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TS inter 1st year Maths Blueprint

Education, TS Inter Maths / Nayini Satyanarayana Reddy

TS inter1st year Maths Blueprint TS Inter 1st year Maths Blueprint: : These blueprints were designed by the ‘Basics in Maths’ team. These to-dos help the TS intermediate first-year Maths students fall in love with mathematics and overcome their fear. These blueprints cover all the topics of the TS I.P.E first-year maths syllabus and help in I.P.E exams. Maths IA Two-Mark Questions & Solutions Maths IB Two Marks Questions & Solutions Maths – IA Concept Maths – IB Concept TS inter 1st year Maths Blueprin Maths – IIA Concept Maths – IIB Concept YouTube

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TS Inter second year Maths 2B Concept

Education, TS Inter Maths, TS Inter Maths 2B / Nayini Satyanarayana Reddy

TS Inter second year TS Inter second year: This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear. These notes cover all the topics covered in the TS I.P.E second year maths 2B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations. TS Inter second year 1. CIRCLES Circle: In a plane, the set of points that are at a constant distance from a fixed point is called a circle. ∗ The fixed point is called the centre (C) of the circle and the constant distance is called the radius(r) of the circle Unit circle: If the radius of the circle is 1 unit, then that circle is called the unit circle. Point Circle: A circle is said to be a point circle if its radius is zero. A point circle contains only one point in the centre of the circle. • ∗ The equation of the circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2 ∗ The equation of the circle with centre origin and radius r is x2 + y2 = r2 ⇒ x2 + y2 = r2 is called standard form of the circle. The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, f, g, h and c are real numbers, represent a circle iff (i) a = b ≠ 0 (ii) h = 0 and (iii) g2 + f2 + c ≥ 0 ∗ The general equation of the circle is x2 + y2 + 2gx + 2fy + c = 0 It’s centre c = (– g, – f) and radius ∗ The equation of the circle passing through origin is x2 + y2 + 2gx + 2fy = 0. ∗ The equation of the circle whose centre on the x-axis is x2 + y2 + 2gx + c = 0. ∗ The equation of the circle having centre on y-axis is x2 + y2 + 2fy + c = 0. ∗ The circles which have the same centre are called concentric circles. ∗ The equation of the circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is x2 + y2 + 2gx + 2fy + k = 0. ∗ The length of the intercept made by a circle x2 + y2 + 2gx + 2fy + c = 0 on x -axis is if g2 – c > 0 y -axis is if f2 – c > 0 Note: – (a) if g2 – c = 0, then A1 A2 = 0 ⇒ the circle touches the x- axis at only one point. (b) if f2 – c = 0, then B1 B2 = 0 ⇒ the circle touches the y- axis at only one point. (c) if g2 – c < 0, then the circle does not meet the x- axis. (d) if f2 – c < 0, then the circle does not meet the y- axis. ∗ The equation of the circle having the line segment joining A (x1, y1) and B (x2, y2) as a diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 0. ∗ Let A, B be any two points on a circle then, The line is called the secant line of the circle. The line segment is called the chard of the circle. AB is called the length of the chord. ∗ A chord passing through the centre is called the diameter of the circle. ∗ The angle subtended by a chord on the circumference of at any point is equal. The perpendicular bisector of a chord of a circle is asses through the centre of the circle. ∗ The angle in a semicircle is 900. ∗ The equation of the circle passing through three non-collinear points A (x1, y1), B (x2, y2), C (x3, y3) is Where ci = − (x2 + y2) and i = 1,2,3 ∗ centre of the circle is Parametric form: If P (x, y) is a point on the circle with centre (h, k) and radius r, then X = h + r cosθ, y = k + r sinθ 0 ≤ θ ≤ 2π. ⇒ A point n the circle x2 + y2 = r2 is taken as (r cosθ, r sinθ) and simply denoted by θ. Note: If the centre of the circle is the origin, then the parametric equations are x = r cosθ, y = r, 0 ≤ θ ≤ 2π. The point (h + rcosθ1, k + r sin θ1) is referred to as the point θ1 on the circle having the centre (h, k) and radius r. Notations: S = x2 + y2 + 2gx + 2fy + c S1 = xx1 + yy1 + g(x +x1) + f (y +y1) + c S11 = x12 + y12 + 2gx1 +2fy1 + c S12 = x1x2 + y1y2 + g(x1 + x2 ) + f (y1 + y2) + c Position of a point with respect to the circle: A circle divides the plane into three parts. 1. The interior of the circle 2. The circumference which is the circular curve. 3. The exterior of the circle. Power of point: Les S = 0 be a circle with radius ‘r’ and centre ‘C’ and P (x1, y1) be a point on the circle, then CP – r2 is called the power of point ‘P’ concerning S = 0. The power of point P (x1, y1) w.r.t. S = 0 is S11. •Let S = 0 be a circle in a plane and P

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TS Inter Second Year Maths 2A concept

Education, TS Inter Maths, TS Inter Maths 2A / Nayini Satyanarayana Reddy

TS Inter Second Year Maths 2A concept TS Inter Second Year Maths : This note is designed by ‘Basics in Maths’ team. These notes to do help the TS intermediate second year Maths students fall in love with mathematics and overcome the fear. These notes cover all the topics covered in the TS I.P.E second year maths 2A syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations. 1. COMPLEX NUMBERS • The equation x2 + 1 = 0 has no roots in real number system. ∴ scientists imagined a number ‘i’ such that i2 = − 1. Complex number: if x, y are any two real numbers then the general form of the complex number is z = x + i y; where x real part and y is imaginary part. ∗ z = x + iy can be written as (x, y) ∗If z1 = x1 + i y1, z2 = x2 + i y2, then ∗ z1 + z2 = (x1 + x2, y1 + y2) = (x1 + x2) + i (y1 + y2) ∗ z1 − z2 = (x1 − x2, y1 − y2) = (x1 − x2) + i (y1 − y2) ∗ z1∙ z2 = (x1 x2 −y1 y2, x1y2 + x2y1) = (x1x2 −y1 y2) + i (x1y2 +x2 y1) ∗ z1/ z2 = (x1x2 + y1 y2/x22 +y22, x2 y1 – x1y2/ x22 +y22) = (x1x2 + y1 y2/x22 +y22) + i (x2 y1 – x1y2/ x22 +y22) Multiplicative inverse of complex number: Multiplicative inverse of complex number z is 1/z. z = x + i y then 1/z = x – i y/ x2 + y2 Conjugate complex number: The complex numbers x + iy, x – iy are called conjugate complex numbers. The sum and product of two conjugate complex numbers are real. If z1, z2 are two complex numbers then Modulus and amplitude of complex number: Modulus: – If z = x + iy, then the non-negative real number is called the modulus of z and it is denoted by or ‘r’. Amplitude: – The complex number z = x + i y is represented by the point P (x, y) on the XOY plane. ∠XOP = θ is called amplitude of z or argument of z. ∗ x = r cosθ, y = r sinθ ⇒ x2 + y2 = r2 cos2θ + r2 sin2θ = r2 (cos2θ + sin2θ) = r2(1) ⇒ x2 + y2 = r2 ⇒ r = and = r. ∗ Arg (z) = tan−1(y/x) ∗ Arg (z1.z2) = Arg (z1) + Arg (z2) + nπ for some n ∈ { −1, 0, 1} ∗ Arg(z1/z2) = Arg (z1) − Arg (z2) + nπ for some n ∈ { −1, 0, 1} Argand plane: The plane containing all complex numbers is called the Argand plane. This was introduced by the mathematician Gauss (1777-1855), who first thought that complex numbers can be represented as a two-dimensional plane. The square root of a complex number: 2.DE- MOIVER’S THEOREM De- Moiver’s theorem: For any integer n and real number θ, (cosθ + i sinθ) n = cos nθ + i sin nθ. → cos α + i sin α can be written as cis α → cis α.cis β= cis (α + β) → 1/cisα = cis(-α) → cisα/cisβ = cis (α – β) ⟹ (cosθ + i sinθ) -n = cos nθ – i sin nθ ⟹ (cosθ + i sin θ) (cosθ – i sin θ) = cos2θ – i2 sin2θ = cos2θ + sin2θ = 1. → cosθ + i sin θ = 1/ cosθ – i sin θ and cosθ – i sin θ = 1/ cosθ + i sin θ ⟹ (cosθ – i sin θ) n = (1/ (cosθ –+i sin θ)) n = (cosθ + i sin θ)-n = cos nθ – i sin nθ nth root of a complex number: let n be a positive integer and z0 ≠ 0 be a given complex number. Any complex number z satisfying z n = z0 is called an nth root of z0. It is denoted by z01/n or ⟹ let z = r (cosθ + i sin θ) ≠ 0 and n be a positive integer. For k∈ {0, 1, 2, 3…, (n – 1)} let . Then a0, a1, a2, …, an-1 are all n distinct nth roots of z and any nth root of z is coincide with one of them. nth root of unity: Let n be a positive integer greater than 1 and Note: The sum of the nth roots of unity is zero. The product of nth roots of unity is (– 1) n – 1. The nth roots of unity 1, ω, ω2, …, ωn-1 are in geometric progression with common ratio ω. Cube root of unity: x3 – 1 = 0 ⇒ x3 = 1 x =11/3 3.QUADRATIC EXPRESSIONS Quadratic Expression: If a, b, c are real or complex numbers and a ≠ 0, then the expression ax2 + bx + c is called a quadratic expression in variable ‘x’. ∎ A complex number α is said to be a zero of the quadratic expression ax2 + bx + c if aα2 + bα + c = 0. Quadratic Equation: If a, b, c are real or complex numbers and a ≠ 0, then ax2 + bx + c = 0 is called a quadratic equation in variable ‘x’. ∎ A complex number α is said to be root or solution of the quadratic equation ax2 + bx + c = o if aα2 + bα + c = 0. The roots of a quadratic equation: ∎ The zeroes of the quadratic expression ax2 + bx + c are same as the roots of quadratic equation ax2 + bx + c = o.

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TS Inter Maths 1B Concept

Education, TS Inter Maths, TS Inter Maths1B / Nayini Satyanarayana Reddy

TS Inter Maths 1B Concept Ts Inter Maths 1B Concept: designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate first-year Maths students fall in love with mathematics and overcome the fear. These notes cover all the topics covered in the TS I.P.E first year maths 1B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations. 0.COORDINATE GEOMETRY( BASICS) Distance between two points A(x1, y1), B(x2, y2) is distance between a point A(x1, y1) to the origin is The midpoint of two points A(x1, y1), B(x2, y2) is If P divides the line segment joining the points A(x1, y1), B(x2, y2) in the ratio m:n then the coordinates of P are Area of the triangle formed by the vertices A (x1, y1), B (x2, y2) and C (x3, y3) is 1. LOCUS Locus: The set of points that are satisfying a given condition or property is called the locus of the point. Ex:- If a point P is equidistant from the points A and B, then AP =BP Ex 2: – set of points that are at a constant distance from a fixed point. here the locus of a point is a circle. • In a right-angled triangle PAB, the right angle at P and P is the locus of the point, then AB2 = PA2 + PB2 •Area of the triangle formed by the vertices A (x1, y1), B (x2, y2), and C (x3, y3) is 2.CHANGE OF AXES Transformation of axes: When the origin is shifted to (h, k), without changing the direction of axes then •To remove the first degree terms of the equation ax2 + 2hxy + by2 +2gx +2fy+ c = 0, origin should be shifted to the point •If the equation ax2 + by2 +2gx +2fy+ c = 0, origin should be shifted to the point Rotation of axes: When the axes are rotated through an angle θ then •To remove the xy term of the equation ax2 + 2hxy + by2 = 0, axes should be rotated through an angle θ is given by 3.STRAIGHT LINES Slope:- A-line makes an angle θ with the positive direction of the X-axis, then tan θ is called the slope of the line. It is denoted by “m”. m= tan θ • The slope of the x-axis is zero. • Slope of any line parallel to the x-axis is zero. • The y-axis slope is undefined. • The slope of any line parallel to the y-axis is also undefined. • The slope of the line joining the points A (x1, y1) and B (x2, y2) is Slope of the line ax + by + c = 0 is Types of the equation of a straight line: Equation of x- axis is y = 0. Equation of any line parallel to the x-axis is y = k, where k is the distance from above or below the x-axis. Equation of y- axis is x = 0. Equation of any line parallel to y-axis is x = k, where k is the distance from the left or right side of the y-axis. Slope- intercept form The equation of the line with slope m and y-intercept c is y = mx + c. Slope point form: The equation of the line passing through the point (x1, y1) with slope m is y – y1 = m (x – x1) Two points form: The equation of the line passing through the points (x1, y1) and (x2, y2) ’ is Intercept form: The equation of the line with x-intercept a, y-intercept b is • The equation of the line ∥ el to ax +by + c = 0 is ax +by + k = 0. • The equation of the line ⊥ler to ax +by + c = 0 is bx −ay + k = 0. Note: – If two lines are parallel then their slopes are equal m1 = m2 If two lines are perpendicular then product of their slopes is – 1 m1 × m2 = – 1 The area of the triangle formed by the line ax + by + c = 0 with the coordinate axes is The area of the triangle formed by the line with the coordinate axes is Perpendicular distance (Length of the perpendicular): The perpendicular distance from a point P (x1, y1) to the line ax + by + c = 0 is • The perpendicular distance from origin to the line ax + by + c = 0 is Distance between two parallel lines: •The distance between the parallel lines ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0 is Perpendicular form or Normal form: The equation of the line which is at a distance of ‘p’ from the origin and α (0≤ α ≤ 3600) is the angle made by the perpendicular with the positive direction of the x-axis is x cosα + y sinα = p. Symmetric form: The equation of the line passing through point P (x1, y1) and having inclination θ is Parametric form: if P (x, y) is any point on the line passing through A (x1, y1) and making inclination θ, then x = x1 + r cos θ, y = y1 + r sin θ where ‘r’ is the distance from P to A. • The ratio in which the line L ≡ ax + by + c = 0 divide the line segment joining the points A (x1, y1), B (x2, y2) is

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Ts Inter Maths IA Concept

TS Inter Maths, TS Inter Maths 1A / Nayini Satyanarayana Reddy

1. Functoins

Set: A collection of well-defined objects is called set. Ordered pair: Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b). Cartesian product: Let A and B are two non- empty sets. The Cartesian product of A and B is denoted by A × B and is defined as set of all ordered pairs (a, b) where a ϵ A and b ϵB cartesion product Rel

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TS 10th Class Maths Concept (T/M)

Education, TS SSC / Nayini Satyanarayana Reddy

10 వ తరగతి గణితం నోట్స్ 10 వ తరగతి గణితం 10 వ తరగతి గణిత శాస్త్రాన్ని అధ్యయనం చేయడం అంటే, పిల్లలు తమ స్వంత అభ్యాసానికి బాధ్యత వహిస్తారు మరియు సమస్యలను పరిష్కరించడానికి భావనలను వర్తింపజేయడం నేర్చుకుంటారు. ఈ విషయం . ఈ గమనికలు విద్యార్థులకు గణితంను ఇస్టపడేలా మరియు భయాన్ని అధిగమించడానికి సహాయపడతాయి. 1. వాస్తవ సంఖ్యలు మనం ముందు తరగతులలో వివిధ రకాలైన సంఖ్యలను గురించి తెలుసుకున్నాము .అంటే సహజ సంఖ్యలు, పూర్ణాంకాలు, పూర్ణ సంఖ్యలు, కరణీయ , అకరణీయ సంఖ్యలను గురించి నేర్చుకున్నాము . అకరణీయ సంఖ్యలు : p,q లు పూర్ణ సంఖ్య లై , q ≠ 0 అయిన సందర్భం లో రూపం లో రాయగల సంఖ్య లను అకరణీయ సంఖ్యలు అంటారు . దీనిని Q తో సూచిస్తారు . ఉదా :- మొదలగునవి. కరణీయ సంఖ్యలు : రూపం లో రాయలేని సంఖ్యలను కరణీయ సంఖ్యలు అంటారు . దీనిని QI లేదా S తో సూచిస్తారు . ఉదా :- మొదలగునవి. వాస్తవ సంఖ్యలు : అకరణీయ , కరణీయ సంఖ్యల సమూహాన్ని వాస్తవ సంఖ్యలు అంటారు . కింది పటములో మనం వీటిని చూడ వచ్చు. భాగహార శేష నిధి : a, b అనే ధన పూర్ణాంకాలు ఇచ్చినప్పుడు a = b q + r, 0≤ r <b అయ్యే విధంగా ఏకైక జత పూర్ణాంకాలు q ,r లు వ్యవస్తితం అవుతాయి. ఇది అందరికి తెలిసినప్పటికీ యూక్లిడ్ పుస్తకాల సంకలనం లోని 7 వ పుస్తకం లో మొట్టమొదటగా నమోదు చేయడం జరిగింది. ఈ భాగహార శేషనిధి మీద యూక్లిడ్ భాగహార శేష నిధి ఆధారపడి ఉంది. యూక్లిడ్ భాగహార శేషనిధి కేవలం ధన పూర్ణ సంఖ్యల పైనే నిర్వచించ బడినా , దానిని అన్ని శూన్యేతర పూర్ణ సంఖ్యలకు అనువర్తింప చేయవచ్చు . యూక్లిడ్ భాగహార శేషనిధి ఉపయోగించి గ . సా . భా ను కనుక్కోవడం : రెండు ధన పూర్ణ సంఖ్యల సామాన్య కారాణాంకాలలోని అతి పెద్ద కారణాo న్కాన్ని గ .సా. భా అంటారు . ఉదా:- 9 , 24 ల గ . సా .భా కనుక్కోవడం దీనిని 24 = 9×2 + 18 గా రాయవచ్చు 9 , 24 కన్నా పెద్దది కావున 24 ను 9 చే భాగిస్తే శేషం 6 వస్తుంది పై దానిలో ని భాజకం 9 మరియు 6 పై యూక్లిడ్ న్యాయాన్ని అనువర్తింప చేయగా 9 = 6 ×1 + 3 గా రాయవచ్చు పై దానిలో ని భాజకం 6 మరియు శేషం 3 పై యూక్లిడ్ న్యాయాన్ని అనువర్తింప చేయగా దానిని 6 = 3 ×2 + 0 గా రాయవచ్చు పై దాని లో శేషం సున్నా వచ్చింది కావున 9 , 24 ల గ . సా .భా 3 అవుతుంది. ప్రాథమిక అంకగణిత సిద్ధాంతం : ప్రతి సంయుక్త సఖ్యను ప్రదానానంకముల లబ్దంగా రాయవచ్చు మరియు ప్రధాన కారణాంకాల క్రమం ఏదైనప్పటికీ ఈ కారణాంకాల లబ్దం ఏకైకం . ఒక సంయుక్త సంఖ్య x ను x = p 1 p 2 ….p n అని రాయవచ్చు . దీనిలో p 1 , p 2, …., p n ఆరోహణ క్రమం లో రాయబడిన ప్రధానాంకాలు , అంటే p 1≤ p 2 ≤….≤ p n. ఈ సందర్భం లో ఒకే రకమైన ప్రదానంకములు వాడినచో వాటిని ప్రధానాంకాల ఘా తాoకాలుగా రాస్తాము . ఒకసారి మనం ఈ సంఖ్యలు ఆరోహణ క్రమంలో ఉన్నాయని భావిస్తే . అప్పుడు లబ్దం ఏకైకం . ఉదా :- 360 = 3×3×2× 2 × 2 × 5 = 32 × 23 × 5 ప్రధాన కారణాంకాల లబ్ద పద్ధతి ద్వారా గా. సా . భా మరియు కా . సా . గు కనుక్కోవడం; 9 , 24 ల గ . సా .భా మరియు కా. సా . గు. కనుక్కోవడం 9 యొక్క ప్రధాన కారణాంకాలు = 3 × 3 = 32 24 యొక్క ప్రధాన కారణాంకాలు = 2 × 2 ×2 × 3 = 23 ×31 9 , 24 ల గ . సా .భా = 31 = 3 ( సంక్యల యొక్క సామాన్య కారణాంకంల కనిష్ఠ ఘాతాల లబ్ధం ) 9 , 24 ల కా. సా . గు.= 32× 23 = 9×8 = 72 (సంఖ్యల యొక్క కారణాంకంల గరిష్ఠ ఘాతాల లబ్ధం) అకరణీయ సంఖ్యలు మరియు వాటి దశాంశ రూపాలు : x అనేది ఒక అకరణీయ సంఖ్య మరియు దీని ధశాంశ రూపం ఒక అంతమయ్యే దశాంశము ,అయినప్పుడు x ను p, q లు పరస్పర ప్రధా నాంకములు అయివున్న p /q రూపం లో వ్యక్త పరచవచ్చు . మరియు q యొక్క ప్రధాన కారాణాంకాల లబ్దం 2m 5 n అగును , n ,m లు ఋణేతర పూర్ణ సంఖ్యలు . పై దాని విపర్యయం ఇలా ఉంటెుంది • n ,m లు ఋణేతర పూర్ణ సంఖ్యలు మరియు q యొక్క ప్రధాన కారాణాంకాల లబ్దం 2m 5 n కలిగినటువంటి అకరణీయ సంఖ్య x = p /q అయిన, xయొక్క ధశాంశ రూపం ఒక అంతమయ్యే దశాంశము అగును , • n ,m లు ఋణేతర పూర్ణ సంఖ్యలు మరియు q యొక్క ప్రధాన కారాణాంకాల లబ్దం 2m 5 n రూపంలో లేకుంటే , అకరణీయ సంఖ్య x = p /q అయిన, xయొక్క ధశాంశ రూపం ఒక అంతంకాని దశాంశము అగును. ఉదా :- కరణీయ సంఖ్యలు :- • p, q లు కరణీయ సంఖ్యలు మరయు q ≠ 0 అయిన p /q రూపం లో రాయలేని సంఖ్యలను కరణీయ సంఖ్యలు అంటారు . • ప్రతీ కరణీయ సంఖ్య ధశాంశ రూపం ఒక అంతంకాని దశాంశము అగును. ప్రవచనం: p అనేది ఒక ప్రధాన సంఖ్య మరియు a ఒక ధనపూర్ణ సంఖ్య అయితే “ a2 ను p నిశ్శేషంగా భాగిస్తే a ను p నిశ్శేషంగాభాగిస్తుంది. ఘాతాలు : • a n ను ఘాతాంక రూపం అంటాము. a ను భూమి అని , n ను ఘాతము అని అంటారు. (i) (ii) (iii) ( am)n = amn (iv) a0 = 1 సంవర్గమానాలు:- x మరియు aలు ధనపూర్ణసంఖ్యలై a ≠1 అయివుండి ax = n అయిన x = అగును. 2. సమితులు • గణిత పరిశోధనలలో సమితి వాదాన్ని ‘ జార్జి కాంటర్’ అభివృద్ధి పరిచారు. సమితి: సునిర్విచిత వస్తువుల సముదాయాన్ని సమితి అంటారు. • సునిర్విచితం అనగా : 1 . సమితిలోని వస్తువులన్నిటికి ఒకే విధమైన సామాన్య పోలిక లేదా ధర్మం కలిగి ఉండాలి . 2 . ఏదైనా ఓకే సమితికి చెందినది, లేనిది నిర్దారించే టట్లు ఉండాలి. • సమితి పేరును ఇంగ్లీష్ వర్ణమాల లోని పెద్ద అక్షరాలతో సూచిస్తారు. ఉదాహరణకు A, B, … మొదలగునవి. • ఏదైనా ఓకే వస్తువు ఒక సమితికి చెందితే దాన్ని వస్తువులు/ మూలకాలు అంటారు . చెందినది (belongs to) అని తెలపటానికి మనం ∈ గుర్తు తో సూచిస్తాము.సమితికి చెందినది అయితే దానిని ∉ చే సూచిస్తాము. • జాబితా రూపం లేదా రోస్టర్ రూపం : సమితికి చెందిన మూలకాలన్నిటిని ‘కామ’ (,) తో వేరు చేసి ప్లవర్ బ్రాకెట్ { } లో ఉంచితే వచ్చే రూపాన్ని జాబితా రూపం లేదా రోస్టర్ రూపం అంటారు. ఉదా :- A = {1, 2, 3, 4}, B = { a, e, I, o, u}. • సమితి నిర్మాణ రూపం లేదా లాక్షణిక రూపం : సమితి లోని మూలకాన్ని x ( లేక y, z మొదలగు ఏవైన గుర్తులు ) గా సూచించి , x ప్రక్కన : లేదా / (colon ) ఉంచి ఆ సమితి కి చెందిన మూలకాల యొక్క లక్షణాలు లేదా ధర్మాలను రాసి ప్లవర్ బ్రాకెట్ { } ఉంచితే వచ్చే రూపాన్ని సమితి నిర్మాణ రూపం లేదా లాక్షణిక రూపం అంటారు . : లేదా / గుర్తులను such that అని చదువుతాము . ఉదా :- A = { x/ x ఒక సరి సంఖ్య మరియి x ∈N }, B = { y : y ఒక ప్రధాన సంఖ్య మరియు x < 10 }. సమితులు – రకాలు

TS 10th Class Maths Concept (T/M) Read More »

TS 8th Class Maths Concept

Education, TS State VI - IX / Nayini Satyanarayana Reddy

viii Class Maths || TS 8th Class Maths Concept viii Class Maths TS 6th Maths Concept TS 7th Maths Concept TS 8th Maths Concept TS 9th Maths Concept TS 10th Maths Concept viii Class Maths Studying maths in VIII class successfully meaning that children take responsibility for their own learning and learn to apply the concepts to solve problems. This notes is designed by the ‘Basics in Maths team’. These notes to do help students fall in love with mathematics and overcome fear. 1. RATIONAL NUMBERS • Natural numbers: All the counting numbers starting from 1 are called Natural numbers. 1, 2, 3… Etc. • Whole numbers: Whole numbers are the collection of natural numbers. 0, 1, 2, 3 … • Integers: integers are the collection of whole numbers and negative numbers. ….., -3, -2, -1, 0, 1, 2, 3…. • Rational numbers: The numbers which are written in the form of p/q, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q. Properties of Rational numbers • Natural numbers: 1.Closure property:- 2.Commutative property:- 3. Associative property:- • Whole numbers: 1. Closure property:- 2.Commutative property:- 3. Associative property:- • Integers: 1. Closure property:- 2.Commutative property:- 3. Associative property:- • Rational numbers: 1. Closure property:- 2.Commutative property:- 3. Associative property:- Additive identity:- 1 + 0 = 0 + 1 = 1, 3/2 + 0 = 0 + 3/2 = 3/2 • For any rational number ‘a’, a + 0 = 0 + a • 0 is the additive identity. Additive inverse:- 2 + (-2) = (-2) + 2 = 0, 5 + (-5) = (-5) + 5 = 0 • For any rational number ‘a’, a+ (-a) = (-a) + a = 0 • Additive inverse of a = -a and additive inverse of (-a) = a Multiplicative identity:- 2 × 1 = 1 × 2 = 2, 6 ×1/6 = 1 × 1/6 = 1/6 • For any rational number ‘a’, a × 1 = 1 × a = a • 1 is the multiplicative identity. Multiplicative inverse:- 2 × 1/2 = 1/2 × 2 =1 For any rational number ‘a’, a × 1/a = 1/a × a = 1 • multiplicative inverse of a =1/a • Multiplicative inverse of 1/a= a. Distributive property:- For any three rational numbers a, b and c, a × (b + c) = (a × b) + (a × c) 3/2×(5/3+1/5)=(3/2×5/3)+(3/2×1/5) Representing rational numbers on a number line: Ex: represent 29/6 on a number line this lies between 4 and 5 Divide the number line between 4 and 5 into 6 equal parts. Mark 5th part counting from 4. The role of zero: • If 0 is added to any rational number, then the rational number remains the same. For any rational number ‘a’ a + 0 = a = 0 + a • 0 is the additive identity. • Natural numbers do not have an additive identity. Additive inverse:- for any rational number ‘a’ a + (-a) = 0 = (-a) + a 3 + (-3) = 0, 10 + (-10) = 0 • additive inverse of ‘a’ is ‘-a’ and additive inverse of ‘-a’ is ‘a’ The role of 1: • If 1 is multiplied to any rational number, then the rational number remains same. For any rational number ‘a’ a × 1 = a = 1 × a 1 is the multiplicative identity. Multiplicative inverse:- 3 × 1/3 = 1 = 1/3 × 3 for any natural number ‘a’ a × 1/a = 1 = 1/a ×a • Multiplicative inverse of ‘a’ is ‘1/a’ and multiplicative inverse of ‘1/a’ is ‘a’ Distributive property: For any 3 rational numbers a, b and c, a (b + c) = ab + ac Ex:- 1/3 (2/5 + 1/5) = 1/3(3/5) = 3/15 1/3× 2/5 + 1/3 × 1/5 = 2/5 + 1/5 = 3/5 Inserting rational numbers between given two numbers: • There are infinitely many rational numbers between given two numbers. • We have two methods to find rational numbers between two numbers. First method: – First we have to convert given rational numbers as the same denominator and write the rational numbers which come between given numbers. Second method: – if a and b any given rational numbers then a/bis a rational number between a and b. The decimal representation of rational numbers The decimal expansion of rational is either terminating or non-terminating repeating decimal. Note:-Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non-terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals. Terminating decimals: Consider a rational number = 0.75 Non-terminating decimals: Consider a rational number 2-3 2/3 = 0.66666… 2/3 = 2. LINEAR EQUATIONS IN ONE VARIABLE Equations: An algebraic equation is the equality of algebraic expressions involving variables and constants. It has an equality sign. The expression on the left of the equality sign is called the LHS (Left Hand Side) and right of the equality is called RHS (Right Hand Side) of the equation. In an equation, the value of RHS and LHS are equal. This happens to be true only for certain values of the variable. This value is called the solution of the equation. Linear equations: If the degree of the equation is 1, then it is called a linear equation. Ex: 2x – 3 = 5, x = 3y, 5x + 3y = 3 and so

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TS 9th Class Math Concept

Education, TS State VI - IX / Nayini Satyanarayana Reddy

ix Class Maths Concept ix Class Maths 6th Maths Concept 7th Maths Concept 8th Maths Concept 9th Maths Concept 10th Maths Concept TS IX CLASS MATHS CONCEPT Studying maths in IX class successfully means that children take responsibility for their own learning and learn to apply the concepts to solve problems. This note is designed by the ‘Basics in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear. 1.REAL NUMBERS Rational numbers:- The numbers which are written in the form of, where p, q are integers and q≠ 0 are called rational numbers. Rational numbers are denoted by Q. ex:- 3/2, 3/5, 2, 1 and so on Natural numbers, Whole numbers, and Integers are rational numbers. The rational numbers do not have a unique representation. Representation of rational number: Represent To find a rational number between given numbers: Mean method:- A rational number between two numbers a and b is Ex:- insert two rational number between 1 and 2 To find a rational number in a single step:- Ex:- insert two rational number between 1 and 2 To find two rational numbers, we 1 and 2 as rational numbers with same denominator 3 (∵ 1 + 2 = 3) The decimal form of rational numbers: Note:- Every rational number can be expressed as a terminating decimal or non-terminating repeating decimal. Converting decimal form into a fraction: Terminating decimals:- (i) 1.2 = 12/10 = 6/5 (ii) 1.35 =135/100 = 135/100 = 27/20 Non-Terminating repeating decimals:- Irrational numbers: The numbers which are not written in the form of, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S. Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal. Ex:- Calculation of square roots: There is a reference of irrationals in the calculation of square roots in Sulbha Sutra. Procedure to find value: Representing irrational numbers on a number line: Ex:- Locate on a number line At ‘O’ draw a unit square OABC on a number line with each side 1 unit in length. By Pythagoras theorem OB2 = OA2 + AB2 = 12 + 12 OB2 = 2 OB = Using a compass with centre O and radius OB, draw an arc on the right side to O intersecting the number line at the point The location of is now at k. Note:- If a and b are two positive rational numbers such that ab is not a perfect square, this an irrational number lies between a and b. Real numbers The collection of all rational and irrational numbers is called real numbers. Real numbers cover all the points on the number line. Every real number is represented by a unique point on the number line. Ex:- are some examples of real numbers. Representing real numbers on the number line through successive magnifications:- locating 2. 775 on a number line Operation on real numbers The sum, difference, product and quotient of irrational numbers need not be an irrational number. Irrational numbers are not closed under addition, subtraction, multiplication, and division. For any two real numbers a and b Rationalizing the denominator: Rationalizing factor(R.F):-If the product of two irrational numbers is rational, then each of the two is the rationalizing factor to others. The rationalizing factor of a given irrational number is not unique. It is convenient to use the simplest of all R.F.s of given irrational number. Note:- Law of exponents for real numbers: 2. POLYNOMIALS AND FACTORIZATION Polynomial: An algebraic expression in which the variables involved have only whole number powers is called a polynomial. Ex: x2 , x3 + 1, x2 + xy + y2 and so on. Polynomials in one variable: The polynomials which are in the form of (a constant) × (some power of variable) are called polynomials in one variable. Ex: 2x, 4x, 3×2 + 1 and so on. Degree of the polynomial: The degree of a term is the sum of the exponent of its variable factors. The degree of the polynomial is the highest power of its variable term. Ex: degree of 3×2 + 2×3 + 1 is 3 degree of 5x2y3 + 2xy + 3×3 is 5 a polynomial in one variable x of degree n is anxn + an-1xn-1 + …+a1x + a0. Where a0, a1…an are constants and an ≠ 0. Types of polynomials: According to no. of terms: No. of non-zero terms Name of the polynomial Examples Terms 1 Monomial 3x 3x 2 Binomial -2 x + 7 -2x, 7 3 Trinomial 5×2 + 4x + 2 5×2, 4x, 2 More than 3 Multinomial 6×3 – 5×2 + 7x – 3 6×3, -5×2, x, -3 According to a degree: Degree of the polynomial Name of the polynomial Example Not defined Zero polynomial 0 0 Constant polynomial -12, 4, 7 etc. 1 Linear polynomial 2x+3, x – 3 etc. 2 Quadratic polynomial 2×2 + 3x + 1, x2 – 4 etc. 3 Cubic polynomial 3×3 – 4×2 + 2x + 6 4 Bi quadratic polynomial 4×4 + 2×3 + 45×2 +9x + 7 Zero of the polynomial: Let p(x) be a polynomial, if p(x) = 0 then, x is the zero of the polynomial p(x). Ex: p(x) = 2x – 2 P(1) = 2(1) – 2 = 2 – 2 = 0 ∴ 1 is the zero of the polynomial. Zero of the linear polynomial in one variable: Linear polynomial Zero of the polynomial x+ a – a x – a a ax + b -b/a ax – b b/a Dividing polynomials: If p(x) is divided by g(x), then there exists quotient polynomial q(x) and remainder r(x) such that p(x) = q(x) × g(x) + r(x) this

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2nd Year Maths Blue Print | TS Inter

Education, TS Inter Maths / Nayini Satyanarayana Reddy

2nd-Year Maths Blueprint for Inter 2nd Year Maths Blue Print 2nd Year Maths blueprints designed by the ‘basicsinmaths’ team. These to-dos help the TS intermediate second-year Maths students fall in love with mathematics and overcome their fear. These blueprints cover all the topics of the TS I.P.E second-year maths syllabus and help in I.P.E exams. These blueprints designed by ‘’ team. These to-dos help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear. These blueprints cover all the topics of the TS I.P.E second year maths syllabus and help in I.P.E exams. 2nd Year Maths Blue Print Visit my YouTube channel: Click on the Logo

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