JEE MAIN - Mathematics (2008)

1
The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?
Одговорити
(D)
a = 3, b = 4
2
Let $$f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr } } \right.$$

Then which one of the following is true?
Одговорити
(C)
$$f$$ is differentiable at x = 0 but not at x = 1
3
Let $$f:N \to Y$$ be a function defined as f(x) = 4x + 3 where
Y = { y $$ \in $$ N, y = 4x + 3 for some x $$ \in $$ N }.
Show that f is invertible and its inverse is
Одговорити
(D)
$$g\left( y \right) = {{y - 3} \over 4}$$
4
It is given that the events $$A$$ and $$B$$ are such that
$$P\left( A \right) = {1 \over 4},P\left( {A|B} \right) = {1 \over 2}$$ and $$P\left( {B|A} \right) = {2 \over 3}.$$ Then $$P(B)$$ is :
Одговорити
(B)
$${1 \over 3}$$
5
It is given that the events $$A$$ and $$B$$ are such that
$$P\left( A \right) = {1 \over 4},P\left( {A|B} \right) = {1 \over 2}$$ and $$P\left( {B|A} \right) = {2 \over 3}.$$ Then $$P(B)$$ is :
Одговорити
(B)
$${1 \over 3}$$
6
A die is thrown. Let $$A$$ be the event that the number obtained is greater than $$3.$$ Let $$B$$ be the event that the number obtained is less than $$5.$$ Then $$P\left( {A \cup B} \right)$$ is :
Одговорити
(C)
$$1$$
7
The solution of the differential equation

$${{dy} \over {dx}} = {{x + y} \over x}$$ satisfying the condition $$y(1)=1$$ is :
Одговорити
(D)
$$y = x\,\ln x + x$$
8
The area of the plane region bounded by the curves $$x + 2{y^2} = 0$$ and $$\,x + 3{y^2} = 1$$ is equal to :
Одговорити
(D)
$${4 \over 3}$$
9
Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x=cy+bz,$$ $$y=az+cx,$$ and $$z=bx+ay.$$ Then $${a^2} + {b^2} + {c^2} + 2abc$$ is equal to :
Одговорити
(D)
$$1$$
10
Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal entries of $$a$$. Assume that $${a^2} = I.$$
Statement-1 : If $$A \ne I$$ and $$A \ne - I$$, then det$$(A)=-1$$
Statement- 2 : If $$A \ne I$$ and $$A \ne - I$$, then tr $$(A)$$ $$ \ne 0$$.
Одговорити
(D)
statement - 1 is true, statement - 2 is false.
11
How many real solutions does the equation
$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?
Одговорити
(B)
$$1$$
12
Suppose the cubic $${x^3} - px + q$$ has three distinct real roots
where $$p>0$$ and $$q>0$$. Then which one of the following holds?
Одговорити
(A)
The cubic has minima at $$\sqrt {{p \over 3}} $$ and maxima at $$-\sqrt {{p \over 3}} $$
13
The value of $$cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right)$$ is :
Одговорити
(A)
$${{6 \over 17}}$$
14
The non-zero vectors are $${\overrightarrow a ,\overrightarrow b }$$ and $${\overrightarrow c }$$ are related by $${\overrightarrow a = 8\overrightarrow b }$$ and $${\overrightarrow c = - 7\overrightarrow b \,\,.}$$ Then the angle between $${\overrightarrow a }$$ and $${\overrightarrow c }$$ is :
Одговорити
(D)
$$\pi $$
15
A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at :
Одговорити
(B)
$$(1,0)$$
16
A focus of an ellipse is at the origin. The directrix is the line $$x=4$$ and the eccentricity is $${{1 \over 2}}$$. Then the length of the semi-major axis is :
Одговорити
(A)
$${{8 \over 3}}$$
17
The point diametrically opposite to the point $$P(1, 0)$$ on the circle $${x^2} + {y^2} + 2x + 4y - 3 = 0$$ is :
Одговорити
(C)
$$(-3, -4)$$
18
The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is :
Одговорити
(D)
-4
19
The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is
Одговорити
(B)
- 12
20
How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
Одговорити
(D)
$$7.{}^6{C_4}.{}^8{C_4}$$
21
In a shop there are five types of ice-cream available. A child buys six ice-cream.
Statement - 1: The number of different ways the child can buy the six ice-cream is $${}^{10}{C_5}$$.
Statement - 2: The number of different ways the child can buy the six ice-cream is equal to the number of different ways of arranging 6 A and 4 B's in a row.
Одговорити
(A)
Statement - 1 is false, Statement - 2 is true
22
The quadratic equations $${x^2} - 6x + a = 0$$ and $${x^2} - cx + 6 = 0$$ have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is
Одговорити
(D)
2
23
STATEMENT - 1 : For every natural number $$n \ge 2,$$ $$${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n .$$$
STATEMENT - 2 : For every natural number $$n \ge 2,$$, $$$\sqrt {n\left( {n + 1} \right)} < n + 1.$$$
Одговорити
(B)
Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for statement - 1
24
Let R be the real line. Consider the following subsets of the plane $$R \times R$$ :
$$S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right\}$$
$$T = \left\{ {(x,y): x - y\,\,\,is\,\,an\,\,{\mathop{\rm int}} eger\,} \right\}$$,
Which one of the following is true ?
Одговорити
(D)
T is an equivalence relation on R but S is not
25
The conjugate of a complex number is $${1 \over {i - 1}}$$ then that complex number is :
Одговорити
(C)
$${{ - 1} \over {i + 1}}$$
26
If the straight lines $$\,\,\,\,\,$$ $$\,\,\,\,\,$$ $${{x - 1} \over k} = {{y - 2} \over 2} = {{z - 3} \over 3}$$ $$\,\,\,\,\,$$ and$$\,\,\,\,\,$$ $${{x - 2} \over 3} = {{y - 3} \over k} = {{z - 1} \over 2}$$ intersects at a point, then the integer $$k$$ is equal to
Одговорити
(A)
$$-5$$
27
The line passing through the points $$(5,1,a)$$ and $$(3, b, 1)$$ crosses the $$yz$$-plane at the point $$\left( {0,{{17} \over 2}, - {{ - 13} \over 2}} \right)$$ . Then
Одговорити
(C)
$$a=6,$$ $$b=4$$