Integrals

JEE Main 2024 (Online) 27th January Morning ShiftMCQMedium📈 Topic in 67% papers+4−1

Ques. 1JEE Main 2024, 27 January (Morning Shift)

If (a, b) be the orthocentre of the triangle whose vertices are (1,2),(2,3) and (3,1), and I1= ab x (4x-x2 )dx, I2= ab (4x-x2 )dx, then 36 I1I2 is equal to :

Answer

(2) 72

Solution

Slope of AB= 3-22-1=1 altitude from C(3,1): y-1=-(x-3), i.e. x+y=4

[figure]

Orthocentre H(a, b) lies on this altitude a+b=4

King rule: x a+b-x=4-x, and 4x-x2=x(4-x) is unchanged

I1= ab(4-x) (4x-x2 )dx

Adding the two forms: 2I1=4I2 I1I2=2

36 I1I2=72

Answer: (2)

Keep practising — Integrals

Ques. 2JEE Main 2024, 27 January (Evening Shift)

The integral (x8-x2 )dx (x12+3x6+1 ) -1 (x3+ 1x3 ) is equal to :

e ( | -1 (x3+ 1x3 ) | )1/3+C
e ( | -1 (x3+ 1x3 ) | )+C
e ( | -1 (x3+ 1x3 ) | )1/2+C
e ( | -1 (x3+ 1x3 ) | )3+C

Ques. 3JEE Main 2024, 27 January (Evening Shift)

For 0<a<1, the value of the integral 0 dx1-2a x+a2 is :

2 +a2
2 -a2
1-a2
1+a2

Ques. 4JEE Main 2024, 27 January (Evening Shift)

Let f(x)= 0x g(t) e ( 1-t1+t )dt, where g is a continuous odd function. If - /2 /2 (f(x)+ x2 x1+ex )dx= ( )2- , then is equal to .

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