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 .

Hint : p +q = - p1

p+q = - p

pq = q1

pq = qp =1 , q = -2or 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 .

( Ans. The zeroes of the given polynomial are 13 ,

73 and the value of k is

−53

Pair of linear equations in two variablesH.O.T. Questions

(2-marks each)

1. Is the system 2 x+3 y−9=0and 4 x+6 y−18=0 consistent? [Yes,Consistent]

2. For what value k the system kx− y−2=0 and 6 x−2 y−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:

2 x+3 y+8=0 ,4 x+5 y+14=0 [x= -1, y= -2]

(3-marks each)

1. Solve the following pair of equations graphically:

2 x+3 y=12;2 y−1=x [ x = 3, y = 2 ]

2. For what value of ¿b , will the following pair of equations have infinitely many solutions?

x+2 y=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. [6 Km/hr and

4 Km/hr]

5. Solve ; 7 x−2 yx y

=5 ; 8 x+7 yx y

=15 [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 5 x− y=5 and 3 x− y=3 . Determine the co-ordinates

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∧ y : 35x+ y

+ 14x− y

=19 , 14x+ y

+ 35x− 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. Kuldeep2. Jasmer Singh3.4. Rita Sharma

HOTS

INTRODUCTION TO TRIGONOMETRY

1. If 2cosθ-sin θ= x and cosθ-3sin θ = y .Prove that 2x2+y2-2xy = 5.

2. Prove that sinAsecA+tanA−1

+ cosAcosecA+cotA−1 = 1

3. Prove that 2(sin6θ+cos6θ) - 3(sin4θ+cos4θ) +1 = 0

4. Evaluate sec2 (900−θ )−cot2θ2(sin2250+sin2650)

+ 2cos2600 tan2280 tan2620

3 ( sec2430−cot 2470 ) + cot 400

tan 500

Ans = 53

STATISTICS

(HOT QUESTIONS)

Q1.Calculate the mean, median and mode of the following distributions:

No. of goals: 0 1 2 3 4 5

No. of matches: 2 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) 118-126

127-135

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) 150-155

156-160

161-165 166-170 171-175 176-180 181-185

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 (2) 36 (3) 100 arrows

(4)12m (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 :- CirclesQ 1 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  

 

Q 2 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.  

Q 4 ABC is right-angled at B such that BC = 6 cm and AB = 8 cm.

Find the radius of its incircle.  Answer :- 2cm.

  

Q 5 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 CIRCLEHOTS

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 1- cm and semi-circle are drawn with each side of the square as diameter (use π=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 Akhnoor2. 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 VOLUMEQues 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)