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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
...
Two godowns, A and B, have a grain storage capacity of 100 quintals and 50
quintals respectively
...
The costs of transportation
per quintal from godowns to the shops are given in the following table :
From
Transportation cost per quintal
To

A

B

D

6

4

E

3

2

F

2
...
The Company is to supply oil to three petrol pumps D,E, F whose

requirements are 4500L ,3000L and 3500L , respectively
...
1 per ltr
...

2
...

4
...

6
...

8
...

10
...

12
...

14
...
profit = Rs
...
cost is Rs
...
profit = Rs
...
profit = Rs
...
profit = 16 monetary
units
...
profit = Rs
...
profit = Rs
...
cost = Rs
...
cost = Rs
...
37
4 units of A , 2 units of B , min
...
160/50/3 km at 25 km /h and 40/3 km at 40 km/h
40 of first class, 160 of economy class, Max
...
cost = Rs
...

From B : 50 quintal, 0 quintal, 0 quintal to D, E, F resp
...


Min
...
44,000
From A : 500 L, 3000 L, 3500 L to D, E, F resp
...


CHAPTER-13
PROBABILITY
(4 marks)
1
...
The respective probabilities of their appointments are 0
...
3, 0
...
The
probabilities that research facilities will be enhanced by these people if they are
appointed are 0
...
7, and 0
...
If the research facilities are enhanced,
what is the probability that it was due to the appointment of C
...
:- 4/13
2
...

Find the probability that one is a spade and other is a queen of red colour
...
If A and B are two independent events, show that
4
...
If the occurrence of an odd number in a
single die is considered a success, find the probability that there are odd numbers
of successes
...
:- ½
5
...
Find the probability distribution of number of spades
...
:-

0

1

703/1700

2

741/1700

234/1700

3
22/17000

6
...
Determine the distribution
...
A problem in statistics is given to three students A, B and C, whose chances of
solving it are ½, 1/3 and ¼ respectively
...

Ans
...
From a well shuffled pack of 52 cards, 3 cards are drawn one by one with
replacement
...


Ans
...
Find the probability distribution of the number of face cards when two cards are
drawn without replacement from a well shuffled deck of 52 cards
...
Find the probability distribution of number of doublets in 4 throws of a pair of
dice
...
Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by
one without replacement
...
A speaks truth in 60% of the cases and B in 90% of the cases
...
A factory has three machines I,II and III which produce 30%, 50% and 20%
respectively of the total items of the same variety
...
An item is picked up at random and found
to be defective
...

Ans
...
A random variable X has the following probability distribution:X

0

1

2

3

4

5

6

7

P(X)

0

K

2K

2K

3K

K2

2K2

7K2+K

(i)
Find K (ii) Evaluate P (X<3) , P(X≥6) and P(0 ...
Find the mean of binomial distribution B(4,1/3)
...
A die is thrown five times, if getting an odd number is a success, find the
probability of getting at least four success
...
Probabilities of solving a specific problem independently A and B are
respectively
...
Find the
probability that
i
...


Exactly one of them solves the problem
...
A bag contains (2n+1) coins
...
A coin is picked up at random from the bag
and is tossed
...
Determine the

value of n
...
n= 10
19
...
The engines function
independently of each other with respective probabilities

and
...
Let X denote the event that
the ship is operational and X1 , X2 and X3 denote respectively the events that the
engines E1 , E2 and E3 are functioning then find P


...


20
...
Odds in favour of
A hitting the target (in a single try) are 3:2 and the odds in favour of B hitting the
target (in a single try) are 4:3
...

Ans
...
A and B throw a pair of dice turn by turn, the first who get the sum on dice 9 is
awarded
...

Ans
...
A fair coin is tossed, if head appears then it tossed again, but if a tail appears ,a
dice is rolled
...
2/9
3
...
Find the mean and variance of the probability
distribution of the number of heads
...
2,1
4
...
How many times must he fire so
that the probability of his hitting the target at least once is greater than 2/3?
Ans
...
A man is known to speak truth 3 out of 4 times
...
Find the probability that it is actually a six
...
3/8
6
...
3/16
7
...
What is the probability that he/she will pass in
computer science if it is known that he/she has passes in mathematics?
Ans
...
By examining a chest X–ray, the probability that T
...
is detected is when a
person is actually suffering from TB is 0
...
The probability that the doctor
diagnoses incorrectly that a person has T
...
on the basis of chest X-ray is 0
...
In
a certain city, 1 person in 1000 suffers from T
...
A person is selected at random
random and is diagnosed to have TB
...
A,B and C shot to hit a target
...
5/6
10
...
From the remaining cards of the pack,
two cards are drawn and are found to be hearts
...

Ans
...
If the people having HIV, 90%of the tests detect the disease but 10% go
undetected
...
From A large population of which only 0
...
What is the probability that a person actually has HIV?

12
...
If she gets 5 or 6, she tosses a coin 3 times and notes
number of heads
...
If she obtained exactly one head, what is
the probability that she threw a 1, 2, 3 or 4 with the die?
Ans
...
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls
...
The ball so drawn is found to be red in colour
...

14
...
On the
envelope just two consecutive letters TA are visible
...
A die is tossed thrice
...

Find mean and variance of the probability distribution of number of successes
...
:- Mean = ½, Variance = 5/12
16
...
One ball at random is transferred from the first bag to second bag and
then a ball is drawn from the second bag
...

Ans
...
In bag A there are 5 white and 8 red balls, in bag B, 7 white and 6 red balls and in
bag C, 6 white and 5 red balls
...
Find the
probability that all the three balls are of same colour
...
:- 450/1859
18
...
Find
their respective probabilities of winning, if A starts first
...
:-6/11, 5/11
19
...
Let the probability P that none of the
events E1 ,E2 or E3 occurs satisfy the equations (

)P =

and (

All the given probabilities are assumed to lie in the interval (0,1)
...



...
6
20
...
Two balls are drawn from the bag and both balls are
found white
...

Ans

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Title: HOTS questions mathematics (Can you solve it?)
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