aieee 2012 question paper and solution
TRANSCRIPT
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7/31/2019 AIEEE 2012 Question Paper and Solution
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(Pg. 1)
Questions and Solutions
PAPER - 1 : MATHEMATICS, PHYSICS & CHEMISTRY
PART- A : MATHEMATICS
1. The equation sinx sinxe e 4 0 has :
(1) infinite number of real roots (2) no real roots
(3) exactly one real root (4) exactly four real roots
1. (2)
esin x
e-sin x
4 = 0
Let esin x = t
t1
t4 = 0
t2
4t 1 = 0
t =4 16 4
2
t = 2 5
esin x
= 2 + 5 Not possible [2.7 < e < 2.8]
esin x
= 2 5 Not possible [Never Negative]
2. Let a and b be two unit vectors. If the vectors c a 2b and d 5a 4b are perpendicular
to each other, then the angle between a and b is :
(1)6
(2)2
(3)3
(4)4
2. (3)
c d = 0
(a 2b) (5a 4b) = 0
5 6a b 8 = 0
a b =1
2
a b cos =1
2
cos =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) 9/7 (2) 7/9 (3) 2/9 (4) 9/2
Test Booklet
Code - C
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AIEEE 2012 Paper and Solution (2)
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3. (3)
34R
3= 4500
R3
= 3 1125 R = 15m
dv
dt= 2
4 dr3 r
3 dt= 72
drdt
= 218r (1)
Also, 4500 72 49 = 34
r3
r = 9m
dr
dt=
18
81=
2
9m/minute
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
3 3 3
k 1
k (k 1) n , 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.
4. (2)
Statement 2:n
3 3
K 1
K (K 1)
2 3n(n 1) n(n 1)
2 2= n
3
Statement 2 is correct
Statement 1:
(13
03) + (2
31
3) + (3
32
3) + (20
319)
3= 8000
Statement 1 is correct
and statement 2 explain statement 1
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.
5. (1)
6. If the integral
5tan x
dx x a n sin x 2cos x k tan x 2 then a is equal to :
(1) 1 (2) 2 (3) 1 (4) 2
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(3) VIDYALANKAR : AIEEE 2012 Paper and Solution
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6. (4)
5tanxdx
tanx 2= x + a ln | sin x 2 cos x | + K
Differentiating on both side
5tanx
tanx 2= 1 +
a[cosx 2sinx]
sinx 2cos
5sinx
sinx 2cosx =
sinx 2cosx a(cosx 2sinx)
sinx 2cosx
Equating co-efficient of both side
sinx
5 1 2a ,
cosx
0 2 a
a = 2
7. Statement 1 : An equation of a common tangent to the parabola 2y 16 3 x and the ellipse
2x2
+ y2
= 4 is y = 2x + 2 3 .
Statement 2 : If the line4 3
y mx ,m
(m 0) is a common tangent to the parabola
2y 16 3 x and the ellipse 2x
2+ y
2= 4, then m satisfies m
4+ 2m
2= 24.
(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.
7. (2)
Put y = 2 24 3
mx in 2x y 4m
2
2 4 32x mx
m= 4 2 2
2
482 m x 8 3 x 4
m= 0
y =4 3
mxm
is a tangent, discriminant of the above quadratic equation must be zero.
28 3 =
2
2
484 2 m 4
m
m4 + 2m2 24 = 0 (m2 + 6) (m2 4) = 0m = 2
Statement (2) is a correct explanation of statement (1).
8. Let
1 0 0
A 2 1 0 .
3 2 1
If u1 and u2 are column matrices such that Au1 =
1
0
0
and Au2 =
0
1 ,
0
then
u1 + u2 is equal to :
(1)
1
10
(2)
1
11
(3)
1
10
(4)
1
11
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AIEEE 2012 Paper and Solution (4)
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8. (4)
AU1 =
1 0 0 a
2 1 0 b
3 2 1 c
=
1
0
0
a
2a b
3a 2b c
=
1
0
0
a = 1; b = 2; c = 1
U1 =
1
2
1
AU2 =
1 0 0 x
2 1 0 y
3 2 1 z
=
0
1
0
x
2x y
3x 2y z
=
0
1
0
x = 0; y = 1; z = 2
U2 =
0
1
2
U1 + U2 =
1
2
1
+
0
1
2
=
1
1
1
9. If n is a positive integer, then2n 2n
3 1 3 1 is :
(1) an irrational number
(2) an odd positive integer
(3) an even positive integer
(4) a rational number other than positive integers
9. (1)2n 2n
3 1 3 1
1 3 2n 1
2n 1 2n 3
C C C2 2n 3 2n 3 ......2n 3
which is Irrational Number
10. If 100 times the 100th
term of an AP with non zero common difference equals the 50 times its
50
th
term, then the 150
th
term of this AP is :(1) 150 (2) 150 times its 50th
term
(3) 150 (4) zero
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(5) VIDYALANKAR : AIEEE 2012 Paper and Solution
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10. (4)
Let first term a and common difference d
100 [a + 99 d] = 50 [a + 49 d]
a = 149 d
Now,
T150 = a + 149 d
T150 = 0
11. In a PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to :
(1)5
6(2)
6(3)
4(4)
3
4
11. (2)
P + Q + R =
3 sin P + 4 cos Q = 6 . (i)
4 sin Q + 3 cos P = 1 . (ii)
squaring and adding (i) and (ii)
9 + 16 + 24 sin (P + Q) = 37
sin (P + Q) =1
2
P + Q =6
;5
6
if P + Q =6
0 < sin P