Maths2C Unit 1 PDF

# Unit-1: Introduction of Linear Equation **P. No.:** A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. **Ex) 2x + 1 = 0** The standard form of a linear equation in two variables is of the form Ax + By = C where x...

# Unit-1: Introduction of Linear Equation **P. No.:** A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. **Ex) 2x + 1 = 0** The standard form of a linear equation in two variables is of the form Ax + By = C where x and y are variables and A, B, and C are constants. ## Intercepts and Slopes The slope indicates the steepness of a line and the intercept indicates the location where it intersects an axis. The slope and the intercept define the linear relationship between two variables and can be used to estimate an average rate of change. (ढलान एक रेखा की ढलान को रंगित करता है, और अवरोधन उस स्थान को इंगित करता है जहाँ यह एक अदा को काटता है ढलान व अवरोधून दो चर के बीच सेवक संबंध को परिभाषित करते की और परिवर्तन की औसत दर का अनुमान लगाने के लिए इसका उपयोग किया जा सकता है।) Usually, this relationship can be represented by the equation y = bx + c where c is the y-intercept and b is the slope. ## Slope of a One Variable Linear Equation **1. If θ is the angle which a line makes with the positive direction of the x-axis then its slope is defined by the tangent of an angle θ i.e. m = tan θ.** **2. Slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is:** $m = \frac{y_2 - y_1}{x_2 - x_1}$. ## Slope Intercept Form or Tangent Form: If θ is the inclination to x-axis and intercept on y-axis be C, intercept cuts from the positive direction of y-axis, then the equation of the line is y = mx + C, where m = tan θ. It is called Slope of the line. ## The Point Slope Form: The equation of a line which passes through a point (x₁, y₁) and whose slop is m is given by: (y - y₁) = m(x - x₁) ## Intercept Form: If the line cuts off intercepts a and b from the positive directions of x and y axes respectively, then the equation of the line is: $\frac{x}{a} + \frac{y}{b} = 1$. ## Problems **Q.1) Find the equation of a straight line which is parallel to x-axis and passed through the point (5, -3).** **Sol)** Straight line parallel to the x-axis is y = a, where a is an arbitrary constant. Since the line passes through (5, -3), therefore the point (5, -3) will satisfy the equation of the line. a = -3 Therefore, the equation of the required line is: y = -3 y + 3 = 0 **Q.1) Find the equation of the straight line which cut an intercept 3 from the y-axis and makes an angle of 135° from the x-axis.** **Sol)** Equation of straight line in slope intercept form is y = mx + c where m = tan θ = tan 135° = -1 c = 3 y = -1x + 3 x + y - 3 = 0 **Q.2) Find the equation to a line which cuts an intercept of √2 on the negative direction of the y-axis and makes an angle tan^-1(1/√2) with the x-axis.** **Sol)** Here, C = √2 and θ = tan^-1(1/√2) and m = tan θ = 1/√2 Hence the required the equation of the straight line is y = mx + c y = x/√2 + √2 x - √2y + 2 = 0 **Q.3) Find the equation to line which passes through (2, 3) and whose m is 3.** **Sol)** Equation of line passing through one point is (y - y₁) = m(x - x₁) y₁ = 3, x₁ = 2, m = 3 (y - 3) = 3(x - 2) y - 3 = 3x - 6 3x - y - 3 = 0 **Q.4) Find the equation of the straight line which passes through (-4, -5) and parallel to the straight line joining (2, -2) and (4, 8).** **Sol)** The equation of any straight line passing through (-4, -5) is: y + 5 = m(x + 4) Slope of the straight line passing through the points (2, -2) and (4, 8) is: $m = \frac{8 - (-2)}{4 - 2} = 5$. Hence, y + 5 = 5(x + 4) y + 5 = 5x + 20 5x - y + 15 = 0 **Q.5) Find the equation of the line which makes intercepts -4 and 5 on the x and y axes respectively.** **Sol)** x-intercept = a = -4 y-intercept b = 5 Equation of the line in the intercept form is: $\frac{x}{a} + \frac{y}{b} = 1$ $\frac{x}{-4} + \frac{y}{5} = 1$ 5x - 4y + 20 = 0 # System of Equations A system of linear equations is usually a set of two linear equations with two variables. **Ex) x + y = 5 and 2x - y = 1 are both linear equations with two variables.** ## Methods of System of Equations 1. **Graphical Method** 2. **Substitution Method** 3. **Elimination Method** ### Graphical Method In the graphical method: **Case 1: Intersecting Lines** - Only one solution. - a ≠ b₁. - a₂ ≠ b₂ - Consistent **Case 2: Parallel Lines** - No Solution. - a₁ = b₁, a₂ = b₂, c₁ ≠ c₂ - Inconsistent **Case 3: Coincident** - Coincident - Infinitely many solutions a₁ = b₁, a₂ = b₂, c₁ = c₂ - Dependent **Q.1) Solve the system of equations by graphical method**: -2x - y = 0 4x + 3y = 20 **By graphical method**, for equation 2x - y = 0 at x = 0, y = 0 at x = 1, y = 2 at x = 2, y = 4 for equation 4x + 3y = 20 at x = 0, y = 6.66 at x = 5, y = 0 Hence, the lines are intersecting at (2, 4) **Q.2) Solve the system of equations by substitution method**: x + 3y = 6 2x - 3y = 12 **Sol)** From equation 1: x = 6 - 3y By equation 2: 2(6 - 3y) - 3y = 12 12 - 6y - 3y = 12 -9y = 12 - 12 y = 0 By equation 3: x = 6 Hence, x = 6 and y = 0 **Q.1) Find their solution graphically**: 10 students of BTech took part in a mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz. **Sol**: Let the number of boys be x Number of girls be y x + y = 10 y = x + 4 Hence x = 3 y = 7 **Q.2) Find the value of variables which satisfy the following equation**: 2x + 5y = 20 3x + 6y = 12 **Sol)** Using elimination method: 2x + 5y = 20 -> (1) 3x + 6y = 12 -> (2) Multiplying (1) by 3 and (2) by 2: 6x + 15y = 60 -> (3) 6x + 12y = 24 -> (4) Subtracting (4) from (3): 3y = 36 y = 12 By (1): 2x + 5 * (12) = 20 2x = 20 - 60 x = -20 **Q.3) Find the value of variables which satisfies the equation**: 3x + 2y = 19 x + y = 8 **Sol)** Using elimination method: 3x + 2y = 19 -> (1) x + y = 8 -> (2) Multiply equation (2) by 3: 3x + 3y = 24 Subtracting (2) from (1): x= 5 Put the value of y in (2): x = 8 - 15 x = -3 Hence x = 3, y = 5 # Problem Set **Q.1) Determine the solution using the substitution method.** Find the number if one number is three times of the other number. Also, given that the difference between two numbers is 26. **Ans:** 13 and 39. **Q.2) Solve the linear equation using the substitution method.** a) 11p + 15q = -23 7p - 29 = 20 b) 7p + 69 = 3800 3p + 5q = 1750 **Ans:** a) p = 2, q = -3 b) p = 500, q = 50 **Q.3) Solve the following system of equations graphically.** a) 2x - y + 1 = 0 x - 2y + 8 = 0 b) 2x - y - 6 = 0 2x - 2y + 2 = 0 c) 2x + y - 1 = 0 4x + 2y - 2 = 0 **Ans:** a) x = 2, y = 5 b) No solution c) Infinitely many solutions, x = -1, 0 , 2, 3, ... y = 3, 1, -3, -5, ... **Q.4) Determine the value of K for which the given system of equation has many solutions.** 5x - ky = 15 x + 5y = 15 **Ans**: K = -5 **Q.5) Determine the solution using the elimination method** The sum of a two-digit number and the number obtained by reversing the digits is 88. If the digits of the number differ by 2, find the number. How many such numbers are there? **Ans**: 35 and 53. **Q.6) Solve the system of linear equations using the elimination method**. a) 2x + y = 3 6x - y = 9 b) 3u + 2t = 8 5u + 9t = 2 **Ans:** a) x = 3/2, y = 0 b) t = -2, u = 4 # Exponentials ((घातीय)) 1. a^o = 1 2. a^m = √a^m 3. (a^m)ⁿ = a^mn 4. a^m / aⁿ = a^{m - n} 5. a^m * b^m = (ab)^m 6. a^m * aⁿ = a^{m + n} 7. a^m / b^m = (a/b)^m 8. (a^m)ⁿ = (aⁿ)^m 9. a^m/n = √a^m # Logarithm Rules 1. log_b(MN) = log_bM + log_bN 2. log_b(M/N) = log_bM - log_bN 3. log_b(M^k) = klog_bM 4. log_b(1) = 0 5. log_b(b) = 1 6. log_b(a^k) = k **Q.1) Solve the equation Exponential equation**: a) 3x + 2 + 3x = 10 / 81 **Sol)** 2 * 3^x + 3^x = 10 / 81 3^x * (2 + 1) = 10 / 81 3^x * 3 = 10^0.3-4 x = -4 b) 5 / 4^x - 4^-x = 3 / 8 Let A = 4^x 2A² - 3A - 2 = 0 2A² -2A + A - 2 = 0 2A(A - 2) + 1(A - 2) = 0 (A - 2)(2A + 1) = 0 A = 2, A = -1/2 4x = 2 Hence 4x = 2 2x = 2 **Q.2) Solve the Exponential equation 3^x * 4^2x = 15** **Sol)** Given equation: 3^x * 4^2x = 15 3^x * 916^x = 15 48^x = 15 x = log₄₈(15) **Q.3) If 4^k = 5k + 3 then k =?** 4^k = 5^k * 5³ 4^k / 5^k = 5³ = 125 (4/5)^k = 125 log_(4/5)(125) = k **Q.4) Solve an Exponential equation**: 5^x *6^x = 150 **Sol)** 5^{x + 1} = 150 5^x(5 + 1) = 150 6* 5^x = 150 5^x = 25 = 5² x = 2 **Q.5) Solve 2^x * 3^{5 - x} = 4 and x + y = 5** **. Sol)** Equation: 2^x * 3^5-x = 4 xlog2 + (5-x)log3 = log4 x(log2 - log3) = log4 - 5log3 x = (log4 - 5log3) / (log2 - log3) x = 2log2 - 5log3 / log2 - log3 x = 2*0.3010 - 0.4771 / 0.3010 - 0.4771 x = 0.602 - 2.3855 / -0.1761 x = -1.7835 /-0.1761 x = 10.1295 **. Also,** y = 4x and y = 2^2x + 6 y = e^4x - (1) y = e^2x + 6 - (2) From (1) and (2): e^4x = e^2x + 6 (e^2x)² = e^2x + 6 (e^2x)² - e^2x - 6 = 0 Let e^2x = K K² - K - 6 = 0 K² - 3K + 2K - 6 = 0 K(K - 3) + 2(K - 3) = 0 (K - 3)(K + 2) = 0 K = 3, -2. e^2x = 3 log_e(e^2x) = log_e(3) 2xlogex = log_e(3) 2x = log_e(3) x = log_e(3) / 2 **Q.1) 4^2x - 20 * 4^x + 64 = 0** **Sol)** Let a = 4^x a² - 20a + 64 = 0 (a - 16)(a - 4) = 0 a = 16, a = 4 4^x= 16 4^x = 4² x = 2 **Q.2) 3^2x - 3^2x-1 = 18** 3^2x - 3^2x * 3^-1 = 18 3^2x (1 - 1/3) = 18 3^2x (2/3) = 18 3^2x = 18 * (3/2) 3^2x = 27 = 3³ 2x = 3 x = 3 / 2 **Q.1) Evaluate the expressions** a) log₄(12) - log₄(36) + log₄(192) **Sol)** log₄(12 / 36) + log₄(192) log₄ (1/3) + log₄(192) log₄((1/3) * 192) = log₄(64) = log₄(4³) = 3 b) log₇(x² - 4x + 4) - log₇(x - 2) **Sol)** log₇(x² - 4x + 4) / log₇(x - 2) log₇((x - 2)²) / log₇(x - 2) log₇(x - 2)² * log₇(x - 2)^-1 = log₇(x - 2) c) log₂√x = √log₂x log₂(x^1/2) = (log₂x)^1/2 log₂(x)^1/2 = (log₂x)^1/2 log₂x = (log₂x)^1/2 log_xx = m * log_xx # Substitution Let u = log₂x u = 1 / 2 (u²) = (1/2)² 4u² - 9u = 0 u(4u - 9) = 0 u = 0, u = 9/4 log₂x = 0, log₂x = 9/4 x = 2⁰ = 1 x = 2^9/4 **Q.1) 2log₃(3) + log₃(27)** **Sol)** 2log₃(3) + log₃(3³) 2 * 1 + 3 * 1 **Q.2) Solve the logarithmic equation for x**. log₃(4x - 3) - log₃(x + 4) = 1 **Sol)** log₃((4x-3) / (x + 4)) = 1 (4x-3) / (x + 4) = 3¹ 3x + 12 = 4x - 3 x = 15 # Polynomials and Polynomial Operations A polynomial is defined as an expression which is composed of variables, constants and exponents that are combined using mathematical operations such as addition, subtraction, multiplication, and division. **Ex. Q⁴ + 3Q³ - 2Q² + Q + 1 = 0** There are four main polynomial operations which are: 1. **Addition of Polynomial** 2. **Subtraction of Polynomial** 3. **Multiplication of Polynomial** 4. **Division of Polynomial** **Ex. Q.1) Find the sum of two polynomials: 5x² + 3x²y + 4xy - 6y², 3x² + 7x²y - 2xy + 4xy² - 5** **Sol**) 5x² + 3x² + 10x²y + 2xy + 4xy² - 6y² - 5 **Q.2) Find the difference of two polynomials: 7x³ + 28² + 3x + 9 and 5x² + 28 + 1** **Sol**) 7x³ - 3x² + 8 + 8 **Q.3) Solve (6x - 3y) * x(2x + 5y)** **Sol)** 12x² + 2xy² - 15y² **Q.4) Solve x² + 7x + 12** **Sol)** x² + 7x + 12 = (x + 3)(x + 4) = (x + 4) -(x + 3) ## Factorizations Method **Q.1) x² - 42x + 216 = 0** **Sol)** x² - 36x - 6x + 216 = 0 x(x - 36) - 6(x - 36) = 0 (x - 36)(x - 6) = 0 x = 36, x = 6 # Introduction to Quadratic Equation A quadratic equation is the equation of the 2nd degree. where a x² + bx + c = 0, here a, b, c are constants and x here is an unknown variable. ## Quadratic Formula $x = \frac{-b ± √(b² - 4ac)}{2a}$ ## Nature of Roots D = b² - 4ac - D > 0 - two distinct real roots - D = 0 - two equal real roots - D < 0 - no real roots **Q.1) 2x² - 5x + 3 = 0** 2x² - 2x + 3x + 3 = 0 2x(x - 1) - 3(x - 1) = 0 (x - 1)(2x - 3) = 0 x = 1, 3/2 **Q.2) Find the roots of the equation 2x² - 5x + 3 = 0 using factorization** **Sol)** Given: 2x² - 5x + 3 = 0 2x² - 2x - 3x + 3 = 0 2x(x - 1) - 3(x - 1) = 0 (2x - 3)(x - 1) = 0 x = 2, 3/2 **Q.3) 2x² + x - 300 = 0 using factorization** **Sol)** 2x² - 24x + 25x - 300 = 0 2x(x -1 2) + 25(x - 12) = 0 (x - 12)(2x + 25) = 0 x = 12, -25/2 **Q.4) Solve the quadratic equation 2x² + x - 528 = 0 using the quadratic equation formula** **Sol)** If we compare it with the standard equation ax² + bx + c = 0 a = 2, b = 1, and c = -528 Using the quadratic equation formula: $x = \frac{-b ± √(b² - 4ac)}{2a}$ x = -1 ± √(1² - 4 * 2 * -528) / (2 * 2) x = -1 ± √4225 / 4 x = -1± 65 / 4 x = 64 / 4 or x = -66/4 x = 16 or x = -33/2 **Q.1) Divide (4x²y² - 8xy³ + 16xy²) ÷ (4x²y) **Q.2) Add (4x² - 12xy + 9y²) + (25x² + 4xy - 32y³) **Q.3) Subtract (14x³y² - 5xy + 14y) - (7x³y² - 8xy + 10y) **Q.4) Multiply (4x - 7xy) * (2y + 3y) # Functions Let A and B be two non-empty sets. A function f from A to B is a set of ordered pairs ∈ A × B with the property that for each element 'a' in A there is a unique element 'b' in B such that (a, b) ∈ f. It is denoted by f : A → B A | | f | | 1 2 3 -- -- -- -- B | | | 9 b c d **Domain:** A = {1, 2, 3} **Co-domain:** B = {a, b, c, d} **Range:** {a, b} # Practice Questions **1) Find the slope of a line whose coordinates are (2, 7) and (8, 1)?** **2) If the slope of a line passing through the points (4, b) and (2, 9) is 3, then what is the value of b?** **3) Find the equation of the line that has a slope of 2/3 and contains the point (2, -1).** **4) Find the equation of the line that has a slope of 2/3 and a y-intercept of (0, 4).** **5) Find the slope and y-intercept of the line 4x - 7y + 1 = 0.** **6) Find the equation of a straight line passing through (- 9, 5) and inclined at an angle of 120° with the positive direction of the x-axis.** **7) Find the equation of the line which makes intercepts (-3, 0) and (0, 2) on the x and y axes respectively.** **8) Solve graphically the system of equations:** a) x + y = 3, 3x - 2y = 4 b) 2x + 4y = 10, 3x + 6y = 12 c) 3x - y = 2, 9x - 3y = 6 **9) Solve the following pair of equations by substitution method: x + 2y = 3, 7x - 15y = 2** **10) Solve the following pair of equations by elimination method: 3x - 4y = 2000, 7x - 3y = 2000** **11) Solve log₂(5) + log₂(x) = log₂(2x + 20)** **12) Solve log(x + 2) + log(x - 1) = 1** **13) Solve log₃(2) + 2log₃(x) = log₃(7x - 3)** **14) Solve log(x) + log(x - 8) = 1 ** **15) Solve the exponential equations:** a) 27 * 27^{2x - 3} = 81^{3x - 5} b) 3^{2x - 2} * 3^{x - 8} = 0 c) e^2x - 5e^x + 6 = 0 **16) Solve:** a) (x² - 2x² - 19x + 20) ÷ (x - 1) b) (4x² - 12xy + 9y²) + (25x² + 4xy - 32y²) c) (10ab + 15ac - 25bc + 5) + (49b - 8bc - 12) d) Subtract 7x³y² - 8xy + 10y from 14x³y² - 5xy + 14y e) Multiply 5x²y² (2x² + 5xy - 10) **17) Find the roots of the equation using factorization:** a) 4x² + 9x + 2 = 0 b) 4x² - 2x - 20 = x² + 9x **18) Find the roots of the equation using the quadratic formula:** a) 2x² - 8x - 24 = 0 b) 2x² - 14x - 13 = 0 c) x + 12 = 12 / (x + 1) # Answers 1) m = -1 2) b = -3 3) x - 3y = 5 4) 2x - 3y = -12 5) m = 4/7 and b = 2/7 6) √3x - y + 9√3 + 5 = 0, 2x - 3y + 6 = 0 7) x = 2 and y = 1 或者 No solution 或者 many solutions 8) a) x = 2, y = 1. b) No solution, c) many solutions. 9) x = 49/19, y = 13/19 10) x = 2000, y = 4000 11) x = 5 12) x = -4 or x = 3 13) x = 4/2, x =3 14) x = 9 15) a) x = 7/3, b) x = log 3 or x= log 2, c) x = 2 or x = 4 16) a) x² - x - 20. b) 29x² - 8xy - 23y², c) 10ab + 15ac - 33bc - 7, d) 7x³y² + 3xy - 4y, e) 10x⁴y² + 25x³y³ - 50x²y² 17) a) x = -2, -1/4. b) x = -4/3, 5 18) a) x = 6, -2. b) x = 7 ± 5√3, c) x = 0, -13