JEE Advance - Mathematics (1992)

1
A lot contains $$50$$ defective and $$50$$ non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events $$A, B, C$$ are defined as
$$A=$$ (the first bulbs is defective)
$$B=$$ (the second bulbs is non-defective)
$$C=$$ (the two bulbs are both defective or both non defective)
Determine whether
(i) $$\,\,\,\,\,$$ $$A, B, C$$ are pairwise independent
(ii)$$\,\,\,\,\,$$ $$A, B, C$$ are independent
إجابة
(A)
A, B, C are pairwise independent but A, B, C are dependent
2
A unit vector coplanar with $$\overrightarrow i + \overrightarrow j + 2\overrightarrow k $$ and $$\overrightarrow i + 2\overrightarrow j + \overrightarrow k $$ and perpendicular to $$\overrightarrow i + \overrightarrow j + \overrightarrow k $$ is ...........
إجابة
A
B
3
India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points $$0,$$ $$1$$ and $$2$$ are $$0.45, 0.05$$ and $$0.50$$ respectively. Assuming that the outcomes are independent, the probability of India getting at least $$7$$ points is
إجابة
(B)
$$0.0875$$
4
The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle $${x^2} + {y^2} = 9$$is
إجابة
(D)
$$\left( {{1 \over 2}, - {2^{{1 \over 2}}}} \right)\,$$
5
$${\rm{z }} \ne {\rm{0}}$$ is a complex number
Column I

(A) Re z = 0
(B) Arg $$z = {\pi \over 4}$$
Column II

(p) Re$${z^2}$$ = 0
(q) Im$${z^2}$$ = 0
(r) Re$${z^2}$$ = Im$${z^2}$$
إجابة
(A)
(A) - q, (B) - p
6
Show that the value of $${{\tan x} \over {\tan 3x}},$$ wherever defined never lies between $${1 \over 3}$$ and 3.
إجابة
(A)
The range of the expression is (-∞, 1/3] ∪ [3, ∞).
7
Let $$\alpha \,,\,\beta $$ be the roots of the equation (x - a) (x - b) = c, $$c \ne 0$$. Then the roots of the equation $$(x - \alpha \,)\,(x - \beta ) + c = 0$$ are
إجابة
(C)
a, b
8
The expansion $${\left( {x + {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5}$$ is a polynomial of degree
إجابة
(C)
7
9
If $$\sum\limits_{r = 0}^{2n} {{a_r}{{\left( {x - 2} \right)}^r}\,\, = \sum\limits_{r = 0}^{2n} {{b_r}{{\left( {x - 3} \right)}^r}} } $$ and $${a_k} = 1$$ for all $$k \ge n,$$ then show that $${b_n} = {}^{2n + 1}{C_{n + 1}}$$
إجابة
(D)
${b_n} = {}^{2n+1}{C_{n+1}}$
10
Let $$p \ge 3$$ be an integer and $$\alpha $$, $$\beta $$ be the roots of $${x^2} - \left( {p + 1} \right)x + 1 = 0$$ using mathematical induction show that $${\alpha ^n} + {\beta ^n}.$$
(i) is an integer and (ii) is not divisible by $$p$$
إجابة
A
B
D
11
Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the ratio .........
إجابة
(C)
4:1
12
If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
إجابة
(A)
square
13
Determine all values of $$\alpha $$ for which the point $$\left( {\alpha ,\,{\alpha ^2}} \right)$$ lies insides the triangle formed by the lines $$$\matrix{ {2x + 3y - 1 = 0} \cr {x + 2y - 3 = 0} \cr {5x - 6y - 1 = 0} \cr } $$$
إجابة
(A)
$$\alpha \in \left( { - {3 \over 2}, - 1} \right) \cup \left( {{1 \over 2},1} \right)$$
14
In this questions there are entries in columns 1 and 2. Each entry in column 1 is related to exactly one entry in column 2. Write the correct letter from column 2 against the entry number in column 1 in your answer book.
$${{\sin \,3\alpha } \over {\cos 2\alpha }}$$ is

Column $${\rm I}$$
(A) positive

(B) negative

Column $${\rm I}$$$${\rm I}$$
(p) $$\left( {{{13\pi } \over {48}},{{14\pi } \over {48}}} \right)$$

(q) $$\left( {{{14\pi } \over {48}},\,{{18\pi } \over {48}}} \right)$$

(r) $$\left( {{{18\pi } \over {48}},\,{{23\pi } \over {48}}} \right)$$

(s) $$\left( {0,\,{\pi \over 2}} \right)$$

Options:-
إجابة
(B)
$$\left( A \right) - r,\,\left( B \right) - p$$
15
Let a circle be given by 2x (x - a) + y (2y - b) = 0, $$(a\, \ne \,0,\,\,b\, \ne 0)$$. Find the condition on a abd b if two chords, each bisected by the x-axis, can be drawn to the circle from $$\left( {a,\,\,{b \over 2}} \right)$$.
إجابة
(C)
a^2 > 2b^2
16
Three circles touch the one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is $$4$$. Find the ratio of the product of the radii to the sum of the radii of the circles.
إجابة
(D)
16:1
17
A cubic $$f(x)$$ vanishes at $$x=2$$ and has relative minimum / maximum at $$x=-1$$ and $$x = {1 \over 3}$$ if $$\int\limits_{ - 1}^1 {f\,\,dx = {{14} \over 3}} $$, find the cubic $$f(x)$$.
إجابة
(B)
x^3 + x^2 - x + 2
18
What normal to the curve $$y = {x^2}$$ forms the shortest chord?
إجابة
(E)
The line x + √2y = √2 or the line x - √2y = -√2
19
In this questions there are entries in columns $$I$$ and $$II$$. Each entry in column $$I$$ is related to exactly one entry in column $$II$$. Write the correct letter from column $$II$$ against the entry number in column $$I$$ in your answer book.
Let the functions defined in column $$I$$ have domain $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$

$$\,\,\,\,$$Column $$I$$
(A) $$x + \sin x$$
(B) $$\sec x$$

$$\,\,\,\,$$Column $$II$$
(p) increasing
(q) decreasing
(r) neither increasing nor decreasing
إجابة
A
E
20
Find the indefinite integral $$\int {\left( {{1 \over {\root 3 \of x + \root 4 \of 4 }} + {{In\left( {1 + \root 6 \of x } \right)} \over {\root 3 \of x + \root \, \of x }}} \right)} dx$$
إجابة
A
B
C
21
Sketch the region bounded by the curves $$y = {x^2}$$ and
$$y = {2 \over {1 + {x^2}}}.$$ Find the area.
إجابة
(B)
$$\pi - {2 \over 3}$$
22
Determine a positive integer $$n \le 5,$$ such that $$$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} $$$
إجابة
(C)
3
23
Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively is ...............
إجابة
(A)
1/36