JEE Advance - Mathematics (1983)

1
The unit vector perpendicular to the plane determined by $$P\left( {1, - 1,2} \right),\,Q\left( {2,0, - 1} \right)$$ and $$R\left( {0,2,1} \right)$$ is ...........
Svare
(E)
\pm {{2\widehat i + \widehat j + \widehat k} \over {\sqrt 6 }}
2
A vector $$\overrightarrow A $$ has components $${A_1},{A_2},{A_3}$$ in a right -handed rectangular Cartesian coordinate system $$oxyz.$$ The coordinate system is rotated about the $$x$$-axis through an angle $${\pi \over 2}.$$ Find the components of $$A$$ in the new coordinate system in terms of $${A_1},{A_2},{A_3}.$$
Svare
(C)
A₁' = A₁, A₂' = A₃, A₃' = -A₂
3
The volume of the parallelopiped whose sides are given by
$$\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i + j - k,\,\overrightarrow {OC} = 3i - k,$$ is
Svare
(D)
none of these
4
The points with position vectors $$60i+3j,$$ $$40i-8j,$$ $$ai-52j$$ are collinear if
Svare
(A)
$$a=-40$$
5
If $$X.A=0, X.B=0, X.C=0$$ for some non-zero vector $$X,$$ then $$\left[ {A\,B\,C} \right] = 0$$
Svare
(B)
FALSE
6
The area of the triangle whose vertices are $$A(1, -1, 2), B(2, 1, -1), C(3, -1, 2)$$ is ..........
Svare
(C)
$$\sqrt{13}$$
7
$$A, B, C$$ are events such that
$$P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8$$
$$P\left( {AB} \right) = 0.08,P\left( {AC} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09$$
If $$P\left( {A \cup B \cup C} \right) \ge 0.75,$$ then show that $$P$$ $$(BC)$$ lies in the interval $$0.23 \le x \le 0.48$$
Svare
(B)
0.23 ≤ P(BC) ≤ 0.48
8
Cards are drawn one by one at random from a well - shuffled full pack of $$52$$ playing cards until $$2$$ aces are obtained for the first time. If $$N$$ is the number of cards required to be drawn, then show that $${P_r}\left\{ {N = n} \right\} = {{\left( {n - 1} \right)\left( {52 - n} \right)\left( {51 - n} \right)} \over {50 \times 49 \times 17 \times 13}}$$ where $$2 \le n \le 50$$
Svare
(C)
The probability that the nth card is an ace and exactly one of the first n-1 cards is an ace.
9
Fifteen coupons are numbered $$1, 2 ........15,$$ respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is $$9,$$ is
Svare
(C)
$${\left( {{3 \over {5}}} \right)^7}$$
10
If the letters of the word "Assassin" are written down at random in a row, the probability that no two S's occur together is $$1/35$$
Svare
(B)
FALSE
11
If $$\left( {a + bx} \right){e^{y/x}} = x,$$ then prove that $${x^3}{{{d^2}y} \over {d{x^2}}} = {\left( {x{{dy} \over {dx}} - y} \right)^2}$$
Svare
A
B
C
12
Find the area bounded by the $$x$$-axis, part of the curve $$y = \left( {1 + {8 \over {{x^2}}}} \right)$$ and
the ordinates at $$x=2$$ and $$x=4$$. If the ordinate at $$x=a$$ divides the area into two equal parts, find $$a$$.
Svare
(C)
2√2
13
Evaluate : $$\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx} $$
Svare
(B)
$$\frac{1}{20} \log 3$$
14
The value of the integral $$\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $$ is
Svare
(A)
$$\pi /4$$
15
Evaluate : $$\int {{{\left( {x - 1} \right){e^x}} \over {{{\left( {x + 1} \right)}^3}}}dx} $$
Svare
(A)
${{{e^x}} over {{{left( {x + 1} ight)}^2}}} + C$
16
Find the coordinates of the point on the curve $$y = {x \over {1 + {x^2}}}$$
where the tangent to the curve has the greatest slope.
Svare
(B)
(0, 0)
17
Given positive integers $$r > 1,\,n > 2$$ and that the coefficient of $$\left( {3r} \right)$$th and $$\left( {r + 2} \right)$$th terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then
Svare
(A)
$$n = 2r$$
18
The straight lines $$x + y = 0,\,3x + y - 4 = 0,\,x + 3y - 4 = 0$$ form a triangle which is
Svare
(A)
isosceles
19
The straight line $$5x + 4y = 0$$ passes through the point of intersection of the straight lines $$x + 2y - 10 = 0$$ and $$2x + y + 5 = 0.$$
Svare
(B)
FALSE
20
Given the points $$A\left( {0,4} \right)$$ and $$B\left( {0, - 4} \right)$$, the equation of the locus of the point $$P\left( {x,y} \right)$$ such that $$\left| {AP - BP} \right| = 6$$ is .............
Svare
(B)
$$ {{{y^2}} over 9} - {{{x^2}} over 7} = 1$$
21
Find three numbers $$a,b,c$$ between $$2$$ and $$18$$ such that
(i) their sum is $$25$$
(ii) the numbers $$2,$$ $$a, b$$ are consecutive terms of an A.P. and
(iii) the numbers $$b,c,18$$ are consecutive terms of a G.P.
Svare
(C)
a = 5, b = 8, c = 12
22
The rational number, which equals the number $$2\overline {357} $$ with recurring decimal is
Svare
(C)
$${{2355} \over {999}}$$
23
m men and n women are to be seated in a row so that no two women sit together. If $$m > n$$, then show that the number of ways in which they can be seated is $$\,{{m!(m + 1)!} \over {(m - n + 1)!}}$$
Svare
(A)
Arrange the men in m! ways, then choose n spaces from m+1 spaces and arrange women in n! ways, giving m! * (m+1)Cn * n! = m! * (m+1)! / (m-n+1)!
24
Use mathematical Induction to prove : If $$n$$ is any odd positive integer, then $$n\left( {{n^2} - 1} \right)$$ is divisible by 24.
Svare
(A)
Base Case: n=1, 1(1^2-1) = 0, divisible by 24. Inductive Step: Assume k(k^2-1) is divisible by 24 for odd k. Then (k+2)((k+2)^2-1) = (k+2)(k^2+4k+3) = k^3 + 6k^2 + 11k + 6 = (k^3-k) + 6k^2 + 12k + 6 = k(k^2-1) + 6(k^2 + 2k + 1) = k(k^2-1) + 6(k+1)^2. Since k is odd, k+1 is even, so (k+1)^2 is divisible by 4. Thus 6(k+1)^2 is divisible by 24. Therefore, (k+2)((k+2)^2-1) is divisible by 24.
25
If $${\left( {1 + x} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ..... + {C_n}{x^n}$$ then show that the sum of the products of the $${C_i}s$$ taken two at a time, represented $$\sum\limits_{0 \le i < j \le n} {\sum {{C_i}{C_j}} } $$ is equal to $${2^{2n - 1}} - {{\left( {2n} \right)!} \over {2{{\left( {n!} \right)}^2}}}$$
Svare
(A)
2^(2n-1) - (2n)! / (2 * (n!)^2)
26
The coefficient of $${x^4}$$ in $${\left( {{x \over 2} - {3 \over {{x^2}}}} \right)^{10}}$$ is
Svare
(A)
$${{{405} \over {256}}}$$
27
The vertices of a triangle are
$$\left[ {a{t_1}{t_2},\,\,a\left( {{t_1} + {t_2}} \right)} \right],\,\,\left[ {a{t_2}{t_3},a\left( {{t_2} + {t_3}} \right)} \right],\,\,\left[ {a{t_3}{t_1},\,a\left( {{t_3} + {t_1}} \right)} \right]$$. Find the orthocentre of the triangle.
Svare
(D)
$$\left( { - a,a\left( {{t_1} + {t_2} + {t_3}} \right) + a{t_1}{t_2}{t_3}} \right)$$
28
If $${\left( {1 + ax} \right)^n} = 1 + 8x + 24{x^2} + .....$$ then $$a=..........$$ and $$n =............$$
Svare
(D)
a = 2, n = 4
29
The equation $$2{x^2} + 3x + 1 = 0$$ has an irrational root.
Svare
(B)
FALSE
30
Find all real values of $$x$$ which satisfy $${x^2} - 3x + 2 > 0$$ and $${x^2} - 2x - 4 \le 0$$
Svare
(A)
$$[-1, 1) \cup (2, 4]$$
31
If one root of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the $$n$$-th power of the other, then show that $$${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0$$$
Svare
B
C
E
32
Show that $$$16\cos \left( {{{2\pi } \over {15}}} \right)\cos \left( {{{4\pi } \over {15}}} \right)\cos \left( {{{8\pi } \over {15}}} \right)\cos \left( {{{16\pi } \over {15}}} \right) = 1$$$
Svare
(A)
1
33
Find all solutions of $$4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x$$
Svare
A
B
C
34
Prove that the complex numbers $${{z_1}}$$, $${{z_2}}$$ and the origin form an equilateral triangle only if $$z_1^2 + z_2^2 - {z_1}\,{z_2} = 0$$.
Svare
(D)
The statement is true and equivalent to (z1/z2) being a primitive sixth root of unity or z1 = z2 = 0.
35
The points z₁, z₂, z₃, z₄ in the complex plane are the vertices of a parallelogram taken in order if and only if
Svare
(B)
z₁ + z₃ = z₂ + z₄
36
If $$z = x + iy$$ and $$\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$$ then $$\,\left| \omega \right| = 1$$ implies that, in the complex plane,
Svare
(B)
$$z$$ lies on the real axis
37
The value of $$\tan \left[ {{{\cos }^{ - 1}}\left( {{4 \over 5}} \right) + {{\tan }^{ - 1}}\left( {{2 \over 3}} \right)} \right]$$ is
Svare
(D)
none
38
Show that $$1+x$$ $$In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $$ for all $$x \ge 0$$
Svare
(B)
The inequality is true for all x ≥ 0.
39
If $$y = a\,\,In\,x + b{x^2} + x$$ has its extreamum values at $$x=-1$$ and $$x=2$$, then
Svare
(B)
$$a = 2,b = - {1 \over 2}$$
40
The normal to the curve $$\,x = a\left( {\cos \theta + \theta \sin \theta } \right)$$, $$y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point $$'\theta '$$ is such that
Svare
(C)
it is at a constant distance from the origin
41
$$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then
Svare
(A)
the area of $$\Delta ABC$$ is maximum when it is isosceles
42
If $$x-r$$ is a factor of the polynomial $$f\left( x \right) = {a_n}{x^4} + ..... + {a_0},$$ repeated $$m$$ times $$\left( {1 < m \le n} \right)$$, then $$r$$ is a root of $$\left( x \right) = 0$$ repeated $$m$$ times.
Svare
(B)
FALSE
43
If $$a+b+c=0$$, then the quadratic equation $$3a{x^2} + 2bx + c = 0$$ has
Svare
(A)
at least one root in $$\left[ {0,1} \right]$$
44
The function $$y = 2{x^2} - In\,\left| x \right|$$ is monotonically increasing for values of $$x\left( {x \ne 0} \right)$$ satisfying the inequalities ......... and monotonically decreasing for values of $$x$$ satisfying the inequalities ............
Svare
(C)
$$x \in ( -\frac{1}{2}, 0 ) \cup ( \frac{1}{2}, \infty )$$; $$x \in ( -\infty, -\frac{1}{2} ) \cup ( 0, \frac{1}{2} )$$
45
The larger of $$\cos \left( {In\,\,\theta } \right)$$ and $$In $$ $$\left( {\cos \,\,\theta } \right)$$ If $${e^{ - \pi /2}} < \theta < {\pi \over 2}$$ is ..................
Svare
(A)
$$\cos \left( {In\,\,\theta } \right)$$
46
Find all the solution of $$4$$ $${\cos ^2}x\sin x - 2{\sin ^2}x = 3\sin x$$
Svare
A
B
C
47
If $$\tan \,A = \left( {1 - \cos B} \right)/\sin B,$$ then $$tan2A = tan\,B$$.
Svare
(B)
FALSE
48
The ex-radii $${r_1},{r_2},{r_3}$$ of $$\Delta $$$$ABC$$ are H.P. Show that its sides $$a, b, c$$ are in A.P.
Svare
(A)
Sides a, b, c are in Arithmetic Progression (A.P.)
49
From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is $${15^ \circ }$$. The distance of the boat from the foot of the light house is
Svare
(B)
$$\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right)60$$ metres
50
The derivative of an even function is always an odd function.
Svare
(A)
TRUE
51
Through a fixed point (h, k) secants are drawn to the circle $$\,{x^2}\, + \,{y^2} = \,{r^2}$$. Show that the locus of the mid-points of the secants intercepted by the circle is $$\,{x^2}\, + \,{y^2} $$ = $$hx + ky$$.
Svare
B
C
D
52
The equation of the circle passing through (1, 1) and the points of intersection of $${x^2} + {y^2} + 13x - 3y = 0$$ and $$2{x^2} + 2{y^2} + 4x - 7y - 25 = 0$$ is
Svare
(B)
$$4{x^2} + 4{y^2} + 30x - 13y - 25 = 0$$
53
The centre of the circle passing through the point (0, 1) and touching the curve $$\,y = {x^2}$$ at (2, 4) is
Svare
(C)
$$\left( {{{ - 16} \over 5},{{53} \over {10}}} \right)$$
54
The point of intersection of the line 4x - 3y - 10 = 0 and the circle $${x^2} + {y^2} - 2x + 4y - 20 = 0$$ are ........................and ...................
Svare
A
B
55
The end $$A, B$$ of a straight line segment of constant length $$c$$ slide upon the fixed rectangular axes $$OX, OY$$ respectively. If the rectangle $$OAPB$$ be completed, then show that the locus of the foot of the perpendicular drawn from $$P$$ to $$AB$$ is $${x^{{2 \over 3}}} + {y^{{2 \over 3}}} = {c^{{2 \over 3}}}$$
Svare
(A)
The locus of the foot of the perpendicular drawn from P to AB is x^(2/3) + y^(2/3) = c^(2/3)
56
The coordinates of $$A, B, C$$ are $$(6, 3), (-3, 5), (4, -2)$$ respectively, and $$P$$ is any point $$(x, y)$$. Show that the ratio of the area of the triangles $$\Delta $$ $$PBC$$ and $$\Delta $$$$ABC$$ is $$\left| {{{x + y - 2} \over 7}} \right|$$
Svare
(A)
The problem asks to prove a relationship between the areas of two triangles, PBC and ABC, given the coordinates of their vertices.