}$$
• 20

  The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is

  إجابة

  (A)

  40
• 21

  The sum $$\sum\limits_{i = 0}^m {\left( {\matrix{ {10} \cr i \cr } } \right)\left( {\matrix{ {20} \cr {m - i} \cr } } \right),\,\left( {where\left( {\matrix{ p \cr q \cr } } \right) = 0\,\,if\,\,p < q} \right)} $$ is maximum when $$m$$ is

  إجابة

  (C)

  15
• 22

  The set of all real numbers x for which $${x^2} - \left| {x + 2} \right| + x > 0$$, is

  إجابة

  (B)

  $$( - \infty ,\, - \sqrt 2 ) \cup (\sqrt 2 ,\infty )$$
• 23

  If $${a_1},{a_2}.......,{a_n}$$ are positive real numbers whose product is a fixed number c, then the minimum value of $${a_1} + {a_2} + ..... + {a_{n - 1}} + 2{a_n}$$ is

  إجابة

  (A)

  $$n{(2c)^{1/n}}$$
• 24

  The number of integral values of $$k$$ for which the equation $$7\cos x + 5\sin x = 2k + 1$$ has a solution is

  إجابة

  (B)

  8