JEE Advance - Mathematics (1993)

• 1

  Let $$a, b, c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k$$ and $$c\widehat i + c\widehat j + b\widehat k$$ lie in a plane, then $$c$$ is

  回答

  (B)

  the Geometric Mean of $$a$$ and $$b$$
• 2

  Let $$\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$$ and $$\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k$$ be three vectors. A vector in the plane of $${\overrightarrow b }$$ and $${\overrightarrow c }$$, whose projection on $${\overrightarrow a }$$ is of magnitude $$\sqrt {2/3,} $$ is :

  回答

  A

  C
• 3

  In a triangle $$ABC, D$$ and $$E$$ are points on $$BC$$ and $$AC$$ respectively, such that $$BD=2DC$$ and $$AE=3EC.$$ Let $$P$$ be the point of intersection of $$AD$$ and $$BE.$$ Find $$BP/PE$$ using vector methods.

  回答

  (D)

  8:3
• 4

  Numbers are selected at random, one at a time, from the two- digit numbers $$00, 01, 02 ......, 99$$ with replacement. An event $$E$$ occurs if only if the product of the two digits of a selected number is $$18$$. If four numbers are selected, find probability that the event $$E$$ occurs at least $$3$$ times.

  回答

  (D)

  {{97} \over {{{\left( {25} \right)}^4}}}
• 5

  Find the coordinates of the point at which the circles $${x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36$$ touch each other. Also find equations common tangests touching the circles in the distinct points.

  回答

  (D)

  ((14/5, 8/5), y = 0, 7y - 24x + 16 = 0)
• 6

  If $$A > 0,B > 0\,$$ and $$A + B = \pi /3,$$ then the maximum value of tan A tan B is _______.

  回答

  (C)

  1/3
• 7

  Number of solutions of the equation $$\tan x + \sec x = 2\cos x\,$$ lying in the interval $$\left[ {0,2\pi } \right]$$ is:

  回答

  (C)

  2
• 8

  $$ABCD$$ is a rhombus. Its diagonals $$AC$$ and $$BD$$ intersect at the point $$M$$ and satisfy $$BD$$ = 2$$AC$$. If the points $$D$$ and $$M$$ represent the complex numbers $$1 + i$$ and $$2 - i$$ respectively, then A represents the comp[lex number ..........or..........

  回答

  (A)

  3 - i/2 or 1 - (3/2)i
• 9

  Determine the smallest positive value of number $$x$$ (in degrees) for which $$$\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).$$$

  回答

  (C)

  30°
• 10

  Prove that $$\sum\limits_{r = 1}^k {{{\left( { - 3} \right)}^{r - 1}}\,\,{}^{3n}{C_{2r - 1}} = 0,} $$ where $$k = \left( {3n} \right)/2$$ and $$n$$ is an even positive integer.

  回答

  (A)

  The statement is true due to symmetry properties of binomial coefficients and alternating signs when n is even.
• 11

  Using mathematical induction, prove that
  $${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1/\left( {{n^2} + n + 1} \right)} \right\} = {\tan ^{ - 1}}\left\{ {n/\left( {n + 2} \right)} \right\}$$

  回答

  (C)

  The base case holds and the inductive step can be proven, therefore the statement is true for all n.
• 12

  For $$0 < \phi < \pi /2,$$ if
  $$x = $$$$\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,\,\,\,\,z = \sum\limits_{n = 0}^{} {{{\cos }^{2n}}\phi {{\sin }^{2n}}\phi } } } \infty $$ then

  回答

  B

  C
• 13

  The vertices of a triangle are $$A\left( { - 1, - 7} \right)B\left( {5,\,1} \right)$$ and $$C\left( {1,\,4} \right).$$ The equation of the bisector of the angle $$\angle ABC$$ is ............... .

  回答

  (A)

  x - 7y + 2 = 0
• 14

  A line through $$A (-5, -4)$$ meets the line $$x + 3y + 2 = 0,$$ $$2x + y + 4 = 0$$ and $$x - y - 5 = 0$$ at the points $$B, C$$ and $$D$$ respectively. If $${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)^2},$$ find the equation of the line.

  回答

  (A)

  2x + 3y + 22 = 0
• 15

  [area $$\left( {\Delta {P_1},{P_2},{P_3}} \right)$$]/[area $$\left( {{P_2},{P_3},{P_4}} \right)$$]

  回答

  B

  D
• 16

  The equation of the locus of the mid-points of the circle $$4{x^2} + 4{y^2} - 12x + 4y + 1 = 0$$ that subtend an angle of $$2\pi /3$$ at its centre is.................................

  回答

  (A)

  $$16{x^2} + 16{y^2} - 48x + 16y + 31 = 0$$
• 17

  The locus of the centre of a circle, which touches externally the circle $${x^2} + {y^2} - 6x - 6y + 14 = 0$$ and also touches the y-axis, is given by the equation:

  回答

  (D)

  $${y^2} - 10x - 6y + 14 = 0$$
• 18

  If $$K = \sin \left( {\pi /18} \right)\sin \left( {5\pi /18} \right)\sin \left( {7\pi /18} \right),$$ then the numerical value of K is ______.

  回答

  (B)

  1/8
• 19

  Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the circle $${x^2}\, + \,{y^2} - \,4x - \,6y - 3 = 0$$ cuts the members of the family are concurrent at a point. Find the coordinate of this point.

  回答

  (B)

  (2, 23/3)
• 20

  If in a triangle $$ABC$$, $${{2\cos A} \over a} + {{\cos B} \over b} + {{2\cos C} \over c} = {a \over {bc}} + {b \over {ca}},$$ then the value of the angle $$A$$ is .................... degrees.

  回答

  (D)

  90
• 21

  An observer at $$O$$ notices that the angle of elevation of the top of a tower is $${30^ \circ }$$. The line joining $$O$$ to the base of the tower makes an angle of $${\tan ^{ - 1}}\left( {1/\sqrt 2 } \right)$$ with the North and is inclined Eastwards. The observer travels a distance of $$300$$ meters towards the North to a point A and finds the tower to his East. The angle of elevation of the top of the tower at $$A$$ is $$\phi $$, Find $$\phi $$ and the height of the tower.

  回答

  (C)

  $$\phi = {45^\circ }$$, height = $${150\sqrt 2 }$$ m
• 22

  If $$f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr {37 - x} & {2 < x \le 3} \cr } } \right.$$ then:

  回答

  A

  B

  C
• 23

  Find the equation of the normal to the curve
  $$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$$ at $$x=0$$

  回答

  (B)

  x + y = 1
• 24

  Find all possible real values of $$b$$ such that $$f(x)$$ has the smallest value at $$x=1$$.

  回答

  (A)

  $$b \in (-2, -1) \cup (1, \infty)$$
• 25

  The value of $$\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi } $$ is ..............

  回答

  (C)

  $$\pi \left( {\sqrt 2 - 1} \right)$$
• 26

  The value of $$\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $$ is

  回答

  (D)

  $$\pi /4$$
• 27

  Evaluate $$\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.$$

  回答

  (A)

  $$\frac{1}{6}\log 6 - \frac{1}{10}$$
• 28

  An unbiased die with faces marked $$1,2,3,4,5$$ and $$6$$ is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than $$2$$ and the maximum face value is not greater than $$5,$$ is then:

  回答

  (A)

  $$16/81$$
• 29

  $$E$$ and $$F$$ are two independent events. The probability that both $$E$$ and $$F$$ happen is $$1/12$$ and the probability that neither $$E$$ nor $$F$$ happens is $$1/2.$$ Then,

  回答

  D

  A