JEE Advance - Mathematics (2002)

• 1

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

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (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.

  Trả lời

  (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

  Trả lời

  (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 $$ = n = 1,2,...$$

  Trả lời

  (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

  Trả lời

  (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 $$\,{{{G_1},{G_2}} \over {{H_1},{H_2}}} = {{{A_1} + {A_2}} \over {{H_1} + {H_2}}} = {{(2a + b)\,(a + 2b)} \over {9ab}}$$.

  Trả lời

  (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

  Trả lời

  (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.

  Trả lời

  (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

  Trả lời

  (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.

  Trả lời

  (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

  Trả lời

  (A)

  $$ - {1 \over 2}$$
• 7

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

  Trả lời

  (A)

  isosceles and right angled
• 8

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

  Trả lời

  (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

  Trả lời

  (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
  

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (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)?

  Trả lời

  (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.

  Trả lời

  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

  Trả lời

  (C)

  $$x = 0$$
• 12

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

  Trả lời

  (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

  Trả lời

  (D)

  $$y= x + 2$$
• 13

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

  Trả lời

  (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

  Trả lời

  (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 $$x=1.$$

  Trả lời

  (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

  Trả lời

  (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?

  Trả lời

  (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

  Trả lời

  (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 $$V \le {L^3}\,\,.$$

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (C)

  15
• 22

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

  Trả lời

  (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

  Trả lời

  (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

  Trả lời

  (B)

  8