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PART 2, MOST LIKELY QUESTIONS (2022 EXAM), , Complex Numbers and Quadratic Equations, (1+2+3× 2+5 =14(5Q), ONE MARK QUESTONS, 1., 2., 3., 4., 5., 6., , Express the complex number i9 + i19 in a + ib form, Express the complex number i-39 in a + ib form, Find the multiplicative inverse of –i, Find the modulus of the complex number z = -1-i√3, Find the modulus of the complex number z = -√3 +𝑖, If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find, the values of x and y., , TWO MARKS QUESTONS, 1., , 1, , Express the complex number (-5i)(8 𝑖) in a+ ib form., 3, , 2. Express the complex number (5i)(− 𝑖) in a+ ib form, 5, , 3. Find the multiplicative inverse of 2 – 3i, 4. Find the multiplicative inverse of 4-3i, 5. Find the multiplicative inverse of –i, 6. Find the multiplicative inverse of √5 + 3𝑖, 7. Find the modulus and argument of the complex number z = -1-i√3., 8. Find the modulus and argument of the complex number z = -√3 +𝑖, 9. If x + iy =, , 𝑎+𝑖𝑏, 𝑎−𝑖𝑏, , , prove that x2 + y2 = 1, , THREE MARKS QUESTONS, 1., , 1, , 7, , 1, , 4, , Express the complex number [(3 + 𝑖 3) + (4 + 𝑖 3)]- (− 3 + i)in a + ib, , form., 2. Express the complex number (1 − 𝑖)4 in a + ib form, 3, , 1, , 3. Express the complex number ( + 3𝑖) in a + ib form, 3, , 4. Express, , 5+𝑖√2, 1−𝑖√2, , in the a+ib form., , 5. Find the conjugate of, , (3−2𝑖)(2+3𝑖), (1+2𝑖)(2−𝑖), , 3, 1 25, , 6. Evaluate [𝑖 18 + ( ) ], 7. Reduce (, , 1, , 1−4𝑖, , −, , 𝑖, 2, , 1+𝑖, , 3−4𝑖, , ) ( 5+𝑖 ) to the standard form., , 8. Find the real numbers x and y if (x – iy)(3 + 5i) is the conjugate of, , -6 – 24i, 9., 10., 11., 12., 13., , Solve, Solve, Solve, Solve, Solve, , √5𝑥 2 + 𝑥 + √5 = 0, 2x2 + x + 1 = 0, x2 + 3x + 9 = 0, –x2 + x – 2 = 0, x2 + 3x + 5 = 0, , 17. If (x + iy)3 = u + iv, then show that, , 14. Solve √2𝑥 2 + 𝑥 + √2 = 0, 15. Solve x2 + x +, 16. Solve x2 +, 𝑢, 𝑥, , 𝑣, , 𝑥, √2, , + 𝑦 = 4(𝑥 2 − 𝑦 2 )., , 1, √2, , =0, , + 1=0

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1+𝑖 𝑚, , 18. If (, , 1−𝑖, , ) = 1, then find the least positive integral value of m., , FIVE MARKS QUESTONS, 1. Find the modulus and argument of the complex number, 2. Convert, , 1+3𝑖, , 1−3𝑖, , ., , in the polar form, , 3. Convert, , 1−2𝑖, 1+7𝑖, (2−𝑖)2, , 4., 5., 6., 7., , the, the, the, the, , Convert, Convert, Convert, Convert, , 1+2𝑖, , in the polar form., , complex, complex, complex, complex, , number, number, number, number, , √3 + 𝑖 into polar form, -1 –𝑖 in to polar form, -1 +𝑖 in to polar form, -1 +𝑖 in to polar form, , 8. Convert the complex number −, , 16, 1+𝑖√3, , in to polar form., , LINEAR INEQUALITIES, (1+2+5 =8(3Q)), ONE MARK QUESTONS, 1. Solve 30 x < 200 when, , (i) x is a natural number, (ii) x is an integer., , 2. Solve 5x – 3 < 3x +1 when ) x is an integer., 3. Solve 24x < 100, when (i) x is a natural number. (ii) x is an integer., 4. Solve – 12x > 30, when (i) x is a natural number. (ii) x is an integer., 5.. Solve 5x – 3 < 7, when (i) x is an integer. (ii) x is a real number., 6. Solve 3x + 8 >2, when (i) x is an integer. (ii) x is a real number., 7. Solve 4x + 3 < 6x +7., 8. Solve 4x + 3 < 5x + 7, , 9. Solve 3x – 7 > 5x – 1., , TWO MARKS QUESTONS, , 1. Solve 7x + 3 < 5x + 9. Show the graph of the solutions on number line.., 2. Solve (i) 3x – 2 < 2x + 1 (ii). 5x – 3 > 3x – 5 (iii). 3 (1 – x) < 2 (x + 4) Show the, graph of the solutions on number line, 3. Find all pairs of consecutive even positive integers, both of which are larger, than 5 such that their sum is less than 23.Solution:, 4. Solve the inequalities (i) 3(x – 1) < 2 (x – 3) (ii) 3 (2 – x) >2 (1 – x), 𝑥, 𝑥, 𝑥, 𝑥, 5. Solve (i) 𝑥 + 2 + 3 > 11. (ii) 3 > 2 + 1 (𝑖𝑖𝑖) 2 (2x + 3) – 10 < 6 (x – 2), (iv) 37 – (3x + 5) > 9x – 8 (x – 3), 6. Ravi obtained 70 and 75 marks in first two unit test. Find the minimum, marks he should get in the third test to have an average of at least 60, marks., 7. Find all pairs of consecutive odd positive integers both of which are smaller, than10 such that their sum is more than 11., , FIVE MARKS QUESTONS, 1. Solve the following system of inequalities graphically

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5x + 4y  40 , x  2 and y  3 ., 1. Solve the following system of inequalities graphically, x + 2y  8 ,2x + y  8 ,x > 0 and y > 0., 3. Solve the following system of inequalities graphically:, 1) 5x + 4y  20, x  1, y  2 2) 3x + 4y  60, x +3y  30, x  0, y  0, 3) 2x + y  4, x + y  3, 2x – 3y  6, 4) x – 2y  3, 3x + 4y  12, x  0 , y  1, 5) 4x + 3y  60, y  2x, x  3, x, y  0, 6) 3x + 2y  150, x + 4y  80, x  15, y  0, x  , ) x + 2y  10, x + y  1, x – y  0, x  0, y  0., , PERMUTATIONS AND COMBINATIONS, (1+2+3+5=11(4Q)), , ONE MARK QUESTONS, 1. Evaluate (i) 5 ! (ii) 7 ! (iii) 7 ! – 5! (iv) 8 ! (v) 4 ! – 3 !, 7!, 12!, 8!, 2. Compute (i) 5! (ii) 10! (2!) (iii) 6! (2!)., 3. If, 4. If, , 1, 8!, 1, , 1, , X, , 1, , X, , + 9! = 10! find 𝑥., , + 8! = 9! find 𝑥., 7!, , 5. If nC9 = nC8, find nC17., 6. If nC8 = nC2, find nC2, , TWO MARKS QUESTONS, 1. Find the number of 4 letter words, with or without meaning, which, can be formed out of the letters of the word ROSE, where the, repetition of the letters is not allowed., 2. Given 4 flags of different colours, how many different signals can be, generated, if a signal requires the use of 2 flags one below the other?, 3. How many 2 digit even numbers can be formed from the digits, 1, 2, 3, 4, 5 if the digits can be repeated?, 4. Find the number of different signals that can be generated by, arranging at least 2 flags in order (one below the other) on a vertical, staff, if five different flags are available., 5. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, and 5 assuming that repetition of the digits is allowed?, 6. How many 4-letter code can be formed using the first 10 letters of the, English alphabet, if no letter can be repeated?, 7. How many 5-digit telephone numbers can be constructed using the, digits 0 to 9 if each number starts with 67 and no digit appears more, than once?, 8. How many words, with or without meaning, can be formed using all, the letters of the word EQUATION, using each letter exactly once?, 9. Find n if 𝑛−1𝑃3 : 𝑛𝑃4 = 1: 9., 10. Find r, 5 4𝑃𝑟 = 6 5𝑃𝑟−1 .

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THREE MARKS QUESTONS, 1. Find the number of arrangements of the letters of the word, INDEPENDENCE. In how many of these arrangements, (i) do the words, start with P (ii) do the words begin with I and end in P?, 2. How many words, with or without meaning can be made from the letters, of the word MONDAY, assuming that no letter is repeated, if.(i) 4 letters, are used at a time, (ii) all letters are used at a time,, 3. In how many of the distinct permutations of the letters in MISSISSIPPI do, the four I’s not come together?, 4. In how many ways can the letters of the word PERMUTATIONS be, arranged if the (i) words start with P and end with S,, (ii) vowels are all together,, 5. How many words, with or without meaning, can be formed using all the, letters of the word EQUATION at a time so that the vowels and, consonants occur together?, 7. If the different permutations of all the letter of the word EXAMINATION, are listed as in a dictionary, how many words are there in this list before, the first word starting with E ?, 8. In how many ways can the letters of the word ASSASSINATION be, arranged so that all the S’s are together ?, 9. Find the number of different 8-letter arrangements that can be made, from the letters of the word DAUGHTER so that, (i) all vowels occur together (ii) all vowels do not occur together., , FIVE MARKS QUESTONS, 1. A committee of 3 persons is to be constituted from a group of 2 men and, 3 women. In how many ways can this be done? How many of these, committees wouldconsist of 1 man and 2 women?, 2. What is the number of ways of choosing 4 cards from a pack of 52, playing cards? In how many of these (i) four cards are of the same suit,, (ii) four cards belong to four different suits,(iii) are face cards,(iv) two are, red cards and two are black cards,(v) cards are of the same colour?, 3. Find the number of ways of selecting 9 balls from 6 red balls, 5 white, balls and 5blue balls if each selection consists of 3 balls of each colour., 4. In how many ways can one select a cricket team of eleven from 17 players, in which only 5 players can bowl if each cricket team of 11 must include, exactly 4bowlers?, 5. In how many ways can a student choose a programme of 5 courses if 9, Courses are available and 2 specific courses are compulsory for every, student?, 6. A group consists of 4 girls and 7 boys. In how many ways can a team of, 5 members be selected if the team has (i) no girl ? (ii) at least one boy and, one girl ? (iii) at least 3 girls ?, 6. A committee of 7 has to be formed from 9 boys and 4 girls. In how many, ways can this be done when the committee consists of:, (i)exactly 3 girls ? (ii) atleast 3 girls ? (iii) atmost 3 girls ?, 8. The English alphabet has 5 vowels and 21 consonants. How many words, With two different vowels and 2 different consonants can be formed from, The alphabet ?, 9. In how many ways can 5 girls and 3 boys be seated in a row so that no, two boys are together?

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BINOMIAL THEOREM, (1+3+5=09(3Q)), , ONE MARK QUESTONS, , 1.Expand (1 + 𝑎)𝑛 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ 𝑁., , 2. Expand (𝑖) (1 − 2𝑥)4 , (ii)using Binomial theorem (101)4, 3. Which term is middle terms in (𝑥 + 2𝑦)8 ., 4. Write the general term in the expansion of (𝑥 + 2𝑦)12 ., , 5. Write the indices of a and b is n in every term of the expansion (𝑎 + 𝑏)𝑛 ., 6.Write the number of terms in the expansion of (𝑎 + 𝑏)𝑛 ., 7. Find the coefficient of 𝑥 𝑟 in the expansion of (1 + 𝑥)𝑛 ., 8. Write the binomial coefficients in the expansion of (1 + 𝑥)𝑛 ., 9. Write the general term in the expansion of (𝑥 2 − 𝑦)12 ., , THREE MARKS QUESTONS, 1. Find the 4 term in the expansion of (𝑥 − 2𝑦)12 ., 𝑡ℎ, , 3 4, , 2, , 𝑥 5, , 2. Expand 1. (𝑥 2 + 𝑥) . 2. (1– 2𝑥)5 3. (𝑥 + 2) 4. ((2𝑥 − 3)5 ., 3. Using binomial theorem, evaluate each of the following:, 1. (96)3 2. (102) 5 3 . (101)4 ., 4. Find (𝑎 + 𝑏)4 – (𝑎 − 𝑏)4 . Hence, evaluate (√3 + √2)4 – (√3 − √2)4, 5. Find ( 𝑥 + 1)6 + ( 𝑥 − 1)6 . Hence or otherwise evaluate ( √2 + 1)6 + ( √2 − 1)6 ., 1, , 18, , 6. Find the 13𝑡ℎ term in the expansion of (9𝑥 − 3 𝑥) ., √, , 7. Find the coefficient of 𝑥 𝑖𝑛 (𝑥 + 3)8 ., 5, , 10, , 𝑥, , 8. Find the middle term in the expansion of (3 + 9𝑦) ., 9. Find the middle term in the expansion of (3 −, , 𝑥3, , 7, , ) ., 6, 10. Find a positive value of m for which the coefficient of 𝑥 2 in the expansion, (1 + 𝑥)𝑚 is 6., , 11. Find the term independent of 𝑥 ∈ 𝑡ℎ𝑒𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑜𝑓, 3, , ( √𝑥 +, , 1, 3, , 2 √𝑥, , 18, , ), , , 𝑥 > 0., , 12. Find an approximation of (0.99)5 using the first three terms of its expansion., , FIVE MARKS QUESTONS, 1. State and prove Binomial Theorem for any positive integer n., , SEQUENCES AND SERIES, (1+ 3× 2+ 4 + 5 =16(5Q), ONE MARK QUESTONS, 1., 2., 3., 4., , What is the 20, , Find the, , 𝑡ℎ, , 7th, , term of the sequence defined by 𝑎𝑛 = (n – 1) (2 – n) (3 + n) ?, , term of the sequence an =(−1)𝑛 𝑛3, , The 𝑛𝑡ℎ 𝑜𝑓 𝑎 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 an = 4n – 3,then find; a17,, The 𝑛𝑡ℎ 𝑜𝑓 𝑎 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 an = 4n – 3,then find;

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𝑛(𝑛−2), , 5. Find the 20th term of the sequence an =, 6. Find the 7th term of the sequence an =, , x2, , 2n, , 𝑛+3, , ., , 7. Find the nth terms of the G.P. 5, 25,125,…, 8. Write the first three terms in each of the sequences defined by, 𝑛−3, , the following (i) an = 2n + 5, (ii) an =, , 4, , ., , THREE MARKS QUESTONS, 1. an A.P. if, term is n and the nth term is m, where m ¹ n, find the pth term., 2. In an A.P., the first term is 2 and the sum of the first five terms is one-fourth, mth, , of the next five terms. Show that 20th term is –112., 3. If the sum of first p terms of an A.P. is equal to the sum of first q terms and p and, q are distinct, then find the sum of first (p + q) terms ., 1, , 1, , 4. In an A.P. if pth term is 𝑞 and qth term is 𝑝, show that the sum of first pq terms is, 1, (pq+1),, 2, , where p≠q, , 5. Insert 6 numbers between 3 and 24 such that the resulting sequence isan A.P., 6. If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the, last term., , 7. In a G.P., the 3rd term is 24 and the 6th term is 192.Find the 10th term0, 3 3, , 8. In the G.P. 3,2 , 4,… if Sn=, , 3069, 512, , , find the value of n., , 9. The sum of first three terms of a G.P. is, , 13, 12, , and their product is -1. Find the, , common ratios and the terms., , 10. Find the sum of the sequence 7, 77, 777, 7777, ... to n terms., 11. Insert three numbers between 1 and 256 so that the resulting sequenceis a G.P., 12. The sum of first three terms of a G.P. is, , 39, and, 10, , their product is 1. Find the common, , ratios., , 13. Find the 12th term of the G.P,whose 8th term is 192 and common ratio is 2., 14. The 5th, 8th and 11th terms of a G.P. are p, q and s respectively. Show that q2=ps., 15. Find a G.P. for which sum of the first two terms is – 4 and the fifth term is, 4 times the third term., , 16. Find the sum to n terms of the sequence, 8, 88, 888, 8888… ., 17. If the sum of three numbers in A.P., is 24 and their product is 440, find the, numbers., , FOUR MARKS QUESTONS, 1. Find the sum to n terms of the series :, , 5+11+19+29+41+...., , 2. Find the sum to n terms of the series whose nth term is n (n+3)., 3. Find the sum to n terms of the series: 1x2 + 2x3 + 3x4 + 4x5 + ...., 4. Find the sum to n terms of the series: 1x2x3 + 2x3x4 + 3x4x5 + ...

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5. Find the sum to n terms of the series: 3x12 + 5x22 + 7x32 + ...., 6. Find the sum to n terms of the series: 12 + (12 + 22 ) + (12 + 22 + 32 )+ ...., 7. Find the sum to n terms of the series:, , 1, 1, 1, + 2𝑥3 + 3𝑥4+, 1𝑥2, , ..., , FIVE MARKS QUESTONS, 1. If A.M. and G.M. of two positive numbers a and b are 10 and 8,, respectively, find the numbers., 2. If A and G be A.M. and G.M., respectively between two positive numbers,, prove that the numbers are A ± √𝐴2 − 𝐺 2 ., 3. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively,, then, obtain the quadratic equation., 4. If A and G be A.M. and G.M., respectively between two positive numbers,, prove that A≥ 𝐺., , STRAIGHT LINES, (1+ 2× 2+ 5+ 6 =16(5Q), 1. Find the slope of the lines:, , ONE MARK QUESTONS, , (a) Passing through the points (3, – 2) and (–1, 4),, (b) Passing through the points (3, – 2) and (7, – 2),, (c) Passing through the points (3, – 2) and (3, 4),, (d) Making inclination of 60° with the positive direction of x-axis., 2. Find the slope of the line, which makes an angle of 30° with the positive, direction of y-axis measured anticlockwise, 3. Find the angle between the x-axis and the line joining the points, (3,–1) and (4,–2)., 4. Write the equations for the i) x-axis ii) y-axis iii) line passing through (0, 0), with slope m., 5.Find the slopes of the line (i) x + 7y = 0, (ii) 6x + 3y – 5 = 0,, 6. Find the y-intercept of the line (i) x + 7y = 0, (ii) 6x + 3y – 5 = 0,, 7. Find the x-intercept and y-intercept of the line (i) 3x + 2y – 12 = 0,(ii) 4x – 3y = 6,, 8. Find the equations of the lines parallel to axes and passing through (– 2, 3)., 9. Equation of a line is 3x – 4y + 10 = 0. Find its (i) slope, (ii) x - andy-intercepts., 10.Find the distance between parallel lines 15x + 8y – 34 = 0and 15x + 8y + 31 = 0., , TWO MARKS QUESTONS, 1. Find the equation of the line through (– 2, 3) with slope – 4 ii) point (– 4, 3) with, slope ½ ., 2. Write the equation of the line through the points (1, –1) and (3, 5) ii) (–1, 1) and, (2, – 4)., 3. Find the equation of the line passing through (2, 2√3)and inclined with the xaxis at an angle of 75o.

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4. Find the equation of the line, which makes intercepts –3 and 2 on the, x- and y-axes respectively., 5. Find the equation of the line intersecting the x-axis at a distance of 3 units to the, left of origin with slope –2., 6. The vertices of D PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the, median through the vertex R., 7. Find the equation of a line that cuts off equal intercepts on the coordinate axes, and passes through the point (2, 3)., 2𝜋, , 8.Find equation of the line through the point (0, 2) making an angle 3 with the, positive x-axis.0, 9. P (a, b) is the mid-point of a line segment between axes. Show that equation, of the line is, , 𝑥, 𝑎, , 𝑦, , + 𝑏 = 2., , 10. Reduce the equation 3x + y − 8 = 0 into normal form. Find the values, of p and w., 11. Find the equation of a line perpendicular to the line x − 2y + 3 = 0, andpassing through the point (1, – 2)., 12. Find equation of the line parallel to the line 3x − 4y + 2 = 0 and passing, through the point (–2, 3)., 13. Find the equation of the right bisector of the line segment joining the points (3, 4), and (–1, 2)., 14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the, line 3x – 4y – 16 = 0., 15. Find the values of q and p, if the equation x cos q + y sinq = p is the normal form, of the line √3 x + y + 2 = 0., 16. Find the equation of the line whose perpendicular distance from the origin is 4 units and the, angle which the normal makes with positive direction of x-axis is 15°., , FIVE MARKS QUESTONS, 1. Derive the equation of the line in normal form., 2. Derive an expression for the perpendicular distance between a point and a, line., 3. Derive an expression for the acute angle between two lines having slopes, m1 and m2 ., And hence (i)Find the distance of the point (3, -5) from the line 3x-4y-26=0., ii)Find the angle between the lines √3x + y =1 and 𝑥 + √3𝑦 =1., iii)Find the angle between the lines x - √3y + 5 = 0 and √3𝑥 − 𝑦 + 7 = 0., iv)The slope of a line is double of the slope of another line. If tangent of the angle, between them is 1/3 find the slopes of the lines., 𝜋, , v)If the angle between two lines is 4 and slope of one of the lines is ½ find, the slope of the other line., , SIX MARKS QUESTONS

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1. Derive the equation of the line in normal form, and hence, a) Find the equation of line perpendicular distance from the origin is 5 units, andthe angle made by the perpendicular with the positive x-axis is 30°., b) Reduce the following equations into normal form. Find their perpendicular, distances from the origin and angle between perpendicular positive x-axis., (a)x – 3y + 8 = 0,, , (b) y – 2 = 0, (c) x – y = 4., , c)Reduce the equation √3x + y − 8 = 0 into normal form. Find the values of p and w., 2. Derive an expression for the perpendicular distance between a point and a, line and hence, a)Find the distance of the point (3, -5) from the line 3x-4y-26=0., b) Find the distance of the point (–1, 1) from the line 12(x + 6) = 5(y – 2)., 3. Derive an expression for the acute angle between two lines having slopes m1, and m2 and hence, a)Find the angle between the lines √3x + y =1 and 𝑥 + √3𝑦 =1., b)Find the angle between the lines x - √3y + 5 = 0 and √3𝑥 − 𝑦 + 7 = 0., c)The slope of a line is double of the slope of another line. If tangent of the angle, between them is 1/3 find the slopes of the lines., d)If the angle between two lines is, the slope of the other line., , 𝜋, 4, , and slope of one of the lines is ½ find