REAL NUMBERS
HOTS
Q1.Show that one and only one of n,n+2 and n+4 is divisible by 3.
Q2.Use Euclid’s division Lemma to show that the cube of any positive integer is of the form
9m,9m+1 or 9m+8.
Q3.Check whether 6𝑛 can end with the digit 0 for any Natural number n.
Q4.A book seller has 420 science books and 130 arts stream books. He wants to stack them in
such a way that each stack has the same number and they take up the least area of the surface:
(i)What is the maximum number of science books that can be placed in each stack for this
purpose?
(ii)What is the mathematical concept used to solve the above problem?
(iii)If the book seller make a stack, then which kinds of quality are shown by the book seller?
Q5.Show that √2+√3 is an irrational number.
HOTS QUESTIONS OF PLOYNOMIAL
1. Find p and q if p and q are zeros of the quadratic polynomials x2 + px + q .
𝑝
Hint : p +q = - 1
p+q = - p
𝑞
pq = 1
pq = q
p =1 , q = -2
or p=0 , q =0
2. If two zeros of the polynomials x4 – 6x3 – 26x2 +138x -35 are 2-√3 and 2+√3 , find all
the zeros .
( Ans. Zeros p(x) are 2-√3 and 2+√3 , -5 and 7)
3. On dividing 3x3+4x2+5x-13 by a polynomial g(x), the quotient and remainder were 3x+10
and 16x-43 respectively. Find the polynomial g(x).
(Ans. X2-2X+3 )
4. For which values of a and b , are the zeros of g(x) = x3+2x2+a, also the zeros of the
polynomial f(x) =x5-x4-4x3+3x2+3x+b ? Which zeros of f(x) are not the zeros of g(x)?
(Ans. 1 and 2 are the zeros of g(x) which are not the zeros f(x) and this happens
when a= -2 , b= -2)
5. If one zero of the polynomial 3x2- 8x –( 2k + 1) is seven times the other , find both zeroes
of the polynomial and the value of k .
1
( Ans. The zeroes of the given polynomial are 3 ,
7
3
and the value of k is
−5
3
Pair of linear equations in two variables
H.O.T. Questions
(2-marks each)
1. Is the system 2𝑥 + 3𝑦 − 9 = 0 and 4𝑥 + 6𝑦 − 18 = 0 consistent? [Yes,Consistent]
2. For what value k the system 𝑘𝑥 − 𝑦 − 2 = 0 and 6𝑥 − 2𝑦 − 3 = 0 have no solution.
[K=3, k≠ 4]
3. Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio
becomes 4:5. Find the numbers.
[40 and 48]
4. Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which
y = mx + 3.
[x = -2, y=5, m=-1]
5. Solve the following pair of equations by cross multiplication method:
2𝑥 + 3𝑦 + 8 = 0, 4𝑥 + 5𝑦 + 14 = 0
[x= -1, y= -2]
(3-marks each)
1. Solve the following pair of equations graphically:
2𝑥 + 3𝑦 = 12 ; 2𝑦 − 1 = 𝑥
[ x = 3, y = 2 ]
2. For what value of 𝑎𝑛𝑑 𝑏 , will the following pair of equations have infinitely many
solutions?
𝑥 + 2𝑦 = 1
(𝑎 − 𝑏)𝑥 + (𝑎 + 𝑏)𝑦 = 𝑎 + 𝑏 − 2
[a = 3 , b= 1]
3. The sum of the digits of a two digit number is 9. Also nine times the number is twice the
number obtained by reversing the order of the number. Find the number.
[18]
4. Ritu can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find her
speed of rowing in still water and speed of the current.
4 Km/hr]
7 𝑥−2𝑦
5. Solve ;
=5;
𝑥𝑦
8 𝑥+7 𝑦
𝑥𝑦
= 15
[6 Km/hr and
[x = 1, y=3]
(4- marks each)
1 Places A and B are 100 km apart on a highway. One car starts from A and another from B
at the same time. If the cars travel in the same direction at different speeds, they meet
in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds
of the two cars?
[60 Km & 40
Km]
2 A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go
40 km upstream and 55 km down-stream. Determine the speed of the stream and that
of the boat in still water.
[8 Km/hr & 3
Km/hr]
3 2 women and 5 men can together finish an embroidery work in 4 days, while 3
women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish
the work, and also that taken by 1 man alone.
[18 days & 36 days]
4 Draw the graphs of the equations 5𝑥 − 𝑦 = 5 and 3𝑥 − 𝑦 = 3 . Determine the coordinates of the vertices of the triangle formed by these lines and the y- axis.
[Co-ordinates: (1,0), (0,-3),(0,-5)]
5 Solve for 𝑥 𝑎𝑛𝑑 𝑦 ∶
35
𝑥+𝑦
+
14
𝑥−𝑦
= 19,
14
𝑥+𝑦
35
+ 𝑥−𝑦 = 37
[x = 4, y = 3]
SIMILAR TRIANGLES
(HOTS QUESTION)
1. ABC is a right-angled triangle, right-angled at A. A circle is inscribed in it. The
lengths of the two sides containing the right angle are 6cm and 8 cm. Find the
radius of the in circle.
(Ans: r=2
2. ABC is a triangle. PQ is the line segment intersecting AB in P and AC in Q such
that PQ parallel to BC and divides triangle ABC into two parts equal in area. Find
BP: AB.
3. In a right triangle ABC, right angled at C, P and Q are points of the sides CA and
CB respectively, which divide these sides in the ratio 2: 1.
Prove that: (i) 9AQ2= 9AC2 +4BC2
(ii) 9BP2= 9BC2 + 4AC2
(iii) 9(AQ2+BP2) = 13AB2
4. In an equilateral triangle ABC, the side BC is trisected at D.
Prove that 9AD2 = 7AB2
5. P and Q are the mid points on the sides CA and CB respectively of triangle ABC
right angled at C. Prove that 4(AQ2 +BP2) = 5AB2
6. Find the length of the second diagonal of a rhombus, whose side is 5cm and one of
the diagonals is 6cm.
(Ans: 8cm)
GROUP:1. Kuldeep
2. Jasmer Singh
3.
4. Rita Sharma
HOTS
INTRODUCTION TO TRIGONOMETRY
1. If 2cos 𝜃-sin 𝜃 = x and cos 𝜃-3sin 𝜃 = y .Prove that 2x2+y2-2xy = 5.
2. Prove that
3. Prove that
4. Evaluate
𝑠𝑖𝑛𝐴
𝑠𝑒𝑐𝐴+𝑡𝑎𝑛𝐴−1
+
𝑐𝑜𝑠𝐴
𝑐𝑜𝑠𝑒𝑐𝐴+𝑐𝑜𝑡𝐴−1
= 1
2(sin6𝜃+cos6𝜃) - 3(sin4𝜃+cos4𝜃) +1
𝑠𝑒𝑐 2 (900 −𝜃)−𝑐𝑜𝑡 2 𝜃
2(𝑠𝑖𝑛2 250 +𝑠𝑖𝑛2 650 )
+
= 0
2𝑐𝑜𝑠 2 600 𝑡𝑎𝑛2 280 𝑡𝑎𝑛2 620
3(𝑠𝑒𝑐 2 430 −𝑐𝑜𝑡 2 470 )
+
5
Ans = 3
𝑐𝑜𝑡400
𝑡𝑎𝑛500
STATISTICS
(HOT QUESTIONS)
Q1.Calculate the mean, median and mode of the following distributions:
No. of goals:
0
No. of matches:
2
1
2
3
4
5
4
7
6
8
3
Ans- Mean= 2.77, median=3, mode=4
Q-2- The mean of the following distribution is 50.if the sum of the frequencies is 120, find the values of
f1 & f2.
Xi
10
30
50
70
90
Fi
17
f1
32
f2
19
Ans. f1=28, f2=24
Q3. The length of 40 leaves of a plant are measured correct to the nearest millimeter, the data obtained
is represented in the following table:
Length(in mm)
118126
127135
136-144
145-153
154-162
163-171
172-180
No. of leaves
3
5
9
12
5
4
2
Find the median length of the leaves.
Ans. 146.75
Q4. The following table shows the distribution of the heights of a group of factory worker.
Height(in cm)
150155
156160
161-165
166-170
171-175
176-180
181185
No. of workers
6
12
18
20
13
8
6
Convert the distribution to a more than cumulative frequency distribution and draw its ogive. Hence
obtain the median height from the graph.
Ans. 166.5
HOTS
CHAPTER: 4
QUADRATIC EQUATIONS
1. Out of a group of swans, 7/2 times the square root of the total number are lying on the shore of
a pond. The two remaining ones are swinging in water. Find the total numbers of swans
2. one-fourth of a herd of camels was seen in the forest. Twice the square root of the herd had
gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total
number of camels.
3. The angry Arjun carried some arrows for fighting with Bheeshm. With half the arrows, he cut
down the arrows thrown by Bheeshm on him and with six others arrows he killed the rath driver
of Bheeshm. With one arrow each he knocked down respectively the rath, flag and the bow of
Bheeshm . Finally, with one more than four times the square root of arrows he laid bheeshm
unconscious on an arrow bed. Find the total number of arrows Arjun had.
4. A peacock is sitting on the top of a pillar , which is 9 meter high. From a point 27m away from
the bottom of a pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake
peacock pounces on it. If their speeds are equal, at what distance from the whole is the snake
caught?
5. A pole has to be erected at a point on the boundary of a circular park of diameter 13m in such a
way that the difference of its distances from two diametrically opposite fixed gates A & B on the
boundary is 7m. Is it the possible to do so? If yes, at what distances from the two gates should
the pole be erected?
Answers: (1) 16
(4)12m
(2) 36
(3) 100 arrows
(5) at a distance of 5m from the gate B.
HOTS QUESTIONS
Q1)
Find the sum of all natural number’s between 100 & 1000 which are multiples of 7?
Ans) 70336
Q2)
200 logs are stacked in the following manner : 20 logs in the bottom row ,19 logs in the
next row ,18 in the row next to it & so on. In how many rows the 200 logs are placed
and how many logs are in the top row?
Ans)
16 rows: 5 logs on the top row.
Q3)
The sum of n terms of a progression is 3n2 + 4n. Is this progression in A.P. If so find the
A .P & the sum of its rth term.
Ans)
7,13,19,25,…. : (3r2+4r).
Q4)
A theif runs away from a Police Station with a uniform speed of 100 m/min.After a
minute a Policeman runs behind the theif to catch him. He goes at a speed of 100
m/min in the first minute and increases his speed 10m each succeeding minute . After
how many minutes,the Policeman will catch the thief?
Ans)
5 minutes
Prepared By :- Group IX
Group Leader :- Satish Sharma
Meera Gupta
Prabjeet Kaur
Shabha
(CONSTRUCTION) --------------- HOTS
1.Draw a circle circumscribing a triangle ABC .construct a pair of tangents from a point
outside the circle.
2.let ABC be a right triangle in which AB=6cm, BC=8cm and ˂B=90 .BD is the
perpendicular from B on AC.The circle through B,C D is drawn . Construct a
tangents from a to this circle .
Q.3. Draw a right triangle ABC in which BC = 12 cm , AB = 5cm and ˂B=900.
Construct a triangle similar to it and of scale factor ⅔. Is the new triangle
also a right triangle? .
Q.4 Draw a triangle ABC with side BC = 7cm, <B = 450 , <A = 1050 . Then, construct
a triangle whose sides are 4/3 times the corresponding sides of a triangle
ABC.
Q.5 Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure
the two parts?
HOTS
Chapter :- Circles
Q1
In figure, AQ and AR are tangents from A to the circle with
centre O. P is a point on the circle. Prove that AB + BP = AC +
CP
Q2
In figure, if AB = AC, prove that BE = EC.
Q 3 Prove that the tangents drawn at the ends of a diameter of a
circle are parallel.
Q4
ABC is right-angled at B such that BC = 6 cm and AB = 8 cm.
Find the radius of its incircle.
:- 2cm.
Q5
Answer
Two tangents TP and TQ are drawn to a circle with centre O
from an external point T. Prove that PTQ = 2 OPQ.
Prepared By
Group :- XII
Group Leader :- Amit Kumar ( KV Nangal Bhur)
Kashmiri Lal Kyashap ( KV No.- 1, Akhnoor)
Rakesh Sharma ( KV No. 1 , Jammu)
Rakesh Kumar ( KV No.2 , Jammu)
AREA RELATED TO CIRCLE
HOTS
Q1.
In the given figure, ABC is a quadrant of a circle of radius 14 cm and a semicircle is
drawn with BC as diameter. Find the area of the shaded region. (Answer = 98 cm2)
Q2.
The area of an equilateral triangle is 17320.5 cm2. Taking each angular point as center a
circle is described with radius equal to half the length of the side of triangle as shown in
figure. Find the area of triangle not included in circle.(
3  1.73205 )
(Answer 1620.5 cm2)
Q3.
Find the area of the shaded region in the figure given here, if BC = BD = 8 cm, AC = AD =
15 cm, and O is ( Answer = 106.87) the centre of the circle.
Q4.
Find the area of the region shown shaded in the given figure.
Q5.
In the given figure, find the area of the shaded design, where ABCD is a square of side 1cm and semi-circle are drawn with each side of the square as diameter (use   3.14 )
(Answer 57 cm2)
GROUP NO. 14
GROUP LEADER
Mr. Lakhbir Singh TGT (Maths) KV No.1 Udhampur
GROUP MEMBERS
1. Mr. Ashok Kumar Bhasin TGT (Maths), KV No. 2 Akhnoor
2. Mr. Vijay Kumar Kashyap TGT (Maths), KV Bantalab
3. Mr. Peerzada Nayeem TGT (Maths), KV Bandipora (Contractual)
4. Mr. Vikas Sharma TGT (Maths), KV Chenani (Contractual)
GROUP 15
HOTS OF CHAPTER SURFACE AREA AND VOLUME
Ques 1. A cylindrical bucket 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is
emptied on the ground and a conical heap is formed. if the height of conical heap is 24 cm, find the
radius and slant height of the heap.
Ans :12√13 cm
Ques 2. Water in a canal 6 m wide and 1.5 m deep is flowing with a speed 10 km/hr. How much area will
it irrigate in 30 mintues,if 8 cm of standing water is needed? Also find the area in hectares.
Ans : 5625000 m2, 56.25 hec.
Ques 3. A farmer connects a pi[pe of internal diameter 20 cm from a canal into a cylindrical tank in her
field , which is 10 m in diameter and 2 m deep. If water flows through pipe at the rate of 3 km/h , in how
much time will the tank be filled?
Ans : 100 minutes
Ques 4. A toy is in the form of a cone of radius 3.5 cm mounted on a hemi-sphere of same radius. The
total height of the toy is 15.5 cm. Find the total surface area of the toy.
[Take π = 3.14]
Ans :214.5 cm3
Ques 5.Marbles of the 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm ,containing some
water. Find the number of marbles that should be dropped into the beaker so that water rises by 5.6
cm?
Ans: 150
Self Evaluation/Hots
Q.1 A dice has its Six faces marked 0,1,1,1, 6, 6. Two such dice are thrown together and total
score recorded.
(A) How many different scores are possible?
(Ans. 5 scores)
(B) What is the probability of getting a total of seven?
(Ans. 1/3 )
Q2. A child game has 8 triangles of which three are blue and rest are red and ten squares of
which six are blue and rest are red. One piece is lost at random. Find the probability of that is
(A) A square
(Ans. 5/9,)
(B) A triangle of red colour.
(Ans. 5/18)
Q.3 A group of scientific men, reported 11705 boys and 11527 girls. If this is a fair sample from
the general population. What is the probability that a child to be born will be boy?
(Ans. 11705/23232)
Q4.The probability of selecting a red ball at random from a jar that contain only red, blue and
orange ball is 1/4. The probability of selecting a blue ball at random from the same jar is 1/3. If
this jar contains 10 orange balls, then what is the total number of balls in the jar?
(Ans. 24)
Q.5 There are 33 cards of same size in a bag on which numbers 1 to 33 are written. One card is
taken out of the bag at random. Find the probability that the number on the selected card is
not divisible by 3
(Ans.2/3)