JEE Advance - Mathematics (1994)

1
The circle $${x^2} + {y^2} = 1$$ cuts the $$x$$-axis at $$P$$ and $$Q$$. Another circle with centre at $$Q$$ and variable radius intersects the first circle at $$R$$ above the $$x$$-axis and the line segment $$PQ$$ at $$S$$. Find the maximum area of the triangle $$QSR$$.
Odpoveď
(D)
$$\frac{4\sqrt{3}}{9}$$ sq. units
2
Find the indefinite integral $$\,\int {\cos 2\theta {\mkern 1mu} ln\left( {{{\cos \theta + \sin \theta } \over {\cos \theta - \sin \theta }}} \right)} {\mkern 1mu} d\theta $$
Odpoveď
(A)
$$\frac{\sin 2\theta}{2} ln\left(\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}\right) - \frac{1}{2} ln(\sec 2\theta) + C$$
3
The value of $$\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$$ is ...........
Odpoveď
(C)
1/2
4
Show that $$\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v} $$ where $$n$$ is a positive integer and $$\,0 \le v < \pi .$$
Odpoveď
(C)
The integral of |sin(x)| from 0 to nπ + v is 2n + 1 - cos(v).
5
In what ratio does the $$x$$-axis divide the area of the region
bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {x^2} - x?$$
Odpoveď
(A)
121:4
6
Find the equation of such a curve passing through $$(0,k).$$
Odpoveď
(D)
x = \pm \int{\frac{dy}{\sqrt{k^2 - y^2}}} + C
7
If two events $$A$$ and $$B$$ are such that $$P\,\,\left( {{A^c}} \right)\,\, = \,\,0.3,\,\,P\left( B \right) = 0.4$$ and $$P\left( {A \cap {B^c}} \right) = 0.5,$$ then $$P\left( {B/\left( {A \cup {B^c}} \right)} \right.$$$$\left. \, \right] = $$ ............
Odpoveď
(A)
1/4
8
Let $$A, B, C$$ be three mutually independent events. Consider the two statements $${S_1}$$ and $${S_2}$$
$${S_1}\,:\,A$$ and $$B \cup C$$ are independent
$${S_2}\,:\,A$$ and $$B \cap C$$ are independent
Then,
Odpoveď
(A)
Both $${S_1}$$ and $${S_2}$$ are true
9
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $$2, 3,4,.....12$$ is picked and the number on the card is noted. What is the probability that the noted number is either $$7$$ or $$8$$?
Odpoveď
(B)
0.2436
10
A unit vector perpendicular to the plane determined by the points $$P\left( {1, - 1,2} \right)Q\left( {2,0, - 1} \right)$$ and $$R\left( {0,2,1} \right)$$ is ............
Odpoveď
(B)
$$ -{{2\widehat i + \widehat j + \widehat k} \over {\sqrt 6 }} $$
11
Let $$\overrightarrow p $$ and $$\overrightarrow q $$ be the position vectors of $$P$$ and $$Q$$ respectively, with respect to $$O$$ and $$\left| {\overrightarrow p } \right| = p,\left| {\overrightarrow q } \right| = q.$$ The points $$R$$ and $$S$$ divide $$PQ$$ internally and externally in the ratio $$2:3$$ respectively. If $$OR$$ and $$OS$$ are perpendicular then
Odpoveď
(A)
$$9{q^2} = 4{q^2}$$
12
Let $$\alpha ,\beta ,\gamma $$ be distinct real numbers. The points with position
vectors $$\alpha \widehat i + \beta \widehat j + \gamma \widehat k,\,\,\beta \widehat i + \gamma \widehat j + \alpha \widehat k,\,\,\gamma \widehat i + \alpha \widehat j + \beta \widehat k$$
Odpoveď
(B)
form an equilateral triangle
13
The vector $$\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$$ is
Odpoveď
A
C
D
14
If the vectors $$\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,$$ are not coplanar, then prove that the vector
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) + \left( {\overrightarrow a \times \overrightarrow c } \right) \times \left( {\overrightarrow d \times \overrightarrow b } \right) + \left( {\overrightarrow a \times \overrightarrow d } \right) \times \left( {\overrightarrow b \times \overrightarrow c } \right)$$ is parallel to $$\overrightarrow a .$$
Odpoveď
(B)
The vector is parallel to $$\overrightarrow a $$.
15
Let $$n$$ be a positive integer such that $$\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.$$ Then
Odpoveď
(D)
$$4 < n < 8$$
16
Which one of the following curves cut the parabola $${y^2} = 4ax$$ at right angles?
Odpoveď
(D)
$${x^2} = 4ay$$
17
Let $$0 < x < {\pi \over 4}$$ then $$\left( {\sec 2x - \tan 2x} \right)$$ equals
Odpoveď
(B)
$$\tan \left[ {{\pi \over 4} - x} \right]$$
18
If $$\omega \,$$ is an imaginary cube root of unity then the value of $$\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}$$ is
Odpoveď
(C)
$${1 \over {\sqrt 2 }}$$
19
Suppose Z₁, Z₂, Z₃ are the vertices of an equilateral triangle inscribed in the circle $$\left| Z \right| = 2.$$ If Z₁ = $$1 + i\sqrt 3 $$ then Z2 = ......., Z₃ =..............
Odpoveď
(A)
$$ - 2,,,1 - i\sqrt 3 $$
20
Let $$2{\sin ^2}x + 3\sin x - 2 > 0$$ and $${x^2} - x - 2 < 0$$ ($$x$$ is measured in radians). Then $$x$$ lies in the interval
Odpoveď
(D)
$$\left( {{\pi \over 6},\,2} \right)$$
21
The number of points of intersection of two curves y = 2 sin x and y $$ = 5{x^2} + 2x + 3$$ is
Odpoveď
(A)
0
22
If p, q, r are + ve and are on A.P., the roots of quadratic equation $$p{x^2} + qx + r = 0$$ are all real for
Odpoveď
(B)
$$\left| {{p \over r} - 7} \right| \ge 4\sqrt 3 $$
23
Let $$p,q \in \left\{ {1,2,3,4} \right\}\,$$. The number of equations of the form $$p{x^2} + qx + 1 = 0$$ having real roots is
Odpoveď
(C)
7
24
Let $$n$$ be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of $${\left( {1 + x} \right)^n}$$ are in A.P., then the value of $$n$$ is ................
Odpoveď
(C)
7
25
If $$x$$ is not an integral multiple of $$2\pi $$ use mathematical induction to prove that : $$$\cos x + \cos 2x + .......... + \cos nx = \cos {{n + 1} \over 2}x\sin {{nx} \over 2}\cos ec{x \over 2}$$$
Odpoveď
(C)
The statement can be proven using mathematical induction.
26
Let $$n$$ be a positive integer and $${\left( {1 + x + {x^2}} \right)^n} = {a_0} + {a_1}x + ............ + {a_{2n}}{x^{2n}}$$
Show that $$a_0^2 - a_1^2 + a_2^2...... + {a_{2n}}{}^2 = {a_n}$$
Odpoveď
(E)
None of the above
27
A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to included in a committee? In how many of these committees? In how may of these committees
(a) The women are in majority?
(b) The men are in majority?
Odpoveď
B
C
D
28
If $$In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)$$ are in A.P., then
Odpoveď
(D)
$$a,\,b,\,c$$ are in H.P.
29
The locus of a variable point whose distance from $$\left( { - 2,\,0} \right)$$ is $$2/3$$ times its distance from the line $$x = - {9 \over 2}$$ is
Odpoveď
(A)
ellipse -
30
The equations to a pair of opposites sides of parallelogram are $${x^2} - 5x + 6 = 0$$ and $${y^2} - 6y + 5 = 0,$$ the equations to its diagonals are
Odpoveď
(C)
$$4x + y = 13,\,y = 4x - 7$$
31
The circles $${x^2} + {y^2} - 10x + 16 = 0$$ and $${x^2} + {y^2} = {r^2}$$ intersect each other in two distinct points if
Odpoveď
(C)
2 < r < 8
32
The equation $$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$ represents
Odpoveď
(C)
a point if $$k = 0$$
33
The curve $$y = a{x^3} + b{x^2} + cx + 5$$, touches the $$x$$-axis at $$P(-2, 0)$$ and cuts the $$y$$ axis at a point $$Q$$, where its gradient is $$3$$. Find $$a, b, c$$.
Odpoveď
(A)
a = -1/2, b = -3/4, c = 3
34
The function defined by $$f\left( x \right) = \left( {x + 2} \right){e^{ - x}}$$
Odpoveď
(D)
decreasing in $$\left( { - 1,\infty } \right)$$ and increasing in $$\left( { - \infty , - 1} \right)$$
35
Let $$P$$ be a variable point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ with foci $${F_1}$$ and $${F_2}$$. If $$A$$ is the area of the triangle $$P{F_1}{F_2}$$ then the maximum value of $$A$$ is ..........
Odpoveď
(C)
$$abc$$
36
Let $$C$$ be the curve $${y^3} - 3xy + 2 = 0$$. If $$H$$ is the set of points on the curve $$C$$ where the tangent is horizontal and $$V$$ is the set of the point on the curve $$C$$ where the tangent is vertical then $$H=$$.............. and $$V=$$ .................
Odpoveď
(A)
H = φ, V = {(1,1)}
37
If we consider only the principle values of the inverse trigonometric functions then the value of
$$\tan \left( {{{\cos }^{ - 1}}{1 \over {5\sqrt 2 }} - {{\sin }^{ - 1}}{4 \over {\sqrt {17} }}} \right)$$ is
Odpoveď
(D)
$${3 \over {29}}$$
38
Consider the following statements connecting a triangle $$ABC$$
(i) The sides $$a, b, c$$ and area $$\Delta $$ are rational.

(ii) $$a,\tan {B \over 2},\tan {c \over 2}$$ are rational.

(iii) $$a,\sin A,\sin B,\sin C$$ are rational.
Prove that $$\left( i \right) \Rightarrow \left( {ii} \right) \Rightarrow \left( {iii} \right) \Rightarrow \left( i \right)$$
Odpoveď
A
B
C
D
39
Let $${A_1},{A_2},........,{A_n}$$ be the vertices of an $$n$$-sided regular polygon such that $${1 \over {{A_1}{A_2}}} = {1 \over {{A_1}{A_3}}} + {1 \over {{A_1}{A_4}}}$$, Find the value of $$n$$.
Odpoveď
(C)
7
40
A tower $$AB$$ leans towards west making an angle $$\alpha $$ with the vertical. The angular elevation of $$B$$, the topmost point of the tower is $$\beta $$ as observed from a point $$C$$ due west of $$A$$ at a distance $$d$$ from $$A$$. If the angular elevation of $$B$$ from a point $$D$$ due east of $$C$$ at a distance $$2d$$ from $$C$$ is $$\gamma $$, then prove that $$2$$ tan $$\alpha = - \cot \beta + \cot \gamma $$.
Odpoveď
(A)
The problem can be solved using trigonometric relationships in right-angled triangles.
41
If the lengths of the sides of triangle are $$3, 5, 7$$ then the largest angle of the triangle is
Odpoveď
(C)
$${2\pi \over 3}$$
42
A circle is inscribed in an equilateral triangle of side $$a$$. The area of any square inscribed in this circle is ..............
Odpoveď
(D)
$${{{a^2}} over 6}$$ sq. unit
43
In a triangle $$ABC$$, $$AD$$ is the altitude from $$A$$. Given $$b>c$$, $$\angle C = {23^ \circ }$$ and $$AD = {{abc} \over {{b^2} - {c^2}}}$$ then $$\angle B = $$.................
Odpoveď
(C)
113°
44
If $$y = {\left( {\sin x} \right)^{\tan x}},$$ then $${{dy} \over {dx}}$$ is equal to
Odpoveď
(A)
$${\left( {\sin x} \right)^{\tan x}}\left( {1 + {{\sec }^2}x\,\log \,\sin \,x} \right)$$
45
Through the vertex $$O$$ of parabola $${y^2} = 4x$$, chords $$OP$$ and $$OQ$$ are drawn at right angles to one another . Show that for all positions of $$P$$, $$PQ$$ cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of $$PQ$$.
Odpoveď
(A)
The line PQ always passes through the point (4, 0), and the locus of the midpoint of PQ is y^2 = 2(x - 4).
46
Let $$E$$ be the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ and $$C$$ be the circle $${x^2} + {y^2} = 9$$. Let $$P$$ and $$Q$$ be the points $$(1, 2)$$ and $$(2, 1)$$ respectively. Then
Odpoveď
(D)
$$P$$ lies inside $$C$$ but outside $$E$$
47
The point of intersection of the tangents at the ends of the latus rectum of the parabola $${y^2} = 4x$$ is ...... .
Odpoveď
(C)
(-1, 0)