# Project #13176 - none

1. The equation esinx ? e?sinx ? 4 = 0 has :

(1) infinite number of real roots

(2) no real roots

(3) exactly one real root

(4) exactly four real roots

Ans. (2)

Sol. Let esinx = t

? t2 ? 4t ? 1 = 0

? t =

2

4 ? 16 ? 4

? t = esinx = 2 ± 5

? esin x = 2 ? 5 , esin x = 2 + 5

esin x???? 2 ? 5 < 0, ? sinx = ln(2 + 5 ) > 1

so rejected so rejected

hence no solution

2. Let a? and b ? be two unit vectors. If the vectors b ? 2 a? c ? ?

? and b ? 4 a? d ? 5 ?

?

are perpendicular to each other,

then the angle between a? and b ? is :

(1)

6

?

(2)

2

?

(3)

3

?

(4)

4

?

Ans. (3)

Sol. c

? = b ? 2 a? ?

d ?

= b ? 4 ? a? 5

c.d

? ?

= 0

? ) b ? 4 ? a? 5 ).( b ? 2 a? ( ? = 0 ? 5 + b ? . a? 6 ? 8 = 0

? b ? . a? =

2

1

? ? =

3

?

3. A spherical balloon is filled with 4500? cubic meters of helium gas. If a leak in the balloon causes the gas to

escape at the rate of 72? cubic meters per minute, then the rate (in meters per minute) at which the radius

of the balloon decreases 49 minutes after the leakage began is :

(1)

7

9

(2)

9

7

(3)

9

2

(4)

2

9

Ans. (3)

Sol. V =

3

4

?r3 4500 ? =

3

4?r3

dt

dV

= 4?r2 ?

?

?

??

?

dt

dr

45 × 25 × 3 = r3

r = 15 m                    4. Statement-1 : The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + .... + (361 + 380 + 400)

is 8000.

Statement-2 : ?

?

? ? ?

n

k 1

(k3 (k 1)3 ) n3 , for any natural number n.

(1) Statement-1 is false, Statement-2 is true.

(2) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.

(3) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.

(4) Statement-1 is true, statement-2 is false.

Ans. (2)

Sol. Tn = (n ? 1)2 + (n ? 1)n + n2 =

? ?

(n ? 1) ? n

(n ? 1)3 ? n3

=n3 ? (n ? 1)3

T1 = 13 ? 03

T2 = 23 ? 13

?

T20 = 203 ? 193

S20 = 203 ? 03 = 8000

5. The negation of the statement

?If I become a teacher, then I will open a school?, is :

(1) I will become a teacher and I will not open a school.

(2) Either I will not become a teacher or I will not open a school.

(3) Neither I will become a teacher nor I will open a school

(4) I will not become a teacher or I will open a school.

Ans. (1)

Sol. Let p : I become a teacher

q : I will open a school

Negation of p ? q is ~ (p ??q) = p ^ ~ q

i.e. I will become a teacher and I will not open a school.

6. If the integral ? tan x ? 2

5 tan x

dx = x + a ?n |sin x ? 2 cos x| + k, then a is equal to :

(1) ? 1 (2) ? 2 (3) 1 (4) 2

Ans. (4)

Sol. dx

tan x ? 2

5tan x ? = dx

sin x ? 2cos x

5sin x ? = dx

 Subject Mathematics Due By (Pacific Time) 09/24/2013 12:00 am
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