1 Real Numbers 2. Sets 3. Polynomials 4. Pair of Linear Equations in Two Variables 5. Quadratic Equations 6. Progressions 7. Coordinate Geometry 8. Similar Triangles 9. Tangents and Secants to a Circle 10. Mensuration 11. Trigonometry 12. Applications of Trigonometry 13. Applications of Trigonometry 14. Probability 15. Statistics
- Problem Solving :- Using concepts and procedures to solve mathematical problems like following
a. Kinds of problems: Problems can take various forms-puzzles, word problems, pictorial problems, procedural problems, reading data, tables, graphs etc.
Reads problems. * Identifies all pieces of information/data
Separates relevant pieces of information. Understanding what concept is involved. Recalling of (synthesis of) concerned procedures, formulae etc. Selection of procedure. * Solving the problem. Verification of answers of raiders, problem based theorems.
b. Complexity: The complexity of a problem is dependen
- Making connections (as defined in the connections section). * Number of steps. Number of operations. Context unra
Reasoning Proof :- Reasoning between various steps (involved invariably conjuncture
Understanding and making mathematical generalizations and conjectures. Understands and justifies procedures. * Examining logical arguments. * Understanding the notion of proof. * Uses inductive and deductive logic. Testing mathematical
Communication conjectures :- Writing and reading, expressing mathematical notations (verbal and symbolic forms).
Example: 3 + 4 = 7; n+ni+n Sum of angles in triangle 180°
*Creating mathematical expressions
Connecting concepts within a mathematical domain-for example relating adding to multiplication, parts of a whole to a ratio, to division. Patterns and symmetry, measurements and space. Making connections with daily life. Connecting mathematics to different subjects, Connecting concepts of different mathematical domains like data handling and arithmetic or arithmetic and space. *Connecting concepts to multiple procedures
Connecting concepts within a mathematical domain-for example relating adding to multiplication, parts of a whole to a ratio, to division. Patterns and symmetry, measurements and space. Making connections with daily life. Connecting mathematics to different subjects, Connecting concepts of different mathematical domains like data handling and arithmetic or arithmetic and space. *Connecting concepts to multiple procedures
Ramu says, “If log_10(x) = 0 value of x =0^ prime prime . Do you agree with him? Give reason.
log 10^ X = 0 10 ^ 0 = x x =1\&x ne0 [** log_a(N) = x Rightarrow a ^ x = N ] .. I don’t agree with Ramu
- Determine ‘x’ so that 2 is the slope of the line passing through A(- 2, 4) and B(x, – 2)
Sol. Given slope of the line passing through A(- 2, 4) and B (x, – 2) is 2. Here x_{1} = – 2 y_{1} = 4 x_{2} = x y_{2} = – 2 Slope of overline AB = (y_{2} – y_{1})/(x_{2} – x_{1}) = 2 . (- 2 – 4)/(x – (- 2)) =2 Rightarrow -6 x+2 =2 tong Rightarrow – 6 = 2(x + 2) Rightarrow – 6 = 2x + 4 2x=-10 Rightarrow x = – 10/2 = – 5
x = – 5
- -3, 0 and 2 are the zeroes of the poly- nomial p(x) = x ^ 3 + (a – 1) * x ^ 2 + bx + c find a and c.
Sol. Given solution a * x ^ 3 + b * x ^ 2 + cx + d Given equation p(x) = x ^ 3 + (a – 1) * x ^ 2 Given roots-3, 0, 2 bx + c Sum of the roots = α + β + γ = – 3 + 0 + 2 = (- (a – 1))/1 -1=-a+1 Rightarrow – 1 – 1 = – a a = 2 Product of the roots alphabetagamma = (- d)/a -b a
(-3) (0)(2) = -c 1 Rightarrow c=0
a 2 and c = 0
Find the discriminant of the quadratic equation 3x ^ 2 – 5x + 2 = 0 and hence write the nature of its roots
Sol. Given quadratic equation 3x ^ 2 – 5x + 2 = 0 Given quadratic equation compare with a * x ^ 2 + bx + c = 0 Here a = 3 b = – 5 c = 2 Therefore, the discriminant b ^ 2 – 4ac = (- 5) ^ 2 – 4(3)(2) = 25 – 24 = 1 > 0 . The quadratic equation has distinct and real roots
Find the 11th term the of the A.P.: : 15, 12,
. Here
a_{1} = 15 a_{2} = 12 d = a_{2} – a_{1} = 12 – 15 = – 3 a_{n} = a + (n – 1) * d a_{11} = 15 + (11 – 1)(- 3) = 15 – 30 a_{11} = – 15
If chi = {1, 2, 3} B = {3, 4, 5} then find A – B and B – A
If chi = {1, 2, 3} B = {3, 4, 5} then find A – B and B – A
- Write any two linear polynomials hav- ing one term and three terms.
Sol. Linear polynomial one term = 2x. Linear polynomial three terms = x + y + z
- If A{x: x is a factor of 12} and B=\ x / x is a factor of 6}, then find A cup B and A cap B
Sol. Given A = {x: x is a factor of 12} = {1, 2, 3, 4, 6, 12} B={x: x is a factor of 6) = {1, 2, 3, 6}
A cup B= {1, 2, 3, 4, 6, 12} cup\ 1,2,3,6\ = {1, 2, 3, 4, 6, 12} A cap B= {1, 2, 3, 4, 6, 12} cap\ 1,2,3,6\ = {1, 2, 3, 6}
- Find the roots of quadratic equation x ^ 2 + 4x + 3 = 0 by “Completing Square method”.
Sol. Given quadratic equation is x ^ 2 + 4x + 3 = 0 x ^ 2 + 4x = – 3 x ^ 2 + 2x * 2 = – 3 Now LHS is of the form a ^ 2 + 2ab where b = 2 Adding b ^ 2 = 2 ^ 2 on both sides, we get x ^ 2 + 2(x)(2) + (2) ^ 2 = – 3 + (2) ^ 2 (x + 2) ^ 2 = – 3 + 4 (x + 2) ^ 2 = 1 Tomar x + 2 = plus/minus 1 x + 2 = 1 x + 2 = – 1 x = 1 – 2 = – 1 ; x = – 1 – 2 = – 3 -1, -3 are the roots of the given Q.E
- For what value of ‘m’ in the following, mx + 4y = 10 and 9x + 12y =30 s psi s- tem of equations will have no solu- tion? Why?
Given equations compare with a_{1}x + b_{1}z + c₁ = 0 and a_{2}x + b_{2}y + c_{2} = 0 are parallel If a_{1}/a_{2} = b 1 b 2 ne c 1 c 2 a_{1} = m_{1} a_{2} = 9 here given b_{1} = 4 b_{2} = 12 c_{1} = – 10 c_{2} = – 30 a_{1}/a_{2} = b 1 b 2 Rightarrow m 9 = 4 12 Rightarrow m=3 . If m = 3 then the above system will have no solution
- For what value of ‘m’ in the following, mx + 4y = 10 and 9x + 12y =30 s psi s- tem of equations will have no solu- tion? Why?
Given equations compare with a_{1}x + b_{1}z + c₁ = 0 and a_{2}x + b_{2}y + c_{2} = 0 are parallel If a_{1}/a_{2} = b 1 b 2 ne c 1 c 2 a_{1} = m_{1} a_{2} = 9 here given b_{1} = 4 b_{2} = 12 c_{1} = – 10 c_{2} = – 30 a_{1}/a_{2} = b 1 b 2 Rightarrow m 9 = 4 12 Rightarrow m=3 . If m = 3 then the above system will have no solution
- If x ^ 2 + y ^ 2 = 10xy prove that 2 * log(x + y) = log(x) + log(y) 2 * log(2) + log(3)
Sol. Given equation x ^ 2 + y ^ 2 = 10xy Adding 2xy on both sides x ^ 2 + y ^ 2 + 2xy = 10xy + 2xy (x – y) ^ 2 = 12xy Applying log on both sides log(x + y) ^ 2 = log(12xy) 2 * log(x + y) = log(12) + log(x) + log(y) .. 2 * log(x + y) = log(4) + log(3) log(x) + log(y) Hence proved.
- Shashanka said that (x + 1) ^ 2 = 2(x – 3) is a quadratic equation. Do you agree?
Sol. (x + 1) ^ 2 = 2(x – 3) x ^ 2 + 1 + 2x = 2x – 6 x ^ 2 + 1 + 2x – 2x + 6 = 0 x ^ 2 + 7 =0 Rightarrow x^ 2 0.x+7=0 Yes, this is a quadratic equation
- Use division algorithm to show that the square of any positive integer is of the form 5m or 5m + 1 or 5m + 4 where ‘m’ is a whole number.
Sol. a = bq + r, 0 <= r < b b = 5sor = 0, 1, 2, 3, 4
