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HOTS QUESTIONS FOR BRIGHT STUDENTS
CLASS - XII

SUB : MATHEMATICS

CHAPTER 7
INTEGRATION
(1 Mark Questions)
Find the value of the following integrals
Q1

Cosec ax dx

Q2

sin x/sin[x-a] dx

Q3

 (logx) /x2 dx

Q4

 (x+ cos6x)/(3x2+sin6x) dx

Q5

√1-4x-x2 dx

Q6

 cos2x/(sinx +cosx)2 dx

Q7

 dx/(ex+ e-x) dx

Q8

 (1+tan2 x)/(1+tanx) dx

Q9

∫ (sin2x cos2x ) / √(9-cos42x ) dx

Q10(a)

3



dx/(1+x2) dx

1

Q10(b)


...


–

23)
e2

24)


0

 1
1
[
- 
log x  log x

2


 ] dx


25)
Answers:

INTEGRATION

(1 Mark Questions)
Q1

1/a log | cosec ax-cot ax | +c

Q2

(Hint:Put x-a=t)
(x-a) Cos a + Sin a log|Sin (x-a)| +c

Q3

Hint: Write in the form ∫ log
...

-1/x log x- 1/x +c or -1/x [log x+1] +c

Q4

1/6 log |3x2 + Sin 6x |+c

Q5

5/2 [Sin-1 (x+2)/√5] + (x+2)(√1-4x-x2)/2 +c

Q6

log | Sin x + Cos x | +c

Q7

tan-1 (ex) +c

Q8

log |1+t an x| +c

Q9

-1/4 Sin-1 [1/3 Cos22x] +c

Q10

 /12
(4 Mark Questions)

Q11 0
Q12 Hint: put x=cos 2 t ,the question will be transformed to Q
...

Find the area bounded by the curve x=a(θ - sin θ), y=a(1-cos θ) 0    2
...

4

Sketch the graph y=|x - 1|
...

Find the area bounded by the curve (1- x 2)1/2 , line x=y and the positive

Q7

x-axis
...


x=0 and x=  / 2
...

Sketch the rough graph of y= 4 x  1 , 1≤x≤3 and evaluate the area between
the curve, x -axis and the line x=3
...
and the x-axis
...






Q12
...


Find the area of the region bounded by the curves y = x  1 and
y = – x  1 +1
...


Find the area of the region {(x,y) : 0  y  x2+1 ,0  y  x+1, 0  x 2}

Q15
...
units

Q3

3  / 2 sq
...
units

Q5

9 sq
...
units

Q7

1/2 sq
...
units

Q9

16 2 /3 sq
...
units

Q11

3/2 sq units
...
units
 4 2

Q13

1/2 sq units

Q14

23/6 sq units

Q15

 a2
4



2a 2
Sq
...

3

CHAPTER-9
DIFFERENTIAL EQUATIONS
(1 Mark Questions)
Q1
...


Q2
...

Q3

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