JEE Advance - Mathematics (1982)

1
Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx} $$
Хариулт
(C)
$$\frac{3}{\pi} + \frac{1}{\pi^2}$$
2
Show that $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.} $$
Хариулт
(B)
Let $I = int_0^pi xf(sin x) dx$. Substitute $u = pi - x$, then $I = int_0^pi (pi - u)f(sin(pi - u)) du = int_0^pi (pi - x)f(sin x) dx = piint_0^pi f(sin x) dx - int_0^pi xf(sin x) dx = piint_0^pi f(sin x) dx - I$. Hence $2I = piint_0^pi f(sin x) dx$, so $I = rac{pi}{2}int_0^pi f(sin x) dx$.
3
For any real $$t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}$$ is a point on the
hyperbola $${x^2} - {y^2} = 1$$. Show that the area bounded by this hyperbola and the lines joining its centre to the points corresponding to $${t_1}$$ and $$-{t_1}$$ is $${t_1}$$.
Хариулт
A
C
D
4
If $$A$$ and $$B$$ are two events such that $$P\left( A \right) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left( {{{\overline A } \over {\overline B }}} \right)$$ is equal to
Хариулт
(C)
$${{1 - P\left( {A \cup B} \right)} \over {P\left( {\overline B } \right)}}$$ (Here $$\overline A $$ and $$\overline B $$ are complements of $$A$$ and $$B$$ respectively).
5
$$A$$ and $$B$$ are two candidates seeking admission in $$IIT.$$ The probability that $$A$$ is selected is $$0.5$$ and the probability that both $$A$$ and $$B$$ are selected is atmost $$0.3$$. Is it possible that the probability of $$B$$ getting selected is $$0.9$$ ?
Хариулт
(C)
No, because P(A ∩ B) ≤ P(A) and P(A ∩ B) ≤ P(B), therefore P(B) cannot be 0.9.
6
For non-zero vectors $${\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },$$ $$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right).\overrightarrow c } \right| = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|$$ holds if and only if
Хариулт
(D)
$$\overrightarrow a \,.\,\overrightarrow b = \overrightarrow b \,.\,\overrightarrow c = \overrightarrow c \,.\,\overrightarrow a = 0$$
7
$${A_1},{A_2},.................{A_n}$$ are the vertices of a regular plane polygon with $$n$$ sides and $$O$$ is its centre. Show that
$$\sum\limits_{i = 1}^{n - 1} {\left( {\overrightarrow {O{A_i}} \times {{\overrightarrow {OA} }_{i + 1}}} \right) = \left( {1 - n} \right)\left( {{{\overrightarrow {OA} }_2} \times {{\overrightarrow {OA} }_1}} \right)} $$
Хариулт
(B)
The summation can be simplified using vector properties and the regularity of the polygon.
8
Find all values of $$\lambda $$ such that $$x, y, z,$$$$\, \ne $$$$(0,0,0)$$ and
$$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow k } \right)x + \left( {3\overrightarrow i - 3\overrightarrow j + \overrightarrow k } \right)y + \left( { - 4\overrightarrow i + 5\overrightarrow j } \right)z$$
$$ = \lambda \left( {x\overrightarrow i \times \overrightarrow j \,\,y + \overrightarrow k \,z} \right)$$ where $$\overrightarrow i ,\,\,\overrightarrow j ,\,\,\overrightarrow k $$ are unit vectors along the coordinate axes.
Хариулт
A
B
9
The area bounded by the curves $$y=f(x)$$, the $$x$$-axis and the ordinates $$x=1$$ and $$x=b$$ is $$(b-1)$$ sin $$(3b+4)$$. Then $$f(x)$$ is
Хариулт
(C)
$$\sin \left( {3x + 4} \right) + 3\left( {x - 1} \right)\cos \left( {3x + 4} \right)$$
10
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the numbers of words which have at least one letter repeated are
Хариулт
(A)
69760
11
The inequality |z-4| < |z-2| represents the region given by
Хариулт
(D)
none of these
12
Without using tables, prove that $$\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right) = {1 \over 8}.$$
Хариулт
(B)
Using trigonometric identities, we can show that (sin 12°)(sin 48°)(sin 54°) = 1/8 is correct
13
$$mn$$ squares of equal size are arranged to from a rectangle of dimension $$m$$ by $$n$$, where $$m$$ and $$n$$ are natural numbers. Two squares will be called ' neighbours ' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares.Show that this is possible only if all the numbers used are equal.
Хариулт
(B)
If the numbers are not all equal, a contradiction arises from considering the maximum value.
14
Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution.
Хариулт
(A)
The function $$f(x) = e^{\sin x} - e^{-\sin x} - 4$$ is always greater than 0, thus no real solution exists.
15
The coeffcient of $${x^{99}}$$ in the polynomial (x -1) (x - 2)...(x - 100) is ..............
Хариулт
(B)
-5050
16
If $$2 + i\sqrt 3 $$ is root of the equation $${x^2} + px + q = 0$$, where p and q are real, then (p, q) = (..........,....................).
Хариулт
(B)
(-4, 7)
17
The number of real solutions of the equation $${\left| x \right|^2} - 3\left| x \right| + 2 = 0$$ is
Хариулт
(A)
4
18
Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built at
Хариулт
(C)
town A
19
If p, q, r are any real numbers, then
Хариулт
(B)
min (p, q) = $${1 \over 2}\left( {p + q - \left| {p - q} \right|} \right)$$
20
The largest interval for which $${x^{12}} - {x^9} + {x^4} - x + 1 > 0$$ is
Хариулт
(D)
$$ - \infty < x < \infty $$
21
The larger of $${99^{50}} + {100^{50}}$$ and $${101^{50}}$$ is ..............
Хариулт
(B)
${101^{50}}$
22
The sum of the coefficients of the plynomial $${\left( {1 + x - 3{x^2}} \right)^{2163}}$$ is ...............
Хариулт
(A)
-1
23
Prove that $${7^{2n}} + \left( {{2^{3n - 3}}} \right)\left( {3n - 1} \right)$$ is divisible by 25 for any natural number $$n$$.
Хариулт
(C)
The statement is true for all natural numbers n. Use induction with base case n=1 and showing divisibility by 25 for the inductive step.
24
In a certain test, $${a_i}$$ students gave wrong answers to atleast i questions, where i = 1, 2,..., k. No student gave more than k wrong answers. The total number of wrong answers given is.....................................
Хариулт
(C)
$$\sum_{i=1}^{k} a_i$$
25
If $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5},$$ then
Хариулт
(B)
$${\rm I}m\left( z \right) = 0$$
26
Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4; and then the men select the chairs from amongst the remaining. The number of possible arrangements is
Хариулт
(D)
none of these
27
The value of the expression $$\,{}^{47}{C_4} + \sum\limits_{j = 1}^5 {^{52 - j}\,{C_3}} $$ is equal to
Хариулт
(C)
$${}^{52}{C_4}$$
28
The third term of a geometric progression is 4. The product of the first five terms is
Хариулт
(B)
4⁵
29
If $$x,\,y$$ and $$z$$ are $$pth$$, $$qth$$ and $$rth$$ terms respectively of an A.P. and also of a G.P., then $${x^{y - z}}\,{y^{z - x}}\,{z^{x - y}}$$ is equal to :
Хариулт
(C)
$$1$$
30
Does there exist a geometric progression containing $$27, 8$$ and $$12$$ as three of its terms? If it exits, how many such progressions are possible ?
Хариулт
(E)
Yes, and there are infinitely many such geometric progressions.
31
$$y = {10^x}$$ is the reflection of $${\log _{10}}\,x$$ in the line whose equation is ...........
Хариулт
(D)
y = x
32
The set of lines $$ax + by + c = 0,$$ where $$3a + 2b + 4c = 0$$ is concurrent at the point ..........
Хариулт
(A)
(3/4, 1/2)
33
If A and B are points in the plane such that PA/PB = k (constant) for all P on a given circle, then the value of k cannot be equal to ..........................................
Хариулт
(B)
1
34
$$A$$ is point on the parabola $${y^2} = 4ax$$. The normal at $$A$$ cuts the parabola again at point $$B$$. If $$AB$$ subtends a right angle at the vertex of the parabola. Find the slope of $$AB$$.
Хариулт
(C)
$$\pm \sqrt{2}$$
35
If $$y = f\left( {{{2x - 1} \over {{x^2} + 1}}} \right)$$ and $$f'\left( x \right) = \sin {x^2}$$, then $${{dy} \over {dx}} = ..........$$
Хариулт
(A)
$\frac{2 + 2x - 2x^2}{(x^2 + 1)^2} \sin \left( \frac{2x - 1}{x^2 + 1} \right)^2$
36
Let $$f$$ be a twice differentiable function such that
$$f''\left( x \right) = - f\left( x \right),$$ and $$f'\left( x \right) = g\left( x \right),h\left( x \right) = {\left[ {f\left( x \right)} \right]^2} + {\left[ {g\left( x \right)} \right]^2}$$

Find $$h\left( {10} \right)$$ if $$h(5)=11$$
Хариулт
(C)
11
37
A vertical pole stands at a point $$Q$$ on a horizontal ground. $$A$$ and $$B$$ are points on the ground, $$d$$ meters apart. The pole subtends angles $$\alpha $$ and $$\beta $$ at $$A$$ and $$B$$ respectively. $$AB$$ subtends an angle $$\gamma $$ and $$Q$$. Find the height of the pole.
Хариулт
(C)
$$\frac{d}{\sqrt{\cot^2 \alpha + \cot^2 \beta - \cot \alpha \cot \beta \cot \gamma}}$$
38
If $$f(x)$$ and $$g(x)$$ are differentiable function for $$0 \le x \le 1$$ such that $$f(0)=2$$, $$g(0)=0$$, $$f(1)=6$$; $$g(1)=2$$, then show that there exist $$c$$ satisfying $$0 < c < 1$$ and $$f'(c)=2g'(c)$$.
Хариулт
B
D
39
If $$a{x^2} + {b \over x} \ge c$$ for all positive $$x$$ where $$a>0$$ and $$b>0$$ show that $$27a{b^2} \ge 4{c^3}$$.
Хариулт
(D)
The inequality holds only if $$27ab^2 \ge 4c^3$$