CHAPTER-1. SETS. Q.4 Write down the proper subsets of { a, b, Q.5 Write down the power set of { 5,6,7 }? Verify the following result :

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1 CHAPTER-. SETS Q. Write the following sets in roster form (i) A = { : is an integer and 5 5 } (ii) B = { : is a natural number and < < 4} (iii) C= { : is a two- digit natural number such that sum of digit is 9} (iv) E = { : N and + 6 < 0} (v) H ={ : is an integer, 9} Q. Write the following sets in set builder form (i) {,,,5,6,5,0 } (ii) { I, N, D, A } Q. Write down the subset of {,,,4 } (iii) Q.4 Write down the proper subsets of { a, b, c } Q.5 Write down the power set of { 5,6,7 } Q.6 For the set A = {,,,4,5 }, B = {,4,6,8 }, Verify the following result : if U = {,,...9,0 } 4 5 6,,,,, c (i) A B = A B (ii) ( ) c c c A B = A B (iii) ( ) c c c A B = A B Q.7 If A = { 4,5,8, }, B = {, 4,6,9} and A = { 4,5,8, }, C = {,, 4,7,8,0 }, then find (i) A ( B A) (ii) A ( C B) If A,B,C are any three sets which are subsets of universal set U, then prove the following : Q.8 A ( B C) = ( A B) ( A C) Q.9 A ( B C) = ( A B) C Q.0 ( ) c c c A B = A B Q. Prove that A B = A B iff A = B c c Q. Prove that A B = B A Q. If n( U ) = 80, n( A) = 5, n( B) = 46and n( A B ') = 5. By using Venn diagram, find n( A B) Q.4 n a survey of 500 students the following information has produced: 85 take Economics, 95 take Mathematics, 5 take History, 45 take Economics and History, 70 take Economics and Mathematics, 50 take Mathematics and History, 50 do not take any of three subject. How many students take all three subjects How many take eactly one of three subjects Q.5 In a group of 50 persons, the number of persons like products A,B and C were found to be as follow : Product A=7,Product B=, Product C = 5, Products A and B = 9, Products B and C = 4, Products A and C = 5,Products A,B and C =. Find the number of persons who like (i) Product A only (ii) Product B only (iii) Product C only (iv) Products B and C but not A (v) Products A and C but not A (vi) Products A and B but not C (vii)at least one of the three products (viii) None of three products.

2 Q. Find and y, if Q. If { 5,6} CHAPTRR-. RELATIONS AND FUNCTIONS 5 +, y =, 4,5,6, A = and B = { } show that A B B A and y ( 4,6,9,0 ), form the set of all ordered pairs such that y 5 Q. If (,,,4 ),,,,5,,, A B C A B A C Q.4 Let A = B = { } C = { ) verify that (i) ( ) = ( ) ( ) (ii) A ( B C) = ( A B) ( A C) f, p q y = f = q p Q.5 Let ( ) = then show that ( ) Q.6 If ( ), prove that f ( y) = f + f = 0 Q.7 Let f = {(, ),(, ),( 0, ),(, ) } be a linear function from Z into Z. Find ( ) Q.8 Find the domain of Q.9 f ( ) Q. f ( ) Find the domain and range of following functions : = 9 = = 4 Q. f ( ) Q.0 f ( ) = 5 Q. f ( ) 5 Q.4 ( ) = f + = f = 4 +

3 CHAPTER- TRIGONOMETRIC FUNCTIONS o o Q. Prove that ( ) ( ) tan A + tan 60 + A tan 60 A = tan A Q. Solve the equation tan + tan + tan = tan tan tan Q. Find the value of (i) sin 5 o (ii) tan ( 0 o ) (iii) 5 cot 4 π (iv) cos ec ( 05 o ) π π 7π 4π Q.4 Evaluate : sin + sin + sin + sin Q.5 Prove that : cos9 o sin 9 o + o = tan 54 o o cos9 sin 9 π cot A Q.6 Evaluate : tan Q.7 Prove that : cos A = cot A Q.8 Solve for, sec( 90 o A) sin A tan ( 90 o A) cos( 90 o A) = + θ θ Q.9 Find the value of other t-ratios in each of the following : Find cot.tan 5 (i) cos θ =, θ. in quadrant II (ii) cot θ, θ sin A = cos A B + cos B cos A B cos A cos B Q.0 ( ) ( ) sin ( B C) sin ( C A) sin ( A B) Q. + + = 0 = in quad II.(iii) cos B cosc cosc cos A cos Acos B Q. tana tan 9A tan 4A = tana tan 9A tan 4A tan y + tan y z + tan z = tan y tan y z tan z Q. ( ) ( ) ( ) ( ) ( ) ( ) 4 tan θ, θ o o o Q.4 sin0 sin 0 sin 50 sin 60 o = 6 π π sec8 tan 8 Q.5 cos + cos + cos + = 0 Q.6 = sec 4 tan Q.7 If sin A = +, prove that sin A = Q.8 cos A = + + cos 4 cos A cos A sin A + sin A Q.9 + = cos A sin A π π Q.0 cos + cos + + cos = Q. If tan ( A + B) = n tan ( A B), show that ( n + ) sin A = ( n ) sin B + k Q. If sin = k sin ( + y), then prove that tan ( + y) = tan y k cosφ e Q. If cos θ =, show that ecosφ θ + e φ tan = ± tan e ***** = in III.

4 CHAPTER- 4 PRINCIPLE OF MATHEMATICAL INDUCTION By using the principle of mathematical induction, prove the following for all n N. Q n = n( n + ) Q. n( n + )( n + ) n = 6 Q. n( n + )( n + ) n( n + ) = Q.4 n n n. = ( n ) + Q.5 n( n + ) = n( n + )( n + ) 4( n + )( n + ) 5 7 n + Q = ( n + ) 4 9 n Q is divisible by n Q.8 4 n is multiple of 9 Q.9 n n is divisible by 9 Q.0 n( n + )( n + 5) is divisible by Q. n 7 ( n ) + < + n Q n > CHAPTER-5 COMPLEX NUMBERS AND QUADRATIC EQUATION 4

5 Epress the following in the form + iy : l + i Q. ( 5 + 6i) + ( + i) ( i) Q. 9 9 Q. Q i 4 + i Q i + 6i + i 6i + 4i + i Find the multiplicative inverse of the following comple numbers : Q.6 ( 5 7i) Q.7 ( + i) i Q.8 z = 7i and z = i, then verify that (i) z + z = Re( z ) (ii) z + z = z + z (iii) zz = z z z + z + Q.9 If z = i, z = + i, find z z + i Q.0 Find the modulus of comple number ( + i )( + i ) Write polar form also. Q. Solve Q. Solve = = Q. Simply : ( 5 7i)( 5 7i) ( 7i) Q.4 Find the values of and y,if ( )( ) + i + iy + i = i Q.5 Let z = ( i) and z = + i. Find Im z z Q.6 If ( ) y + iy = a + ib, then prove that 4( a b ) = a + b Q.7 Find the value of , if = 4 + 7i a ai a ai Q.8 Show that : + =, where a R a + ai a + + ai a + Find the square root of following comple numbers : Q i Q Q. i Q. i l CHAPER-6 LINEAR INEQUALITIES 5

6 Solve the following inequalities for real : + Q. 4 + Q. Q ( ) > Q.4 7 ( 5) 9 8( ) Q.5 ( ) Q.7 Q.9 Q < < Q Q.0 Find all pair of consecutive even natural numbers both of which are larger than 5, such that their sum is less than + Q. 7 < Q. 7, + > 8 < 5 Q. > 0 Q > Q.5 5 7, < + Solve the following system of inequalities graphically : y + > y + and 0 Q.6 0, 8 0 y Q.7 + y 8, > y, > 0and y > 0 Q.8 + y 0, + y 4, > 0 and y < Q.9 + y, y, y 4 and Q.0 y 5, 4 y 4, 5y 5, 4, y * CHAPTER-7 PERMUTIONS AND COMBINATIONS Q. If n + = 60 n, find n Q. If + =, find Q. Find r, if. Pr = Pr Q.4 Find n, if n n P = 00 P Q.5 n n If P 0 = P, find n Q.6 Sandy has 5 shirts, 4 pants, pairs of socks and kinds of shoes. In how many different ways can he dress himself (He does not know about colour combinations!) Q.7 in how many ways can captain and vice captain be chosen out of players Q.8 In how many ways can 5 children be arranged in a row such that two boys Ajay and Sachin (i) always sit together (ii) never sit together Q.9 How many words can be formed by taking 4 letters at a time out of the letters of the word : (i) MATHEMATICS (ii) EXAMINATION (iii) EXPRESSION Q.0 Find the number of ways in which (i) a selection (ii) an arrangement of 4 letters can be made from the letters of the word INFINITE 6

7 CHAPTER-8 BINOMIAL THEOREM Q. Write down the number of terms in the epansion of made( a) ( a) Epand the following : Q. Q Q.4 Simplify ( a) ( a) , hence evaluate ( ) ( ) 4 4 Q.5 Find the value of : ( ) ( a a a a ) Q.6 (.04 ) n Q.7 Using binomial theorem, prove that 6 5n is always by 5, where n N Q.8 Write the general term in the epansion of ( y) 6 Q.9 Find the 0 th term in the epansion of : + Q.0 Find the th term from the end in the epansion of 9 Q. y in the epansion of y y + 5 Q. Find the term involving a b in the epansion of ( a b) 4 ( a + b) Q. Find the coefficient of in the epansion of ( + ) ( ) Find the term independent of in the following epansions (if any) 0 Q.4 Q Find the middle terms in : Q.6 0 Q.7 Q.8 Prove that there is no term involving 6 b a in, 0 5 Q.9 If the coefficients of, and in the epansion of ( ) n + are in A.P., then show that n 9n = *** CHAPTER-9 SEQUENCES AND SERIES 7

8 Q. If the ( m + n) th term of a G.P. is p and [ m n] th term is q, show that nth terms are pq and q p. p m n m th and Q. Find the sum of n terms of the series whose n th term is n + n + + n Q. Find the nterms of the series Q.4 Find κ, so that, κ, 5 are in A.P. Q.5 Find the 0 th terms of G.P Q.7 In a G.P. if a = 4and 6 9,,,, Q.6 Insert two G.M. between and 8 a = find 0 th term Q.8 Evaluate ( + κ ) Q.9 If a, b, c are p th, q th and r th terms respectively of an A.P., prove that (i) p ( b c) + q( c a) + r ( a b) = 0 (ii) a( q r) + b( r p) + c( p q) = 0 Q.0 Find the 0 th terms from the end of the sequence, 8,,, 5 Q. Solve the equation = 55 Q. If the sum of n terms of two arithmetic series are in ratio : (i) ( 4 4 n) : ( n + 5 ), find the ratio of 8 th terms (ii)( + n) : ( + n), find the ratio of 7 th terms. Q. The n th term of an A.P. is p and the sum of first n terms is s. Prove that the first term is s pn Q.4 If p th n term of a G.P. is P and q th term is Q, prove that n th term is Q.5 If the first terms of a G.P. is 5 and the sum of first three is, find the common ratio Q.6 In a G.P. the first term is 7, the last term is 448, and the sum is 889. Find the common ratio Q.7 If S, S, Sare sum of first n, n,n terms of a G.P., show that S ( ) ( ) S S = S S Q.8 If Sn represents the sum of n terms of a G.P. whose first term and common ratio are a and r respectively, prove that na ar S + S + S S = r 5 n ( r ) ( r) n Q.9 If G, G are two G.M s. between a and b, show that G G (i) G G = ab (ii) + = a + b G G S S S S be the sums of n terms of A.P. whose first terms are,,..., p and Q.0 If,,,..., p whose common difference are,,5,..., ( p ) respectively, show that np S + S + S Sp = ( np + ) b c c a a b Q. If a, b, c are in A.P. and, y, z are in G.P., prove that. y. z = ***** κ= P Q n q n p p q 8

9 CHAPTER-0 STRAIGHT LINES Q. Reduce he straight line + y 8 = 0 to normal form and hence find p. Q. Find the value of κ for which the line ( κ ) ( 4 κ ) y + κ 7κ + 6 = 0 (i) parallel to -ais (ii) parallel to y -ais (iii) passing through origin Q. Find the equation of the line through the point ( 0,) making an angle π with positive of line parallel to it and crossing the y-ais at a distance of units below the origin Q.4 P( a, b) is the mid- point of a line segment between aes. Show that equation of the y + = a b line is Q.5 Find the equation of a line perpendicular to the line + 5y + 7 = 0 and with y- intercept Q.6 Find the equation of a line parallel to y + 8 = 0and passing through the point (, ) Q.7 Find the eq. of a straight line perpendicular to = y + 5 and with -intercept 4 Q.8 Show that the product of perpendicular on the line cosθ + sinθ = from the points ( a ) b,0 b ± is Q.9 Find equation of the line mid way between the parallel lines 9 6y y + 6 = 0 a y b + = and 9

10 CHAPTER- CONIC SECTIONS y Q. Length of the latus rectum of the ellipse + = is a b Find the centre and radius of following circle Q. + y + = 4 b a Q. + y 8 + 0y = 0 Find the equation of parabolas satisfying the following conditions : Q.4 Focus (,0 ),directri = Q.5 Verte ( 0,0 ), passing through (,) and ais is along -ais Q.6 Focus (, 4 ), directri + y = 0 Q.7 Find the equation of the circle : (i) whose centre is ( 0, 4) and which touches the -ais. (ii) whose centre is (,4) and which touches the y-ais. Q.8 Find the equation of circles which touch both the aes and touch the line = a Q.9 Find the equation circles which pass through two points on -ais at distances of 4 units from the origin and whose radius is 5 units In each of the parabolas : Q.0 y = 4 Q. = 4y Q. If the parabola y = 4 p passes through the point (, ), find the length of latus rectum and the coordinates of the focus Find the equation to the ellipse referred to its aes as coordinates aes. Q. whose major ais = 8, eccentricity = Q.4 which passes through the point (,) and has eccentricity 5 Q.5 Find the equation of the ellipse whose eccentricity is and whose foci are ( ) ±,0 Find the equation to the hyperbola referred to its aes as coordinate aes if Q.6 The distance between the foci is 5 and conjugate ais is Q.7 The distance between foci is 6 and eccentricity is Q.8 Find the equation of the ellipse with foci at ( ± 5,0) and 5 6 = 0as one directri Q.9 Vertices at ( ± 5,0 ), foci at ( ± 4,0) Q.0 Foci at ( ±,0 ), passing through( 4, ) 0

11 CHAPTER- INTRODUCTION TO THREE DIMENSIONAL GEOMETRY Find the distance between the following point : Q. ( 0,, ) and (,,) Q. (,, 4) and (,, ) Prove that the following points are collinear. Q. (,, );( 0,, );(, 4, ) Q.4 (, 4, );( 4,4, );(,0,) Q.5 Show that the points ( 4,,4 ),( 0,, ) and (,0, 4) are the vertices of an equilateral triangle Q.6 Show that the points (,, ),(,, ),(,,) and ( 4,7,6) are the vertices of a parallelogram Q.7 Show that the points (,, ),(, 4, ),(,5,5 ) and (,5,5) are the vertices of a square Q.8 If A and B be the points (,4,5) and (,, 7) respectively, find the equation of the set of point P such that PA + PB = κ, where κ is a constant Q.9 Find the ratio in which the line joining the points (,4, ) and (,5,4 ) is divided by the y -plane Q.0 Prove that the points (,, ),( 5, 4,0 ),(,, ) and ( 0,6, ) taken in order from a parallelogram Q. Find the coordinates of the points which trisect PQ, given that P is ( 4,, 6) and Q is ( 0, 6, 6) Q. The centroid of a triangle ABC is ( 7,,5). If the coordinates of A and B are (,6, 4) and ( 4,,). Find the coordinates of C

12 ( ) CHAPTER- LIMITS AND DERIVATIVES Q. ( ) + 5 Q. 0 tan sin 0 sin cot cos π / cos sin 4 tan 4 0 Q. Q.5 Q.7 sin sin 0 sin tan sin 0 tan sin 0 cos π / sin Q.4 Q.6 Q.8 Find the derivative of the following (0-) 4 Q.9 f ( ) = Q.0 f ( ) ( a b) n Q. 6 + Q. ( ) Q. ( 5 + ) Q.4 Q.5 + Q.6 m n < 0 f = n + m 0 n + m > f eist Q.7 If ( ) Q.8 0 ( ) + sin = + r p + q + s ( ) + For what integers m and n both f ( ) and Q.9 Find the derivative of following using first principle. Q.0 π π Q. sin 0 sin ( ) Find the derivative w.r.t.. Q. + Q. p + q + r a + b Q.4 sin Q.5 sin Q.6 cos Q.7 sec

13 CHAPTER-5 STATISTICS Q. Find the mean deviation from the mean as well as from median for the series,,, 4, 4,5,6,7 Q. Calculate the mean deviation about the mean from the following data: i f i Q. Find the mean deviation about from the mean as well as from the median for the following series : 0 4 Frequency 8 Q.4 Find the mean deviation from mean for the following distribution: Class Frequency 4 Q.5 Class Frequency Q.6 Find the mean and variance for the following distribution : Class Frequency Q.7 The annual rainfall (in cm) at different places was as follow. Year Rainfall (in cm) Estimate the mean rainfall and the standard deviation Q.8 Calculate the standard deviation from the following data : Mid-point Frequency Q.9 Find the mean, variance and standard deviation for the following data using cut method Class Frequenc y

14 CHAPTER-6 PROBABILITY Q. What is the chance that a non- leap year should have fifty three Sunday Q. The odds in favour of occurrence of an event are 5 :. Find the probability that it will occur Q. What are the odds in favour of getting a spade if card is drawn from a well shuffled deck of cards Q.4 Apair of dice is thrown. Find the probability of getting a doublet Q.5 A pair of dice is thrown. Find the probability of getting a sum of 0 or more, if 5 appears on the first die Q.6 In a single throw of three dice, find the probability of getting : (i) a total of 5 (ii) a total of at most 5 (iii) a total of at least 5 (iv) the same number on all the dice. Q.7 In a single throw of two dice, find the probability of getting a total of 0 or Q.8 (i) What is probability that the numbers selected from the numbers,,,., 0 is prime number You may assume that each of the 0 numbers is equally likely to be selected. (ii) What is the probability that a number selected from,,,., 5 is a prime number if each of the 5 numbers is equally likely to be selected Q.9 In a single throw of two dice, find (i) P (an odd number on one die and 6 on the other) (ii) P (a number >4 on each dice) (iii) P (a total of ) (iv) P (a total of 8) (v) P (a total of ) Q.0 Two dice are thrown. Find the probability of getting an odd number on one and a multiple of on the other Q. In a single thrown of two dice, find the probability that the neither a doublet nor a total of 9 will appear Q. A card is drawn from a well shuffled deck of 5 cards. Find the probability of drawing : (i) a black king (ii) a jack, queen, king or an ace (iii) a card, which is neither a heart nor a king (iv) a spade or club. 4

15 Q. Evaluate p + cos 0 q sin SECTION- Q. Solve the equation z = z + i, where z is a comple number + 7i Convert ( ) i into polar from Sn Q. Let Sn be the sum of first n terms of an A.P. If Sn = Sn then prove that 6 S = n Find the sum to n terms : Q.4 Find the co-ordinates of the point which is equidistant from four points ( 0,0,0 ),( a,0,0), ( 0, b,0) and ( 0,0,c ) Q.5 The letters of the word SOCIETY are placed at random in a row. What is the probability that three vowels come together k + Q.6 If θ + φ = α and tanθ = k tan φ, prove that sinα = sin ( θ φ ) k For sin,cos and tan if 5 Q.7 Evaluate 0 sin + Q.8 Find the domain and range of the function f ( ) = + Q.9 If α and β be two distinct roots satisfying the equation a cosθ + bsin θ = c, show that ab tan ( α + β ) = a b Prove that tan 8 ( )( ) o = n n 7n Q.0 Prove by P.M.I. that + + is a natural number for all n N 5 5 n n+ Use P.M.I. prove that is divisible by 9 for all n N Q. Solve > Q. Verify by method of contradiction that p : is irrational Q. Prove that : cos α = sin β + 4 cos( α + β ) sinα sin β + cos ( α + β ) If a cosφ + b θ a b φ cos θ =, prove that cos = tan a + bcosφ a + b * 5

16 SECTION-II Q. Prove that (i) o o o cos tan 70 = tan 0 + tan 50. (ii) tan ( 60 ) tan ( 60 ) + o + A o A = cos A Q. cosφ e θ + e φ If cos θ =, show that tan = ± tan ecosφ e Q. Find the equation of the circle passing through the vertices of the triangle whose sides are + y =, 4y = 6 and y = 0 Q.4 q r r p p q If p, q, r are in A.P., while, y, z are in G.P., prove that. y, z = Q.5 ( + y) sec( + y) sec Evaluate: y 0 y Q.6 sin β If tanα = tan β, show that tan ( α β ) = 5 cos β Q.7 If S, S, S are sum of, n n and n terms of a G.P. show that S ( S S ) ( S S ) = Find the sum of the series to n terms Q.8 Find the modulas and argument of the comple number i z = π π cos + i sin Q sin β Psin β Q.9 If tan α =, prove that tan ( β α ) = P + Q cos β Q + P cos β o o Prove that sin A sin ( 60 A) sin ( 60 + A) = sin A 4 β α Q.0 If α and β are different comple numbers with β =, then find αβ Q. If S be the sum, P the product and R the sum of the reciprocal of n terms in a G.P., prove that n S P = R The sum of two numbers is 6 times their G.M. Show that the numbers are in the ratio + : Q. Find the mean deviation from the mean for the following data : : f i i : P Q. If pth term of a G.P. is P and its qth term is Q. Prove that the nth term is Q b c c a a b If a, b, c, are in A.P. and, y, z are in G.P. Prove that. y. z = cos Q.4 Evaluate 0 + n Q.5 If sinθ = nsin ( θ + α ), show that tan ( θ + α ) = tanα n 6 n q n p p q

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QUESTION BANK. Class : 10+1 & (Mathematics)

QUESTION BANK. Class : 10+1 & (Mathematics) QUESTION BANK Class : + & + (Mathematics) Question Bank for + and + students for the subject of Mathematics is hereby given for the practice. While preparing the questionnaire, emphasis is given on the