TIME: TEST NO. - 1 MARKS

TIME: TEST NO. - 1 MARKS:

ALL THE BEST DEEPAK SIR 9811291604

Time : 2h REAL NUMBER.-1 each question 4marks



Time : 2h REAL NUMBER.-2 each question 4marks

1. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify

your answer.(exemplar)

2. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural

number. Justify your answer.

3. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in

any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

4. The numbers 525 and 3000 are both divisible only by 3, 5, 15, and 25 and 75. What is

HCF (525, 3000)? Justify your answer.

5. Explain why 3 × 5 × 7 + 7 is a composite number.

6. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

7. Without actually performing the long division, find if 987/10500 will have terminating or non-

terminating (repeating) decimal expansion. Give reasons for your answer.

8. A rational number in its decimal expansion is 327.7081. What can you say about the prime

factors of q, when this number is expressed in the form p/q? Give reasons.

9. Can 6n have 5 as its factor explain?

10. Show that every positive odd integer is of the form (6 q +1) or (6q+ 3) or (6q + 5) for some

integer q.



1. Show that the square of an odd positive integer is of the form 8m + 1, for some whole number m.

2. Prove that √2 + √3 is irrational.

3. Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.

4. Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

5. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

6. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

7. Show that the square of any odd integer is of the form 4q + 1, for some integer q.

8. If n is an odd integer, then show that n2 – 1 is divisible by 8.

9. Prove that if x and y are both odd positive integers, then x2 + y2 is even but not divisible by 4.

10. Use Euclid’s division algorithm to find the HCF of 441, 567, 693.

11. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving

remainders 1, 2 and 3, respectively.

12. Prove that √3 + √5 is irrational.

13. Show that 12n cannot end with the digit 0 or 5 for any natural number n.

14. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45

cm, respectively. What is the minimum distance each should walk so that each can cover the same distance

in complete steps?

15. Prove that √ p + √q is irrational, where p, q are primes.

16. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

17. Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any

positive integer

18. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.

19. Prove that one of any three consecutive positive integers must be divisible by 3.

20. For any positive integer n, prove that n3 – n is divisible by 6.



1. The values of the remainder r, when a positive integer a is divided by 3 are 0 and 1 only. Justify

your answer

2. Can the number 6n, n being a natural number, end with the digit 5? Give reasons.

3. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify

your answer.

4. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or

false? Give reasons.

5. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or

false”? Justify your answer.

6. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural

number. Justify your answer.

7. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any

form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

8. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525,

3000)? Justify your answer.

9. Explain why 3 × 5 × 7 + 7 is a composite number.

10. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

11. Without actually performing the long division, find if 987/10500 will have terminating or non-

terminating (repeating) decimal expansion. Give reasons for your answer.

12. A rational number in its decimal expansion is 327.7081. What can you say about the prime

factors of q, when this number is expressed in the form p/q ? Give reasons.



Time : 2h LINEAR EQUATION IN TWO VERIABLES-1 each question 4marks

1. For which values of p and q, will the following pair of linear equations have infinitely many

solutions? 4x + 5y = 2 ; (2p + 7q) x + (p + 8q) y = 2q – p + 1.

2. Solve the following pair of linear equations: 21x + 47y = 110; 47x + 21y = 162

3. Draw the graphs of the pair of linear equations x – y + 2 = 0 and 4x – y – 4 = 0. Calculate the

area of the triangle formed by the lines so drawn and the x-axis.4

4. For which value(s) of λ , do the pair of linear equations λx + y = λ2 and x + λy = 1 have

(i) no solution? (ii) infinitely many solutions? (iii) a unique solution?4

5. For which value(s) of k will the pair of equations kx + 3y = k – 3 ; 12x + ky = k have no solution?4

6. For which values of a and b, will the following pair of linear equations have infinitely many solutions? x

+ 2y = 1 (a – b)x + (a + b)y = a + b – 2 4

7. The angles of a cyclic quadrilateral ABCD are A = (6x + 10)°, B = (5x)° C = (x + y)°, D = (3y – 10)°

Find x and y, and hence the values of the four angles.4

8. The angles of a triangle are x, y and 40°. The difference between the two angles x and y is 30°. Find x

and y. 4

10. Two years ago, Salim was thrice as old as his daughter and six years later, he will be four years older

than twice her age. How old are they now? 4

11. The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be

equal to the sum of the ages of his children. Find the age of the father.4

12. 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.4

13. There are some students in the two examination halls A and B. To make the number of students equal in

each hall, 10 students are sent from A to B. But if 20 students are sent from B to A, the number of students in

A becomes double the number of students in B. Find the number of students in the two halls. 4

14. A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an

additional charge for each day thereafter. Meha paid Rs 22 for a book kept for six days, while Unnati paid

Rs 16 for the book kept for four days. Find the fixed charges and the charge for each extra day.4

15. In a competitive examination, one mark is awarded for each correct answer while ½ mark is deducted

for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she

answer correctly? 4




1) Jamila sold a table and a chair for Rs 1050, thereby making a profit of 10% on the table and 25% on

the chair. If she had taken a profit of 25% on the table and 10% on the chair she would have got Rs 1065.

Find the cost price of each.

2) It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for

4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled.How long would it take

for each pipe to fill the pool separately?[ 20, 30]

3) Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour if she

travels 2 km by rickshaw, and the remaining distance by bus. On the other hand, if she travels 4 km by

rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and

of the bus.

4) A person, rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km

upstream as in going 40 km downstream. Find the speed of the stream.

5) Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour ifshe

travels 2 km by rickshaw, and the remaining distance by bus. On the other and, if she travels 4 km by

rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and

of the bus.

6) A person, rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km

upstream as in going 40 km downstream. Find the speed of the stream. A motor boat can travel 30 km

upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the

speed of the boat in still water and the speed of the stream.

7) A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting

5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

8) A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket

as on a full ticket. One reserved first class ticket from the station A to B costs Rs 2530. Also, one reserved

first class ticket and one reserved first class half ticket from A to B costs Rs 3810. Find the full first class

fare from station A to B, and also the reservation charges for a ticket.

9) A shopkeeper sells a saree at 8% profit and a sweater at 10% discount, thereby, getting a sum Rs

1008. If she had sold the saree at 10% profit and the sweater at 8% discount, she would have got Rs 1028.

Find the cost price of the saree and the list price (price before discount) of the sweater.

10) Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of

8% per annum and 9% per annum, respectively. She received Rs 1860 as annual interest. However, had

she interchanged the amount of investments in the two schemes, she would have received Rs 20 more as

annual interest. How much money did she invest in each scheme?

11) Vijay had some bananas, and he divided them into two lots A and B. He sold the first lot at the rate

of Rs 2 for 3 bananas and the second lot at the rate of Re 1 per banana, and got a total of Rs 400. If he had

sold the first lot at the rate of Re 1 per banana, and the second lot at the rate of Rs 4 for 5 bananas, his total

collection would have been Rs 460. Find the total number of bananas he had .

12) It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for

4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled.How long would it take

for each pipe to fill the pool separately?




1. Taxi charges in a city consist of fixed charges per day and the remaining depending upon the

distance travelled in kilometres. If a person travels 110 km, he pays Rs 690, and for travelling 200

km, he pays Rs 1050. Find the fixed charges per day and the rate per km.

2.HadAjita scored 10 more marks in her mathematics test out of 30 marks, 9 times these marks

would have been the square of her actual marks. How many marks did she get in the test?

3. For which values of p and q, will the following pair of linear equations have infinitely many

solutions? 4x + 5y = 2 ; (2p + 7q) x + (p + 8q) y = 2q – p + 1.

4.Solve the following pair of linear equations: 21x + 47y = 110; 47x + 21y = 162

5. Draw the graphs of the pair of linear equations x – y + 2 = 0 and 4x – y – 4 = 0. Calculate the

area of the triangle formed by the lines so drawn and the x-axis.

6.For which value(s) of λ , do the pair of linear equations λx + y = λ2 and x + λy = 1 have

(i) no solution? (ii) infinitely many solutions? (iii) a unique solution?

7. For which value(s) of k will the pair of equations kx + 3y = k – 3 ; 12x + ky = k have no

solution?

8. Solve for x and y

1/ 2[2x+3y] + 12/7[3x-2y] = ½ , 7/ [2x+3y] + 4/[3x-2y] = 2

9. Solve for p and q [p+q ] / pq =2 and [p-q ] / pq =6

10. Solve for x and y ,

2/[3x+2y] + 3/[3x-2y]=17/5 ;

5/[3x+2y] + 1/[3x-2y]=2

11.Solve for x and y .

6 /[x+y] = 7/ [x-y] + 3 ;

1/ 2[x+y] = 1/ 3[x-y]

12. Solve for x and y ;

2/√x + 3/√y =2 ;

4/√x - 9/√y =2 ;




1) If from twice the greater of two nos., 20 is subtracted, the result is the other no. If from

twice the smaller no., 5 is subtracted, the result is the greater no. Find the nos.

2) 27 pencils and 31 rubbers together costs Rs. 85 while 31 pencils and 27 rubbers together

costs Rs. 89. Find the cost of 2 pencils and 1 rubber.

3) The area of a rectangle remain the same if its length is increased by 7 cm and t he breadth is

decreased by 3 cm. The area remains unaffected if length is decreased by 7 cm and the

breadth is increased by 5 cm. Find length and breadth.

4) A two digit no. is obtained by either multiplying the sum of the digits by 8 and adding 1; or

by multiplying the difference of the digits by 13 and adding 2. Find the no. How many such

nos. are there.

5) A no. consists of three digits whose sum is 17. The middle one exceeds the sum of other two

by 1. If the digits are reversed, the no. is diminished by 396. Find the no.

6) A boatman rows his boat 35 km upstream and 55 km down stream in 12 hours. He can row

30 km. upstream and 44 km downstream in 10 hours. Find the speed of he stream and that of

the boat in still water. Hence find the total time taken by the boat man to row 50 cm

upstream and 77 km downstream.

7) Ashok covers 60 km in 1½ hours with the wind and 2 hours against the wind. Find the speed

of the Ashok and speed of the wind.

8) The distance between school and metro station is 300 m. Kartikay starts running from

school towards metro station, while Ashu starts running from metro station to school. They

meet after 4 minutes. Had Kartikay doubled his speed and Ashu reduced his speed to third

of the original they would have met one minute earlier. Find their speeds.

9) Puru chase Vinayak who is 5 km ahead. Vinayak is travelling at a speed of 80 km/h and

Puruchase at an average speed of 90 km/h. After how much time Puru met Vinayak.

10) In a unit-test the no. of hose that passed and the no. of these that failed were in the ratio 3:1.

had 8 more appeared and 6 less passed, the ratio of passes to failures would have been 2:1.

Find how many appeared?

11) Form a pair of linear equations for : The sum of the numerator and denominator of fraction

is 3 less than twice the denominator. If the numerator and denominator both are decreased

by 1, the numerator becomes half the denominator.

12) Amar gives Rs. 9000 to some athletes of a school as scholarship every month. Had there

been 20 more athletes each would have got Rs. 160 less. Form a pair of linear equations for

this.




1. Find the value of k so that the equations x + 2y = – 7, 2x + ky + 14 = 0 will represent

concident lines.

2. Give linear equations which is coincident with 2 x + 3y - 4 = 0

3. What is the value of a for which (3, a) lies on 2x – 3y = 5

4. The sum of two natural nos. is 25 of their difference is 7. Find the nos.

5. Dinesh in walking along the line joining (1, 4) and (0, 6), Naresh is walking along the line

joining (3, 4,) and (1,0). Represent on graph and find the point where both of them cross

each other. (graphically)

6. Solve the pair or linear equations x – y = 2 and x + y = 2. Also find p if p = 2x + 3

7. For what value of K the following system of equation are parallel.

2x + Ky = 10 3x + (k + 3) y = 12

8. Form a pair of linear equations for the following situation assuming speed of boat in still

wateras ‘x’ and speed of stream ‘y’ : A boat covers 32 km upstream and 36 km downstream

in 7 /hours. It also covers 40 km upstream and 48 km downstream in 9 hours.

9. Check graphically whether the pair of linear equations 3x + 5y = 15, x – y = 5 is consistent.

10. For what value of p the pair of linear equations(P + 2) x – (2 p + 1)y = 3 (2p – 1) , 2x – 3y =

7 has unique solution.

11. Find the value of K so that the pair of linear equations :

(3 K + 1) x + 3y – 2 = 0 (K2 + 1) x + (k–2)y – 5 = 0 is inconsistent.

12. Solve x – y = 4, x + y = 10 and hence find the value of p when y = 3 x –p

13. Determine the value of K for which the given system of o linear equations has infinitely

many solutions: Kx + 3y = K – 3 , 12x + Ky = K

14. find the values of and for which and following system of linear equations has infinite no of

solutions : 2x + 3y = 7 2 x + y = 28.

15. Solve for x and y : [ x +1 ]/ 2 + [y-1 ]/3 = 8 , [ x +1 ]/ 3 + [y-1 ]/2 = 8

16. Solve for x and y 139x + 56y = 641 , 56x + 139y = 724

17. Solve for x and y, 5 / [x + y ] + 1/ [x – y ] =2 , 15 / [x + y ] - 5/ [x – y ] = -2

18. Solve for x and y 37x + 43y = 123 43x + 37y = 117

19. Check graphically whether the pair of eq. 3x + 2y – 4 = 0 and 2x – y – 2 = 0 is consistent.

Also find the coordinates of the points where the graphs of the lines of equations meet the y-

axis.




MULTIPLE CHOICE QUESTIONS

1. Every linear equation in two variables has ___ solution(s). (a) no (b) one (c) two (d) infinitely many

2. a1/a2 = b1/b2=c1/c2 is the condition for

(a) intersecting lines (b) parallel lines (c) coincident lines (d) none

3. For a pair to be consistent and dependent the pair must have

(a) no solution (b) unique solution (c) infinitely many solutions (d) none of these

4. Graph of every linear equation in two variables represent a ___

(a) point (b) straight line (c) curve (d) triangle

5. Each point on the graph of pair of two lines is a common solution of he lines in case of ___

(a) Infinitely many solutions (b) only one solution (c) no solution (d) none of these

6. Which of he following is the solution of the pair of linear equations 3x – 2y = 0, 5y – x = 0

(a) (5, 1) (b) (2, 3) (c) (1, 5) (d) (0, 0)

7. One of the common solution of ax + by = c and y-axis is _____

(a) (0, c/b) (b) (0,b/c ) (c) , 0 , (c/ b ) (d) (0, c/ b)

8. If the value of x in the equation 2x – 8y = 12 is 2 then the corresponding value of y will be

(a) –1 (b) +1 (c) 0 (d) 2

9. The pair of linear equations is said to be inconsistent if they have

(a) only one solution (b) no solution (c) infinitely many solutions. (d) both a and c

10. On representing x = a and y = b graphically we get ____

(a) parallel lines (b) coincident lines (c) intersecting lines at (a, b) (d) intersecting lines at (b, a)

11. How many real solutions of 2x + 3y = 5 are possible

(a) no (b) one (c) two (d) infinitely many

12. The value of k for which the system of equation 3x + 2y = – 5, x – ky = 2 has a unique solutions.

(a) K = 2/ 3 (b) K 2/3 (c) K = -2 /3 (d) K - 2/3

13. If the lines represented by the pair of linear equations 2x + 5y = 3, 2(k + 2) y + (k + 1) x = 2k

are coincident then the value of k is ____ (a) –3 (b) 3 (c) 1 (d) –2



Time : 2h POLYNOMIALS-1 each question 4marks

Section-A

1. The zeroes of the quadratic polynomial x2 + 99x + 127 are

(A) both positive (B) both negative (C) one positive and one negative (D) both equal

2. The zeroes of the quadratic polynomial x2 + k x + k, k ≠ 0,

(A) cannot both be positive (B) cannot both be negative (C) are always unequal (D) are always equal

3. If the zeroes of the quadratic polynomial ax2 + bx + c, c ≠ 0 are equal, then

(A) c and a have opposite signs (B) c and b have opposite signs (C) c and a have the same sign (D) c and b

havethe same sign

4. If one of the zeroes of a quadratic polynomial of the form x2+ax + b is the negative of the other, then it

(A) has no linear term and the constant term is negative.

(B) has no linear term and the constant term is positive.

(C) can have a linear term but the constant term is negative.

(D) can have a linear term but the constant term is positive.

5. The number of polynomials having zeroes as –2 and 5 is

(A) 1 (B) 2 (C) 3 (D) more than 3 Section-B

1. Find the zeroes of 2x3 – 11x2 + 17x – 6.

2. Find the quadratic polynomial, the sum and the product of whose zeroes are 1/2, and –2 .

3. Find the values of m and n for which x = 2 and –3 are zeroes of the polynomial: 3x2 – 2mx + 2n.

4. Check whether x2 + 4 is factor of x4 + 9x2 + 20

Section-C

5. Divide the polynomial (x4 + 1) by (x – 1) and verify the division algorithm.

6. Find all zeroes of x4 – 3x3 – 5x2 + 21x – 14, if two of its zeroes are √7 and – √7

7. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4

respectively, find g(x).

8. Given that √2 is a zero of the cubic polynomial 6x3 + √2 x2 – 10x – 4 √2 , find its other two zeroes.

9. Find k so that x2 + 2x + k is a factor of 2x4 + x3 – 14 x2 + 5x + 6. Also find all the zeroes of the two

polynomials.



Time : 2h POLYNOMIALS-2 each question 4marks

1. The graph of y=f(x) is given below. Find the number of zeroes of f(x)

2. Write the zeroes of the polynomial x2 -2x + 4.

3. Find a quadratic polynomial, the sum and product of whose zeroes are 0 and √5 respectively.

4. Find the quadratic polynomial, the sum and product of whose zeroes are 4 and 1, respectively

5. If a and b are zeros of the quadratic polynomial f(x)= x2-5x+4, find value of 1/α + 1/β-2βα

6. Find the zeroes of the quadratic polynomial 4 √3 x2 + 5 x - 2 √3 and verify the relationship

between the zeroes and the coefficients.

7. Find the zeroes of the quadratic polynomial 4u2 + 8u and verify the relationship between the

zeroes and the coefficients

8. Find the quadratic polynomial, the sum and product of whose zeroes are √2 and √3 respectively.

9. If α and β are the zeros of the given quadratic polynomial f(x)= 5x2 - 7x + 1, find the

value 1/β + 1/α

10. Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the

Coefficients

11. Find the remainder when p(x)= x3-6x2+2x-4 when divided by 1 - 2x.

12. Find the remainder when x51+51 is divided by (x+1).

13. Find all the integral zeros of x3 -3x2 - 2x + 6

14. Obtain all zeros of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeros are √5/√3 and - √5/√3

15. If (x - 2) and [x – ½ ] are the factors of the polynomials qx2 + 5x + r prove that q = r

16. If the zeroes of the polynomial are 3x2 − 5x + 2 are a+ b and a- b, find a and b.

17. On dividing 2x2 + 3x + 1 by a polynomial g(x), the quotient and the remainder were 2x-1 and 3

respectively. Find g (x).



TIME : 2h CH 1-2-3 EACH QUESTION 4MARKS

1. Find the zeros of the polynomial f(x) = 2x2 + 5x -12 and verify the relation between its

zeros and coefficients.

2. Find the zeros of the polynomial f(x) = x2 - 2 and verify the relation between its zeros and

coefficients.

3. It being given that 3 and -3 are two zeros of the polynomial x4 + x3 -11x2 -9 x +18, find all

the zeros of the given polynomial.

4. Find all the zeros of the polynomial (2x4 - 3x3 - 3x2 + 6x -2), it being given that two of its

zeros are √2 and -√2.

5. 5x – 6y + 30 = 0, 5x + 4y – 20 =0, also find the vertices of the triangle formed by these two

lines and x-axis.

6. 2x + y = 6, 2x – y + 2 =0, and shade the region bounded by these lines and x-axis. Find the

area of shaded region.

7. Find the values of ‘a’ and ‘b’, foe which the following system of linear equations has

infinite numbers of solutions ; 2x + 3y = 7, ( a + b + 1)x + (a + 2b + 2)y = 4 ( a + b)

+ 1

8. Find the value of ‘a’ and ‘b’ for which the following system if linear equations of has

infinite many solutions: (2a – 1)x – 3y =5, 3x + ( b – 2)y = 3

9. There are two classrooms A and B. If 10 students are sent from A to B, the number of

students in each room becomes the same. If 20 students are sent from B to A, the number of

students in A becomes double the number of students in B. Find the number of students in

each room. 100in a ,80 in b

10. Points A and B are 70 km apart on a highway. A ear starts from A and another car starts

from B simultaneously. If they travel in the same direction, they meet in 7 hours. But, if

they travel towards each other, they, meet in 1 hour. Find the speed of each car. 40, 30 km/h

[cbse 2007c]

11. A train covered a certain distance at a uniform speed. If the train had been 5 kmph faster, it

would have taken 3 hours less than the scheduled time. And, if the train were slower by 4

kmph, it would have taken 3 hours more than the scheduled time. Find the length of the

journey. 1080 km

12. A man travels 370 km, partly by train and partly by car. If he covers 250 km by train and the

rest by car, it takes him 4 hours. But, if he travels 130 km by train and the rest by car, he

takes 18 minutes longer. Find the speed of the train and that of the car. 100 km/h [cbse

2001]

13. Show that any positive integer is of the form 3q or 3q + l or 3q + 2 for some integer q.

14. Show that any positive integer is of the form 5q or 5q + 1 or 5q + 2 or 5q + 3 or 5q + 4 for

some integer q.

15. Find the HCF and LCM of the following integers by prime factorisation method.

(Fundamental Theorem of Arithmetic) (i) 24, 60,150

16. Check whether 5n can end with the digit 0 for any natural number n.

17. Check whether 51n is divisible by 3 for any natural number n.

18. LCM and HCF of two numbers is 415800 and 20 respectively if one of the number is 3300

then find the other number



TIME : 2h CH 1-2-3 EACH QUESTION 4MARKS

1. Find the zeros of the polynomial f(x) = 2x2 + 5x -12 and verify the relation between its zeros

and coefficients.

2. Find the zeros of the polynomial f(x) = x2 - 2 and verify the relation between its zeros and

coefficients.

3. It being given that 3 and -3 are two zeros of the polynomial x4 + x3 -11x2 -9 x +18, find all

the zeros of the given polynomial.

4. Find all the zeros of the polynomial (2x4 - 3x3 - 3x2 + 6x -2), it being given that two of its

zeros are √2 and -√2.

5. Obtain all zeros of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeros are √5/√3 and - √5/√3

6. Show that any positive integer is of the form 3q or 3q + l or 3q + 2 for some integer q.

7. Show that any positive integer is of the form 5q or 5q + 1 or 5q + 2 or 5q + 3 or 5q + 4 for

some integer q.

8. Check whether 5n can end with the digit 0 for any natural number n.

9. Check whether 51n is divisible by 3 for any natural number n.

10. LCM and HCF of two numbers is 415800 and 20 respectively if one of the number is 3300

then find the other number 11. Using Euclid’s division algorithm, find which of the following pairs of numbers are co-prime: 231, 396 12. A part of monthly hostel charges in a college are fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 25 days, he has to pay Rs 1750 as hostel charges whereas a student B, who takes food for 28 days, pays Rs 1900 as hostel charges. Find the fixed charges and the cost of the food per day. Rs 500 fixed charges, rs 50 for food . 13. There are two classrooms A and B. If 10 students are sent from A to B, the number of students in each room becomes the same. If 20 students are sent from B to A, the number of students in A becomes double the number of students in B. Find the number of students in each room. 14. Points A and B are 70 km apart on a highway. A ear starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 7 hours. But, if they travel towards each other, they, meet in 1 hour. Find the speed of each car. 15. Find the value of k for which each of the following systems of linear equations has an infinite

number of solutions: 2x+ 3y = 7, (k-l)x + (k + 2)y = 3k.

16. Solve the following system of linear equations graphically:

x-y+1 =0, 3x + 2y -12 = 0. Calculate the area bounded by these lines and the x-axis.



TIME : 2h SIMILER TRIANGLE-1 EACH QUESTION 4MARKS

1. 1.In the given figure, AABC and ADEF are similar. The area of AABC is 9 sq. cm and the

area of ADEF is'16 sq. cm. If BC=2.1 cm, find the length of EF.

2. In the given figure 2, considering triangles BEP and CPD, prove that BP xPD = EPx PC.

3. ABC is an isosceles triangle right angled at B. Two equilateral triangles are constructed with

side BC and AC as shown in figure. Prove that ar(BCD)= 1/2arACE. 2

4. Any point O, inside AABC, is joined to its vertices. From a point D on AO, DE is drawn so

that DE II AB and EFII BC as shown in figure. Prove that DFII AC

5. In the given fig., D is a point on the side BC of ABC such that angle ADC = angle BAC.

Prove that CA/CD = CB/CA

6. In the given fig., ABCD is a trapezium in which AB II DC. The diagonals AC and BD

intersect atO. Prove that =

7. In AABC, ADPERPENDECULER BC, prove thatAB2 + CD2=AC2 + DB2.

Fig 1

fig no 2

Fig no 3

Fig no 4

fig 5

FIG NO 6

FIG NO 7

FIG NO 8



TIME : 1.5h linear eq SIMILER TRIANGLE-1 EACH QUESTION 4MARKS




1) An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the

same time, another aeroplane leaves the same airport and flies due west at a speed of

1200 km per hour. How far apart will be the two planes after1 ½ hours? Q11 ncert

2) A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a

tower casts a shadow 28 m long. Find the height of the tower .

3) The perpendicular from A on side BC of a tri ABC intersects BC at D such that DB = 3 CD

Prove that 2 AB2 = 2 AC2 + BC2.( fig 1)

4) In an equilateral triangle ABC, D is a point on side BC such that BD =13BC. Prove that 9

AD2 = 7 AB2

5) In Fig.2, O is a point in the interior of a triangle ABC, OD ⊥BC, OE ⊥ AC and OF ⊥ AB.

Show that

(i) OA 2+ OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2,

(ii) AF2 + BD2+ CE2 = AE2 + CD2 + BF2

6) BL and CM are the two median of triangle right angle at A. prove that 4(BL2+ CM2)=5BC2.

7) Theorem 6.8 : In a right triangle, the square of the hypotenuse is equal to the

sum of the squares of the other two sides.

8) D, E and F are respectively the mid-points of sides AB, BC and CA of Δ ABC. Find the

ratio of the areas of Δ DEF and Δ ABC.

9) Example 9 : In Fig. 6.43, the line segment XY is parallel to side AC of Δ ABC and it

divides the triangle into two parts of equal areas. Find the ratio AX /AB .

10) Theorem 6.6 : The ratio of the areas of two similar triangles is equal to the square of the

ratio of their corresponding sides.

11) D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that CA2 = CB.CD.

12) ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show

that AO /CO = BO/ DO

Fig1 fig2 fig3/q6




1) Fig. 6.60, AD is a median of a triangle ABC and AM ⊥BC. Prove that

➢ AC2= AD2 + BC . DM + (BC2/2 )

➢ AB2 = AD2 – BC . DM + (BC2/2 )

➢ AC2 + AB2 = 2 AD2 +1/2 BC2

2) In Fig. 6.59, ABC is a triangle in which ∠ABC < 90° and AD ⊥BC. Prove that

AC2 = AB2 + BC2–2 BC . BD

3) The perpendicular from A on side BC of a ABC intersects BC at D such that DB = 3 CD

(see Fig. 6.55). Prove that 2 AB2 = 2 AC2 + BC2 .

4)

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TIME: TEST NO. - 1 MARKS