21.
The mean of the coefficients of x,x 2 , ...... x 7 in the
binomial expansion of (2 + x) 9 is _ __
Official Ans. by NTA (2736)
Sol. Coefficient of
Of x 2
-
Of x 7
--
-
X
9
C 2 27
9
C 7 • 22
= 9Cl 28
9c1. 2s + 9C2· 21 ..... + 9c1. 22
Mean=-~--~-----7
9 9
9 - 9C • i1
1
+
2)
C
•
2
(
0
8
-
9C
9
=--~--------7
9
9
3 -2 -18-1
=----7
= 19152 = 2736
7
20.
Let x1,
x2 ....,
x,oo be in an arithmetic progression,
with x 1 = 2 and their mean equal to 200. If
y,
=i(x, -
i),1:5i:5100, then the mean of y,. y2,
......, y,oo is.
(I) 10101.50
(2) 10051.50
(3) 10049.50
(4) 10100
Offl<ial Ans. by NTA (3)
Sol.
Mean
= 200
I00(2x2+99d)
2
100
200
4+99d=400
Y; =i(xi-i)
=i( 2+( i-1)4-i) =3i 2 -2i
LY
Mean=--'
100
1 ~3-2 2·
=-L-1-1
100 ;.,
=10049·50
13.
Area of the
region
{(x,y):x'+(y-2)2 S4,
x' 2'2y}is
(I) 211-~
(2)
8
(3) 11+3
16
(4) 2n+3
3
71-~
3
Oflidal Ans. by NTA (I)
Sol.
2
x'+(y-2) !>22 and x 2 ~2y
Solving circle and parabola simultaneously :
2
2y+y -4y+4=4
y2 -2y=0
y= 0,2
Put y = 2 in x2 = 2y
(2,2) and (-2,2)
~J-
I
Required area = 2
x = ±2
I
=2x2---,r•2'=4-,r
4
, '
l
• 2
J
[J~dx-( 4- ,r
=2[~:-4+,r~
=2[~+,r-4~
=2[,r-f~
= 2,r-~
6
2.
If equation of the plane that contains the point
(-2,3,5) and is perpendicular to each of the planes
2x + 4y+5z = 8and3x -2y+ 3z = 5 is
ax+/3y+yz+97=0then a+l3+y=
(I) 18
(2) 17
(3) 16
(4) 15
Official Ans. by NTA (4)
Sol. The equation of plane through (-2,3,5) is
a(x+2) + b(y-3) + c (z-5) = 0
it is perpendicular to 2x+4y+5z=8 & 3x-2y+3z=5
2a+4b+5c= 0
3a-2b+3c =0
a
-b
c
1-: !I =1: !I =1: _:1
a
b
c
-=-=22 9 -16
Equation of Plane is
22( X + 2)+9(y-3)-J6( z-5) = 0
22x+9y-16z+97 =0
Comparing with ax+ py + yx + 97 = 0
We get a +P +r = 22+9-16= 15
6.
Let a, b, c be three distinct real numbers, none
equal to one. If the vectors ai + j + k, i + bj + k
and
are
i+j+ck
coplanar,
1
1
1
- - + - - + - - is equal to
1-a 1-b 1-c
(I) I
(2)- 1
(3)-2
(4) 2
Official Ans. by NTA (I)
a 1
Sol.
b
1 =0
1
C
C2
-C,, C3
a 1-a
1-a
b-1
0
0
c-1
=0
a(b-1 )(c-1 )-(I-a) (c-1 )+( I-a)( 1-b) = 0
a(l-b)(l-c)+(l-a) (1-c)+(l-a)(l-b) = 0
a
1
1
--+--+--=0
1-a 1-b 1-c
1
1-a
1
1-a
1
1-b
1
1
1-b 1-c
1
1-c
then
20.
Let the equation of plane passing through the line
of intersection of the planes x + 2y + az = 2 and
x-y+z=3be5x-11y+bz=6a-l.
Force Z,
if the distance of this plane from the point
. 2r:, then a+b.1s equaI to
(a, -c,c) 1s
va
C
(I) -2
(2) 2
(3) -4
(4) 4
Official Ans. by NTA (3)
Sol.
(x + 2y+az-2)+ A(x -y + z-3) = 0
I + A 2 - A a + A 2 + 31..
--=--=--=-5
-11
b
6a-l
A = _7_
2' a = 3, b = 1
= 15a + I Jc+ be - 6a + I
c=-1
a+b =3+1 =-4
C
-1
19.
Let y = y 1(x) and y = yiCx) be the solution curves of
the differential equation dy = y + 7 with initial
dx
conditions y1(0)=0,y2(0)=1 respectively. Then the
curves y = y1(x) and y = yiCx) intersect at
(I) Two points
(2) no point
(3) infinite number of points
(4) one point
Official Ans. by NTA (2)
Sol.
dy
dx
dy
dx
1.F. =e-x
ye-x = -7e-x +c
y=-7+cex
-7+7ex =-7+8ex
=O
No solution
18.
Let S1, s2, s3, ..... ,s 10 respectively be the sum to 12
terms of 10 A.P.s whose first terms are I, 2, 3, .... , 10
and the common differences are I, 3, 5, .... , 19
10
respectively. Then Is; is equal to
i=l
(I) 7380
(2) 7220
(3) 7360
(4) 7260
Official Ans. by NTA (4)
Sol.
Sk = 6(2k + (11)(2k -1))
Sk =6(2k+22k-l 1)
Sk =144k-66
10
10
1
k=l
Isk =144Lk-66x)O
=144x 10x11_660
2
= 7920-660
= 7260
16.
Let a=i+4]+2k,b=3i-2]+7k and
c=2i-]+4k. Ifavector iisatisties iixb=cxb
and
d.a = 24,
then
Id12 is equal to
(I) 413
(2) 423
(3) 323
(4) 313
Official Ans. by NTA (1)
Sol.:
cixb=cxb
ci =c+Ab
Also ii.a= 24
24
A= 24_-i.c = 24-6 = 2
b.i
9
= sf-s]+ tsk
2
1a1 = 64+25+324 = 413
10.
The area of the region enclosed by the curve
f(x) = max{sin x, cos x} , -it
x
1t and
the x-axis
is
(I)
+ 1)
(2)
(3)
(4) 4
Official Ans. by NTA (4)
Sol.
-7t
Area=
L sin x dx I+ IfJ.cosx dx I+J~cosx dx + J;sinxdx
-]fl:
I
-1t
II:
4
2
4
=4
4