JEE Advance - Mathematics (1982)

1
Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx} $$
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(C)
$$\frac{3}{\pi} + \frac{1}{\pi^2}$$
2
Show that $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.} $$
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(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 $${t_1}$$.
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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
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(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 $$0.9$$ ?
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(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
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(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
$$\sum\limits_{i = 1}^{n - 1} {\left( {\overrightarrow {O{A_i}} \times {{\overrightarrow {OA} }_{i + 1}}} \right) = \left( {1 - n} \right)\left( {{{\overrightarrow {OA} }_2} \times {{\overrightarrow {OA} }_1}} \right)} $$
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(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.
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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
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(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
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(A)
69760
11
The inequality |z-4| < |z-2| represents the region given by
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(D)
none of these
12
Without using tables, prove that $$\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right) = {1 \over 8}.$$
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(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.
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(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.
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(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 ..............
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(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) = (..........,....................).
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(B)
(-4, 7)
17
The number of real solutions of the equation $${\left| x \right|^2} - 3\left| x \right| + 2 = 0$$ is
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(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
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(C)
town A
19
If p, q, r are any real numbers, then
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(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
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(D)
$$ - \infty < x < \infty $$
21
The larger of $${99^{50}} + {100^{50}}$$ and $${101^{50}}$$ is ..............
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(B)
${101^{50}}$
22
The sum of the coefficients of the plynomial $${\left( {1 + x - 3{x^2}} \right)^{2163}}$$ is ...............
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(A)
-1
23
Prove that $${7^{2n}} + \left( {{2^{3n - 3}}} \right)\left( {3n - 1} \right)$$ is divisible by 25 for any natural number $$n$$.
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(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.....................................
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(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
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(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
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(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
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(C)
$${}^{52}{C_4}$$
28
The third term of a geometric progression is 4. The product of the first five terms is
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(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 :
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(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 ?
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(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 ...........
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(D)
y = x
32
The set of lines $$ax + by + c = 0,$$ where $$3a + 2b + 4c = 0$$ is concurrent at the point ..........
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(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 ..........................................
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(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 $$AB$$.
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(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 $${{dy} \over {dx}} = ..........$$
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(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 $$h(5)=11$$
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(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.
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(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 $$f'(c)=2g'(c)$$.
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B
D
39
If $$a{x^2} + {b \over x} \ge c$$ for all positive $$x$$ where $$a>0$$ and $$b>0$$ show that $$27a{b^2} \ge 4{c^3}$$.
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(D)
The inequality holds only if $$27ab^2 \ge 4c^3$$