IITIAN's MASTER PACKAGE for JEE ADVANCED PDF
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This document contains practice problems and solutions for IIT JEE Advanced. Topics covered include algebra, trigonometry, coordinate geometry, calculus, vectors, and 3-D geometry, along with physics and chemistry. It seems to be a collection of questions and solutions to help students prepare for the exam.
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IITIAN’s M ASTER PACKAGE for JEE ADVANCED A n IITian’ s Pras hiks han K endra Co mpilat io n ! ! ! CONT ENT S MAT H EMAT I CS Sr. No. Topic Page No. 1. ALGEBRA...
IITIAN’s M ASTER PACKAGE for JEE ADVANCED A n IITian’ s Pras hiks han K endra Co mpilat io n ! ! ! CONT ENT S MAT H EMAT I CS Sr. No. Topic Page No. 1. ALGEBRA 1 2. TRIGONOMETRY 24 3. 2-D CO-ORDINATE GEOMETRY 41 4. CALCULUS 61 5. VECTORS & 3-D GEOMETRY 88 PH YSI CS 1. MECHANICS 97 2. FLUID MECHANICS 131 3. HEAT 140 4. ELECTROMAGNETISM 152 5. WAVE MECHANICS 181 6. RAY & WAVE OPTICS 189 7. MODERN PHYSICS 200 CH EMI ST RY 1. PHYSICAL CHEMISTRY – 1 209 2. PHYSICAL CHEMISTRY – 2 222 3. PHYSICAL CHEMISTRY - 3 246 4. ORGANIC CHEMISTRY - 1 258 5. ORGANIC CHEMISTRY – 2 275 6. ORGANIC CHEMISTRY - 3 311 7. INORGANIC CHEMISTRY - 1 317 8. INORGANIC CHEMISTRY - 2 327 ANSWER-SHEETS 335 The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 1 CHAPTER 1 ALGEBRA Topics Coverage : Quadratic Equations & Expressions Sequences & Series Determinants & M atrices Binomial Theorem Complex Numbers Permutations & Combinations Theory of Probability SECTION – 1 : OBJECTIVE QUESTIONS (M ultiple Correct Answ ers Ty pe) LEVEL - 1 Q.1 If the equation ax 2 bx c 0 , a, b, c R have non-real roots, then : (a) c ( a b c) 0 (b) c ( a b c ) 0 (c) c ( 4a 2b c ) 0 (d) none of these Q.2. If the quadratic equation ax 2 bx c 0 ( a 0) has sec 2 and cos ec 2 as its roots R , then which of the following must hold good ? (a) b c 0 (b) b 2 4ac 0 (c) c 4a (d) 4a b 0 Q.3 Let a, b, c Q satisfying a b c. Which of the following statement(s) hold true for the quadratic polynomial f ( x ) ( a b 2c ) x 2 (b c 2a) x (c a 2b) ? (a) The mouth of the parabola y f (x) opens upwards (b) Both roots of the equation f ( x) 0 are rational. (c) The x coordinate of vertex of the graph is positive (d) The product of the roots is always negative Q.4 If roots of ax 2 bx c 0 are and and 4a 2b c 0, 4a 2b c 0 and c 0 , then possible value/values of [ ] [ ] is/are ____ ( where [.] represents greatest integer function) (a) 2 (b) 1 (c) 0 (d) 1 ALGEBRA. 2 Algebra Q.5 The roots of equation x 5 40 x 4 x 3 x 2 x 0 are real and in GP. If the sum of their reciprocals is 10, then can be ___ 1 (a) 32 (b) 32 1 (c) 32 (d) 32 Q.6 The real numbers x1 , x 2 , x3 satisfying the equation x 3 x 2 x 0 are in A.P. The intervals in which and lie are : 1 1 (a) , (b) , 3 27 1 1 (c) , (d) , 3 27 343 Q.7 Let a and b be positive integers. The value of xyz is 55 or according to a, x, y, z and 55 b are in AP or HP. Then, ____ b (a) a1 b 2 50 (b) 7 a (c) a b 8 (d) a 2 b 2 25 1 1 1 Q.8 For the series, S 1 (1 2) 2 (1 2 3) 2 (1 2 3 4) 2 .... (1 3) (1 3 5) (1 3 5 7) (a) 7th term is 16 (b) 7th term is 18 505 405 (c) sum of first 10 terms is (d) sum of first 10 terms is 4 4 n Q.9 If r (r 1) (2r 3) an 4 bn 3 cn 2 dn e , then r 1 (a) a b d c (b) e 0 (c) a, b 2 / 3, c 1 are in A.P. (d) (b d ) / a is an integer Q.10 Let S1 , S 2 ,... be squares such that for each n 1 , the length of a side of S n equals the length of a diagonal of S n 1.. If the length of a side of S1 is 10 cm, then for which of the following values of n is the area of S n less than 1 sq.cm? (a) 7 (b) 8 (c) 9 (d) 10 Q.11 Let (n) sum of all the divisors of the natural number n. Then ( 2 9.3 2 ) is divisible by : (a) 11 (b) 15 (c) 33 (d) 31 The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 3 Q.12 Which of the following statements is/are true about square matrix A of order n ? (a) ( A) 1 is equal to A 1 where n is odd only (b) If A n O , then I A A 2 ..... A n1 ( I A) 1 (c) If A is skew-symmetric matrix of odd order, then its inverse does not exist. (d) ( AT ) 1 ( A 1 ) T holds always Q.13 The value of n C1 n 1C 2 n 2 C3 ........ n m 1C m is equal to (a) m n C n 1 (b) m n C n 1 (c) m C1 m 1C 2 m 2C 3 ..... m n 1C n (d) m n C 1 m Q.14 For the expansion ( x sin p x 1 cos p )10 , ( p R ), (a) the greatest value of the term independent of x is 10! / 2 5 (5!) 2 (b) the least value of sum of coefficient is zero (c) the greatest value of sum of coefficient is 32 (d) the least value of the term independent of x occurs when p (2n 1) , nZ 4 Q.15 In the expansion of (71 / 3 111 / 9 ) 6561 , (a) there are exactly 730 rational terms (b) there are exactly 5831 irrational terms (c) the term which involves greatest binomial coefficients is irrational (d) the term which involves greatest binomial coefficients is rational n4 Q.16 Let (1 x 2 ) 2 (1 x) n ak x k. If a1 , a 2 and a3 are in arithmetic progression, then the k 0 possible value/values of n is/are : (a) 5 (b) 4 (c) 3 (d) 2 Q.17 The value of x , for which the 6th term in the expansion of 7 1 log 2 x 1 2 (9 7 is 84 is equal to: (1 / 5) log 2 (3 x 1 1) 2 (a) 4 (b) 3 (c) 2 (d) 1 m 30 20 p Q.18 If f (m) 30 i m i where p Cq , then i 0 q (a) Maximum value of f (m) is 50 C 25 (b) f (0) f (1) ..... f (50) 250 (c) f (m) is always divisible by 50 50 (1 m 49 ) 50 (d) The value of ( f ( m)) 2 100C50 m0 ALGEBRA. 4 Algebra 11 Q.19 The integer just below 53 7 2 711 is (a) divisible by exactly 4 prime factors (b) divisible by exactly 3 prime factors (c) is divisible by 7 (d) has 53 as its only two digit prime factor 50 C5r a Q.20. If 50 29 43 where a I is not a multiple of 47 and b I then which of the following Cr b is true? (a) r is a prime number (b) r is a composite number 50 C5r (c) r has two different integer values (d) exponent of prime number 3 in 50 is 2 Cr Q.21 Number of ways in which 200 people can be divided in 100 couples is: (200) ! (a) (b) 1 3 5 ......... 199 100 2 (100) ! 101 102 200 ( 200) ! (c) ....... (d) 2 2 2 (100) ! Q.22 If a seven-digit number made up of all distinct digits 8, 7, 6, 4, 3, x and y is divisible by 3, then : (a) maximum value of x y is 9 (b) maximum value of x y is 12 (c) minimum value of xy is 0 (d) minimum value of x y is 3 Q.23 If 10 ! 2 p. 3 q. 5 r. 7 s , then (a) 2q p (b) pqrs 64 (c) number of divisors of 10 ! is 280. (d) number of ways of putting 10 ! as a product of two natural numbers is 135 Q.24 Kanchan has 10 friends among whom two are married to each other. She wishes to invite five of them for a party. If the married couples refuse to attend separately, then the number of different ways in which she can invite five friends is: (a) 8 C 5 (b) 2 8C 3 (c) 10 C 5 2 8 C 4 (d) none of these Q.25 Let f (n) be the number of regions in which n coplanar circles can divide the plane. If it is known that each pair of circles intersect in two different points and no three of them have common point of intersection, then : (a) f (20) 382 (b) f (n) is always an even number (c) f 1 (92) 10 (d) f (n) can be odd The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 5 Q.26 Number of ways of selecting three integers from {1, 2, 3,….., n} if their sum is divisible by 3 is: (a) 3( n / 3 C 3 ) ( n / 3) 3 if n 3k , k N (b) 2( ( n 1) / 3C3 ) ( ( n 2) / 3C3 ) (( n 1) / 3) 2 (n 2), if n 3 k 1, k N (c) 2( ( n 1) / 3C3 ) ( ( n 2) / 3C3 ) (( n 1) / 3) 2 (n 2), if n 3 k 2, k N (d) independent of n Q.27 A is a set containing n elements. A subset P1 of A is chosen. The set A is reconstructed b replacing the elements of P1. Next, a subset of P2 of A is chosen and again the set is reconstructed by replacing the elements of P2. In this way, m ( 1) subsets P1 , P2 ,......., Pm is: (a) ( 2 m 1) n if P1 P2 ..... Pm (b) 2 mn if P1 P2 ...... Pm A (c) 2 mn if P1 P2 ........ Pm (d) ( 2 m 1) n if P1 P2 .......... Pm A 3i 1 Q.28 Let and P { n : 1, 2, 3,........}. Further H 1 z C : Re z and 2 2 1 H 2 z C : Re z . where, C is the set of all complex numbers. 2 If z1 P H 1 , z 2 P H 2 and O represents the origin, then z1 O z 2 (a) / 2 (b) / 6 (c) 2 / 3 (d) 5 / 6 Q.29 If | z 1| 1, then (a) arg (( z 1 i) / z ) can be equal to / 4 (b) ( z 2) / z is purely imaginary number (c) ( z 2) / z is purely real number (d) If arg ( z ) , where z 0 and is acute, then 1 2 / z i tan 2 Q.30 If z1 , z 2 be two complex number ( z1 z 2 ) satisfying | z12 z 22 | | z1 2 z 1 z 2 z 2 | , then z z (a) 1 is purely imaginary (b) 1 is purely real z2 z2 (c) | arg z1 arg z 2 | (d) | arg z1 arg z 2 | 2 Q.31 If z1 , z 2 , z 3 , z 4 are represented by the vertices of a rhombus taken in the anticlockwise order then : (a) z1 z 2 z 3 z 4 0 (b) z1 z 2 z 3 z 4 z z4 z z2 (c) amp 2 (d) amp 1 z1 z 3 2 z3 z 4 2 ALGEBRA. 6 Algebra 3z 6 3i Q.32 Let the complex numbers z of the form x iy satisfy arg and | z 3 i | 3. 2 z 8 6i 4 Then the ordered pairs ( x, y ) are: 4 2 4 2 (a) 4 , 1 (b) 4 , 1 5 5 5 5 (c) (6, 1 ) (d) (0, 1 ) Q.33 P ( z1 ) , Q ( z 2 ) , R ( z 3 ) and S ( z 4 ) are four complex numbers representing the vertices of a rhombus taken in order on the complex plane, then which one of the following is/are correct? z z4 z z4 z z4 (a) 1 is purely real (b) arg 1 arg 2 z 2 z3 z2 z4 z3 z 4 z z3 (c) 1 is purely imaginary (d) it is not necessary that | z1 z 3 | | z 2 z 4 | z2 z4 Q.34 A rectangle of maximum area is inscribed in the circle | z 3 4i | 1. If one vertex of the rectangle is 4 4i, then another adjacent vertex of this rectangle can be: (a) 2 4i (b) 3 5i (c) 3 3i (d) 3 3i Q.35 If z1 5 12 i and | z 2 | 4 , then (a) maximum ( | z1 iz 2 | ) 17 (b) minimum ( | z1 (1 i) z 2 | ) 13 4 2 z1 13 z1 13 (c) minimum (d) maximum 4 4 4 3 z2 z2 z2 z2 Q.36 If | z 3 | min {| z 1 |, | z 5 |} then Re(z ) equals to: 5 7 (a) 2 (b) (c) (d) 4 2 2 Q.37. A bag contains four tickets marked with one ticket is drawn at random from the bag. Let denote the event that ith digit on the ticket is 2. Then (a) and are independent (b) and are independent (c) and are independent (d) are independent Q.38. Let be a set containing elements. If two subsets and of are picked at random, the probability that and have the same number of elements is 2n Cn 1 1,3, 5.....(2n 1) 3n (a) (b) (c) (d) 2 2n 2n Cn 2n. n! 4n The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 7 Q.39. Three six faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is is ( k 2 )( k 1 ) k ( k 2) 1 k2 (a) (b) (c) k 1 C 2 (d) 432 432 216 432 Q.40. A card is selected at random from cards numbered as. An event is said to have occurred. If product of digits of the card number is. If card is selected times with replacement each time, then the probability that the event occurs exactly three times is 2 2 3 2 3 97 3 97 (a) 5 C 3 (b) 5 C 3 100 100 100 100 3 3 0.3 9.7 (c) 5 C 3 (d) 10 (0.03) 3 (0.97) 2 100 100 Q.41. Three numbers are chosen at random without replacement from. The probability that minimum of the chosen number is or their maximum is , cannot exceed 11 11 11 11 (a) (b) (c) (d) 30 40 50 60 OBJECTIVE QUESTIONS (M ultiple Correct Answ ers Ty pe) LEVEL - 2 Q.1. Let f ( x ) x 2 bx c & g( x ) f ( x 2 ) , then (a) If c 0 b , then g( x ) 0 has two real roots. (b) If b c 0 , then g( x ) 0 has no real solutions. (c) If 0 b c then g( x ) 0 has no real solutions. (d) If b , c odd integers then f ( x ) 0 has no rational roots. Q.2. If 2 A & A B are integers then : (a) f ( x ) Ax 2 Bx 1 is an integer whenever x is an integer. (b) f ( x ) Ax 2 Bx ( A B) is an integer whenever x is an integer (c) f ( x) 2 Bx 2 Ax B A is an integer whenever x is an integer (d) f ( x ) ( A B) x 2 ( A B ) x 1 is an integer whenever x is an integer Q.3. If a, b, c Q , then which of the following equations has rational roots : 2 (a) ax bx c 0 , where if a b c 0 2 (b) (a c b ) x 2cx (b c a ) 0 2 2 2 2 2 2 (c) abc x 3a cx b cx 6 a 4 ab 2b 0 (d) ( a b c ) x 2 ( a c b ) x ( b c a ) 0 ALGEBRA. 8 Algebra 9n 2 5n 12 Q.4. If , , are roots of x 3 2 x 2 3x 1 0 , then value of Vn , ( n N ) is less 2 than (a) 2 (b) 3 (c) 4 (d) 5 Q.5. If a, b, c are in AP and A, B, C are in GP (common ratio 0 ). Then which of the following is/are correct. A B C c (a) , , are in HP if common ratio of GP is a b c a a b c (b) , , are in HP if common ratio of GP is equal to common difference of AP. A B C A2 B 2 C 2 c (c) , , are in HP if common ratio of GP is a b c a a b c (d) , , are in HP if common ratio of GP is equal to square root of common A2 B 2 C 2 difference of AP. Q.6 A sequence a1 , a2 ,...., an..... of integers is defined by an1 an2 an 1 and a1 2 then: (a) an is a prime number n N (b) an is an odd number n N , n 2 (c) H.C.F. of ( ai , a j ) = 1 for i j (d) H.C.F. of ( ai , ai 2 ) 2 i N a n21 Q.7. Consider a sequence {a n } with a1 2 & a n for all n 3 , terms of the sequence a n2 being distinct. If a 2 & a 5 are +ve integers and a 5 162 , then the possible value (s) of a 5 can be : (a) 162 (b) 64 (c) 32 (d) 2 Q.8. a1 , a 2 , a 3 ,........ are distinct terms of an A.P. We call ( p, q, r ) an increasing triad if a p , a q , a r are in G.P. where p, q, r N such that p q r. If ( 5, 9, 16) is an increasing triad, then which of the following option is/are correct (a) if a1 is a multiple of 4 then every term of the A.P. is an integer (b) ( 85, 149, 261) is an increasing triad 1 1 (c) If the common difference of the A.P. is , then its first term is. 4 3 (d) ratio of the (4k 1)th term and 4k term can be 4. th The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 9 Q.9. Let A, B, C be three sets of complex number as defined below : A { z :| z 1 | 2 Re( z ) } , B {z :| z 1 | 1} and z 1 C z : 1 z 1 Then which of the following are correct (a) Area of region bounded by A B C is 3 (b) Area of region bounded by A B C is 2 3 (c) The real part of the complex no. in region A B C and having maximum 3 amplitude is 2 (d) The number of points having integral coordinates in region A B C is 6 Q.10. z1 , z 2 , z 3 are three non zero distinct points satisfying | z 1 | 1 & z 22 z1 z 3 , then z3 z 2 z 1 z (a) is purely imaginary (b) Arg 2 2 Arg 1 z 2 z3 2 z1 1 z2 z 1 z 1 1 1 1 1 1 (c) Arg 2 2 Arg 3 (d) z1 1 z1 z 2 z 3 z1 z 2 z1 z 3 Q.11. Line OQ is angle bisector of angle O of right angle triangle OPR, right angled at P. Point Q is such that ORQP is concyclic. If point O is origin and points P, Q, R are represented by z 22 the complex numbers z 3 , z 2 , z1 respectively. If then (R is circumradius of OPR ) z1 z 3 (a) Angles of OPR are , , (b) Angles of OPR are , , 6 3 2 4 4 2 (c) Area of OPR is 2 2 R 2 (d) Area of OPR is 2R 2 Q.12. If z1 , z 2 , z 3 , z 4 are complex numbers in an Argand plane satisfying z1 z 3 z 2 z 4. A complex number ' z ' lies on the line joining z1 and z 4 such that z z2 z z2 Arg Arg 3 . It is given that | z z 4 | 5, | z z 2 | | z z 3 | 6 then, z1 z 2 z z2 (a) Area of the triangle formed by z , z1 , z 2 is 3 7 sq. units. 15 7 (b) Area of the triangle formed by z , z 3 , z 4 is sq.units 4 27 7 (c) area of the quadrilateral formed by the points z1 , z 2 , z 3 , z 4 taken in order is sq.units 2 27 7 (d) area of the quadrilateral formed by the points z1 , z 2 , z 3 , z 4 taken in order is sq.units 4 ALGEBRA. 10 Algebra Q.13. Let A1 , A2 ,.... A7 be a polygon and a1 , a 2 ,....a 7 be the complex numbers representing vertices A1 , A2 ,........ A7. If, | a1 | | a 2 | ...... | a 7 | = R, then 2 a a i j 1 i j 7 (a) greater than 30R 2 (b) has minimum value as 35R 2 (c) has its minimum value in (25 R 2 , 45 R 2 ) 2 (d) is less than 45R Q.14. Let Pn be the point on the complex plane with origin O expressed by : i i i Pn (1 i ) 1 1 .........1 ( n 1, 2, 3,...) then correct option(s) is(are) 2 3 n Pn 1 n2 n 1 (a) tan arg P (b) area of OPn Pn 1 n 1 n 2 n3 P 1 (c) area of OPn Pn 1 (d) tan arg n 1 n 1 2 Pn 1 2 4 2 Q.15. Let A 2 and B, C are 2 2 matrices such that ABC , a a 2 1 3 Let ( BC ) T [ p ij ] 22 and p 21 1 , then the value of a can be : 3 1 1 (a) 0 (b) (c) (d) 2 2 2 Q.16. Which of the following is/are correct ? (a) If A is a n n matrix such that aij (i 2 j 2 5ij ). ( j i ) i and j then trace (A) = 0 (b) If A is a n n matrix such that aij (i 2 j 2 5ij ). ( j i ) i and j then trace (A) 0 (c) If P is a 3 3 orthogonal matrix, , , are the angles made by a straight line with sin 2 sin . sin sin . sin OX, OY, OZ and A sin . sin sin 2 sin . sin and Q = P T AP , then sin . sin sin . sin sin 2 PQ 6 P T 32 A (d) If matrix A [ aij ] 33 and matrix B [ bij ]33 where aij a ji 0 and bij b ji 0 and j then A 6 B 7 is a singular matrix. Q.17. A is a matrix of order 3 3 and aij is its elements of i th row and j th column. If aij a jk a ki 0 holds for all 1 i, j , k 3 , then (a) A is a non-singular matrix (b) A is a singular matrix (c) aij is equal to zero (d) A is a skew symmetric matrix 11, j 3 The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 11. 3 x 2 ( x 2) 2 5x 2 2x Q.18. Let A 1 , B [a b c ] C 5 x 2 2x ( x 2) 2 be three given matrices 6x 2x ( x 2) 2 5 x 2 Where a, b, c and x R given t r ( AB ) t r (c ) x R , where t r ( A) denote trace of A. Then which of the following are correct ? (a) abc7 ln x ln p (b) if 2 cx ax b dx q where p and q are coprime then p q 29 0 (c) abc6 ln x ln p (d) if 2 cx ax b dx q where p and q are coprime then p q 27 0 Q.19. The product of 13 5 matrix 5 13 matrix contains a variable entry x in exactly 2 places. If D (x ) is the determinant of the matrix product such that D (0) 1, D(1) 1 & D( 2) 7 which of the following is/are correct ? (a) D(2) 3 (b) D(1) 3 (c) D(3) 7 (d) D(1) 2 Q.20. Let A aij be a matrix of order 3 3 , where x , if i j , x R ai j 1 , If i j 1 0 , other wise Then which of the following is (are) correct (a) For x 2, A is a diagonal matrix (b) A is a symmetric matrix (c) For x 2 , det A is equal to 6 (d) Let f ( x) det( A) then f (x ) has both maxima and minima Q.21. A and B play a game in which they alternately call out positive integers less than or equal to n , according to the following rules. A goes first and always calls out an odd number, B always calls out an even number and each player must call out a number which is greater then the previous number (except for A’s first turn). The game ends when one player cannot call out a number, then which of the following is/are correct ? (a) For n 6 , number of possible games is 8. (b) For n 6 , number of possible games is 10. (c) For n 10 , number of possible games is 55. (d) For n 8 , number of possible games is 21. ALGEBRA. 12 Algebra Q.22. Consider all 10 digit numbers formed by using all the digits 0,1 , 2, 3, …., 9 without repetition such that they are divisible by 11111, then : (a) the digit in tens place for smallest number is 6. (b) the digit in tens place for largest number is 3 (c) total numbers of such numbers is 3456 (d) total numbers of such numbers is 4365 Q.23. Each of 2010 boxes in a line contains one red marble and for 1 k 2010, the box is the k th position also contain k white marbles. A child begins at the first box and successively drawn a single marble at random from each box in order. He stops when he first draws a red marble. Let P(n) be the probability that he stops after drawing exactly n marbles. The 1 possible value(s) of n for which p (n) is : 2010 (a) 44 (b) 45 (c) 46 (d) 47 Q.24. Let m and n be the number of red and black balls in an urn. A ball is drawn at random and is put back into the urn along with 5 additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. If probability that the ball drawn is red is 1 then correct options with possible values of m n can be : 5 (a) 10 (b) 12 (c) 15 (d) 18 Q.25. Let n be an integer n 3. Let p1 , p 2 ,..... p n be a regular n sided polygon inscribed in a circle. Three points pi , p j , p k are randomly chosen, where i, j, k are distinct integer’s between 1 and n. If p (n) denotes probability that pi p j p k is obtuse angle triangle then which of the following are correct ? 1 3 (a) p (5) (b) p(8) 2 7 3 1 (c) p (6) (d) p (7) 10 2 Q.26 From A pack of cards, a man is drawing cards one by one without replacement. Consider the events : E1 : The event that the 10th card drawn is second heart card. E2 : The event that the 8th card drawn is third ace. E3 : The event that the 9th card drawn is first king. E4 : The event that the 12th card drawn is club ‘6’ 39 C8 594 (a) P( E1 ) 3 (b) P( E 2 ) 51 7735 C9 48 4( C8 ) 1 (c) P( E 3 ) (d) P( E 4 ) 32 52 9( C9 ) The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 13 Q.27 If three numbers are chosen randomly from the set { 1, 3, 3 2 ,........,3 n } without replacement, then the probability that they form an increasing geometric progression is : 3 3 (a) if n is odd (b) if n is even 2n 2n 3n 3n (c) n is even (d) n is odd 2(n 2 1) 2(n 2 1) Q.28. A Rational number is selected at random from the set of all rational numbers from the interval (2010, 2011) all whose digits after the decimal point are non-zero and are in the decreasing order, then 36 (a) The probability that it has exactly seven digits after the decimal point is. 511 256 (b) The probability that it contains the digit 3 after the decimal point is 511 63 (c) The probability that the last digit after the decimal point is atleast 4 is 511 32 (d) The probability that the last digit after the decimal point is 4 is. 511 Q.29. Let a, b, c be integers and a 2 b 2 c 2 2. a b c x 1 Let A b c a . Consider the system of simultaneous equations A y 0 c a b z 0 1 (a) The probability that the system of equations has unique solution is 2 (b) The number of triplets (a, b, c ) for which the system of equations has infinitely many solutions is 6. (c) If a 0 , the number of ordered pairs (b, c) for which the system of equations has no solution is 2. (d) The number of elements in the range of ab bc ca is 2. ALGEBRA. 14 Algebra SECTION – 2 : SUBJECTIVE QUESTIONS LEVEL - 1 Q.1. A quadratic equation with integral coefficients has two prime numbers as its roots. If the sum of the coefficients of the equation is prime, then the sum of the roots is ________ Q.2. Let f ( x) ax 2 bx c. if f (x) is symmetric about the line x 2 and the difference between the largest and smallest values of the function f ( x) ax 2 bx c in the interval x [0, 3] is 4, then find the value of (a 2 b 2 ). Q.3. If p , q , r and s are non-zero numbers such that ‘ r ’ and ‘ s ’ are roots of x 2 px q = 0 and p ,.q are the roots of x 2 rx s 0 , then find the value of ( 4 p q 2r s ). Q.4. Let a, b, c, d be four distinct real numbers in A.P. Find the smallest positive value of k satisfying 2( a b ) 4 k (b c ) 2 (c a ) 3 2( a d ) (b d ) 2 (c d ) 3 Q.5. Let S n and n be the sum of first n terms of 2 A.P' s , whose r th term are Tr and t r S 2n 5 T m respectively. If n then 11 1 where m1 and m 2 are coprime. Find the value n 3n 2 t10 m2 m m1 of 2 2 Q.6. z1 , z 2 are the roots of quadratic equation z 2 az b 0 ( a, b are complex). If | z1 z 2 | | z1 | | z 2 | then there exists positive real number such that a2 b , whose minimum value is ________ Q.7. If z x iy is a complex number satisfying ( z z ) 12 4 | z | 2 then maximum value of 2 Re( z ) lm( z) is _________ Q.8. Let a k when k 0, 1, 2, 3, 4,.........., 253 are 254th roots of unity then the unit digit of 253 2 2 z z k 2 is ________ (where z e 7 k 0 Q.9. If w is the imaginary cube root of unity, then the number of ordered pairs of integers ( a, b) such that | aw b | 1 is _____ Q.10 Let A and B be two non-singular matrices such that A I , B 3 I and AB BA 2 , where I is the identity matrix, then the least value of k such that A k I , is _____ x x y x y z Q.11 If 2 x 3 x 2 y 4 x 3 y 2 z 64 , then the real value of x is ____ 3 x 6 x 3 y 10 x 6 y 3z The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 15 0 1 1 Q.12 Let X be the solution set of the equation A I ,where A 4 3 4 and I is the x 3 3 4 corresponding unit matrix and x N, then the minimum value of (cos x sin x ), R. a b c Q.13 Given a matrix A b c a , where a, b, c are real positive numbers, abc 1 and c a b AT A I , then find the value of a 3 b 3 c 3. Q.14 10 different objects are arranged around a circle. In how many ways can 3 objects be selected when no two of the selected objects are consecutive ? Q.15 If the number of selections of 6 different letters that can be made from the words SUMAN and DIVYA so that each selection contains 3 letters from each word is N 2 , then find the value of N. Q.16 Number of permutations of 1, 2, 3, 4, 5, 6, 7, 8 and 9 taken all at a time are such that the digit 1appearing somewhere to the left of 2 3 appearing to the left of 4 and 5 somewhere to the left of 6, is k 7 ! Then the value of k is _____ Q.17. Seven people leave their bags outside a temple and returning after worshipping picked one bag each at random. In how many ways at least one and at most three of them get their correct bags ? Q.18 The number of distinct throws which can be thrown with n six-faced normal dice, which are indistinguishable among themselves is f(n). Find f(6) ? Q.19 Consider the set of eight vectors V { ai bj ck ; a, b, c {1, 1}}. Three non-coplanar vectors can be chosen from V in 2 p ways. Then p is _____ Q.20 Find the number of positive unequal integral solutions of the equation x y z 12 Q.21 12 boys and 2 girls are to be seated in a row such that there are atleast 3 boys between the 2 girls. The number of ways this can be done is 12! , then find the value of . Q.22 The number of triangles whose angular points are at the angular points of a given polygon of n sides, but none of whose sides are the sides of the polygon is TN. Find T10 ? ALGEBRA. 16 Algebra Q.23 An ordinary cubical dice having six faces marked with alphabets A, B, C, D, E and F is thrown n times and the list of n alphabets showing up are noted. The total number of ways in which among the alphabets A, B, C, D, E and F only three of them appear in the list Tn. T6 Find ? 10 Q.24 The members of a chess club took part in a round robin competition in which each player plays with other once. All members scored the same number of points, except four juniors whose total score were 17.5. How many members were there in the club? Assume that for each win a player scores 1 point, ½ for a draw, and zero for losing. Q.25 Find the number of ways in which we can choose 3 squares on a chess board such that one of the squares has its two sides common to other two squares. Q.26 Find the number of four-digit numbers that can be made with the digits 1, 2, 3, 4 and 5 in which at least two digits are identical. Q.27 Find the total number of six-digit natural numbers that can be made with the digits 1, 2, 3, 4, if all digits are to appear in the same number at least once. Q.28 In the decimal system of numeration the number of six-digit numbers in which the sum of k the digits is divisible by 5 is k. Find ? 125 Q.29 If the difference of the number of arrangements of three things from a certain number of dissimilar things and the number of selections of the same number of things from them exceeds 100, then find the least number of dissimilar things. Q.30 In how many ways can a team of 6 horses be selected out of a stud of 16 , so that there shall always be three out of A B C A’ B’ C’, but never A A’, B B’ or C C’ together. Q.31 Two packs of 52 cards are shuffled together. the number of ways in which a man can be dealt 26 cards so that he does not get two cards of the same suit and same denomination is n C k 2 k. Find ( n k ). Q.32 n is selected from the set { 1, 2, 3, ….., 10} and the number 2 n 3 n 5 n is formed. Find the total number of ways of selecting n so that the formed number is divisible by 4. Q.33 A man has three friends. Find the number of ways he can invite one friend everyday for dinner on six successive nights so that no friend is invited more than three times. Q.34 The number of three-digit numbers having only two consecutive digits identical is N, then the value of (N/2)1/2 is _____ Q.35 A bag contains 10 different balls. Five balls are drawn simultaneously and then replaced and then seven balls are drawn. The probability that exactly three balls are common to the two drawn is p , then the value of 12 p is ____ The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 17 Q.36 A fair coin is tossed 5 times, then the probability that no two consecutive heads occur is m P (expressed in lowest form) find the value of (m n). n a b Q.37 The entries in a two-by-two determinant are integers that are chosen randomly and c d independently, and, for each entry, the probability that the entry is odd is p. If the probability that the value of the determinant is even is ½, then the find the value of [2 p ]. Q.38 There are two packs A and B of 52 playing cards. All the four aces from the pack A are removed whereas from the pack B, one ace, one king, one queen and one jack is removed. One of these two packs is selected randomly and two cards are drawn from it, and found to m be a pair (i.e. both have same rank e.g. two 9’s or two king etc.). If Q (expressed in n lowest form) denotes the probability that the pack A was selected, find (m n). Q.39 A person flips 4 fair coins and discards those which turn up tails. He again flips the m remaining coin and then discards those which turn up tails. If P (expressed in lowest n form) denotes the probability that he discards atleast 3 coins, find the value of (m n). Q.40 In a tournament, team X, plays with each of the 6 other teams once. For each match the probabilities of a win, draw and loss are equal. If the probability that team X, finishes with p more wins than losses can be expressed as rational q in their lowest form, find ( p q ). SUBJECTIVE QUESTIONS LEVEL - 2 Q.1. Let f ( x ) ax 2 bx c , where a 0, a , b , c are integers and f ( 1 ) 1, 6 f ( 3 ) 8 and 18 f ( 5 ) 22. Then the number of solutions of equation f ( x ) e x is ____ Q.2. The three different polynomials x 2 ax b, x 2 x ab and ax 2 x b have exactly one common zero. Where a, b are non-zero real numbers. Find the value of a 2b. Q.3 Let p( x ) x 5 x 2 1 4 have roots x1 , x 2 , x3 , x 4 and x 5. g ( x ) x 2 2 , then the value of g ( x1 ) g ( x 2 ) g ( x3 ) g ( x 4 ) g ( x5 ) 30 g ( x1 x 2 x3 x 4 x5 ) is ____ Q.4 Consider the polynomials P ( x ) x 6 x 5 x 3 x 2 x and Q ( x ) x 4 x 3 x 2 1. Given that x1 , x 2 , x3 and x 4 are the roots of the equation Q( x) 0. The value of P( x1 ) P( x 2 ) P( x3 ) P( x 4 ) is ______ ALGEBRA. 18 Algebra Q.5. Let , , , are zeroes of P ( x) 5 x 4 px 3 qx 2 rx s ( p , q , r , s R ) and , , are zeroes of Q ( x) x 3 9 x 2 ax 24 ( ). If , , (taken in that order) are in arithmetic progression and , , , (taken in that order) are in harmonic progression, P(1) then find the value of. Q(1) Q.6. If x 3 ax 2 bx c 0 has the roots 2 3 4 , 2 3 4 and 2 3 4 where , , are the roots of x 3 x 2 1 0 then the value of | a b c | is equal to_______ Q.7. Let f ( x) ax 4 bx 2 3 x 7 and f (4) 2286 and f (4) N. If K be the number of ways in which the number N can be resolved as a product of two divisors which are relatively prime then K-7. Q.8. Let equation x 3 px 2 qx q 0 where p, q R {0} has 3 real roots 1 1 1 k , , in H.P. and the minimum value of is then the value of k is ? 2 2 2 12 Q.9. Let a1 , a 2 , a 3 , a 4 , a 5 be five terms of an increasing geometric progression such that a1 , a 2 , a 3 , a 4 , a 5 N , and a 5 100. Then the number of possible geometric progression (is/are) ______ Q.10. If r th , ( r 2) th , (r 6) th terms of an increasing A.P. are last three terms of a G.P. and there lies exactly 2 p 1 number of terms of that G.P. in the interval ( p, 2018 p) ( p N ) then maximum value of p is _________ Q.11. If 8 harmonic means are inserted between two numbers a and b ( a b ) such that 5 arithmetic mean of a and b is times equal to geometric mean of a and b , then 4 H8 a b H is equal to ____ 8 n 1 r 1 r Q.12. The value of is ______ n k (2n 2k 1) (2n k 1) k 1 The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 19 n3 1 1 2 4 and N ( 1 n ) Q.13. Let L 1 ; M n3 1 1 n 3 n2 n2 n 3 1 2 n Then find the value of L1 M 1 N 1 99 10 n n 1 Q.14. If P , then [P] is : {where [.] us GIF} 99 10 n n 1 12 2 2 32 500 2 Q.15. Let S ...... . If where [.] denotes the greatest integer function 1.3 3.5 5.7 999.1001 then the number of total divisors of [S ] is _____ Q.16. Let z1 and z 2 be two fixed point in argand plane z is a variable point such that z z1 z z2 arg 1 and arg 1 2 z 2 z1 z z2 tan 1 / 2 3 If then locus of z is branch of hyperbola of eccentricity equal tan 2 / 2 2 to_______ Q.17. Consider the complex number z satisfying | z 3 | | z | | z 3 | 12.If the greatest integer less than or equal to | z | is ‘ a ’ and the least integer greater than or equal to is ‘ b ’, then ‘ a b ’ is _______ Q.18. Complex numbers a, b and c are zeroes of polynomials P ( z ) z 3 qz r (where q, r R) and | a 2 | | b 2 | | c 2 | 250. If the points corresponding to a, b and c in h2 complex plane are the vertices of a right angled triangle with hypotenuse h, then . 125 Q.19. ABC and BDE are two equilateral triangles such that D is the mid-point of BC. AE intersect BC at F. 2 | z z a | max area (FED 2 | z z d | | z a z c | and 2 k | z z a | min area (AFC Then, find the value of k. (Where z represents complex number, z a , z d & z c are the affixes of A, D & C respectively) 1 1 1 1 1 1 1 1 Q.20 Let , , , , , , , are vertices of a1 2i a 2 2i a3 2i a 4 2i a5 2i a 6 2i a 7 2i a8 2i regular octagon. If the area of octagon is A, then the value of 8 2 A is (where a j R for j = 1, 2, 3, 4, 5, 6, 7, 8 and i 1 ) ______ ALGEBRA. 20 Algebra Q.21 The polynomial p( x ) (1 x x 2 ...... x 17 ) 2 x 17 has 34 complex roots of the form z k rk [ Cos ( 2 ak ) i sin ( 2 ak ) ] where k 1,2 ,3.....34 with 0 a1 a2 a3........ a4 1 & y k 0. Given that a1 a2 a3 a4 a5 m / n (where m & n are co-primes), find the value of (m + n) 2x 3 y 3x y x 2y Q.22. If x 3 z 3x 2 z ( x y z ) for all x, y, z R , then the value of is 2x z y 2 z 2 y 3z 3 y z equal to _____ Q.23. If , , are the roots of x 3 x 2 2 x 1 0 , then the value of is equal to____ 2 2 2 2 2 2 2 2 2 2 2 2 Where Q.24. If A is an idempotent matrix and I is an identity matrix of the same order, then the value of n, where n N such that A I I 127 A. n Q.25. A is a square matrix of order n Maximum number of distinct entries if A is a triangular matrix. m Maximum number of distinct entries if A is a diagonal matrix p Minimum number of zeros if A is a triangular matrix. If 5 P 2m then the value of n is ___________ Q.26. Let A be a 2 3 matrix whereas B be a 3 2 matrix. If det(AB) = 4 then the value of det(BA) is ______ Q.27. Let A and B be two non-singular square matrices such that B I and AB BA If 2 k A 4 B 1 A 4 B k then value of is _______(Where [.] = GIF and I is an identity matrix). 3 Q.28. Let A [a ij ]33 be a matrix such that AA T 4 I and aij 2cij 0 where cij is the cofactor of aij i & j, I is the unit matrix of order 3 and A T is the transpose of the matrix A.If a11 4 a12 a13 a11 1 a12 a13 a 21 a 22 4 a 23 5 a 21 a 22 1 a 23 a31 a32 a33 4 a31 a32 a33 1 =0 a Then, where a and b are coprime positive integers then the value of a b is b _______ The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 21 Q.30. Let A be a square matrix of order 3 satisfies A 3 6 A 2 12 A 81 0 & B= A-2I. If 1 | A | 8 & | adj ( I 2 A ) | K , then [K ] ____________ Where [.] represents G.I.F. Q.31. Consider A and B as 2 2 matrices with determinant equal to 1, then tr ( AB ) tr ( A). tr ( B ) tr ( AB 1 ) 2 is ________ Q.32. The ordered triplet ( , , ) satisfies following equations x y 3z 1, x y z 1, ax (a b) y 6 z 1, (a, b R ) If 2 2 2 is minimum, then minimum value of a 2 b 2 is _____ k 2 r n C r k 1C r 1 Q.33. The value of r 1 (where n k ) is equal to ____ k n k r 1 n Cr C n1 r 0 m (m 1) (r 1)m r 1 Q.34. The value of m m is equal to ______ r 1 r mCr 2 n Q.35. If f ( x) a 0 a1 x a 2 x ...... a n x .... and f ( x) bo b1 x b2 x 2 .... bn x n ..... If 1 x a 0 1 and b1 3 and b10 k 11 1 , then k is _______ (Given that a 0 , a1 , a 2 ,....... are in G.P.) Q.36. The coefficient of x2012 in the expansion (x + 1) (x² + 2) (x4 + 4) …… (x1024 + 1024) is--------- 1 14 Q.37. The value of (1) k (15 k ) 17 C k is _______ 3 k 0 200 Q.38. Consider two polynomial f (x) and g (x) given by f ( x) r x r and r 0 200 g ( x) r xr r 0 such that r 1 100 r 200 and f ( x 1) g ( x). 200 Let A r , then the remainder when A is divided by 15 is equal to____ r 100 24 A19 1 Q.39 Let (3x² + 2x + c)12 = A x r 0 r r and A 5 27 then c is---------- Q.40. Number of all five digit numbers of the form 34 x 5 y ( x and y are digits) divisible by 36, is ____ ALGEBRA. 22 Algebra N Q.41. Let N be a natural number. If its first digit (from the left) is deleted, it gets reduced to. 29 Find the sum of all the digits of N. Q.42. There are “ n ” married couples at a party each person shakes hand with every person other 1 1 than her or his spouse then number of handshakes is f (n) , ? n 4 f ( n) Q.43. For a positive integer n, let an be the remainder when n is divided by 3 and bn be the remainder when n2 is divided by 3. Find the value of positive integer m such that m (a n 1 n 2 bn1 2an ) = 2016 Q.44. Using the digits 0, 1, 2, 3 & 4, the number of ten digit sequences can be written so that the k difference between any two consecutive digits is 1, is equal to k , then is equal to 72 _____ Q.45. 40 slips are placed in a box, each bearing a number 1 to 10 with each number entered on 4 slips. 4 slips are drawn from the box at random without replacement. The probability that 10 k C2 two of the slips bear a number ‘a’ & other two bear a number ‘ b ’ ( a) is where 9 40 C4 k ________ Q.46. A coin is tossed n times. The probability of getting at least one head is greater than that of 5 getting at least two tails by. Then n is _____ 32 Q.47. A tosses 2 fair coins & B tosses 3 fair coins after game is won by the person who throws greater number of heads. In case of a tie, the game is continued under identical rules until someone finally wins the game. The probability that A finally wins the game is K/11, then K is _____ Q.48. If a and b are selected at random from the range of y ( a, b are distinct positive integers). Then the probability of selecting distinct ordered pairs ( a, b) of prime , numbers from the 147 K range of y , where y x 0 , is. Find K_____ 1 15 x 5 x Q.49. A and B are two persons standing in the frame as shown in figure. A can take a step eastward or northward with equal probabilities while B can take a step westward or southward with equal probabilities. In each move they can step only to an adjoining square (having a common side). Both A and B take 3 steps at random. The probability that they will meet in a shaded square is p/q where p and q are co-prime then ( p q ) / 3 is ______ The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 23 N B W E A S Q.50 A North-South white line is drawn in a school ground. At first, you are 5 meter due west from a point A on the line. You toss a coin, and if it is head move 1 meter east. If it is tail, move 1 meter north. Note that once you reach the line you must stop. Let Pn be the probability test that you stop at the point n meters north from A. Find the sum of values of n for which Pn is maximum. Q.51. Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. The probability that no two ants m mn arrive at the same vertex is where m and n are co-prime then is? n 3 ALGEBRA. 24 Trigonometry CHAPTER 2 TRIGONOMETRY Topics Coverage : Trigonometrical Ratios & Identities Trigonometrical Equations Triangle Trigonometry Inverse Trigonometric Functions. SECTION – 1 : OBJECTIVE QUESTIONS (M ultiple Correct Answ ers Ty pe) LEVEL - 1 Q.1 If cos is the geometric mean between sin and cos , where 0 , , then 2 cos 2 is equal to 2 (a) 2 sin (b) 2 cos 2 4 4 2 2 (c) 2 sin (d) 2 cos 4 4 If 0 and 81sin 81cos 30 , then is 2 2 Q.2 (a) 30 (b) 60 (c) 120 (d) 150 Q.3 Which of the following quantities are rational ? 11 5 9 4 (a) sin sin (b) cosec sec 12 12 10 5 2 4 8 (c) sin 4 cos 4 (d) 1 cos 1 cos 1 cos 8 8 9 9 9 5 Q.4 In which of the following sets the inequality sin 6 x cos 6 x holds good? 8 3 5 3 7 9 (a) , (b) , (c) , (d) , 8 8 8 8 4 4 8 8 The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 25 Q.5 Which of the following inequalities hold true in any triangle ABC ? A B C 1 A B C 3 3 (a) sin sin sin (b) cos cos cos 2 2 2 8 2 2 2 8 A B C 3 A B C 9 (c) sin 2 sin 2 sin 2 (d) cos 2 cos 2 cos 2 2 2 2 4 2 2 2 4 Q.6 For which of the following hold(s) good? 7 (a) tan tan 2 tan 3 tan 3 tan 2 tan (b) cosec cosec 2 cosec 4 (c) cos cos 2 cos 3 1 / 2 (d) 8 cos cos 2 cos 4 1 Q.7 Which of the following do/ does not reduce to unity ? sin 180 A cot 90 A cos 360 A cos ec A (a) tan 180 A tan 90 A sin A sin A tan90 A cos A (b) sin 180 A cot A sin 90 A sin 24 cos 6 sin 6 66 (c) sin 21 cos 39 cos 51 sin 69 cos 90 A sec Atan 180 A (d) sec 360 Asin180 Acot 90 A Q.8 The expression tan 4 x 2 tan 2 x 1 cos 2 x when x / 12 can be equal to (a) 4 2 3 (b) 4 2 1 (c) 16 cos 2 12 (d) 16 sin 2 12 Q.9 a 2 sin 2a 1 cos 2a 1 if tan is 3 4 2a 2a (a) (b) (c) (d) 4 3 a 1 2 a 1 2 Q.10 Solve 4 sin 4 x cos 4 x 1 , then x is equal to n Z 2 2n (a) n (b) n sin 1 (c) (d) 2n 5 3 4 Q.11 A general solution of the equation, tan 2 cos 2 1 is n Z (a) n (b) 2n (c) n (d) 2n 4 4 4 4 TRIGONOMETRY. 26 Trigonometry for x 0, , then 1 Q.12 If sin x cos x y y 3 (a) x (b) y 0 (c) y 1 (d) x 4 4 Q.13 sin 3 cos 6 x x 2 11, 0 4 , x R , holds for (a) no values of x and (b) one value of x and two values of (c) two values of x and two values of (d) two points of values of x, Q.14 For the smallest positive values of x and y the equation, 2sin x sin y 2 cosx y 3 , has a solution then, which of the following is/ are true ? x y x y 1 (a) sin 1 (b) cos 2 2 2 (c) Number of ordered pairs ( x , y ) is 2 (d) Number of ordered pairs ( x , y ) is 3 Q.15 For the equation 1 2 x x 2 tan 2 x y cot 2 x y . (a) exactly one value of x exists (b) exactly two values of x exist (c) y 1 n / 4, n Z (d) y 1 n / 4, n Z Q.16 If x y 2 / 3 and sin x / sin y 2 , then (a) the number of values of x 0, 4 are 4 (b) number of values of x 0,4 are 2 (c) number of values of y 0, 4 are 4 (d) number of values of y 0, 4 are 8 1 2 For 0 x 2 , then 2 cos ec 2 Q.17 x y y 1 2 2 (a) is satisfied by exactly one value of y. (b) is satisfied by exactly two values of x. (c) is satisfied by x for which cos x 0 (d) is satisfied by x for which sin x 0 Q.18 If the equation sin 2 x a sin x b 0 has only one solution in 0, then, which of the following statements are correct ? (a) a , 1 2, (b) b , 01 1, (c) a 1 b (d) None of these The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 27 Q.19 If sin x 2 2 x b 2 for all the real values of x 1 and 0, / 2 / 2, then the possible real values of b is/are ____ (a) 2 (b) 3 (c) 4 (d) 5 Q.20 If the tangents of the angles A and B of a triangle ABC satisfy the equation abx 2 c 2 x ab 0 , then (a) tan A a / b (b) tan B b / a (c) cos C 0 (d) sin 2 A sin 2 B sin 2 C 2 Q.21 In a triangle ABC if 2a 2 b 2 2b 2 c 2 a 4 b 4 c 4 ,then angle B is equal to ____ (a) 45 (b) 135 (c) 120 (d) 60 Q.22 If in a triangle ABC, a, b, c and angle A are given and c sin A a c , then ____ (a) b1 b2 2c cos A (b) b1 b2 c cos A (c) b1b2 c 2 a 2 (d) b1b2 c 2 a 2 Q.23 CF is the internal bisector of angle C of ABC , then CF is equal to ____ 2ab C ab C (a) cos (b) cos ab 2 2ab 2 b sin A (c) (d) None of these Sol.19 C sin B 2 Q.24 The sides of a ABC satisfy the equation 2a 2 4b 2 c 2 4ab 2ac. Then, ____ (a) the triangle is isosceles (b) the triangle is obtuse 7 1 (c) B cos 1 (d) A cos 1 8 4 Q.25 If in a triangle sin 4 A sin 4 B sin 4 C sin 2 B sin 2 C 2 sin 2 C sin 2 A 2 sin 2 A sin 2 B , then its angle A is equal to ____ (a) 30 (b) 120 (c) 150 (d) 60 Q.26 The area of a regular polygon of n sides is (where r is inradius, R is circumradius and a is side of the triangle) nR 2 2 sin (b) nr tan 2 (a) 2 n n na 2 (d) nR tan 2 (c) cot 4 n n TRIGONOMETRY. 28 Trigonometry Q.27 If the angles of a triangle are 30 and 45 , and the included side is 3 1 cm , then which options are incorrect (a) Area of the triangle is 1 2 3 1sq.units (b) Area of the triangle is 1 2 3 1sq.units 3 1 (c) Ratio of greater side to smaller side is 2 1 (d) Ratio of greater side to smaller side is 4 3 Q.28 Length of the tangents from A,B,C to the incircle are in A.P., then (a) r1 , r2 , r3 are in H.P. (b) r1 , r2 , r3 are in A.P. 4c 3b (c) a, b, c are in A.P. (d) cos A 2b Q.29 If the sides of a right-angled triangle are in G.P., then the cosines of the acute angle of the triangle are ___ 5 1 5 1 5 1 5 1 (a) (b) (c) (d) 2 2 2 2 Q.30 If , are the roots of the equation 6 x 2 11x 3 0 , then which of the following are real ? (a) cos 1 (b) sin 1 (c) cos ec 1 (d) both cot 1 and cot 1 Q.31 2 tan 1 2 is equal to ___ 1 3 3 (a) cos (b) cos 1 5 5 1 3 1 3 (c) tan (d) cot 2 4 4 Q.32 If , , are the roots of tan 1 x 1 tan 1 x tan 1 x 1 tan 1 3 x , then (a) 0 (b) 1 / 4 (c) 1 (d) max 1 Q.33 If sin 1 x sin 1 w sin 1 y sin 1 z 2 , Then N1 N2 x y D N3 N1 , N 2 , N 3 , N 4 N z w N4 (a) has a maximum value of 2 (b) has a minimum value of 0 (c) 16 different D are possible (d) has a minimum value of -2 The IITian’s Prashikshan Kendra Pvt. Ltd. IITIAN’S MASTER PACKAGE 29 Q.34 Which of the following is a rational number ? 1 3 (a) sin tan 1 3 tan 1 (b) cos sin 1 3 2 4 1 63 1 5 (c) log 2 sin sin 1 (d) tan cos 1 4 8 2 3 Q.35 If f x sin 1 x cos x , then 2 1 2 2 5 2 (a) f x has the least value of (b) f x has the greatest value of 8 8 2 5 2 (c) f x has the least value of (d) f x has the greatest value of 16 4 Q.36 If sin 1 a a2 a3 ..... cos 1 1 b b 2 .... , then 3 9 2 2a 3 3a 2 3 2 (a) b (b) b (c) a (d) a 3a 2a 2 3b 3 2b 2x Q.37 If 2 tan 1 x sin 1 is independent of x , then 1 x2 (a) x 1 (b) x 1 (c) 0 x 1 (d) 1 x 0 Q.38 Which of the following quantities is/ are positive? (a) cos tan 1 tan 4 (b) sin cot 1 cot 4 (c) tan cos cos 5 1 (d) cotsin sin 4 1 OBJECTIVE QUESTIONS (M ultiple Correct Answ ers Ty pe) LEVEL - 2 Q.1. Let cos A cos B x; cos 2 A cos 2B y ; cos 3 A cos 3B z , then which of the following is true : y 1 (a) cos 2 A cos 2 B 1 (b) ( 2 x 2 y 2) cos A cos B 2 4 2 (c) 2 x z 3 x (1 y ) (d) xyz 0 A, B R Q.2. If x sec tan and y cos ec cot , then y 1 y 1 (a) x (b) x y 1 y 1 1 x (c) y (d) xy x y 1 0 1 x TRIGONOMETRY. 30 Trigonometry 2 4 8 Q.3. The value of the expression tan
