130 Important Questions for XI

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1 130 Important Questions for XI E T V A 1

130 Important Questions for XI PREFACE Have you ever seen a plane taking off from a runway and going up and up, and crossing the clouds but just

because all the circumstances are favouring you; the runway and strong tyres are actually your parents and teachers and your efforts are the real

questions of Mathematics strictly according to CBSE curriculum Questions from all the chapters/topics/subtopics are clustered together after

Wish you all the luck Team AVTE For answers log on to wwwavteschoolorg 015 AVTE INDIA PVT LTD Published & Printed 37, AVTE HOUSE, DDA Commercial

2 130 Important Questions for XI PREFACE Have you ever seen a plane taking off from a runway and going up and up, and crossing the clouds but just think again that the real support was given by the runway, strong tires and the thrust of the engine Dear Students, time is now, to take off because all the circumstances are favouring you; the runway and strong tyres are actually your parents and teachers and your efforts are the real thrust which will set you to high esteem, and people will praise your performance About the Booklet: This booklet contains 130 most important questions of Mathematics strictly according to CBSE curriculum Questions from all the chapters/topics/subtopics are clustered together after indepth research and hence it becomes essential study material with which you AVTE can evaluate yourself and hence can improve your performance Wish you all the luck Team AVTE For answers log on to wwwavteschoolorg 015 AVTE INDIA PVT LTD Published & Printed 37, AVTE HOUSE, DDA Commercial Complex, Kailash Colony Ext, Zamrudpur, ND- 48, INDIA Printed in India All rights reserved This book, or parts thereof, may not be reproduced in any form without permission

3 130 Important Questions for XI 3 Important Questions 1 Solve the following system of inequations : 4x 9 3 x, 7x 1 7x x Draw the graphical solution of the following inequations: 3x + 4y 1, 4x + 3y 1, x 0, y 0 3 (a) If p th term of an AP is q and q th term is p, then show that its n th term is p + q - n (b) If b c a c a, b, a b c a b c are in AP, prove that 1, 1, 1 are also in a b c AP 4 Find the GP for which the sum of first two terms is -4 and the fifth term is 4 times the third term 5 Prove that: o sin 75 sin 15 1 o o cos75 cos 15 3 o AVTE Prove that cos cos - cos 3 cos = sin 5 sin 7 Find the equation of the line through the point (, 3) such that the segment of the line intercepted between the axes is bisected at this point 8 If A(10, 4), B(-4, 9), C (-, -1) are the vertices of a ABC, find the equation of the altitude through C 9 Find the angle between the following lines : x a y = 1 and b x b y = 1 a 10 (i) Find the equation of the ellipse with foci (0, ± 4) and e = 4 5, assuming that coordinate axes are the major and minor axes (ii) For the parabola y = -1x, find the co-ordinates of the focus, vertex and the equation of the directrix 11 If A = {x : x N}, B = {3x : x N}, C = {5x : x N}, then find : (i) A B (ii) B C (iii) (A B) C (iv) B'

4 130 Important Questions for XI 4 1 By using Principle of Mathematical Induction, prove that : (n - 1) (n + 1) = n 3 (4n + 6n - 1) n N 13 Find the general solution of the equation: 3 cos x + sin x = 14 Find the equation of the circle passing through the points (1, 1) (-1, ) and whose centre lies on the line x + y 1 = 0 15 If all the letters of the words 'GAAIN' be arranged as in dictionary, what is the fifteenth word? 16 Prove that the co-efficient of x n in the expansion of (1 + x) n is twice the co-efficient of x n in the expansion of (1 + x) n-1 17 Find the length of the major and the minor axes, the coordinates of foci, vertices, eccentricity and the equation of the directrixes for the ellipse x y Find the equation of parabola with vertex at origin, axis along x-axis and passing through the point (3, ) Also draw rough sketch of the parabola 19 The minute hand of a watch is 15 cm long How far does its tip move in 40 minutes? AVTE 0 Prove using the principle of mathematical induction that 41 n - 14 n is a multiple of 7 n N 1 Find the equation of a ellipse, with major axis along the x axis and passing through the points (4, 3) and (-1, 4) Find the equation of the hyperbola with foci (0, ± 3) and vertices 0, 3 If y = cos x + sin x cos x - sin x ; find dy dx 4 Find the coordinates of the point equidistant from four points (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0) 11 5 A (4, 0, 3), B(,, ) and C (-8, 4, 6) are the vertices of a triangle The internal bisector of A meets BC in D Find the coordinate of the point D

5 130 Important Questions for XI 5 3ax + b, x > 1 6 Given the function f(x) = 11, x = 1, Lim x 1 f(x) exists and is equal to the 5ax-b, x < 1 value of the function at x = 1 Find a and b 7 Prove that the following statement is true by direct method If x, y Z are such that x and y are odd, then xy is odd 8 Find the sum of first n terms of the series : In a class of 35 students, 17 have taken mathematics, 10 have taken Mathematics but not Economics Find the number of students who have taken both Mathematics and Economics and the number of students who have taken Economics but not Mathematics, if it is given that each student has taken either Mathematics or Economics or both 30 (i) Write set A in roster form where A = {a n : nn, a n+1 = 3a n and a 1 = } (ii) Write set B in set builder form where B =,,,, (i) Find the domain and range of the real function f(x) = x (ii) Let f = {(1, 1), (, 3) (0, -1) (-1, -3)} be a function from Z to Z defined by AVTE f(x) = ax + b, for some integer a, b Determine a and b 3 Let A and B be two sets such that A B consists of 6 elements If three elements of A B are: (1, 4), (, 6) (3, 6) (i) Find A, B, A B, B A (ii) If A and B are two sets having 3 elements in common, if n(a) = 5, n(b) = 4; Find n(a B) and n[(a B) (B A)] 33 Draw the graph of: y = cos x, 0 x 34 Find the modulus and argument of the complex number 1 i 1 3i 35 If a + ib and a - ib are the roots of 3x x 3 3 0, find a and b and hence prove that a +b = 3 36 Find the number of arrangements of the letters of the word INDEPENDENCE In how many of these arrangements do the words begin with I and end with P?

6 130 Important Questions for XI 6 37 If the lines x + y - 3 = 0, 5x + ky - 3 = 0 and 3x - y - = 0 are concurrent, find the value of k 38 Find n, if the ratio of fifth term from beginning to the fifth term from the end 4 1 in the expansion of 4 3 n is 6 : 1 39 Solve the following system of inequalities graphically: x + y 4, x + y < 11, x + 5y 40 x > 0, y > 0 40 Calculate the mean deviation about median for the following data: Classes Frequency Calculate the mean, variance and standard deviation for the following distribution: Classes Frequency Prove that : cos 6x = 3 cos 6 x - 48 cos 4 x + 18 cos x - 1 AVTE 43 If sin x = 1 4 and x is in II quadrant then find sin x, cos x and tan x 44 Using Principle of Induction prove that n n n n 1 n 4 n 1 n, n N Using Induction prove that: 1, n n n N Find the modulus, argument and multiplicative inverse of z = -1 - i 3 47 Solve for real x: x 1 3x x Find the general solution of : sec x = 1- tan x 49 Draw the graph of sin x Write its domain, range and period 50 IQ of a person is given by the formula : IQ = MA 100 where MA is mental CA age and CA is chronological age If 80 IQ 140 for a group of 1 years old children, find the range of their mental age

7 51 Simplify 130 Important Questions for XI ( a + b) - ( a - b) Hence evaluate (i) How many chords can be drawn through 1 points on a circle? (ii) How many triangles can be formed by joining the vertices of a hexagon? 1 7i 53 Write the polar form of z = i 54 From a class of 5 students, 10 are to be chosen for an excursion party There are students who decide that either both will join or none of them will join In how many ways can they be chosen? 55 Prove that : cos x + cos 3 x cos x Using Induction, prove that : 10 n is divisible by 11, n N 57 Solve: (i) 7x - 10x + 1 = 0 (ii) If x - iy = a ib a ; prove (x c id + y ) = c b 58 In an examination, a question paper consists of 1 questions divided into d AVTE parts ie, part I and part II containing 5 and 7 questions respectively A student is required to attempt 8 questions in all selecting at least 3 from each part In how many ways can a student select the questions? 59 In how many of the distinct permutations of the letters in ASSASSINATION, no two S's come together? 60 (i) Find the rank of the word MOTHER as arranged in a dictionary when the letters of the word are written in all possible orders (ii) Find r, if: 5 P 6 r = P r-1 61 The coefficient of (r - 1) th, r th and (r + 1) th terms in the expansion of (x + 1) n are in the ratio 1 : 3 : 5 Find n and r 6 (i) Prove that : (i) tan 4x = 4 tan x 1 tan x tan x tan x (ii) Show that : cos 8 = cos

8 130 Important Questions for XI 8 63 How many terms of the GP 3, 3, are needed to give the sum 4 51? 64 (i) A letter is chosen at random from the word MISSISSIPPI Find the (ii) probability that no two S s are together Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter Find the probability that atleast one letter is in its proper envelope 65 Find the probability that when a hand of 7 cards is drawn from a well shuffled deck of 5 cards, it contains: (i) all kings (ii) 3 kings (iii) atleast 3 kings 66 Prove that : sin A B tan A tan B sin A B tan A tan B 67 (a) Write the contra-positive of the following statement: If a number is (b) (c) divisible by 9, then it is divisible by 3 Write the negation of the following statement: For every positive real number x, the number (x - 1) is also positive AVTE Write the converse of the following statement: If two integers a and b are such that a > b, then a - b is always a positive integer x y 68 Derive equation of ellipse ie, 1 a b 69 Find the multiplicative inverse of - 3i 70 By using the principle of mathematical induction, prove that, for every natural number, (ab) n = a n b n n N 71 (i) If A (-, 1), B (, 3), C(-, -4) are three points, find the angle between the lines AB and BC (ii) Find the equation of the circle passing through the point (1, -1) and centre at the intersection of the lines x - y = 4 and x + 3y = -7 7 The sum of three numbers in A P is -3 and their product is 8 Find the numbers 73 The foci of an ellipse are (±, 0) and its eccentricity is 1, find its equation

9 130 Important Questions for XI 9 74 Evaluate : 3 lim x0 x 6x 8 x 6x 11x 6 75 Find the probability that a leap year, selected at random, will contain 53 Sundays or 53 Mondays 76 Let A & B are two sets Prove that : (A - B) B = A if and only if B A x 1 77 Find the derivative of with respect to x x 1 78 A bag contains 9 discs of which 4 are red, 3 are blue and are yellow The discs are similar in shape and size Two discs are drawn at random from the bag Calculate the probability that discs will be (i) only one blue (ii) non blue 79 In a survey of 700 students in a college, 180 were listed as drinking Limca, 75 as drinking Mirinda and 95 were listed as drinking both Limca as well as Mirinda Find how many students were drinking neither Limca nor Mirinda 80 If z 1 and z are two complex numbers, show that z 1 + z z 1 + z 81 If a + ib = c i c i, where c is real, prove that a + b = 1 AVTE 8 Draw the graph of the following function: x, if x 0, if x 0 x, if x 0 83 (a) Let R be the relation on the set N of natural numbers defined by R = {(a, b) : a + 3b = 1, a N, b N} Find the domain and range of R (b) Let A = {1,, 3, 4, 6} Let R be the relation on A defined by {(a,b) : a, b A, b is exactly divisible by a} Write R in roster form 84 Solve the following system of inequalities graphically : 5x + 4y 40, x, y 3 85 In how many ways can the letters of the word PERMUTATIONS be arranged if the, (a) Words start with P and end with S (b) Vowels are all together

10 130 Important Questions for XI Find the sum of the following series: to n terms 87 Show that four points A(1,, 3), B (-1, -, -1), C(, 3, ) & D(4, 7, 6) are the vertices of a parallelogram, but not a rectangle 88 Solve the following equation : sin+ sin4 + sin6 = Find the term containing y 3 in the expansion of 1, x y 90 How many numbers greater than can be formed by using the digits 1,, 0,, 4,, 4? 91 Solve graphically: x + y 10, x + y 1, x - y 0, y 0 9 Solve the quadratic equation : 1x - 8 x + 10 = 0 93 Prove by the principle of mathematical induction and binomial theorem that 3 n+ - 8n - 9 is divisible by 64 nn 94 If a, b, c, d are in GP, prove that : (a n + b n ), (b n + c n ), (c n + d n ) are in GP 95 Calculate mean and variance for the following distribution : Classes : Frequency : AVTE 96 For sets X = {1,, 3, 10}, A = {1,, 5, 6}, B = {6, 7}, verify that A - B = A B' = B' - A' 97 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 at most girls? 98 Find the term independent of x in the expansion of 6 3x 1 3x x 99 Find the middle term in the expansion of + 9y Find the equation of the ellipse whose vertices are (0, ± 13) and foci are (0, ± 5) 101 Find the coordinates of the foci, the eccentricity and the length of the latus rectum of the hyperbola, 5y - 9x = If a, b, c are in AP, prove that a, b, c are in AP b c c a a b 10

11 130 Important Questions for XI Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by ZX - plane 104 Using the word necessary and sufficient rewrite the statement The integer n is odd if and only if n is odd Also check whether the statement is true 105 Two students, Anil and Vineet appeared in an examination The probability that Vineet will qualify the examination is 005 and that Anil will qualify the examination is 010 The probability that both will qualify the examination is 00 Find the probability that (a) (b) At least one of them will qualify the examination Both will not qualify the examination 106 In a group of children, 35 play football, out of which 0 play football only, play hockey, 5 play cricket out of which 11 play cricket only 7 play cricket and football but not hockey, 3 play football and hockey but not cricket and 1 play football and cricket both (a) (b) (c) How many play all the three games? How many play cricket and hockey but not football? AVTE How many play exactly one of the three games? x 107 Let f = x, : xr be a function from R into R Determine the domain 1 x and range of f 108 Let R be a relation from Q to Q defined by R = {(a, b) : a, b Q and a - b Z} Show that : (a) (a, a) R for all a Q (b) (a, b) R implies that (b, a) R (c) (a, b) R and (b, c) R implies that (a, c) R 109 If z = x + iy and 1 iz, show that = 1 implies that z is purely real z i 110 The bag X contains 5 red and 3 green balls and bag Y contains 3 red and 5 green balls One ball is drawn from bag X and from bag Y Find the probability that out of 3 balls drawn, are red and 1 is green

12 130 Important Questions for XI In ABC, show that tana + tanb + tanc = tana tanb tanc 11 A person standing at the junction (crossing) of two straight paths represented by the equations: x - 3y - 4 = 0 and 3x + 4y - 5 = 0 wants to reach the path whose equation is 6x - 7y + 8 = 0 in the least time Find the equation of the path that he should follow 113 Find the equation of the line passing through the point of intersection of the lines 4x + 7 y - 3 = 0 and x - 3y + 1 = 0 that has equal intercepts on the axes 114 If the lines y = 3x + 1 and y = x + 3 are equally inclined to the line y = mx + 4, find the value of m 115 An analysis of monthly wages paid to workers in two firms X and Y, belonging to the same industry, gives the following results: Firm X Firm Y No of wage earners Mean of monthly wages s 553 s 553 Variance of the distribution of wages (i) (ii) Which firm X or Y pays larger amount as monthly wages? AVTE Which firm, X or Y, shows greater variability in individual wages? x Find the middle term(s) in the expansion of 3 x 117 Find the sum of the following series up to n terms : 3 11 ; x Insert 6 Arithmetic means between 3 and Find the mean, variance and standard deviation for the following distribution: Ages in years No of persons The mean and standard deviation of 0 observations are found to be 10 and, respectively On rechecking, it was found that an observation 8 was incorrect Calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted (ii) If it is replaced by 16

13 130 Important Questions for XI (a) If sin x = 3 1, cos y = -, where x and y both lie in second quadrant, 5 13 find the value of sin (x + y) (b) Find the value of tan 8 1 (a) Let f (x) = p + qx, if x < 4, if x = q - px, if x > and if lim x f(x) = f(), find p and q (b) Find the derivative of x + 1 x by first principle x 3 13 (a) Examine whether the limit exists or not : lim x3 x 9 x (b) Find the derivative of : n with respect to x cos x 14 Prove by using MI n(3n + 3) = 3n (n + 1) (n + ) n N 15 Find in degrees the angle subtended at the centre of a circle of diameter 50 cm AVTE by an arc of 11 cm 16 If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that p a b 17 If p and q are the lengths of perpendiculars from the origin to the lines x cos y cos = cos and x sec + y cosec = respectively, prove that p + 4q = 18 Find the distance of the line 4x + 7y + 5 = 0 from the point (1, ) measured along the line x y = 0 19 Let S be the sum, P the product and R the sum of reciprocals of n terms in a GP, prove that P R n = S n 130 Find the value of n so that a and b a a n1 n1 n b b n is the geometric mean between ADMISSIONS OPENCLASSES 6 TO 1 Session

DELHI PUBLIC SCHOOL BLUE PRINT WEEKLY TEST CLASS XI (MATHEMATICS)

DELHI PUBLIC SCHOOL BLUE PRINT WEEKLY TEST CLASS XI (MATHEMATICS) S. N0. TYPES OF QUESTIONS NO. OF QUESTION MARKS TOTAL 1. VERY SHT ANSWER 6 1 6 2. SHT ANSWER 5 4 20 3. LONG ANSWER WITH ONE 4 6 24 VALUE