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SAT Algebra Linear Equations:  Step-by-Step Guide to Solving Linear Equations on the Digital SAT (Part 1) Build a Strong Algebra Foundation for a High SAT Math Score Success on the Digital SAT Math section begins with mastering algebra. Among all the algebra topics tested, linear equations are the most fundamental because they appear directly in equation-solving questions and indirectly in linear functions, graph interpretation, systems of equations, coordinate geometry, mathematical modeling, and many real-world word problems. Many students believe algebra is about memorizing formulas. In reality, algebra is about logical thinking. Every equation tells a mathematical story, and every solution follows a sequence of logical steps. Once you understand those steps, even difficult-looking SAT questions become manageable. This guide explains every concept carefully, assuming no prior knowledge beyond basic arithmetic. Each example is solved one step at a time, with an explanation for e...

Show that f: R → R defined as f(x) = x / √(1 + x²) is one-one but not onto.  f: R → R,  f(x) = x / √(1 + x²)  To check One-One / Injective Let x₁, x₂ ∈ R  such that f(x₁) = f(x₂) x₁ / √(1 + x₁²) = x₂ / √(1 + x₂²)  --------[1] Squaring both sides: x₁² / (1 + x₁²) = x₂² / (1 + x₂²) x₁²(1 + x₂²) = x₂²(1 + x₁²) x₁² + x₁²x₂² = x₂² + x₁²x₂² x₁² = x₂² x₁ = ± x₂ [1] is possible only if x₁, x₂ have the same sign x₁ = -x₂, is rejected we have to conclude that x₁ =x₂, Therefore f is one-one. To check Onto / Surjective Let y = x / √(1 + x²) squaring y² = x² / (1 + x²) y²(1 + x²) = x² y² + y²x² = x² y² = x² - y²x² = x²(1 - y²) x² = y² / (1 - y²) For x to be real, RHS ≥ 0 Since y² ≥ 0, we need 1 - y² > 0 ⇒ y² < 1 ⇒ -1 < y < 1 Range of f = (-1, 1) ≠ R Therefore f is not onto. see this video for more explanation  cbse 12th maths old board exam question paper 2025 2026 one to one injective function onto function

  SAT Math Formulas The Digital SAT includes a reference sheet with some geometry formulas, but it does not include everything you'll need. Knowing the most common formulas before test day helps you solve problems more quickly and reduces the chance of making simple mistakes. Exponent Rules For any nonzero number a : a⁰ = 1 a¹ = a aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ (ab)ⁿ = aⁿbⁿ (a/b)ⁿ = aⁿ/bⁿ a⁻ⁿ = 1/aⁿ Remember Multiply → add exponents. Divide → subtract exponents. A negative exponent means take the reciprocal. Radicals √a × √b = √(ab) √a ÷ √b = √(a/b) Examples √49 = 7 √81 = 9 ∛125 = 5 Perfect squares worth memorizing: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 Linear Equations Slope m = (y₂ − y₁)/(x₂ − x₁) Slope-intercept form y = mx + b Point-slope form y − y₁ = m(x − x₁) Standard form Ax + By = C Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes. Coordinate Geometry Distance Formula d = √[(x...

 Find the mean of the following distribution : Class               30 – 40    40 – 50   50 – 60      60 – 70      70 – 80 Frequency             6            13             8               12              11 calculate class mark, mid value or mid -x [x]   then the product f*x Class    f     x     f*x 30-40    6     35    210 40-50    13    45    585 50-60    8     55    440 60-70    12    65    780 70-80    11    75    825 Total  Σf =50,  Σfx= 2840 Mean = Σfx / Σf = 2840 / 50 = 56.8 for more explanation wat...

 A boy  has a collection of balls of different colours. He has a total of 35 balls in his basket out of which seven are black in colour and eight are yellow in colour. Out of remaining balls, some are white and the rest are red. Based on the above, answer the following questions: (a) If the probability of drawing a red ball at random from the basket is three times that of a white ball, then find the number of red balls in the basket. (b) Find the probability of drawing a ball at random from the basket which is either a black or a white ball Total balls = 35 Number of Black balls = 7 Number of Yellow balls = 8 Remaining balls = 35 − 7 − 8 = 20   Let number of white balls = w Let number of red balls = r  Given that  Out of remaining balls, some are white and the rest are red. Remaining balls =  20   w + r  =  20  ---------[1] Given  P(red) = 3 × P(white)   P(red) = r/35 P(white) = w/35   r/35 = 3 × w/35 r...

 A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that : (i) exactly one surgery is successful, (ii) at most two surgeries are successful. probability of independent events cbse 12th maths old board exam question paper 2025 2026 independent events success failure type Given:  P(Success) = 90% = 0.9 = 9/10  P(Failure) = 1 − 0.9 = 0.1 = 1/10 let S debite Success,  F denote Failure Probability that exactly one surgery is successful possibilities SFF, FSF, FFS P(SFF) = 0.9 × 0.1 × 0.1 = 0.009 P(FSF) = 0.1 × 0.9 × 0.1 = 0.009 P(FFS) = 0.1 × 0.1 × 0.9 = 0.009   P(exactly one auccess) = 0.009 + 0.009 + 0.009 = 0.027 = 27/1000  ...