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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
...
What is the order and degree of the differential equation,
Q5
...


(4 Mark Questions)
Q1 Find the equation of the curve that passes through the point (1,2) and satisfies the differential equation,


...


Q5 Solve

Q6 Form the differential equation representing the family of ellipses

Q7 If

, prove that

Q8 Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of
x-axis
...


(6 Marks)
 
 
 
Prove that 2(a  b ) = (a  b )  (a + b )
...



Hence find the area of a parellogram whose diagonals are the vectors a = 3i + j  2 k

and b = i  3 j  4 k

 

  

2
...
b
...
Prove that a   2 b  c
3
...
Prove that
4
cos 2 a  cos 2 b  cos 2   cos 2  
3
 

4
...
b and c are three mutually perpendicular vectors of equal magnitude, find the angle

  
between a and (a  b  c)

1
...


Let a = 3iˆ - ˆj and b = 2iˆ + ˆj - 3kˆ
...
Where c is parallel

to a and d is perpendicular to a
...


Geometrical interpretation : Twice the area of a given parallelogram is equal to the
area of a parallelogram whose adjacent sides are the diagonals of the give is
parallelogram
...
of | | gm
...
unit
2
...


cos 1

5
...
Find the vector equation of the line through the point (5,2,-4) and which is parallel to the
vector 3î + 2ĵ - 8k
...
Find the distance of the plane 2x-3y+4z-6=0 from the origin
...
Find the angle between two planes:
3x - 6y + 2z = 7 and 2x + 2y – 2z = 5
4
...
What angle does it make with the z – axis ?
5
...


(4 marks)
1
...
Find the equation of the plane through the line of intersection of the planes x + y + z = 1
and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z =0
...
Find the vector equation of the line passing through (1,2,3) and parallel to the planes r
...
( 3î + ĵ + k ) = 6
...
Find the vector equation of the plane passing through the points A(1,-2,5) , B(0,-5,-1) and
C(-3,5,0)
...
Find the image of the point (3,-2,1) in the plane 3x - y + 4z = 2
...
If a line makes A,B,C with the coordinate axes, then find the value of cos 2A+
cos 2B+cos 2C
...
-1
2
...
(2ij+k)+3=0
Ans
...
A plane meets the coordinates axes inA, B,C such that the centroid of the
triangle ABC is the point (a,b,c) show that the equation of the plane is
x/a+y/b+z/c=3
4
...
If

be any two vectors, show that

6
...
3√34/17
7
...
D
...
Find the distance of the point (6, 5, 9) from plane passing through the points (3, -1, 2),
(5, 2, 4) and (-1, -1, 6)
...


Find the image of the point (1, 6, 3) in the line

x y 1 z  2

...
(1,0,7)

10
...

2
4
5

CHAPTER-12
LINEAR PROGRAMMING
(6 Marks )
1
...

2
...
It is known that to make a chair requires 5 square
feet of wood and 10 man-hours and yields a profit of Rs
...
80/-
...

3
...
Food P costs Rs
...
and Food Q cost Rs
...
Food P contains
3 units/kg
...
of Vitamin A and 2 unit/kg of Vitamin B
...

4
...
Each product is processed on two
machines M1 and M2
...
on M2 ; product B requires 4 minutes on M1 and 4 min
...
The
machine M1 is available for not more than 8 hrs 20 min
...
during any working day
...
3 and Rs
...
Formulate the problem as a linear
programming problem and find how many products of each type should be
produced by the firm each day in order to get maximum profit
...
A firm manufacturing two types of electric items, A and B, can make a profit of
Rs
...
30 per unit of B
...
The
total supply of these per month is restricted to 210 motors and 300 transforms
...
Formulate the linear programming problem
for maximum profit and solve it graphically
...
A factory uses three different resources for the manufacture or two different
products, 20 units of the resources A, 12 units of B and 16 units of C being
available
...
It is known that the first product gives a profit of 2
monetary units per unit and the second 3
...
How many units of each product should be manufactured for
maximizing the profit/ Solve it graphically
...
A publisher sells a hard cover edition of a text book for Rs
...
00 and a
paperback edition of the same text for Rs
...
00
...
56
...
28
...
9600
...
Both types require 5 minutes of printing time, although hardcover
requires 10 minutes binding time and the paperback requires only 2 minutes
...
How many of each type of book should be produced in order to
maximize profit?
8
...
The
making of a deluxe model requires 2 hrs
...
work
by a semi-skilled man
...
by a semi-skilled man
...
The manufacturers clear profit on deluxe model is Rs
...
10
...

9
...
Each
plant produces three different kinds of soft drinks A, B and C
...

The operating cost per day of running plants at P and Q are respectively
Rs
...
4000
...

10
...
The amount of each nutrient in each of the food (in milligrams per 25
gms) is given in the following table :

Food
F1

F2

0
...
10

Nutrients
Thymine

Phosphorous

0
...
50

Iron

1
...
80

The minimum requirement of the nutrients in the diet are 1
...
50 mg of phosphorous and 10
...
The cost of F 1 is 20 paise per 25
gms while the cost of F2 is 15 paise per 25 gms
...

11
...
The pigs are fed on various products
grown on the farm
...
One unit of products A contains 36 units of X, 3 units of Y, and 20 units of Z
...
The
minimum requirements of X, Y and Z is 108 units, 36 units and 100 units
respectively
...
20 per and product B costs Rs
...

Formulate the above as a linear programming problem to minimize the total
cost, and solve the problem by using graphical method
...
If a youngman rides his motor cycle at 25 km per hour he has to spend Rs 2 per
km
...
per hour, the petrol
cost increase to Rs
...
He has Rs
...
Express this
as a linear programming problem and then solve it graphically
...
An aeroplane can carry a maximum of 200 passengers
...
400 is
made on each first class ticket and a profit of Rs
...
The airline reserves at least 20 seats for first class
...
Determine how many each type of tickets must be sold in order to
maximize the profit for the airline

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