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Page 1
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 1
Date : 5
th
September 2020
Time : 09 : 00 am - 12 : 00 pm
Subject : Maths
Q.1 If the volume of a parallelopiped, whose coterminuous edges are given by the vectors
ˆ ˆ ˆ ˆ ˆ ˆ
a i j nk, b 2i 4j nk ? ? ? ? ? ?
?
?
and
? ?
ˆ ˆ ˆ
c i nj 3k n 0 , ? ? ? ?
?
is 158 cu. units, then:
(1) a
?
.c
?
=17 (2) b
?
.c
?
=10 (3) n=9 (4) n=7
Sol. 2
1 1
2 4 158
1 3
n
n
n
? ?
(12 + n
2
) –(6+n) +n(2n–4)=158
3n
2
–5n + 6 –158 = 0
3n
2
– 5n – 152 = 0
3n
2
– 24n + 19n – 152 = 0
(3n + 19) (n–8) = 0
? n = 8
? ?
b.c 10 ?
? ?
Q.2 A survey shows that 73% of the persons working in an office like coffee, whereas 65%
like tea. If x denotes the percentage of them, who like both coffee and tea, then x
cannot be:
(1) 63 (2) 54 (3) 38 (4) 36
Sol. 4
C ? ?Coffee & T ? ?Tea
n(C) = 73, n(T) = 65
n(coffee) =
73
100
n(tea) =
65
100
? ? ? n T C =
100
x
? ? ? n C T = n(C) + n(T) – x ? 100
= 73 + 65 – x ? 100
? ?x ? 38
? 73 – x ? 0 ? x ? 73
? 65 – x ? 0 ? x ? 65
? ? 38 x 65
, x = 36
x ? min(n(C), n(T)) ? 38 ? x ? 65
Page 2
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 1
Date : 5
th
September 2020
Time : 09 : 00 am - 12 : 00 pm
Subject : Maths
Q.1 If the volume of a parallelopiped, whose coterminuous edges are given by the vectors
ˆ ˆ ˆ ˆ ˆ ˆ
a i j nk, b 2i 4j nk ? ? ? ? ? ?
?
?
and
? ?
ˆ ˆ ˆ
c i nj 3k n 0 , ? ? ? ?
?
is 158 cu. units, then:
(1) a
?
.c
?
=17 (2) b
?
.c
?
=10 (3) n=9 (4) n=7
Sol. 2
1 1
2 4 158
1 3
n
n
n
? ?
(12 + n
2
) –(6+n) +n(2n–4)=158
3n
2
–5n + 6 –158 = 0
3n
2
– 5n – 152 = 0
3n
2
– 24n + 19n – 152 = 0
(3n + 19) (n–8) = 0
? n = 8
? ?
b.c 10 ?
? ?
Q.2 A survey shows that 73% of the persons working in an office like coffee, whereas 65%
like tea. If x denotes the percentage of them, who like both coffee and tea, then x
cannot be:
(1) 63 (2) 54 (3) 38 (4) 36
Sol. 4
C ? ?Coffee & T ? ?Tea
n(C) = 73, n(T) = 65
n(coffee) =
73
100
n(tea) =
65
100
? ? ? n T C =
100
x
? ? ? n C T = n(C) + n(T) – x ? 100
= 73 + 65 – x ? 100
? ?x ? 38
? 73 – x ? 0 ? x ? 73
? 65 – x ? 0 ? x ? 65
? ? 38 x 65
, x = 36
x ? min(n(C), n(T)) ? 38 ? x ? 65
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 2
Q.3 The mean and variance of 7 observations are 8 and 16, respectively. If five observa-
tions are 2,4,10,12,14, then the absolute difference of the remaining two observations
is:
(1) 1 (2) 4 (3) 3 (4) 2
Sol. 4
Var(x) =
? ?
2
2
i
x
x
n
?
?
16 =
2 2 2 2 2 2
1 2 4 5 6 7
64
7
x x x x x x ? ? ? ? ?
?
80 × 7 = x
1
2
+
2 2 2
2 3 7
x x ..... x ? ? ?
Now,
2 2 2 2
6 7 1 5
560 ? ? ? ? x x (x ......x )
2 2
6 7
560 4 16 100 144 196 x x ( ) ? ? ? ? ? ? ?
2 2
6 7
100 x x ? ? ......(1)
Now,
1 2 7
7
x x .... x ? ? ?
= 8
x
6
+ x
7
= 14 ......(2)
from (1) & (2)
(x
6
+ x
7
)
2
– 2x
6
x
7
= 100
2x
6
x
7
= 96 ? x
6
x
7
= 48 ......(3)
Now, |x
6
– x
7
| =
? ?
2
6 7 6 7
4 ? ? x x x x
=
196 192 2 ? ?
Q.4 If 2
10
+2
9
.3
1
+2
8
.3
2
+.....+2.3
9
+3
10
=S-2
11
, then S is equal to:
(1) 3
11
(2)
11
10
3
2
2
? (3) 2.3
11
(4) 3
11
—2
12
Sol. 1
let
S’ = 2
10
+ 2
9
3
1
+ 2
8
3
2
+-----+2.3
9
+ 3
10
3 '
2
?S
= 2
9
× 3
1
+ 2
8
. 3
2
+---- +3
10
11
3
2
?
__________________________________
11
10
' 3
2
2 2
?
? ?
S
S’ = 3
11
–2
11
Now S
’
= S –2
11
S = 3
11
Page 3
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 1
Date : 5
th
September 2020
Time : 09 : 00 am - 12 : 00 pm
Subject : Maths
Q.1 If the volume of a parallelopiped, whose coterminuous edges are given by the vectors
ˆ ˆ ˆ ˆ ˆ ˆ
a i j nk, b 2i 4j nk ? ? ? ? ? ?
?
?
and
? ?
ˆ ˆ ˆ
c i nj 3k n 0 , ? ? ? ?
?
is 158 cu. units, then:
(1) a
?
.c
?
=17 (2) b
?
.c
?
=10 (3) n=9 (4) n=7
Sol. 2
1 1
2 4 158
1 3
n
n
n
? ?
(12 + n
2
) –(6+n) +n(2n–4)=158
3n
2
–5n + 6 –158 = 0
3n
2
– 5n – 152 = 0
3n
2
– 24n + 19n – 152 = 0
(3n + 19) (n–8) = 0
? n = 8
? ?
b.c 10 ?
? ?
Q.2 A survey shows that 73% of the persons working in an office like coffee, whereas 65%
like tea. If x denotes the percentage of them, who like both coffee and tea, then x
cannot be:
(1) 63 (2) 54 (3) 38 (4) 36
Sol. 4
C ? ?Coffee & T ? ?Tea
n(C) = 73, n(T) = 65
n(coffee) =
73
100
n(tea) =
65
100
? ? ? n T C =
100
x
? ? ? n C T = n(C) + n(T) – x ? 100
= 73 + 65 – x ? 100
? ?x ? 38
? 73 – x ? 0 ? x ? 73
? 65 – x ? 0 ? x ? 65
? ? 38 x 65
, x = 36
x ? min(n(C), n(T)) ? 38 ? x ? 65
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 2
Q.3 The mean and variance of 7 observations are 8 and 16, respectively. If five observa-
tions are 2,4,10,12,14, then the absolute difference of the remaining two observations
is:
(1) 1 (2) 4 (3) 3 (4) 2
Sol. 4
Var(x) =
? ?
2
2
i
x
x
n
?
?
16 =
2 2 2 2 2 2
1 2 4 5 6 7
64
7
x x x x x x ? ? ? ? ?
?
80 × 7 = x
1
2
+
2 2 2
2 3 7
x x ..... x ? ? ?
Now,
2 2 2 2
6 7 1 5
560 ? ? ? ? x x (x ......x )
2 2
6 7
560 4 16 100 144 196 x x ( ) ? ? ? ? ? ? ?
2 2
6 7
100 x x ? ? ......(1)
Now,
1 2 7
7
x x .... x ? ? ?
= 8
x
6
+ x
7
= 14 ......(2)
from (1) & (2)
(x
6
+ x
7
)
2
– 2x
6
x
7
= 100
2x
6
x
7
= 96 ? x
6
x
7
= 48 ......(3)
Now, |x
6
– x
7
| =
? ?
2
6 7 6 7
4 ? ? x x x x
=
196 192 2 ? ?
Q.4 If 2
10
+2
9
.3
1
+2
8
.3
2
+.....+2.3
9
+3
10
=S-2
11
, then S is equal to:
(1) 3
11
(2)
11
10
3
2
2
? (3) 2.3
11
(4) 3
11
—2
12
Sol. 1
let
S’ = 2
10
+ 2
9
3
1
+ 2
8
3
2
+-----+2.3
9
+ 3
10
3 '
2
?S
= 2
9
× 3
1
+ 2
8
. 3
2
+---- +3
10
11
3
2
?
__________________________________
11
10
' 3
2
2 2
?
? ?
S
S’ = 3
11
–2
11
Now S
’
= S –2
11
S = 3
11
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 3
Q.5 If 3
2 sin2 ? ?-1
,14 and 3
4-2 sin2 ?
are the first three terms of an A.P. for some ? , then the sixth
term of this A.P. is:
(1) 65 (2) 81 (3) 78 (4) 66
Sol. 4
28 = 3
2sin2 ? – 1
+ 3
4 –2 sin2 ?
28 =
sin 2
sin 2
9 81
3 9
?
?
?
Let 9
sin2 ? ?
= t
28 =
81
3
?
t
t
t
2
– 84t + 243 = 0
t
2
– 81t – 3t + 243 = 0
t(t – 81) – 3(t – 81) = 0
(t – 81) (t – 3) = 0
t = 81, 3
9
sin2 ?
= 9
2
or 3
sin2 ? = 1/2, 2 (rejected)
First term a = 3
2 sin2 ? ?-1
a = 1
Second term = 14
? Common difference d = 13
T
6
= a + 5d
T
6
= 1 + 5 × 13 = 66
Q.6 If the common tangent to the parabolas, y
2
= 4x and x
2
= 4y also touches the circle,
x
2
+y
2
= c
2
, then c is equal to:
(1)
1
2
(2)
1
4
(3)
1
2
(4)
1
2 2
Sol. 3
y = mx +
1
m
x
2
=
1
4 mx
m
? ?
?
? ?
? ?
x
2
–4mx
4
m
? = 0
D = 0
16m
2
+
16
m
= 0
3
1
16
m
m
? ?
?
? ?
? ?
? ?
= 0
m = –1
? y + x =–1
Now,
1
2
?
?|c|
c = ±
1
2
Page 4
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 1
Date : 5
th
September 2020
Time : 09 : 00 am - 12 : 00 pm
Subject : Maths
Q.1 If the volume of a parallelopiped, whose coterminuous edges are given by the vectors
ˆ ˆ ˆ ˆ ˆ ˆ
a i j nk, b 2i 4j nk ? ? ? ? ? ?
?
?
and
? ?
ˆ ˆ ˆ
c i nj 3k n 0 , ? ? ? ?
?
is 158 cu. units, then:
(1) a
?
.c
?
=17 (2) b
?
.c
?
=10 (3) n=9 (4) n=7
Sol. 2
1 1
2 4 158
1 3
n
n
n
? ?
(12 + n
2
) –(6+n) +n(2n–4)=158
3n
2
–5n + 6 –158 = 0
3n
2
– 5n – 152 = 0
3n
2
– 24n + 19n – 152 = 0
(3n + 19) (n–8) = 0
? n = 8
? ?
b.c 10 ?
? ?
Q.2 A survey shows that 73% of the persons working in an office like coffee, whereas 65%
like tea. If x denotes the percentage of them, who like both coffee and tea, then x
cannot be:
(1) 63 (2) 54 (3) 38 (4) 36
Sol. 4
C ? ?Coffee & T ? ?Tea
n(C) = 73, n(T) = 65
n(coffee) =
73
100
n(tea) =
65
100
? ? ? n T C =
100
x
? ? ? n C T = n(C) + n(T) – x ? 100
= 73 + 65 – x ? 100
? ?x ? 38
? 73 – x ? 0 ? x ? 73
? 65 – x ? 0 ? x ? 65
? ? 38 x 65
, x = 36
x ? min(n(C), n(T)) ? 38 ? x ? 65
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 2
Q.3 The mean and variance of 7 observations are 8 and 16, respectively. If five observa-
tions are 2,4,10,12,14, then the absolute difference of the remaining two observations
is:
(1) 1 (2) 4 (3) 3 (4) 2
Sol. 4
Var(x) =
? ?
2
2
i
x
x
n
?
?
16 =
2 2 2 2 2 2
1 2 4 5 6 7
64
7
x x x x x x ? ? ? ? ?
?
80 × 7 = x
1
2
+
2 2 2
2 3 7
x x ..... x ? ? ?
Now,
2 2 2 2
6 7 1 5
560 ? ? ? ? x x (x ......x )
2 2
6 7
560 4 16 100 144 196 x x ( ) ? ? ? ? ? ? ?
2 2
6 7
100 x x ? ? ......(1)
Now,
1 2 7
7
x x .... x ? ? ?
= 8
x
6
+ x
7
= 14 ......(2)
from (1) & (2)
(x
6
+ x
7
)
2
– 2x
6
x
7
= 100
2x
6
x
7
= 96 ? x
6
x
7
= 48 ......(3)
Now, |x
6
– x
7
| =
? ?
2
6 7 6 7
4 ? ? x x x x
=
196 192 2 ? ?
Q.4 If 2
10
+2
9
.3
1
+2
8
.3
2
+.....+2.3
9
+3
10
=S-2
11
, then S is equal to:
(1) 3
11
(2)
11
10
3
2
2
? (3) 2.3
11
(4) 3
11
—2
12
Sol. 1
let
S’ = 2
10
+ 2
9
3
1
+ 2
8
3
2
+-----+2.3
9
+ 3
10
3 '
2
?S
= 2
9
× 3
1
+ 2
8
. 3
2
+---- +3
10
11
3
2
?
__________________________________
11
10
' 3
2
2 2
?
? ?
S
S’ = 3
11
–2
11
Now S
’
= S –2
11
S = 3
11
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 3
Q.5 If 3
2 sin2 ? ?-1
,14 and 3
4-2 sin2 ?
are the first three terms of an A.P. for some ? , then the sixth
term of this A.P. is:
(1) 65 (2) 81 (3) 78 (4) 66
Sol. 4
28 = 3
2sin2 ? – 1
+ 3
4 –2 sin2 ?
28 =
sin 2
sin 2
9 81
3 9
?
?
?
Let 9
sin2 ? ?
= t
28 =
81
3
?
t
t
t
2
– 84t + 243 = 0
t
2
– 81t – 3t + 243 = 0
t(t – 81) – 3(t – 81) = 0
(t – 81) (t – 3) = 0
t = 81, 3
9
sin2 ?
= 9
2
or 3
sin2 ? = 1/2, 2 (rejected)
First term a = 3
2 sin2 ? ?-1
a = 1
Second term = 14
? Common difference d = 13
T
6
= a + 5d
T
6
= 1 + 5 × 13 = 66
Q.6 If the common tangent to the parabolas, y
2
= 4x and x
2
= 4y also touches the circle,
x
2
+y
2
= c
2
, then c is equal to:
(1)
1
2
(2)
1
4
(3)
1
2
(4)
1
2 2
Sol. 3
y = mx +
1
m
x
2
=
1
4 mx
m
? ?
?
? ?
? ?
x
2
–4mx
4
m
? = 0
D = 0
16m
2
+
16
m
= 0
3
1
16
m
m
? ?
?
? ?
? ?
? ?
= 0
m = –1
? y + x =–1
Now,
1
2
?
?|c|
c = ±
1
2
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 4
Q.7 If the minimum and the maximum values of the function f : , R
4 2
? ? ? ?
?
? ?
? ?
, defined by y
? ?
2 2
2 2
sin 1 sin 1
f cos 1 cos 1
12 10 2
? ? ? ? ?
? ? ? ? ? ? ?
?
are m and M respectively, then the ordered pair (m,M)
is equal to :
(1) (0,4) (2) (-4,0)
(3) (-4,4) (4) ? ?
0,2 2
Sol. 2
f( ?) =
2 2
2 2
1 1
1 1
12 10 2
sin sin
cos cos
? ? ? ? ?
? ? ? ? ?
?
C
1
? C
1
–C
2
, C
3
? C
3
+ C
2
2 2
2 2
1 1
1 1
2 10 8
sin sin
cos cos
? ? ? ? ?
? ? ? ? ?
C
2
? C
2
– C
3
2
2
1 1
1 1
2 2 8
sin
cos
? ? ?
? ? ?
1(2cos
2
?–8) + (8+2cos
2
?) –4sin
2
?
f( ?) = 4cos2 ?
Q.8 Let R ? ? . The system of linear equations
2x
1
-4x
2
+
?
x
3
=1
x
1
-6x
2
+x
3
=2
?
x
1
-10x
2
+4x
3
=3
is inconsistent for:
(1) exactly two values of
?
(2) exactly one negative value of
?
.
(3) every value of
?
.
(4) exactly one positive value of
?
.
Sol. 2
Page 5
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 1
Date : 5
th
September 2020
Time : 09 : 00 am - 12 : 00 pm
Subject : Maths
Q.1 If the volume of a parallelopiped, whose coterminuous edges are given by the vectors
ˆ ˆ ˆ ˆ ˆ ˆ
a i j nk, b 2i 4j nk ? ? ? ? ? ?
?
?
and
? ?
ˆ ˆ ˆ
c i nj 3k n 0 , ? ? ? ?
?
is 158 cu. units, then:
(1) a
?
.c
?
=17 (2) b
?
.c
?
=10 (3) n=9 (4) n=7
Sol. 2
1 1
2 4 158
1 3
n
n
n
? ?
(12 + n
2
) –(6+n) +n(2n–4)=158
3n
2
–5n + 6 –158 = 0
3n
2
– 5n – 152 = 0
3n
2
– 24n + 19n – 152 = 0
(3n + 19) (n–8) = 0
? n = 8
? ?
b.c 10 ?
? ?
Q.2 A survey shows that 73% of the persons working in an office like coffee, whereas 65%
like tea. If x denotes the percentage of them, who like both coffee and tea, then x
cannot be:
(1) 63 (2) 54 (3) 38 (4) 36
Sol. 4
C ? ?Coffee & T ? ?Tea
n(C) = 73, n(T) = 65
n(coffee) =
73
100
n(tea) =
65
100
? ? ? n T C =
100
x
? ? ? n C T = n(C) + n(T) – x ? 100
= 73 + 65 – x ? 100
? ?x ? 38
? 73 – x ? 0 ? x ? 73
? 65 – x ? 0 ? x ? 65
? ? 38 x 65
, x = 36
x ? min(n(C), n(T)) ? 38 ? x ? 65
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 2
Q.3 The mean and variance of 7 observations are 8 and 16, respectively. If five observa-
tions are 2,4,10,12,14, then the absolute difference of the remaining two observations
is:
(1) 1 (2) 4 (3) 3 (4) 2
Sol. 4
Var(x) =
? ?
2
2
i
x
x
n
?
?
16 =
2 2 2 2 2 2
1 2 4 5 6 7
64
7
x x x x x x ? ? ? ? ?
?
80 × 7 = x
1
2
+
2 2 2
2 3 7
x x ..... x ? ? ?
Now,
2 2 2 2
6 7 1 5
560 ? ? ? ? x x (x ......x )
2 2
6 7
560 4 16 100 144 196 x x ( ) ? ? ? ? ? ? ?
2 2
6 7
100 x x ? ? ......(1)
Now,
1 2 7
7
x x .... x ? ? ?
= 8
x
6
+ x
7
= 14 ......(2)
from (1) & (2)
(x
6
+ x
7
)
2
– 2x
6
x
7
= 100
2x
6
x
7
= 96 ? x
6
x
7
= 48 ......(3)
Now, |x
6
– x
7
| =
? ?
2
6 7 6 7
4 ? ? x x x x
=
196 192 2 ? ?
Q.4 If 2
10
+2
9
.3
1
+2
8
.3
2
+.....+2.3
9
+3
10
=S-2
11
, then S is equal to:
(1) 3
11
(2)
11
10
3
2
2
? (3) 2.3
11
(4) 3
11
—2
12
Sol. 1
let
S’ = 2
10
+ 2
9
3
1
+ 2
8
3
2
+-----+2.3
9
+ 3
10
3 '
2
?S
= 2
9
× 3
1
+ 2
8
. 3
2
+---- +3
10
11
3
2
?
__________________________________
11
10
' 3
2
2 2
?
? ?
S
S’ = 3
11
–2
11
Now S
’
= S –2
11
S = 3
11
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 3
Q.5 If 3
2 sin2 ? ?-1
,14 and 3
4-2 sin2 ?
are the first three terms of an A.P. for some ? , then the sixth
term of this A.P. is:
(1) 65 (2) 81 (3) 78 (4) 66
Sol. 4
28 = 3
2sin2 ? – 1
+ 3
4 –2 sin2 ?
28 =
sin 2
sin 2
9 81
3 9
?
?
?
Let 9
sin2 ? ?
= t
28 =
81
3
?
t
t
t
2
– 84t + 243 = 0
t
2
– 81t – 3t + 243 = 0
t(t – 81) – 3(t – 81) = 0
(t – 81) (t – 3) = 0
t = 81, 3
9
sin2 ?
= 9
2
or 3
sin2 ? = 1/2, 2 (rejected)
First term a = 3
2 sin2 ? ?-1
a = 1
Second term = 14
? Common difference d = 13
T
6
= a + 5d
T
6
= 1 + 5 × 13 = 66
Q.6 If the common tangent to the parabolas, y
2
= 4x and x
2
= 4y also touches the circle,
x
2
+y
2
= c
2
, then c is equal to:
(1)
1
2
(2)
1
4
(3)
1
2
(4)
1
2 2
Sol. 3
y = mx +
1
m
x
2
=
1
4 mx
m
? ?
?
? ?
? ?
x
2
–4mx
4
m
? = 0
D = 0
16m
2
+
16
m
= 0
3
1
16
m
m
? ?
?
? ?
? ?
? ?
= 0
m = –1
? y + x =–1
Now,
1
2
?
?|c|
c = ±
1
2
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 4
Q.7 If the minimum and the maximum values of the function f : , R
4 2
? ? ? ?
?
? ?
? ?
, defined by y
? ?
2 2
2 2
sin 1 sin 1
f cos 1 cos 1
12 10 2
? ? ? ? ?
? ? ? ? ? ? ?
?
are m and M respectively, then the ordered pair (m,M)
is equal to :
(1) (0,4) (2) (-4,0)
(3) (-4,4) (4) ? ?
0,2 2
Sol. 2
f( ?) =
2 2
2 2
1 1
1 1
12 10 2
sin sin
cos cos
? ? ? ? ?
? ? ? ? ?
?
C
1
? C
1
–C
2
, C
3
? C
3
+ C
2
2 2
2 2
1 1
1 1
2 10 8
sin sin
cos cos
? ? ? ? ?
? ? ? ? ?
C
2
? C
2
– C
3
2
2
1 1
1 1
2 2 8
sin
cos
? ? ?
? ? ?
1(2cos
2
?–8) + (8+2cos
2
?) –4sin
2
?
f( ?) = 4cos2 ?
Q.8 Let R ? ? . The system of linear equations
2x
1
-4x
2
+
?
x
3
=1
x
1
-6x
2
+x
3
=2
?
x
1
-10x
2
+4x
3
=3
is inconsistent for:
(1) exactly two values of
?
(2) exactly one negative value of
?
.
(3) every value of
?
.
(4) exactly one positive value of
?
.
Sol. 2
JEE Main 2020 Paper
5
th
September 2020 | (Shift-1), Maths Page | 5
D =
2 4
1 6 1
10 4
?
?
?
?
?
= 2(3 ? + 2)( ? – 3)
D
1
= –2( ? – 3)
D
2
= –2( ? + 1)( ? – 3)
D
3
= –2( ? – 3)
When ? = 3, then
D = D
1
= D
2
= D
3
= 0
? Infinite many solution
When ? = –2/3 then D
1
, D
2
, D
3
none of them is zero so equations are inconsistent
? ? = –2/3
Q.9 If the point P on the curve, 4x
2
+5y
2
=20 is farthest from the point Q(0, -4), then PQ
2
is
equal to:
(1) 48 (2) 29 (3) 21 (4) 36
Sol. 4
Given ellipse is
2
x
5
+
2
y
4
= 1
Let point P is ? ?
5 2 cos , sin ? ?
(PQ)
2
= 5cos
2
? + 4(sin ? + 2)
2
(PQ)
2
= cos
2
? + 16 sin ? + 20
(PQ)
2
= –sin
2
? + 16 sin ? + 21
= 85 – (sin ? – 8)
2
Will be maximum when sin ? = 1
(PQ)
2
max
= 85 – 49 = 36
Q.10 The product of the roots of the equation 9x
2
-18|x|+5=0 is :
(1)
25
81
(2)
5
9
(3)
5
27
(4)
25
9
Sol. 1
Let |x| = t
9t
2
– 18t +5 = 0
9t
2
– 15t –3t + 5 = 0
(3t –5) (3t–1) = 0
|x| =
5 1
3 3
,
? x =
5 5 1 1
3 3 3 3
, , ,
? ?
? P =
25
81
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