JEE Advance - Mathematics (1988)

1
Urn $$A$$ contains $$6$$ red and $$4$$ black balls and urn $$B$$ contains $$4$$ red and $$6$$ black balls. One ball is drawn at random from urn $$A$$ and placed in urn $$B$$. The one ball is drawn at random from urn $$B$$ and placed in urn $$A$$. If one ball is now drawn at random from urn $$A$$, the probability that it is found to be red is ................
回答
(A)
32/55
2
One hundred identical coins, each with probability, $$p,$$ of showing up heads are tossed once. If $$0 < p < 1$$ and the probability of heads showing on $$50$$ coins is equal to that of heads showing on $$51$$ coins, then the value of $$p$$ is
回答
(D)
$$51/101.$$
3
For two given events $$A$$ and $$B,$$ $$P\left( {A \cap B} \right)$$
回答
B
C
A
4
A box contains $$2$$ fifty paise coins, $$5$$ twenty five paise coins and a certain fixed number $$N\,\,\left( { \ge 2} \right)$$ of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these $$5$$ coins is less than one rupee and fifty paise.
回答
(A)
$$\,1 - {{10left( {N + 2} ight)} over {\binom{N + 7}{5}}}$$
5
The components of a vector $$\overrightarrow a $$ along and perpendicular to a non-zero vector $$\overrightarrow b $$ are ......and .....respectively.
回答
(C)
$$\left( {\frac{{\overrightarrow {a,} .\,\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}}} \right)\overrightarrow b \,\,\text{and}\,\,\overrightarrow a - \left( {\frac{{\overrightarrow a \,.\,\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}}} \right)\overrightarrow b $$
6
Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\overrightarrow r,$$ are vectors defined by the relations $$\overrightarrow p = {{\overrightarrow b \times \overrightarrow c } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow q = {{\overrightarrow c \times \overrightarrow a } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow r = {{\overrightarrow a \times \overrightarrow b } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}$$ then the value of the expression $$\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right),\overrightarrow r $$ is equal to
回答
(D)
$$3$$
7
Let $$OA$$ $$CB$$ be a parallelogram with $$O$$ at the origin and $$OC$$ a diagonal. Let $$D$$ be the midpoint of $$OA.$$ Using vector methods prove that $$BD$$ and $$CO$$ intersect in the same ratio. Determine this ratio.
回答
(B)
2:1
8
Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $$
回答
(D)
$$\frac{1}{2} \left[\log 2 + \frac{\pi}{2} - 1\right]$$
9
If $$P=(1, 0),$$ $$Q=(-1, 0)$$ and $$R=(2, 0)$$ are three given points, then locus of the point $$S$$ satisfying the relation $$S{Q^2} + S{R^2} = 2S{P^2},$$ is
回答
(D)
a straight line parallel to y-axis.
10
$$\,{\left| {a{z_1} - b{z_2}} \right|^2} + {\left| {b{z_1} + a{z_2}} \right|^2} = .........$$
回答
(E)
$$\left( {{a^2} + {b^2}} \right)left( {{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}} \right)$$
11
The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.
回答
(B)
FALSE
12
$$\left| {\matrix{ {1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {1 + {{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {{{\cos }^2}\theta } & {1 + 4\sin 4\theta } \cr } } \right| = 0$$ are
回答
A
C
13
Solve $$\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$$
回答
A
D
14
Total number of ways in which six ' + ' and four ' - ' signs can be arranged in a line such that no two ' - ' signs occur together is.....................................
回答
(C)
35
15
There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is.......................
回答
(B)
9
16
Let $$R$$ $$ = {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}$$ and $$f = R - \left[ R \right],$$ where [ ] denotes the greatest integer function. Prove that $$Rf = {4^{2n + 4}}$$
回答
A
D
17
The sum of the first n terms of the series $${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........$$ is
$$n\,\,{\left( {n + 1} \right)^2}/2,$$ when $$n$$ is even. When $$n$$ is odd, the sum is .............
回答
(B)
$${{{n^2}left( {n + 1} ight)} \over 2}$$
18
Sum of the first n terms of the series $${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$$ is equal to
回答
(C)
$$n + {2^{ - n}} - 1$$
19
If the first and the $$(2n-1)$$st terms of an A.P., a G.P. and an H.P. are equal and their $$n$$-th terms are $$a,b$$ and $$c$$ respectively, then
回答
B
A
D
20
The lines $$2x + 3y + 19 = 0$$ and $$9x + 6y - 17 = 0$$ cut the coordinates axes in concyclic points.
回答
(B)
FALSE
21
The value of the expression $$\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}$$ is equal to
回答
(C)
4
22
Lines$${L_1} = ax + by + c = 0$$ and $${L_2} = lx + my + n = 0$$ intersect at the point $$P$$ and make an angle $$\theta $$ with each other. Find the equation of a line $$L$$ different from $${L_2}$$ which passes through $$P$$ and makes the same angle $$\theta $$ with $${L_1}$$.
回答
(D)
$$\left( {{a^2} + {b^2}} \right)\left( {\ell x + my + n} \right) - 2\left( {a\ell + bm} \right)left( {ax + by + c} \right) = 0$$
23
If the circle $${C_1}:{x^2} + {y^2} = 16$$ intersects another circle $${C_2}$$ of radius 5 in such a manner that common chord is of maximum lenght and has a slope equal to 3/4, then the coordinates of the centre of $${C_2}$$ are.............................
回答
(E)
(9/5, -12/5) or (-9/5, 12/5)
24
If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2}\, = \,{k^2}$$ orthogonally, then the equation of the locus of its centre is
回答
(A)
$$2\,ax\, + \,2\,by\, - \,({a^2}\, + \,{b^2}\, + \,\,{k^2})\, = \,0$$
25
The equations of the tangents drawn from the origin to the circle $${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$$, are
回答
A
C
26
If $${y^2} = P\left( x \right)$$, a polynomial of degree $$3$$, then $$2{d \over {dx}}\left( {{y^3}{{{d^2}y} \over {d{x^2}}}} \right)$$ equals
回答
(C)
$$P\left( x \right)P''\left( x \right)$$
27
If the angles of a triangle are $${30^ \circ }$$ and $${45^ \circ }$$ and the included side is $$\left( {\sqrt 3 + 1} \right)$$ cms, then the area of the triangle is ...............
回答
(B)
$$\frac{\sqrt{3} + 1}{2}$$
28
A sign -post in the form of an isosceles triangle $$ABC$$ is mounted on a pole of height $$h$$ fixed to the ground. The base $$BC$$ of the triangle is parallel to the ground. A man standing on the ground at a distance $$d$$ from the sign-post finds that the top vertex $$A$$ of the triangle subtends an angle $$\beta $$ and either of the other two vertices subtends the same angle $$\alpha $$ at his feet. Find the area of the triangle.
回答
(A)
(d tan β - h)√(h² cot² α - d²)
29
Investigate for maxima and minimum the function $$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$$
回答
B
C
30
Where [ ] denotes the greatest integer function, equals .............
回答
(C)
2 - √2
31
The value of the integral $$\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx} $$ is equal to $$a$$.
回答
(B)
FALSE
32
Find the area of the region bounded by the curve $$C:y=$$
$$\tan x,$$ tangent drawn to $$C$$ at $$x = {\pi \over 4}$$ and the $$x$$-axis.
回答
(D)
$$\frac{1}{2} [\log 2 - \frac{1}{2}]$$ sq. units