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Quick Reference, STD: X 1 Linear Equations in Two Variables p(x) where p(x), q(x) qx The general form of a linear equation in two vari- are polynomials and q(x) 6= 0, is called rational ables x and y is expression or rational algebraic expression ax + by + c = 0, and here a 6= 0, b 6= 0 ⇒ if p(x), q(x) and r(x) are the polynomial Solution of Linear equations in two vari- and q(x) 6= 0, r(x) 6= 0 then ables by the method of Determinants p(x) × r(x) p(x) 1. = Let us consider two equations q(x) × r(x) q(x) ax + by = e . . . (1) cx + dy = f . . . (2) then p(x) p(x) −p(x) p(x) =− = = 2. − q(x) q(x) q(x) −q(x) Dy Dx x= and y = Addition & subtraction of two rational exD D pressions (Denominator are equal) Here, a b p(x) r(x) Let & be any two rational expression then: D = =a×d−b×c c d q(x) q(x) e b Dx = f d a e Dy = c f ∴ An expression of the form 1. p(x) r(x) p(x) + r(x) + = q(x) q(x) q(x) 2. p(x) r(x) p(x) − r(x) = q(x) q(x) q(x) =e×d−b×f =a×f −e×c HCF and LCM of Polynomials An algebric expression of the type a0 + a1 x + a2 x2 + · · · + an−1 xn−1 + an xn is called a polynomial if (i) a0 , a1 , a2 · · · an−1 , an are real numbers (ii) n is an integer which ≥ 0 Addition & subtraction of two rational expressions (Denominator are not equal) p(x) r(x) Let & be any two rational expression then: q(x) s(x) 1. p(x) r(x) p(x) × s(x) + r(x) × q(x) + = q(x) s(x) q(x) × s(x) 2. p(x) × s(x) − r(x) × q(x) p(x) r(x) − = q(x) s(x) q(x) × s(x) Multipication of Rational Expressions Test for divisiblity If ab and dc are two rational numbers, then we we Test for (x − 1):(x − 1) is a factor of polynomial know that their product is given by ab × dc = a×c b×d in x if sum (additon) of all the coefficent of the p(x) r(x) similarly, if & are the two rational exprespolynomial is zero q(x) s(x) sion there product is given by Test for (x + 1):(x + 1) is a factor of polynop(x) r(x) p(x) × r(x) mial in x if sum (additon) of all the coefficent in ⇒ q(x) × s(x) = q(x) × s(x) even power of x is equal to sum (additon) of all Division of Rational Expression the coefficent in odd power of x p(x) r(x) Let & be the two rational expression such q(x) s(x) Relationship between HCF & LCM if there that r(x) 6= is non-zero rational expression then we are two polynomial say p(x) and q(x) then LCM of { p(x) and q(x)}× HCF of { p(x) and q(x)} have p(x) r(x) p(x) s(x) p(x) × s(x) ⇒ ÷ = × = = p(x) × q(x) q(x) s(x) q(x) r(x) q(x) × r(x) Rational Algebraic Expressions we know that a number of the form is called a rational number. m n Quadratic Equations where n 6= 0 The general form of a quadratic equation is ax2 + bx + c = 0, where a,b,c are real number and Compiled by metric, for soft copy mail me at [email protected] Quick Reference, STD: X 2 a 6= 0 n is the number of terms in an AP sol. by perfect square method ⇒Make sure the coefficent of variable with index 2 d is the common difference between the sucis 1 (eg. 5x2 − 4x − 2 = 0, here the coefficent is 5 cessive terms of an AP for x2 hence first we shall divide both side by 5) ⇒Find the 3rd term with formula Sum of the first n terms of an AP 2 1 Let a, a + d, a + 2d, a + 3d, . . . be an AP with total Third term= × coefficent of x 2 n number of terms ⇒ Add third term on both the side a denote first term, tn denote last term of AP sol. by formula method d denotes the common diff. Let ax2 + bx + c = 0 be a quadratic equation, Let us denote sum upto n terms by S , we have n n where where a,b,c are real numbers and a 6= 0 then ⇒ Sn = (a + tn ) solution by√formula method is given by 2 ⇒ tn = a + (n − 1)d −b ± b2 − 4ac n x= ∴ ⇒ Sn = [2a + (n − 1)d] 2a 2 Case1:⇒ b2 − 4ac = 0 i.e b2 = 4ac b b x = − , x = − → both roots are equal 2a 2a Probability 2 Case2:⇒ b − 4ac > 0 i.e b2 > 4ac then the equation has √ two distinct root √ Probability of an event −b + b2 − 4ac −b − b2 − 4ac x= , x= probability of an even A,written as P(A), is defined 2a √ 2a 2 2 Case 3:⇒ b − 4ac < 0 then b − 4ac is not real as Number of outcomes favourable to A number and hence quadractic equation cannot have P(A)= Total number of possible outcomes probability of an impossible event is 0 any real roots probability of a sure event is 1. Probability of any event will lie between 0 and 1, In general for any Imp. Result 2 event A,we have 1 1 ⇒P(A)=1-P(A0 ) 1. x2 + 2 = x + −2 x x ⇒P(A)+P(A0 )=1 2 ⇒P(A0 )=1-P(A) 1 1 2 2. x + 2 = x − +2 Here P(A0 )denotes probability of not happening of x x an event A Arthmetic Progressions(AP) The general form of AP is t, t + d, t + 2d, t + 3d, . . ., here t is the first term and d is common difference, The AP with first term 100 and common difference 50 is 100, 150, 200, 250, 300 . . . Statistics Mean of Raw Data Mean of the values x1 , x2 , x3 . . . , xn is denoted by x̄ and is given by n x 1 , x 2 , x3 . . . , xn 1X x̄ = = xi n n i=1 nth (or the general) term of an AP, nth of an arthemetic progression t, t + d, t + 2d, t + 3d, . . . Assumed Mean Method for Calculating the Mean is given by A=an arbitary constant(usually A is chosen some where in the middle part of the given value) A is tn = t + (n − 1)d , here also called assumed mean tn is the nth term di =The reduced value, di = xi − A and is called t is the first term deviation of xi from A Compiled by metric, for soft copy mail me at [email protected] Quick Reference, STD: X n X d¯ = 3 Similarity fi di i=1 n X & Mean = x̄ = A + d¯ fi Two 4 0 s are said to be similar if 1. Their corresponding angles are equal i=1 2. Their corresponding sides are in the same ratio (i.e sides are propotional) Mean of Grouped Data n X Ratio of area of 40 s Let A1 and A2 be the area, b1 , b2 be the bases and h1 , h2 be the heights of any two 40 s Then the ratio of there area is given as fi xi 1. Direct method x̄ = i=1 n X fi i=1 2. Assumed mean method x̄ = A + d¯ 1. b1 × h1 A1 = A2 b2 × h2 3. Step-deviation method x̄ = A + h.d¯ (h is the width of the class intervals) 2. A1 b1 = , if heights of two 40 s are equal A2 b2 3. A1 h1 = , if bases of two 40 s are equal A2 h2 n X Remember d¯ is calculated as d¯ = fi di i=1 n X fi i=1 Median Formula for computing Median from grouped data N − c.f. × h, where Median=L + 2 f L =Lower boundary of a median class Trigonometry Trigonometric ratios of an acute angle Sin θ Opposite side of angle θ Hypothenuse Cos θ Adjacent side of angle θ Hypothenuse Tan θ Opposite side of angle θ Adjacent side of angle θ Cosec θ Hypotenus Opposite side of angle θ Sec θ Hypotenus Adjacent side of angle θ Cot θ Adjacent side of angle θ Opposite side of angle θ N =Total frequency c.f =cumulative f requency of the class preceding the median class f =f requency of the median class h=Width of the median class Mode Formula for computing Modefrom grouped data fm − f1 ×h Mode = L + 2fm − f1 − f2 L =Lower boundary of a modal class fm =Frequency of the modal class f1 =Frequency of the class coming before the modal class f2 =Frequency fo the class coming after the modal class h=Width of the modal class Instalments ⇒ x n = Pn 1 + R n 100 Pn = Amount of nth instalment. P =Amount borrowed,cash price. R=Rate of interest A=Principle + interest=Amount due x=Amount of installment Relationship between the trignometric ratios Compiled by metric, for soft copy mail me at [email protected] Quick Reference, STD: X 4 sinAcosecA=1 1 sinA= cosecA 1 cosecA= sinA cosAsecA=1 1 secA= cosA 1 cosA= secA tanAcotA=1 1 tanA= cotA 1 cotA= tanA sinA tanA= cosA cotA= cosA sinA Ratios of angle M. of ∠s → 0◦ s ↓ Ratios of ∠ sin θ 0 cos θ tan θ cosec θ sec θ cot θ 1 0 ND 1 ND 3. Right circular cylinder r:radius of the base, h:height (a) Curved Surface Area=2πrh (b) Total Surface Area=2πr(h + r) (c) Volume=πr2 h 4. Cone r:radius of the base h:height, l:slant height ◦ 30 45 1 √2 3 2 √1 3 √1 2 √1 2 2 √2 √3 3 ◦ 1 √ 2 √ 2 1 ◦ 90 3 2 1 √2 1 60 ◦ √ 3 √2 3 2 √1 3 0 ND 1 ND 0 Trigonometric Identities 1. sin2 θ + cos2 θ = 1 2. 1 + tan2 θ = sec2 θ 2 (a) (b) (c) (d) Curved Surface Area=πrl Total Surface Area=2πr(l + r) Volume= 13 πr2 h By pythagoras theorem:l2 = h2 + r2 5. Sphere r:radius (a) Surface Area=4πr2 (b) Volume= 43 πr3 6. Hemisphere r:radius (a) Surface Area=2πr2 (b) Volume= 23 πr3 Co-ordinate Geometry 2 3. 1 + cot θ = cosec θ Distance formula Trigonometric Ratios of Complementary An- Let P (x , y ), Q(x , y ) be any two points 1 1 2 2p gles Distance formula= (x2 − x1 )2 + (y2 − y1 )2 The distance 1. sin(90◦ − θ) = cos θ p of P (x, y) from origin O(0, 0) is given by OP = (x2 + y 2 ) 2. cos(90◦ − θ) = sin θ Section formula for internal division ◦ coordinates of the point P (x, y) which divides a line3. tan(90 − θ) = cot θ segment joining the point A(x1 , y1 ), B(x2 , y2 ) inter4. cot(90◦ − θ) = tan θ nallyin a given ratio m : n are given by mx2 + nx1 my2 + ny1 5. cosec(90◦ − θ) = sec θ P ≡ , m+n m+n 6. sec(90◦ − θ) = cosec θ Section formula for external division Coordinates of the point P (x, y) dividing the seg.joining Surface Areas & Volumes 1. Cuboid l:length, b:breadth, h:heigth (a) Curved Surface Area=2h(l + b) (b) Total Surface Area=2(lb + bh + hl) (c) Volume=lbh 2. Cube a: side of the cube (a) Curved Surface Area=4a2 (b) Total Surface Area=6a2 (c) Volume=a3 points A(x1 , y1 ), B(x2 , y2 ) externally are given by P ≡ mx2 − nx1 my2 − ny1 , m−n m−n Mid-point formula Point P (x, y) is the midpoint of segment joining points B(x2 , y2 ) its coordinates are given by P ≡ x1 + x2 y1 + y2 , 2 2 Coordinates of Centroid If A(x1 , y1 ), B(x2 , y2 ), C(x3 , y3 ) are the vertices of 4ABC, G(x, y) is the centroid then x1 + x2 + x3 y1 + y2 + y3 G≡ , 3 3 A(x 1 , y1 ), Compiled by metric, for soft copy mail me at [email protected]