Final JEE - Main Exam September, 2020/02-09-2020/Morning Session
FINAL JEE–MAIN EXAMINATION – SEPTEMBER, 2020 (Held On Wednesday 02nd SEPTEMBER, 2020) TIME : 9 AM to 12 PM
TEST PAPER WITH SOLUTION
MATHEMATICS 1.
If |x| < 1, |y| < 1 and x ¹ y, then the sum to infinity of the following series (x+y) + (x2+xy+y2) + (x3+x2y + xy2+y3)+.....
x + y - xy (1) (1 - x) (1 - y) (3)
x + y + xy (1 + x) (1 + y)
t r + 1 = 10 C r a10 - r (x)
x + y - xy (2) (1 + x) (1 + y) (4)
10 - r 9 . br
= 10 C r a10 - r br (x)
x
-
r 6
10 - r r 9 6
If tr + 1 is independent of x
x + y + xy (1 - x) (1 - y)
10 - r r - =0 9 6
Þ
r=4
maximum value of t5 is 10 K (given)
Official Ans. by NTA (1) Sol. |x| < 1, |y| < 1, x ¹ y (x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ........ By multiplying and dividing x – y :
Þ
10
C 4 a 6 b4 is maximum
EN
By AM ³ GM (for positive numbers) 1 a 3 a 3 b 2 b2 + + + 6 4 2 2 2 2 ³ æ a b ö4 ç ÷ 4 è 16 ø
(x 2 - y 2 ) + (x3 - y 3 ) + (x 4 - y 4 ) + ...... x-y
Þ a 6 b4 £ 16
(x 2 + x 3 + x 4 + ......) - (y 2 + y 3 + y 4 + ......) = x-y
=
2.
2
3.
(x - y ) - xy(x - y) (1 - x)(1 - y)(x - y)
A
=
2
LL
x2 y2 1-x 1- y = x-y
So, 10 K =
x + y - xy (1 - x)(1 - y)
C 4 16
Þ K = 336
If a function f(x) defined by ìae x + be - x , - 1 £ x < 1 ï 2 f(x) = í cx , 1£ x £ 3 ïax2 + 2cx , 3 < x £ 4 î
be continuous for some a, b, c Î R and f¢(0) + f¢(2) = e, then the value of of a is :
Let a > 0, b > 0 be such that a3 + b2 = 4. If the maximum value of the term independent of x in
(
10
1 9
the binomial expansion of ax + bx
- 16
)
10
10
e e - 3e - 13
(2)
e e + 3e + 13
(3)
1 e - 3e + 13
(4)
e e - 3e + 13
is 10k,
then k is equal to : (1) 176 (2) 336 (3) 352 (4) 84 Official Ans. by NTA (2) Sol. Let tr + 1 denotes 1 æ 1 - ö 9 th r + 1 term of ç ax + b x 6 ÷ ç ÷ è ø
(1)
2
2
2
2
Official Ans. by NTA (4)
Sol.
ìae x + be - x , -1 £ x < 1 ïï f (x) = ícx 2 , 1£ x £ 3 ï 2 ïîax + 2cx, 3 < x £ 4 1
