Mathematics Booklet- SA1
Chap.
No.
Name of the chapter
Page No.
1
Real Numbers
2-3
2
Polynomials
3-5
3
Pair of linear equations in two
variables
5-7
8
Introduction to trigonometry
7-8
14
Statistics
9-11
3
Triangles
11-14
Answers
15-17
Chapter 1 –REAL NUMBERS
1 Mark Questions
1. If two positive integers ‘m’ and ‘n’ can be expressed as m=ab2 and n= a3b; a,b being prime numbers, then
LCM ( m , n ) is :
2. If the least factor of ‘a’ is 3, the least prime factor of ‘b’ is 7, then the least prime factor of (a + b) is :
3. After how many places, the decimal form of 125 will terminate?
24.53
4. Write the condition to be satisfied by ‘q’ so that a rational number p/q has terminating decimal expansion.
5. HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, then the other number is :
6. If a = bq + r in Euclid’s division lemma, then r must satisfy :
7. A series of well defined steps which gives a procedure for solving a type of problem is called _______
8. The unit’s digit of 73 is 3 then what will be the unit’s digit of 711 ?
9. Find the HCF of the smallest composite number and the smallest prime number.
10. If HCF of 65 and 117 is expressible in the form 65m – 117, then the value of ‘m’ is :
2 Mark Questions
11. Find the HCF of 336 and 54 by Euclid’s division algorithm.
12. What is the smallest number by which (√5 - √ 3) be multiplied to make it a rational number? Also find the
number so obtained.
13. Find the largest number which divides 70 and 125, leaving remainder 5 and 8, respectively.
14. Explain why 3 x 5 x 7 + 7 + 2 x 7 is a composite number.
15. Given that HCF (135,225) = 45. Find LCM (135,225).
16. Show that every positive odd integer is of the form 4q + 1 or 4q + 3
17. Prove that there is no natural number for which 8n ends with digit zero
18. Show that 5√2 is irrational
19. Find the smallest number which when increased by 17 is exactly divisible by 520 and 468
20. Find the HCF of LCM of 30, 72 and 432 by prime factorization method.
21. The number 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF ( 525, 3000) ? Justify
your answer.
22. There is a circular path around a sports field. Ankit takes 18 minutes to drive one round of the field, while
Ankita takes 12 minutes for the same. Suppose they both start at the same point, at the same time and go in
the same direction. After how many minutes will they meet again at the starting point?
23. Prove that every positive even integer is of the form 2q and that every positive odd integer is of the form
2q+1, where q is some integer.
24. Write the condition for terminating of a rational number. And hence, find whether the rational number
(13/3125) has a terminating decimal or non-terminating repeating decimal.
25. What is the digit at unit’s place of 9n ?
3 Mark Questions
26. If the HCF of 210 and 55 is expressible in the form 210 X 5 – 55y , then find the value of ‘y’.
27. Show that square of an odd positive integer is of the form 8m + 1, for some whole number m
28. Prove that ( √2 + √ 3 ) is irrational
29. If n is an odd integer, then show that n2 – 1 is divisible by 8
30. Prove that any number of the form 4x + 2 can never be a perfect square.
2|Page
1
31. Prove that (2− √ 5) is an irrational number.
32. Using Euclid’s division, find the largest number that divides 70 and 125, leaving remainder 5 and 8,
respectively.
33. Without actually performing the long division, find if 987/10500 will have terminating or non-terminating
repeating decimal expansion. Give reasons for your answer.
34. Show that there is no positive integer ‘n’ for which √𝑛 − 1 + √𝑛 + 1 is rational.
35. State the following:
(i) Euclid’s Division Lemma with boundary conditions.
(ii) Fundamental Theorem of Arithmetic.
36. Using Euclid’s lemma find the length of the longest tape needed to measure a room of length,
breadth, height of 520 cm, 480 cm, 750 cm respectively.
37. Find the HCF and LCM of 72, 120, 360 using prime factorization method.
38. Prove that product of three consecutive positive integers is divisible by 6.
𝟐−√𝟑
39. If 𝟐+√𝟑 =
𝒙
, then determine whether ‘x’ is rational or irrational.
√𝟑
4 Mark questions
40. Prove that one of every three consecutive positive integers is divisible by 3
41. Show that one and only one of n, , + 2, n+ 4 is divisible by 3.
42. Prove that square of any positive integer is either of the form 3m or 3m+1 for some integer m
43. Prove that square of any positive integer is of the form 5q, 5q+1,5q+4 for some integer, q
44. Prove that √2 is irrational.
45. Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are
stored topic wise and height of each stack is the same. The number of English books is 96, the number of
Hindi books is 240 and the number of Mathematics books is 336. Assuming that books are of same
thickness, determine the number f stacks of English, Hindi and Mathematics books respectively.
46. Prove that √7 is irrational.
47. Prove that if x and y are odd positive integers then x2+y2 is even but not divisible by 4.
48. Prove that n3 – n is divisible by 6, for every positive integer ‘n’
49. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 of 9m
+ 8.
HOTS
50. Prove that if x and y are both odd positive integers then x2 + y2 is even but not divisible by 4. ( 3 Marks)
Chapter 2 - POLYNOMIALS
1 Mark Questions
1.
2.
3.
4.
If a and b are the zeroes of 2x2 + 5x – 10, then the value of ab is ______
If one of the zeroes of the quadratic polynomial (k – 1 )x2 + 2kx + 3 is 1, then the value of k is ____
For what values of k is the polynomial f(x) = 2x3 – kx2 + 5x + 9 exactly divisible by (x+2)?
The graph of y = p(x) is given below. Find the number of zeroes
Y
X
’
3|Page
X
Y’
5. p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that
p(x) = q(x)g(x) + r(x). Here write the condition that r(x) should satisfy.
6. A polynomial of the form ax5 + bx3 + cx2 + dx + e has at most _________ zeroes
7. The sum and product of zeroes of 2(x2 – 1) + 3x – 9 are :
8. Find a quadratic polynomial, the sum and product of its zeroes are 1 and –6 respectively.
9. Find a quadratic equation, whose roots are(1+√5) and (1-√5) .
10. If (x + 2)(2x - 1)(3x - 2) = 0, find the zeroes of the polynomial.
11. If the sum of the zeroes of the polynomial f(x) = 2x3 - 3kx2 + 4x - 5 is 6, then find the value of k.
2 Mark Questions
12. Find the value of p for which x = b is a zero of the polynomial x2 – (a+b) x + p
13. Find the zeroes of the polynomial a2 – a – 12 and verify relationship between the zeroes and the coefficients.
14. If one zero of the polynomial f(x) = 15 x2 + 14 x – k is reciprocal of the other, then what will be the value of
k?
15. If α and β are the roots of the equation ax2 -bx +c = 0, then find the value of
1
𝛼
+
1
𝛽
16. If α and β are the roots of the equation 25x2 - 10x + 1= 0, find the value of α2+ β2.
17. Find a quadratic polynomial, the sum of whose zeroes is -1 and the sum of their reciprocals is 1/6
18. If α and β are the zeroes of the polynomial p(x) = x2 – 16x + 63, then find the value of α4β3 +α3β4
19. If the sum of the squares of the zeroes of a quadratic polynomial x2 –18x + p is 180, find the value of p.
20. If one root of the quadratic equation 2x2 + px + 4 = 0 is ‘2’, then find the other root and also find the value
of ‘p’.
21. Find the value of the quadratic equation 2x2 - 3x - 2 at x = 1 and x = -2.
3 Mark Questions
22. Find a quadratic polynomial whose zeroes are -2/√3 and (√3)/4.
23. For what value of ‘k’, will the equation
𝑥 2 −𝑏𝑥
𝑎𝑥−𝑐
𝑘−1
= 𝑘+1 have roots reciprocal to each other?
24. If α and β are zeroes of p(x) = x2 + px + q, then find a polynomial having 1/ α and 1/ β as its zeroes: q≠0.
25. On dividing x3 + 3x + 2 by a polynomial g(x), the quotient and the remainder are x – 2 and 16 respectively.
Find g(x)
26. What must be subtracted from 4x4 + 2 x3 – 8x2 + 3x – 7 so that it may be exactly divisible by 2x2 + x – 2?
27. If α and β are the roots of the quadratic polynomial 3x2 - 2x – 1 = 0, find the value of
𝛼
𝛽
𝛽
+ 𝛼, without
finding the values of α and β
28. If α and β are the zeroes of a quadratic polynomial such that α + β = 24 and α - β = 8, then find the quadratic
polynomial.
29. If (Z –3) is a factor of Z3 + aZ2 + bZ + 18 and a + b = -7, find a and b.
1
30. If α and β are the zeroes of the quadratic polynomial g(x) = x2 - (a + 12)x + 3(3a + 4), such that (α + β) = 3(
α β), then find the value of a.
31. Check whether the polynomial l(x) = x2 – 5x + 1 is a factor of the polynomial m(x) = 4x4 – 13x3 – 31x2 + 35x
– 10 by dividing m(x) by l(x).
32. Divide -x3 + 3x2 - 3x + 5 by -x2 + x - 1 and verify the division algorithm.
33. If α and β are the zeroes of 2x2 - 6x + 3, then what is the value of
( 𝛼𝛽 + 𝛽𝛼 )+ 3 ( 𝛼1 + 𝛽1 ) + 2 α β
34. Find the zeroes of the quadratic polynomial f(x) = abx2 + (b2 – ac)x - bc and verify the relationship between
the zeroes and the coefficients of the polynomial
4|Page
35. Use the division algorithm to find the quotient q(y) and the remainder r(y) when f(y) = 12y3 + 17y2 – 20y –
10 is divided by g(y) = 3y2 + 2y - 5.
36. Find the G.C.D. and L.C.M. of the following polynomials: p(x) = 6(x - 2)(x2 + x - 6) and, q(x) = 3(x2 + 4x 12).
37. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg
p(x) = deg q(x) (ii) deg q(x) = deg r(x)
(iii) deg r(x ) = 0
4Mark Questions
38. Find the other zeroes of the polynomial x4 – 5x3 + 2x2 + 10x – 8 if it is given that two of its zeroes are – √2
and √2
39. If α and β are the zeroes of the quadratic polynomial 2x2 - 5x +7, find a polynomial whose zeroes are 2 α + 3
β and 3 α + 2 β .
40. Divide ( 3x2 – x3 + 7x + 13) by ( x – 2 – x2 ) and verify the division algorithm.
41. On dividing 3 x3 +x2 + 2x +5 by a polynomial g(x), the quotient and remainder are 3x – 5 and 9x + 10
respectively. Find g(x)
42. If the polynomial x4 + 2 x3 +8x2 + 12x + 18 is divided by another polynomial x2 + 5, the remainder comes
out to be px + q. Find the values of p and q.
43. It is given that √2 and -√2 are two zeroes of the polynomial f(y) = 2y4 - 3y3 - 3y2 + 6y - 2, find all the zeroes
of f(y).
HOTS
44. If α and β are zeroes of p(x) = x2 – 2x + 3, find a polynomial whose zeroes are (α – 1 ) / (α + 1) and
(β – 1 ) / (β + 1).
45. If the polynomial f(x) = x4 - 6x3 + 16x2 - 25x + 10 is divided by another polynomial x2 - 2x + k , the
remainder comes out to be x + a , find k and a.
46. What must be added to the polynomial 3x4 + 5 x3 -7x2 + 5x + 3 so that the resulting polynomial is exactly
divisible by (x2 +3x +1)? The degree of the polynomial to be added must be less than degree of (x2 +3x +1)
Chapter 3-PAIR OF LINEAR EQUATONS IN TWO VARIABLES
1 Mark Questions
1. General form of a linear equation in two variable is ________
2. Geometrical representation of a linear equation in two variables is _________
3. If a pair of linear equations is consistent, then the lines will be ____________
4. Find the number solutions for the pair of linear equations x + 2y + 5 = 0 and -4x – 8y + 1 = 0.
5. The pair of equations y = 2 and y = 7 has ________ solution/s
6. Find the value of ‘k’ for which x + 2y = 5, 3x + ky + 15 = 0 are inconsistent.
7. The value of y in the system of equations (x/a) = (y/b), ax + by = a2 + b2
8. The point of intersection of line -3x + 7y = 3 with x-axis is ____________
9. Find the value of x and y which satisfy the equation x – y = 0; 2x – y = 2 simultaneously.
10. If 4 x-y = 16 and x – 2y = 8 are system of the equations, then the value of x + y is :
11. Find the value of ‘x’ in the following pair of linear equations: 4/x + 5y = 7 and 3/x + 5y = 5.
12. Sum of two numbers in 48 and their difference is 20. Find the numbers
13. If 5x + 7y = 3 and 15x + 21y = k represent coincident lines, then find the value of k.
14. The conditions so that a1x + b1y = c1; a2x + b2y = c2 have unique solution is ________
15. If a pair of equations are consistent, the lines will be _______
5|Page
2 Mark Questions
16. Find the values of α and β for which the following system of linear equations has infinite number of
solutions. 2x + 3y = 7 ; 2αx + (α + β)y = 28
17. The denominator of a fraction is 7 more than the numerator. If 5 is added to each, the value of the resulting
fraction is 1/2. Find the original fraction.
18. Find the values of k for which the system of linear equations has unique solution.
2x + 3y = 7, kx + 9y = 28.
19. Use elimination method to find all possible solutions of the following pair of linear equations:
2x + 3y = 8 and 4x + 6y = 7.
20. Is the pair of equations x + 2y – 3 = 0 and 6y + 3x – 9 = 0 consistent? Justify your answer.
21. Determine the value of u so that the following pair of linear equations have no solution.
(3u + 1)x + 3y - 2 = 0 and (u2 + 1)x + (u - 2)y - 5 = 0.
22. Solve the following pair of linear equations by cross multiplication method
3x + y = - 4 and 6x – 2y = - 4
𝑥
𝑦
𝑥 𝑦
23. Solve the system of linear equations 6 + 2 = 1, 6 - 4 = 1
24. The sum of two numbers is 36 and their difference is 14. Find the numbers.
25. Solve the pair of linear equations by substitution method. x + y = 14, x – y = 4.
26. Solve the systems of linear equations: 0.4x +0.3 y =1.7 and 0.7x - 0.2y = 0.8
3 Mark Questions
27. Write an equation of a line passing through the point representing solution of the pair of linear equations
x + y = 2 and 2x – y = 1. How many such lines can we find?
28. The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a
distance of 10 km, the charge paid is Rs 75 and for a journey of 15 km, the charge paid is Rs 110. How
much does a person have to pay for travelling a distance of 25 km?
29. The path of train A is given by equation x + 2y – 4 = 0 and the path of another train B is given by the
equations 2x + 4y – 12 = 0. Represent this situation geometrically. Also check, whether the two paths can
cross.
30. 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?
31. The sum of the digits of a two digit number is 8 and the difference between the number and that formed by
reversing the digits is 18. Find the number.
32. Solve the pair of linear equations 3x + 5 y – 13 =0 and 2x - 5 y – 7=0 and hence find the value of ‘m’ for
which y = mx–3.
2
3
9
4
9
21
33. Solve the pair of linear equations 𝑥 + 𝑦 = 𝑥𝑦 and 𝑥 + 𝑥 = 𝑥𝑦 where x ≠ 0, y ≠0.
34. Find the values of m and n for which the system of linear equations has infinite number of solutions.
2x + 3y = 7, 2mx + (m+n) y = 28
1
1
35. Solve the following system of equations : 2𝑥 - 𝑦 = -1 and
1
𝑥
1
- 2𝑦 = 8
36. A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was
Rs 15000 after 4 year of service and Rs 18000 after 10 years of service, what was his starting salary and
what is the annual increment?
6|Page
4 Mark Questions
37. A boatman rows his boat 35km upstream and 55km downstream is 12 hours. He can row 30 km upstream
and 44km downstream is 10 hours. Find the speed of the stream and that of the boat in still water. Hence
find the total time taken by the boatman to row 50km upstream and 77 km downstream.
38. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less.
If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.
39. Draw the graphs of the following system of linear equations: 4x +3y – 24 = 0; 2x – y = 2; y + 4 = 0.Obtain
the vertices of the triangle so obtained. Also determine its area.
40. (a)Solve: 217x + 131y = 913 and 131x + 217y = 827
(b) For what value of u the system of equations 3x + 5y = 0 and ux + 10 y = 0 has unique solution.
41. Draw the graphs of the following system of linear equations and find unique solutions: 4x – y – 8 = 0 ; 2x –
3y + 6 = 0. Also determine vertices of triangle formed by the lines and the x-axis.
42. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48
km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.
43. Solve the following system of linear equations graphically: 2x – y – 4 =0 x + y + 1 =0 Find the points
where the line meets the y axis.
44. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter.
Saritha paid Rs 27 for a book kept for seven days while Susy paid Rs 21 for the book she kept for five days.
Find the fixed charge and the charges for each extra day.
45. By the graphical method, find whether the pair of equations 3x + y + 4 = 0 and 6x – 2y + 4 = 0 are
consistent or not. If consistent then find its solution.
46. The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the
number differ by 2, find the number. How many such numbers are there?
HOTS
47. Two pipes running together can fill a cistern in 10/3 minutes. If one pipe takes 3 minutes more than the
other to fill the cistern, find the time in which each would fill the cistern.
48. Solve the following :
(a + b)x + ( a – b )y = a2 – b2 ;
(a – b )x + ( a + b )y = a2 + b2
Chapter 8- INTRODUCTION TO TRIGONOMETRY
1 Mark Questions
1. If sec θ = 5/4 , then what is the value of tan θ ?
2. Express sinA in terms of cotA.
3. The value of
√1−𝑐𝑜𝑠𝜃
√1+𝑐𝑜𝑠𝜃
= _______
Sin2 250 + sin2650 = ____
In a right triangle ABC right angled at B, if sin( A – C) = ½, then angles A, B and C respectively are______
If x = a cos θ and y = b sin θ, then the value of b2x2 +a2y2 is
The value of sin2 50 + sin2 100 + …….+ sin2 850 + sin2 900 is _____
In right triangle PQR right angled at Q, PQ = 8cm, QR = 6cm and PR = 10 cm. Find the value of 25 (sin2θ +
2 cos2 θ - tan θ)
9. The value of tan 10 tan 20 tan 30…… tan 890 is :
10. Find the value of ‘x’ if tan3x = sin 450 . cos 450 + sin 300
2 Mark Questions
11. If sinθ + sin2θ = 1, check the validity of expression: cos2 θ + cos 4 θ = 1.
4.
5.
6.
7.
8.
7|Page
12. If A, B and C are interior angles of ∆ ABC, then show that cot (
𝐶+𝐴
2
) = tan 𝐵2
13. In right triangle ABC, right angled at B, AB = y cm, BC = 10 cm, AC = x cm and < C = 300. Find the values
of x and y
14. ∆ABC is right angled at B and
A=
C. Is cosA = cosC? Justify your answer
0
0
15. Solve the equation when 0 < θ < 90 ; 3 tan2 θ – 1 = 0
16. If 7sin2A+3cos2 A = 4, show that tan A =1/√3.
17. If sin θ = a/b, then find sec θ + tan θ in terms of a and b.
18. Simplify: sec2x(1-sin2x)+secB(sinB/tanB)
19. Evaluate( cos 600 + sin 300 – cot 300 ) / (tan600 + sec 450 – cosec 450)
20. State whether the following expression is True or False. Justify your answer.
sin6 θ + cos 6 θ = 1 – 3 sin2 θ. cos2 θ
21. If A, B are acute angles and sinA= cosB, then find the value of A+B.
22. If 2x=secA and 2/y=tanA, then find the value of 2(x2-1/y2).
23. In ∆ABC, right angled at B , BC = 3 and AC = 6. Determine
BCA and
BAC.
3 Mark Questions
24. If cosec A = √10, then find other five trigonometric ratios.
25. Prove that: cot2A/(1+cosecA)=(1-sinA)/sinA
26. If 7 cosecA-3cotA = 7, prove that 7cotA - 3cosecA = 3.
27. Prove that (SinA+CosecA)2+(CosA+SecA)2=7+tan2A+cot2A.
28. Evaluate cos (400 + θ) – sin (500 – θ ) + (cos2 400 + cos2 500) / (sin2 400 + sin2 500)
29. sec6x(secxtanx)-sec4x(secxtanx) =sec5xtan3x
30. Prove that (cot A – cos A) / ( cot A + cos A) = (cosec A – 1 ) / (Cosec A + 1)
31. Prove that
1
–
1
= 1
–
1
.
sec x – tan x
cos x
cos x
sec x + tan x
32. If cot θ = √7 , show that
𝑐𝑜𝑠𝑒𝑐 2 𝜃−sec2 𝜃
𝑐𝑜𝑠𝑒𝑐 2
𝜃+sec2 𝜃
3
=4
33. If tanA + sinA = m and tanA - sinA = n, prove that (m2 - n2)2 = 16mn.
34. Show that sin1/2xcosx-sin5/2xcosx=cos3xsin1/2x
35. Prove that: secx+tanx=cosx/(1-sinx)
36. Without using trigonometric table, evaluate the following :
𝑐𝑜𝑠530
𝑐𝑜𝑡800
𝑠𝑒𝑐700
2(𝑆𝑖𝑛370 )+ 𝑡𝑎𝑛100 - 3𝑐𝑜𝑠𝑒𝑐200 +
𝑐𝑜𝑠550 𝑐𝑜𝑠𝑒𝑐350
𝑡𝑎𝑛50 𝑡𝑎𝑛250 𝑡𝑎𝑛450 𝑡𝑎𝑛650 𝑡𝑎𝑛850
𝑠𝑖𝑛𝐴+2𝑐𝑜𝑠𝐴
Given 3 cos A – 4 sin A = 0. Evaluate without using table 3𝐴−sin 𝐴 .
4Mark Questions
38. If tan A = n . tan B and sin A = m . sin B, then prove that cos2 A = (m2 – 1) / ( n2 – 1)
39. Prove that (tanθ + sinθ) / (tanθ – sinθ) = ( secθ + 1) / ( secθ – 1) = (1+cosθ) / (1 – cosθ) =tan2 θ / (secθ – 1)2
40. If ( a2 – b2 ) sin θ + 2 ab cos θ = a2 + b2, find the value of tan θ
37.
tan3 θ
cot3 θ
41. Prove that 1+tan2 θ + 1+cot2 θ = sec θ . cosec θ – 2 sin θ . cos θ
tan 𝐴
𝑐𝑜𝑡𝐴
42. Prove that 1−cot 𝐴 + 1−tan 𝐴 = 1 + cosecAsecA
43. If tanA + sinA = m and tanA – sinA = n, prove that ( m2 – n2 )2 = 16mn
1
1
44. If each of α, β and λ is positive acute angle such that sin (α+ β –λ) = 2 , cos (β +λ- α) = 2 and
tan (λ +α+β)=1. Find the value of α, β and λ.
45. If x = r sinAcosC, y = r sinA sinC and z = r cos A, prove that r2 = x2 + y2 + z2.
8|Page
Chapter 14- STATISTICS
1 Mark Questions
1. Construction of a cumulative frequency table is useful in determining the __________
𝑥 𝑥
𝑥
2. 𝑥̅ is the mean of x1, x2, x1, ……….. , xn , then the mean of 𝑘1, 𝑘2, ‘’’’’, 𝑘𝑛 where k ≠0 is ______
3. The mean of n observations is 𝑥̅ . If the first term is increased by 1, second term by 2 and so on, then the
new mean is _________
4. The relationship between mean, median and mode for a moderately skewed distribution is ______
5. Median and mode of distribution are 21.2 and 21.4 respectively. Then, its mean is : _____
6. The median of a frequency distribution is found graphically with the help of _______
7. Find the mode for the following series :
7.5, 7.3, 7.2, 7.2, 7.4, 7.7, 7.7, 7.5, 7.3, 7.2, 7.6, 7.2
8. If 𝑥̅ is the mean of n observations x1, x2, x1, ……….. , xn then ∑ (xi - 𝑥̅ ) ,where i= 1 to n , is equal to _____
9. For what value of k, the mode of the following data is 7?
3,5,6,7,4,6,9,7,6,2k – 1 ,10, 7, 12
2 Mark Questions
10. The mean of five numbers is 18. If one number is excluded, their mean is 16, then what will be the
excluded number?
11. If median = 15 and mean = 16, the mode is ____
12. Find the mean by direct method for the following data:
Classes
Frequency
10-20
4
20-30
8
30-40
10
40-50
12
50-60
10
60-70
4
70-80
2
13. The arithmetic mean of the numbers 7, 11, x, 3, y and 2 is 11. Find the arithmetic mean of x and y.
14. Find the arithmetic mean of the 10 prime numbers.
15. The average mark scored by girls is 68 and that of the boys is 62. The average mark of the whole class is
64. Find the ratio of the girls & boys in the class.
16. The mean, of x – 5y, x – 3y, x – y , x + y , x + 3y & x + 5y is 12. Find the value of x.
17. Find the median class of the following data:
Marks
Obtained
Frequency
0-10
10-20
20-30
30-40
40-50
50-60
8
10
12
22
30
18
3 Mark Questions
18. Find the unknown entries a, b, c, d, e, f in the following distribution of heights of students in a class
Height ( in cm)
135-140
140-145
145-150
150-155
155-160
160-165
9|Page
Frequency
Cumulative frequency
4
b
18
d
e
5
a
11
c
40
46
f
19. Consider the following :
Class interval
0-5
5-10
10-15
15-20
Frequency
10
15
12
20
Find the sum of lower limits of the median class and the modal class.
20. The total number of marks scored by class in test is given below. Find the mean.
Below 20
4
Below 40
12
Below 60
30
Below 80
44
Below 100
50
20-25
9
21. Find the mode for the following frequency distribution of marks obtained by 80 students :
Marks
0-10
10-20
20-30
30-40
No. of students
6
10
12
32
22. The mode of the following frequency distribution is 32. Find the missing frequency in it :
Class
0-10
10-20
20-30
30-40
40-50
50-60
interval
Frequency
8
10
x
16
12
6
23. Find the median from the following distribution.
Class interval
4-6
6-8
8-10
10-12
Frequency
5
4
10
7
24. If the mean of the following data is 20, find the value of p.
x:
f:
15
2
17
3
19
4
21
5p
40-50
20
60-70
7
12-14
4
23
6
4Mark Questions
25. The mean of the following frequency distribution is 62.8 and the sum of ll frequency is 50. Compute the
missing frequencies 𝑓1 and 𝑓2 .
Class
0-20
20-40
40-60
60-80
80-100
100-120
Total
Frequency 5
10
7
8
50
𝑓1
𝑓2
26. The table below gives the distribution of villages under different heights from sea level in a certain
region. Compute the mean height of the region :
Height ( in 200
meters)
No. of
142
villages
600
1000
1400
1800
2200
265
560
271
89
16
27. Draw less than and more than ogives for the following distribution on the same graph and hence find the
median :
Marks
30-39
40-49
50-59
60-69
70-79
80-89
90-99
No. of
14
6
10
20
30
8
12
students
10 | P a g e
28. Find the mean and mode of the following data :
Classes
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Frequency 3
8
10
15
7
4
3
29. During the medical check-up of 35 students of a class, their weights were recorded as follows. Draw less
than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result
by using the formula.
Less than Less than Less than Less than Less than Less than Less than Less than
Wt. in Kg
38
40
42
44
46
48
50
52
No. of
0
3
5
9
14
28
32
35
students
30. The median of the following data is 28.5. Find the values of x and y, if the total frequency is 60.
Class
interval
No. of
students
31.
0-10
10-20
20-30
30-40
40-50
50-60
5
x
20
15
y
5
The frequency distribution of scores obtained by 230 candidates in an engineering entrance test is as
follows :
Scores
400 –
450 –
500 –
550 –
600 –
650 –
700 –
750 450
500
550
600
650
700
750
800
No. of
20
35
40
32
24
27
18
34
candidates
Draw cumulative frequency curves by less than and more than method on the same axes. Also, draw the
two types of cumulative frequency polygons.
Chapter – 3 TRIANGLES
1 Mark Questions
1. In the adjoining figure, LM ║ AB. If AL = x – 3, AC = 2x, BM = x – 2 and BC = 2x + 3, the value of x
is :
C
L
M
B
A
2. Let ∆ABC ~ ∆DEF and their areas be respectively 64 cm2 and 121 cm2. If EF = 15.4 cm, then the value
of BC is _____
3. In ∆LMN, <L = 500 and < N = 600. If ∆LMN ~ ∆PQR, then < Q is equal to :
4. If ∆ABC & ∆DEF are similar such that 2 AB = DE, and BC = 8 cm, then EF = ___ cm.
5. Area of an equilateral triangle with sides 4 cm = ___ __
2 Marks Questions
6. In the adjoining figure,
𝑄𝑅
𝑄𝑆
𝑄𝑇
= 𝑃𝑅 and <TQS = <PRS, show that ∆ PQS ~ ∆ TQR
P
T
Q
11 | P a g e
R
S
7. In the given figure, AD I. BC. Prove that AB 2 + CD 2 = BD2 + AC2
C
D
B
A
8. ABC is an isosceles triangle, right-angled at B. Equilateral triangles ACD and ABE are constructed on
sides AC and AB. Find the ratio between the areas of ∆ ABE and ∆ ACD.
A
D
E
C
B
9. In the adjoining figure, ∆ABC is right- angled at C and DE I AB, AD = 3 cm, DC = 2 cm, BC = 12
cm. Prove that ∆ABC ~ ∆ADE and hence, find the lengths of AE and DE.
A
E
D
B
C
1
10. In ∆PQR, DE ║ QR intersecting PQ and QR in points D and E, respectively. If DE = 3 QR, then show
1
that PD = 2 DQ
11. In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB=AC. If
AD
BC and EF
AC, prove that
ABD
ECF
12. The perimeters of two similar triangles are 36 cm and 48 cm respectively. If one side of the first
triangles is 9 cm, what is the corresponding side of the other triangle?
13. In ∆ABC, D and E are points on the sides AB and AC respectively such that DE ║ BC. If AD = 4x – 3,
AE = 8x – 7, BD = 3x – 1 and CE = 5x – 3, then find x
14. If ∆ABC ~ ∆DEF, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm. Find the perimeter of ∆ABC.
15. ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. Prove
that AE2 + CD2 = AC2 + DE2.
16. Any point X inside the
DEF is joined to its vertices. From a point P in DX, PQ is drawn parallel to
DE meeting XE at Q and QR is drawn parallel to EF meeting XF in R. Prove that PR || DF.
12 | P a g e
3 Marks Questions
17. In the adjoining figure ∆ABC and ∆AMP are 2 right angled triangles at B and M respectively. Prove that
(1) ∆ABC ~ ∆AMP
(2)
𝐶𝐴
𝑃𝐴
=
𝐵𝐶
𝑀𝑃
C
M
A
P
B
18. In the adjoining figure, D and E are points on the sides CA and CB respectively of a ∆ABC, right angled
at C. Prove that AE2 + BD2 = AB2 + DE2
A
D
B
E
A
C
19. 2 isosceles triangles have equal vertical angles and their areas are in the ration 16:25. Find the ratio of
their corresponding heights
20. ABC is an isosceles triangle is which AB=AC=10cm and BC=12. PQRS is a rectangle inside the
isosceles triangle. Given PQ=SR=y cm, PS=QR=2x. Prove that x = 6 -(3/4)y.
21. In the given figure, <AEF=<AFE and E is the mid-point of CA. Prove that BD/CD=BF/CE.
22. ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let P be the length of
perpendicular form C on AB prove that
(i) cp = ab
1
1
1
(ii)𝑝2 = 𝑎2 + 𝑏2
13 | P a g e
4 Marks Questions
23. State and prove Basic proportionality Theorem. In a ∆ABC, D and E are points on sides AB and AC
respectively, such that BD = CE, <B = <C, then show that DE ║ BC using converse of the above
theorem
24. In a right triangle ABC, right angled at C, P and Q are points of the sides CA and CB respectively that
divide these sides in the ratio 2: 1.
Prove that
(i) 9AQ2= 9AC2 +4BC2
(ii) 9BP2= 9BC2 + 4AC2
(iii) 9 (AQ2+BP2) = 13AB2
25. State and prove converse of Pythagoras theorem. Using the theorem, prove that ∆ABC is right angled in
the given figure
A
6 cm
24 cm
D
B
8 cm
26 cm
B
26. Prove that “the ratio of the areas of 2 similar triangles is equal to the squares of the ratio of their
corresponding sides”. Using this theorem, solve the following : If ∆ABC ~ ∆DEF, area (∆ABC) = 64
cm2 , area (∆DEF) = 121 cm2, and EF = 15.4 cm, find BC
27. O is any point in the interior of the ∆ABC. OD I BC, OE I AC and OF I AB. Show that
AF2 + BD2 + CE2 = AE2 + CD2+BF2
1
1
1
28. In the adjoining figure, PA, QB AND RC are each perpendicular to AC. Prove that 𝑥 + 𝑧 = 𝑦
P
R
Q
x
z
y
A
14 | P a g e
B
C
ANSWERS
Chapter 1- Real
numbers
1 Mark questions
a3b2
2
four places
prime factorization of
‘q’ must be of the form
2m X 5n, where m and n
are non-negative
integers
5) 207
6) 0≤ r < b
7) an algorithm
8) 3
9) 2
10) 2
1)
2)
3)
4)
11) 6
12) (√5 + √ 3) ; 2
13) 13
14) given no. is a multiple
of 7
15) 675
16) Prove
17) 5 does not occur in the
prime factorization of 8
3 Marks Questions
26) 19
27) .
28) .
15 | P a g e
12) p = ab
13) zeroes are -3 and 4
14) k = - 15
15) –b/c
16) 2/25
17) k(x2 + x – 6)
18) 4000752
19) P = 72
20) 1, p = -6
21) P(1) = -3, p(-2) = 12
4 Marks Questions
23)
40) take consecutive
integers as n, n+1,n+2
41) .
42) .
43) .
44) .
45) English = 2, Hindi = 5,
Mathematics = 7
46) .
47) .
48) .
49) .
2 Marks question
18)
19) 4663 ( Hint : LCM (520,
468 ) – 17 )
20) 2160
21) 75
22) 36 minutes
23) .
24) Dr. is of the form 2m X
5n; terminating decimal
25) Even power – 1 , odd
power - 9
29) Any positive odd integer
is of the form n = 4q + 1
or 4q + 3
30) .
31) .
32) HCF of 65 and 117 = 13
33) Terminating decimal
expansion
34) .
35) .
36) HCF = 10 cm
37) HCF = 24; LCM = 360
38) .
39) .
HOTS
50) Odd +ve integers x =
2m+1 and y = 2n + 1
Chapter 2 - POLYNOMIALS
1 Mark questions
1)
2)
3)
4)
5)
6)
7)
–5
– 2/3
– 17/4
4
r(x) = 0 or deg r(x) deg q(x)
5 zeroes
– 3/2 , - 11/2
8)
k[x - x - 6]
9)
x2 -2x – 4 = 0
2
10) x = -2 or, x = 1/2 or, x = 2/3
11) k = 4
2 Marks Questions
3 Marks Questions
22) p(x) = 4√3 x2 + 5x - 2√3.
𝑐+1
𝑐−1
24) qx2 + px + 1
25) x2 + 2x + 7
26) 5x – 11
27) -10/3
28) x2 - 24x + 128
29) a = -4, b = -3
30) a = 4
31) Remainder = 28x – 10 ≠0,
not a factor
32) quotient = x - 2, remainder =
3
33) 13
34) –b/a and c/b
35) q(y) = 4y + 3 and r(y) = 5 – 6y
36) GCD = 3 (x-2), L C M = 6 (
x+3)(x+6)(x-2)2
37) (i) p(x) = 6x2 + 3x + 2, g(x) =
2x2 + x, r(x) = 2
(ii) p(x) = 8x3 +6x2 - x + 7,
g(x) = 2x2+ 1, q(x)= 4x + 3,
r(x) = -5x + 4
(iii)p(x) = 9x2+6x + 5, g(x) =
3x+ 2, q(x) = 3x, r(x) = 5
4 Marks Questions
38) 1,4
39) k(x2 -
25
2
x + 41)
40) .
41) x2 + 2x + 1
42) px + q = 2x + 3, p = 2, q = 3
43) √2 , -√2, 1 and 1/2
HOTS
44) 3x2 – 2x + 1
45) K = 5, a = -5
46) -3x – 1 , degree of added
polynomial = 1
Chapter 3-Pair of linear
equations in two variables
1 Mark Questions
1) ax + by + c = 0, where a, b,
c are real numbers, and a
and b are not both zero.
2) Straight line
3) Intersecting or coincident
4) No solution.
5) No solution
6) k = 6
7) y = b
8) (-1,0)
9) X = 2, y = 2
10) - 10
11) x=1/2
12) x = 34 and y = 14
13) k = 9
14)
15)
𝒂𝟏
𝒂𝟐
≠
𝒃𝟏
𝒃𝟐
Intersecting or
coincident
2 Marks Questions
α = 4, β = 8
17) the original fraction is
2/9.
18) k≠6
19) the pair of equations has
no solution
20) Coincident lines, hence
consistent
21) u = -1
22) x = -1, y = - 1
23) x = 6, y = 0
24) x = 25, y = 11
25) x = 4, y = 2
26) x = 2, y = 3
16)
3 Marks Questions
16 | P a g e
27) x = y , infinitely many
28) Rs. 180
29) The path being parallel
cannot cross.
30) Salim’s age = 38 yrs,
daughter’s age = 14yrs.
31) No is 53
32) X = 4, y = 1/5 , m = 4/5
33) X = 1, y = 3
34) m = 4, n= 8
35) x = 1/6 , y = ¼
36) starting salary =Rs
13000
annual
increment = Rs 500.
4 Marks Questions
37) Speed of the boat in still
water = 8 km/hr.
Speed of the stream = 3
km/hr.
38) No. of students = 36
39) A(3,4), B (_1, -4), C (9, -4);
Area = 40 sq. units
40) (a) x = 3, y = 2 (b) u ≠6
41) x= 3, y = 4; (3,4), (2,0), (3,0)
42) the speed of boat in still
water = 10 km/hr and
speed of stream =
2km/hr
43) (0, -4) for the first
equation and (0, -1) for
the second equation.
44) Rs. 15 and Rs. 3
45) x = -1, y = -1
46) the two numbers are 42,
24.
HOTS
47) 5 minutes, 8 minutes
48) x = (a2 + b2 )/ 2a , y = (a2
+ b2 )/ 2a
Chapter 8- INTRODUCTION
TO TRIGONOMETRY
1 Mark Questions
1) ¾
1
2) Sin A = √1+𝑐𝑜𝑡 2
3)
4)
5)
6)
7)
8)
9)
10)
𝐴
cosec θ – cot θ
1
600 , 900 and 300
a2b2
9.5
2/3
1
150
2 Marks Questions
11) .
12) .
13) x =
14)
15)
16)
17)
18)
19)
20)
20√3
3
,y=
10√3
3
Yes
θ = 300
.
√(b+a) / √(b – a )
2
(√3 – 3 ) / 3
False . (sin 800 – sin100=
positive , as θ increases,
value of sin θ increases )
21) A + B = 900
22) ½
23) 600 and 300
3 Marks Questions
24) Sin A = 1/√10, cos A = 3/√10,
tan A = 3, cot A = 3, sec A =
√10/3
25) .
26) .
27) .
28) 1
29) .
30) .
31) .
32) .
33) .
34) .
35) .
36) 1
37) Take
𝑐𝑜𝑠𝐴
4
=
𝑠𝑖𝑛𝐴
4
= k, Ans : 11/9
4 Marks Questions
38) .
39)
18) a = 4, b = 7, c = 29, d = 11, e =
6, f = 51
19) 10 + 15 = 25
20) Mean = 60
21) 36.25
22) x =15
23) 9.2
24) P = 1
𝑎 2 −𝑏2
2𝑎𝑏
4 Marks Questions
41)
42)
43)
44)
.
.
.
45)
.
25)
26)
27)
28)
29)
30)
40)
α = 37.50, β=450 and λ=
52.50
𝑓1 = 8 and 𝑓2 = 12
Mean height = 984.51
69.5
Mean = 32.8, mode = 33.85
Median = 46.5
x = 8, y = 7
Chapter – 3
TRIANGLES
Chapter 14- STATISTICS
1 Mark Questions
1) Median
𝑥̅
2)
𝑘
3) 𝑥̅ +
4)
5)
6)
7)
8)
9)
𝑛+1
2
Mode = 3 median – 2 mean
21.1
Ogives
Mode = 7.2
0
K=4
1 Mark Questions
1)
2)
3)
4)
5)
2 Marks Questions
2 Marks Questions
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
10)
11)
12)
13)
14)
15)
16)
26
13
Mean = 42.2
21.5
12.9
1 :2
x = 12
x=9
11.2 cm
700
EF = 4 cm
Area = 4√3.
.
,
½
DE = 36/13cm and AE = 15 /
13 cm
.
.
12 cm
x=1
Perimeter = 18 cm
.
.
17) 30-40 is the median class.
3 Marks Questions
3 Marks Questions
17) .
18) .
17 | P a g e
19)
20)
21)
22)
.
.
.
.
4 Marks Questions
23)
24)
25)
26)
27)
28)
.
.
AB = 10 cm
BC =11.2 cm
.
.