STUDY MATERIAL – MATHEMATICS (041)
FOR
CLASS – X
TERM I & TERM II
(2014-2015)
PATRON
Mr. SOMIT SRIVASTAVA
DY. COMMISSIONER
KVS RO SILCHER
CHIEF ADVISOR
Mr. R SENTHIL KUMAR
ASSISTANT COMMISSIONER
KVS RO SILCHER
COORDINATOR
Mr. S K SHARMA
PRINCIPAL K V LUMDING
RESOURCE PERSONS
1. Mr. M S SENGAR, PGT(MATHS)
K V LUMDING
PREFACE
In compliance to KVS(RO),Silchar letter no.290350/2014 /KVS(SR) dated 07.07.2014 the responsibility of preparation of Study/ Support Material class X mathematics has been entrusted to the Lumding Cluster.
Cluster Lumding acknowledges the sincere efforts of Sh. S.K Sharma, Principal, KV Lumding, Mr. M .S Sengar, PGT(Maths), KV Lumding, MrPrabir Nandi (Comp. Inst.).
I am confident that the study/support material class x mathematics will directly help the students to understand the concept well and meet quality expectation.
Wish you all the best.
(S K Sharma)
Principal & Cluster I/C
KV Lumding
General Instructions:
1.As per CCE guidelines, the syllabus of mathematics for class x has been divided into two terms.
2. The units specified for each term shall be assessed through both formative and summative assessment.
3. In each term there will be two formative assessment and one summative assessment.
4. Listed Laboratory activities and projects will necessary be assessed through formative assessment.
Value Based Questions: 1. To inculcate knowledge, skills and values through Holistic education.
2. Inclusion of values in assessment scheme will result in greater emphasis to this component of education in teaching learning process.
3. The marks allotted to value based questions is 3-5.
Problem Solving Assessment:
1. The ‘Problem Solving Assessment’ (CBSE-PSA)will be counted towards FA-4 which is 10% of the total assessment of class IX. This assessment will also be carried forwards towards the FA-4 in class X. This score will be reflected in one language (English or Hindi), Mathematics, Science and Social Science. The same score will be reflected in FA-4for class IX and X.
2.The students will have the option to improve their PSA score in class X, as they can sit for the test with class IX students. The best score will be reflected in the final certificate in case of those applying for improvement.
TIPS FOR EFFECTIVE USE OF STUDY MATERIAL :-
- Go through the course of study given in the beginning. Identify the chapter carrying more weight age.
- Suggestive blue print and design of question paper is a guideline for you to have clear picture about the form of the question paper.
- Go through the key points given in the beginning and try to understand the concept with proper example.
- Slow learners may revise the knowledge part first.
- Bright students may emphasize the application part of the question paper.
- Under the active supervision of teachers study habit may be inculcated amongst students.
- Better learning may be promoted in planned way.
- Day wise target may be assigned to the students and reviewed the same by the concern teacher.
- Prepare chapter wise important notes and tricks (i.e. key points) for quick revision for the examination.
- Prepare chapter wise and level wise list of questions.
- Discuss your ‘DOUBTS’ with your teacher and classmates..
12. Go through the questions given in the oral test and quiz in the last of the study material. It will be helpful in understanding the concepts for MCQ.
13. Practice the VALUE BASED questions given in this study material.
14.Discuss the previous years questions of PSA.
DESIGN OF SAMPLE QUESTION PAPER
MATHEMATICS: CLASS X
SA I (2014-15)
TERM I
COURSE STRUCTURE
CHAPTER / MARKS: 90I NUMBER SYSTEM (Real Numbers) / 11
II ALGEBRA
Polynomials, pair of linear equations in two variables. / 23
III GEOMETRY (Triangles) / 17
V TRIGONOMETRY
Introduction to trigonometry, trigonometric identity. / 22
VII STATISTICS / 17
TOTAL / 90
QUESTION PATTERN
TYPE OF QUESTIONS / MARKS PER QUESTION / NUMBER OF QUESTIONS / TOTAL MARKSM. C. Q. / 1 / 4 / 4
S. A. I / 2 / 6 / 12
S. A. II / 3 / 10 / 30
L. A. / 4 / 11 / 44
TOTAL / 31 / 90
BLUE PRINT
CHAPTER / MCQ / SA I / SA II / LA / TOTALNUMBER SYSTEM / 4(2) / 3(1) / 4(1) / 11(4)
ALGEBRA / 1(1) / 4(2) / 6(2) / 12(3) / 23(8)
GEOMETRY / 1(1) / 2(1) / 6(2) / 8(2) / 17(6)
TRIGONOMETRY / 1(1) / 9(3) / 12(3) / 22(7)
STATISTICS / 1(1) / 2(1) / 6(2) / 8(2) / 17(6)
TOTAL / 4(4) / 12(6) / 30(10) / 44(11) / 90(31)
DESIGN OF SAMPLE QUESTION PAPER
MATHEMATICS: CLASS X
SA II (2014-15)
TERM II
COURSE STRUCTURE
CHAPTER / MARKS: 90II ALGEBRA (Contd.) ( Quadratic equations, arithmetic progressions) / 23
III GEOMETRY(Contd.) (Circles, constructions ) / 17
IV MENSURATION (Areas related to Circles, Surface Area & Volumes) / 23
V TRIGONOMETRY (Contd.) (Heights and Distances. ) / 08
VI COORDINATE GEOMETRY / 11
VII PROBABILITY / 08
TOTAL / 90
QUESTION PATTERN
TYPE OF QUESTIONS / MARKS PER QUESTION / NUMBER OF QUESTIONS / TOTAL MARKSM. C. Q. / 1 / 4 / 4
S. A. I / 2 / 6 / 12
S. A. II / 3 / 10 / 30
L. A. / 4 / 11 / 44
TOTAL / 31 / 90
BLUE PRINT
CHAPTER / MCQ / SA I / SA II / LA / TOTALALGEBRA (Contd.) / 1(2) / 3(9) / 3(12) / 7(23)
GEOMETRY(Contd.) / 2(4) / 3(9) / 1(4) / 6(17)
MENSURATION / 1(1) / 2(4) / 2(6) / 3(12) / 8(23)
TRIGONOMETRY (Contd.) / 2(2) / 1(2) / 1(4) / 3(08)
COORDINATE GEOMETRY / 1(1) / 2(6) / 1(4) / 4(11)
PROBABILITY / 2(8) / 3(08)
4(4) / 6(12) / 10(30) / 11(44) / 31(90)
1. If k, 2k – 1 and 2k + 1 are three consecutive terms of an A.P., the value of k is
(A) 2
(B) 3
(C) – 3
(D) 5
2. Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of APB is
(A) 30
(B) 45
(C) 60
(D) 90
3. In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm.
The radius of the circle inscribed in the triangle (in cm) is
(A) 4
(B) 3
(C) 2
(D) 1
4.
5.
6.
7. ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). The length of one of its diagonals is
(A) 5
(B) 4
(C) 3
(D) 25
8.
9. Find the values of p for which the quadratic equation 4x2 + px + 3 = 0 has equal roots.
10. Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.
11. In Figure 1, common tangents AB and CD to the two circles with centres O1and O2 intersect at E. Prove that AB = CD.
12. The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively. Prove that BD = DC.
13. Two different dice are tossed together. Find the probability
(i) that the number on each die is even.
(ii) that the sum of numbers appearing on the two dice is 5.
14. If the total surface area of a solid hemisphere is 462 cm2, find its volume. [ Take = 7/ 22]
15.
16. The sum of the 5thand the 9th terms of an AP is 30. If its 25thterm is three times its 8thterm, find the AP.
17. Construct a triangle with sides 5 cm, 5 . 5 cm and 6.5another triangle, whose sides are 3/5 cm. Now construct times the corresponding sides of the given triangle.
18. The angle of elevation of an aeroplane from a point on the ground is 60o. After a flight of 30 seconds the angle of elevation becomes 30o. If the aeroplane is flying at a constant height of 3000√ 3 m, find the speed of the aeroplane.
19. If the point P(k – 1, 2) is equidistant from the points A(3, k) and B(k, 5), find the values of k.
20. Find the ratio in which the line segment joining the points A(3, – 3) and B(– 2, 7) is divided by x-axis. Also find the coordinates of the point of division.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
***********$$$$$************
INDEX
SLNO / TOPIC / PAGE NO.PART -1
SA-1
1 / Real Numbers
2 / Polynomials
3 / A pair of linear equations in two variables
4 / Triangles
5 / Introduction to Trigonometry
6 / Statistics
7 / Model Question paper SA-1
SA- 2
8 / Quadratic Equation
9 / Arithmetic Progression
10 / Coordinate Geometry
11 / Some Applications of Trigonometry
12 / Circle
13 / Construction
14 / Area Related to Circle
15 / Surface Area and Volume
16 / Probability
17 / Model Question paper SA-2
PART – 2
18 / Activities (Term I)
19 / Activities (Term II)
20 / Projects
21 / Quiz/oral Test
22 / Puzzles
DETAILS OF THE CONCEPTS TO BE MASTERED BY EVERY CHILD OF CLASS X WITH EXERCISE AND EXAMPLES OF NCERT TEXT BOOK
SA-I
SYMBOLS USED
*:-Important Questions, **:- Very important Questions, ***:- Very very important Questions
S.No / TOPIC / CONCEPTS / DEGREE OF IMPORTANCE / References(NCERT BOOK)01 / Real Number / Euclid’s division
Lemma & Algorithm / *** / Example -1,2,3,4
Ex:1.1 Q:1,2,4
Fundamental Theorem of Arithmetic / *** / Example -5,7,8
Ex:1.2 Q:4,5
Revisiting Irrational Numbers / *** / Example -9,10,11
Ex: 1.3 Q:1.2 Th:1.4
Revisiting Rational Number and their decimal Expansion / ** / Ex -1.4
Q:1
02 / Polynomials / Meaning of the zero of Polynomial / * / Ex -2.1
Q:1
Relationship between zeroes and coefficients of a polynomial / ** / Example -2,3
Ex-2.2
Q:1
Forming a quadratic polynomial / ** / Ex -2.2
Q:2
Division algorithm for a polynomial / * / Ex -2.3
Q:1,2
Finding the zeroes of a polynomial / *** / Example: 9
Ex -2.3 Q:1,2,3,4,5
Ex-2.4,3,4,5
03 / Pair of Linear Equations in two variables / Graphical algebraic representation / * / Example:2,3
Ex -3.4 Q:1,3
Consistency of pair of liner equations / ** / Ex -3.2
Q:2,4
Graphical method of solution / *** / Example: 4,5
Ex -3.2 Q:7
Algebraic methods of solution
- Substitution method
- Elimination method
- Cross multiplication method
- Equation reducible to pair of liner equation in two variables
Example-13 Ex:3.4 Q:1,2
Example-15,16 Ex:3.5
Q:1,2,4
Example-19 Ex-3.6
Q :1(ii),(viii),2 (ii),(iii)
04 / TRIANGLES / 1)Similarity of Triangles / *** / Theo:6.1 Example:1,2,3
Ex:6.2 Q:2,4,6,9,10
2)Criteria for Similarity of Triangles / ** / Example:6,7
Ex:6.3 Q:4,5,6,10,13,16
3)Area of Similar Triangles / *** / Example:9 The:6.6
Ex:6.4 Q:3,5,6,7
4)Pythagoras Theorem / *** / Theo:6.8 & 6.9
Example:10,12,14,
Ex:6.5 Q:4,5,6,7,13,14,15,16
05 / Introduction to Trigonometry / 1)Trigonometric Ratios / * / Ex:8.1 Q:1,2,3,6,8,10
2)Trigonometric ratios of some specific angles / ** / Example:10,11
Ex:8.2 Q:1,3
3)Trigonometric ratios of complementary angles / ** / Example:14,15
Ex:8.3 Q:2,3,4,6
4)Trigonometric Identities / *** / Ex:8.4 Q:5 (iii,v,viii)
06 / STATISTICS / CONCEPT 1
Mean of grouped data
- Direct Method
Ex:14.1 Q:1&3
- Assumed Mean Method
- Step Deviation Method
CONCEPT 2
Mode of grouped data / *** / Example:5
Ex:14.2 Q:1,5
CONCEPT 3
Median of grouped data / *** / Example:7,8
Ex:14.3 Q1,3,5
CONCEPT 4
Graphical representation of c.f.(ogive) / ** / Example:9
Ex:14.4 Q:1,2,3
1.Real numbers
( Key Points )
- Euclid’s Division lemma:- Given Positive integers a and b, there exist unique integers q and r satisfying
a = b q +r, where 0≤ r < b, where a, b, q and r are respectively known as dividend, divisor, quotient and remainder.
i.e. Dividend = Divisor Quotient + Remainder
- Euclid’s division Algorithm:- A technique of finding HCF of two positive integers.
- To obtain the HCF of two positive integers say c and d, with c >d, follow the steps below:
Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such that c = d q + r, 0
Step II: If r=0, d is the HCF of c and d. If r division lemma to d and r.
Step III: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF
Ex. Q. Use Euclid’s Division Algorithm to find the HCF of:
•135 and 225
Solution: We start with the larger number 225.
By Euclid’s Division Algorithm, we have
225=135×1+90
We apply Euclid’s Division Algorithm on
Divisor 135 and the remainder 90.
135=90×1+45
Again we apply Euclid’s Division Algorithm on divisor 90 and remainder 45
90=45×2+0
So, H.C.F.(225,90)=45.
- The Fundamental theorem of Arithmetic:-
Every composite number can be expressed (factorised ) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
Ex.:
Ex-Check whether are composite number and justify.
We have, 7×11×13+13=1001+13=1014
1014=2×3×13×13
So, it is the product of more than two prime nos.
2,3 and 13.
Hence, it is a composite number.
7×6×5×4×3×2×1+5=5040+5=5045
5045=5×1009
It is the product of prime factors 5 and 1009.
Hence , it is a composite number.
Ans:-Composite number
Theorem: LET x be a rational number whose decimal expansion terminates. Then x can be expressed in the form ,where p and q are co-prime and the prime factorisation of q is the form of 2m5n, where n, m are non negative integers.
Ex.
Theorem: LET x be a rational number such that the prime factorisation of q is not of the form of 2m5n, where n, m are non negative integers. Then, x has a decimal expansion which is non terminating repeating (recurring).
Ex.
Theorem: For any two positive integers a and b,
HCF (a,b) X LCM (a,b)=a X b
Ex.: 4 & 6; HCF (4,6) = 2, LCM (4,6) = 12; HCF X LCM = 2 X 12 =24
Ans. : a X b = 24
Theorem: Let p be a prime number. If p divides a2 , then p divides a, where ‘a’ is a positive integer
Theorem 4 :√2 is irrational.
Proof : Let √2 be a rational number.
Therefore, √2 = p/q, where p and q are co-primes integers and q ≠ 0.
On squaring both side, we get
2 = p²/q²
p² = 2q²
p² is divisible by 2
Therefore, p is divisible by 2 (by above theorem )
Again, suppose p = 2m
p² = 4m² ( on squaring both side )
2q² = 4m²
i.e q² = 2m²
this means, q² is divisible by 2
Hence, q is divisible by 2.
Since, p and q are both are divisible by 2.
Therefore, p and q both has a common factor 2.
Which gives a contradiction to our assumption.
Therefore, √2 is not rational number
i.e √2 is irrational number.
EASY AND AVERAGE
1. If is a rational number . What is the condition on q so that the decimal representation of is terminating?
Ans.q is form of2m5n
2. The decimal expansion of the rational number will terminates after how many places of decimal ?
Ans.After 4 places of decimal.
3. Find the
Ans.19000
4. Write whether the rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
5.Use Euclid’s division algorithm to find the HCF of 135 and 225.
6.Find the HCF and LCM of 6, 72 and 120 using the prime factorization method.
7.Show that is an irrational number.
8.Find the LCM & HCF of 26 and 91 and verify that product of HCF and LCM is same as the product of numbers given.
Ans.LCM=182, HCF=13
9.Find the HCF and LCM of 6, 72 and 120 using the prime factorization method.
Ans.HCF = 6
LCM = 360
APPLICATION AND HOTS
1.Check whether 7x11x13 +13are composite number.Justify your answer.
Ans.Composite number.
2.Check whether 4n can end with the digit 0, where n is any natural number.
Ans.No, can not end with the digit 0.
3.Given that LCM (26, 169) = 338, write HCF (26, 169 ).]
Ans.13
4.Show that is an irrational number.
5.Show that square of an odd positive integer is of the form 8m + 1, for some integer m.
6. Use Euclid,s division lemma to show that the square of any positive integer is either of
the form 3m or 3m+1 for some integer m
(PROBLEMS FOR SELF EVALUATION/HOTS)
- State the fundamental theorem of Arithmetic.
- Express 2658 as a product of its prime factors.
- Show that the square of an odd positive integers is of the form 8m + 1 for some whole number m.
- Find the LCM and HCF of 17, 23 and 29.
- Prove that is not a rational number.
- Find the largest positive integer that will divide 122, 150 and 115 leaving remainder 5, 7 and 11 respectively.
- Show that there is no positive integer n for which +
- Using prime factorization method, find the HCF and LCM of 72, 126 and 168. Also show that.
2. Polynomials
( Key Points )
Polynomial:
An expression of the form a0 + a1x + a2x2 + ----- + an xn where anis called a polynomial in variable x of degree n.where; a0 ,a1, ----- an are real coefficient and each power of x is a non negative integer.
Ex.:- 2x2 – 5x + 1 is a polynomial of degree 2.
Note: , x + 1/x are not polynomials, because exponent of variable are not non negative integers.
- A polynomial is of the form p(x) = ax + b where a and b are real numbers and a 0, is of degree 1 and called linear polynomial. Ex. 5x -3, 2x etc
- A polynomial is of the form p(x) = ax2 + b x + c where a , b and c are real numbers and
a 0, is of degree 2 and called quadratic polynomial. Ex. 2x2 + x – 1, 1 – 5x + x2 etc
- A polynomial is of the p(x) = ax3 + b x2 + cx + d where a , b, ,c and d are real numbers and a 0, is of degree 3 and called cubic polynomial. Ex. etc.
Zeroes of a polynomial: A real number k is called a zero of polynomial The graph of intersects the X- axis. A polynomial p(x) of degree has at most n zeroes.
- A linear polynomial has at most one zero.
- A Quadratic polynomial has at most two zeroes.
- A Cubic polynomial has at most three zeroes
- The number of times a graph of a polynomial P(x) intersect the X-axis is the number of zeroes of P(x).
For a quadratic polynomial:If α ,βare zeroes of ax2+bx+c= 0 then :
- Sum of zeroes = α +β = -
- Product of zeroes = αβ =
- A quadratic polynomial whose zeroes are α and β, is given by: k｛x2-(α+β)+αβ｝
Division algorithm for polynomials: If g(x),q(x) and r(x) are the polynomials then we can find a polynomials p(x) which satisfy the relation p(x)= g(x)x q(x) +r(x).
This is known as Division Algorithm for polynomials
EASY AND AVERAGE
- In a graph of adjoining figure find the number of zeroes.
Ans. 3.
2. Find the zeroes of the quadratic polynomial 6x2-7x-3 and verify the relationship between the zeroes and the coefficients
Ans. Zeroes are 3/2 & -1/3
3. Find the zeroes of the polynomial P(x)= x2+2x Ans.0,-2
4. Find a quadratic polynomial, whose the sum and product of its zeroes are 5 and -2?
Ans. X2+3x-10
5. Divide P(x) by g(x) and find quotient and remainder, where p(x)=x3-3x2+5x-3, g(x)=x2-2
Ans. Quotientx-3=; Remainder 7x-9
6.Write the degree of zero polynomial.
Ans. Not defined.
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)
Ans. g(x)=x2-x+1
8. Obtain all zeroes of x3+13x2+32x+20.
Ans. -1, -2, -10
APPLICATION AND HOTS
- Form a cubic polynomial with zeroes 3, 2 and -1.
Hints/Ans.
.
- For what value of k, (-4) is a zero of polynomial kx+36.
Ans. k=9
- Give an example of polynomials
Ans.
4. If the zeroes of the polynomial are . Find
Ans.
5. Check whether the polynomial is a factor of
Polynomial
Ans. Remainder=0, Quotient=2t2 + 3t + 4, Given Polynomial is a factor.
6.Obtain all other zeroes of , if two of its zeroes are and
Ans. -1 & -1
(PROBLEMS FOR SELF-EVALUATION)
- Check whether is a factor of
- Find quotient and remainder applying the division algorithm on dividing +2x -4 by
- Find zeros of the polynomial
- Find the quadratic polynomial whose sum and product of its zeros are respectively.
- Find the zeroes of polynomial
- If one of the zeroes of the polynomial 2, find the other root, also find the value of p.
- If are the zeroes of the polynomial +4 show that find the value of k.
- If are the zeroes of the equation
- Pair of linear equations in two variables
(Key Points)
- An equation of the form ax + by + c = 0, where a, b, c are real nos (a 0, b 0) is called a linear equation in two variables x and y.
Ex : (i)x – 5y + 2 =0
(ii)x – y =1
- The general form for a pair of linear equations in two variables x and y is
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
where a1, b1, c1, a2, b2, c2 are all real nos and a1 0, b1 0, a2 0, b2 0.
Examples
- Graphical representation of a pair of linear equations in two variables:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
(i) will represent intersecting lines if
i.e. unique solution. And this type of equations are called consistent pair of linear equations.
Ex: x – 2y = 0
3x + 4y – 20 = 0
(ii)will represent overlapping or coincident lines if
i.e. Infinitely many solutions, consistent or dependent pair of linear equations
Ex: 2x + 3y – 9 = 0
4x + 6y – 18 = 0
(iii)will represent parallel lines if
i.e. no solution and called inconsistent pair of linear equations
Ex: x + 2y – 4 = 0
2x + 4y – 12 = 0
(iv)Algebraic methods of solving a pair of linear equations:
(i)Substitution method
(ii)Elimination Method
(iii)Cross multiplication method
(Level - 1)
- Find the value of ‘a’ so that the point(3,9) lies on the line represented by 2x-3y=5
Ans: a=
- Find the value of k so that the lines 2x – 3y = 9 and kx-9y =18 will be parallel.
Ans: k= 6
- Find the value of k for which x + 2y =5, 3x+ky+15=0 is inconsistent
Ans: k= 6
- Check whether given pair of lines is consistent or not 5x – 1 = 2y, y = +
Ans: consistent
- Determine the value of ‘a’ if the system of linear equations 3x+2y -4 =0 and 9x – y – 3 = 0 will represent intersecting lines.
Ans: a
- Write any one equation of the line which is parallel to 2x – 3y =5
Ans:
- Find the point of intersection of line -3x + 7y =3 with x-axis
Ans: (-1, 0)
- For what value of k the following pair has infinite number of solutions.
(k-3)x + 3y = k
k(x+y)=12
Ans: k= 6
- Write condition sothat a1x + b1y = c1 and a2x + b2y = c2 have unique solution.
Ans:
( Level - 2)
- 5 pencils and 7pens together cost Rs. 50 whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and that of one pen.
Ans: Cost of one pencil = Rs. 3
Cost of one pen = Rs. 5
- Solve the equations:
3x – y = 3
7x + 2y = 20
Ans: x=2, y=3
- Find the fraction which becomes to 2/3 when the numerator is increased by 2 and equal to 4/7 when the denominator is increased by 4
Ans: 28/45
- Solve the equation:
px + qy = p – q
qx – py = p + q
Ans: x = 1, y = -1
( Level - 3 )
- Solve the equation using the method of substitution: