JEE-Mathematics EXERCISE - 01 CHECK YOUR GRASP SELECT THE CORRECT ALTERNATIVE (ONLY ONE CORRECT ANSWER) 1 . The maximum value of the sum of the A.P. 50, 48, 46, 44, .................... is - (A) 325 (B) 648 (C) 650 (D) 652 2 . Let Tr be the rth term of an A.P. for r = 1, 2, 3, ........... If for some positive i nteger s m, n we have Tm 1 & Tn 1 , then T equals - n m mn 1 11 (C) 1 (D) 0 (A) (B) mn mn 3 . The interior angles of a convex polygon are in AP . The smallest angle is 120° & the common difference is 5°. Find the number of sides of the polygon - (A) 9 (B) 16 (C) 12 (D) none of these 4 . The first term of an infinitely decreasing G.P. is unity and its sum is S. The sum of the squares of the terms of the progression is - S S2 S (D) S2 (A) (B) (C) 2S 1 2S 1 2S 5 . A particle begins at the origin and moves successively in the y following manner as shown, 1 unit to the right, 1/2 unit up, 1/4 unit to 1/4 the right, 1/8 unit down, 1/16 unit to the right etc. The length of each move is half the length of the previous move and movement continues in the ‘zigzag’ manner indefinitely. The co-ordinates of the point to which the ‘zigzag’ converges is - NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 1 1/16 0 1/2 x 1/8 (A) (4/3, 2/3) (B) (4/3, 2/5) (C) (3/2, 2/3) (D) (2, 2/5) 100 100 6 . Let a be the nth term of a G.P. of positive numbers. Let a2n = & a2n1 = such that . Then the n n 1 n 1 common ratio of the G.P. is - (A) (B) (C) (D) 7 . If p, q, r in harmonic progression and p & r be different having same sign then the roots of the equation px2 + qx + r = 0 are - (A) real and equal (B) real and distinct (C) irrational (D) imaginary 111 8 . If x > 1, y > 1, z >1 are in G.P., then 1 n x , 1 n y , 1 n z are in - (A) A.P. (B) H.P. (C) G.P. (D) none of above 9. If ln (a + c) , ln (c – a), ln (a – 2b + c) are in A.P., then : E (A) a, b, c are in A.P. (B) a2, b2, c2 are in A.P (C) a, b, c are in G.P. (D) a, b, c are in H.P. 17
JEE-Mathematics 1 0 . If the (m + 1)th , (n +1)th & (r + 1)th terms of an AP are in GP & m, n, r are in HP, then the ratio of the common difference to the first term of the AP is - 1 2 2 (D) none of these (A) (B) (C) n n n 1 1 . The sum of roots of the equation ax2 + bx + c = 0 is equal to the sum of squares of their reciprocals. Then bc2 , ca2 and ab2 are in - (A) AP (B) GP (C) HP (D) none of these 1 2 . The quadratic equation whose roots are the A.M. and H.M. between the roots of the equation, 2x2 – 3x + 5 = 0 is - (A) 4x2 – 25x + 10 = 0 (B) 12x2 – 49x + 30 = 0 (C) 14x2 – 12x + 35 = 0 (D) 2x2 + 3x + 5 = 0 1 3 . If the sum of the first n natural numbers is 1/5 times the sum of the their squares, then the value of n is - (A) 5 (B) 6 (C) 7 (D) 8 1 4 . Suppose p is the first of n(n > 1) AM's between two positive numbers a and b, then value of p is - na b na b nb a nb a (A) n 1 (B) n 1 (C) n 1 (D) n 1 15. If 1 a 1 1 c 1 0 and a, b, c are not in A.P., then - a 2b c 2b (A) a, b, c are in G.P. (B) a, b , c are in A.P. (C) a, b , c are in H.P. (D) a, 2b, c are in H.P. 2 2 35 7 ......... is - 16. The sum to n terms of the series 12 12 22 12 22 32 3n 6n 9n 12n (A) (B) (C) (D) n 1 n 1 n 1 n 1 17. 1 1 1 ........ + to 4 1 1 1 ...... + to is equals to - If 14 90 , then 14 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 24 34 34 54 4 4 8 9 4 (D) none of these (A) (B) (C) 96 45 90 n s 1 8 .If r an3 bn2 cn , then find the value of a + b + c. s 1 r 1 (A) 1 (B) 0 (C) 2 (D) 3 19. If a, b, c are positive numbers in G.P. and log 5c , lo g 3b and log a are in A.P., then a, b, c forms the a 5c 3b sides of a triangle which is - (A) equilateral (B) right angled (C) isosceles (D) none of these SELECT THE CORRECT ALTERNATIVES (ONE OR MORE THAN ONE CORRECT ANSWERS) 2 0 . If sum of n terms of a sequence is given by S = 3n2 – 5n + 7 & t represents its rth term, then - nr (A) t = 34 (B) t = 7 (C) t = 34 (D) t = 40 7 2 10 8 18 E
JEE-Mathematics 21. If 10 harmonic means H , H , H ......... H are inserted between 7 and – 1 , then - 123 10 3 3 1 7 (A) H = –7 (B) H = (C) H = – (D) H = 1 2 7 1 7 10 19 2 2 . If t be the nth term of the series 1 + 3 + 7 + 15 + ........, then - n (A) t + 1 = 32 (B) t = 27 + 1 (C) t = 210 – 1 (D) t = 250 + 1 5 7 10 100 2 3 . Indicate the correct alternative(s), for 0 , if x cos2n , y sin2n and z cos2n sin2n , 2 n0 n 0 n 0 then - (A) xyz = xz + y (B) xyz = xy + z (C) xyz = x + y + z (D) xyz = yz + x NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 BRAIN TEASERS ANSWER KEY EXERCISE-2 Que. 1 2 3 4 5 6 7 8 9 10 Ans. C C A B B A D B D C Que. 11 12 13 14 15 16 17 18 19 20 Ans. A B C A D B A A D A,D Que. 21 22 23 Ans. A,D A,C B,C E 19
JEE-Mathematics BRAIN TEASERS EXERCISE - 02 SELECT THE CORRECT ALTERNATIVES (ONE OR MORE THAN ONE CORRECT ANSWERS) 1 . Consider an A.P. with first term ‘a’ and the common difference d. Let S denote the sum of the first K terms. k S kx Let S x is independent of x, then - (A) a = d/2 (B) a = d (C) a = 2d (D) none of these 2 . Let , , be the roots of the equation x3 + 3ax2 + 3bx + c = 0. If , , are in H.P. then is equal to - (A) – c/b (B) c/b (C) – a (D) a F I 9 r G J3 . (2r 1) H K11 is equal to - r 1 (A) 45 (B) 55 (C) sum of first nine natural numbers (D) sum of first ten natural numbers 4 . For the A.P. given by a1, a2, ............., an, ........, with non-zero common difference, the equations satisfied are- (A) a1 + 2a2 + a3 = 0 (B) a1 – 2a2 + a3 = 0 (C) a1 + 3a2 – 3a3 – a4 = 0 (D) a1 – 4a2 + 6a3 – 4a4 + a5 = 0 5 . If a, a1, a2,.....,a10 , b are in A.P. and a, g1, g2,.....g10 , b are in G.P. and h is the H.M. between a and b, then a1 a2 ..... a10 a2 a3 ..... a9 ....... a5 a6 is - g1 g10 g2g9 g5g6 10 15 30 5 (A) h (B) h (C) h (D) h 6 . The sum of the first n terms of the series 12 + 2.22 + 32 + 2.42 + 52 + 2.62 + ....... is n(n 1)2 , when n is 2 even. When n is odd, the sum is - NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 n2 (n 1) n(n 1)(2n 1) n(n 1)2 n2 (n 1)2 (A) (B) (C) (D) 2 6 2 2 7 . If (1 + 3 + 5 +...+ a) + (1 + 3 + 5 +...+ b) = (1 + 3 + 5 + .... + c), where each set of parentheses contains the sum of consecutive odd integers as shown such that - (i) a + b + c = 21, (ii) a > 6 If G = Max{a, b, c} and L = Min{a, b, c}, then - (A) G – L = 4 (B) b – a = 2 (C) G – L = 7 (D) a – b = 2 8 . If a, b and c are distinct positive real numbers and a2 + b2 + c2 = 1, then ab + bc + ca is - (A) equal to 1 (B) less than 1 (C) greater than 1 (D) any real number 9 . Let p, q, r R+ and 27 pqr (p + q + r)3 and 3p + 4q + 5r = 12 then p3 + q4 + r5 is equal to - (A) 2 (B) 6 (C) 3 (D) none of these 1 0 . The sum of the first 100 terms common to the series 17, 21, 25, ......... and 16, 21, 26, ...............is - (A) 101100 (B) 111000 (C) 110010 (D) 100101 20 E
JEE-Mathematics FG IJ1 1 . If a, b, c are positive such that ab2c3 = 64 then least value of 1 2 3 is - H Ka b c (A) 6 (B) 2 (C) 3 (D) 32 12. If a , a ,..................a R+ and a .a ....... a = 1 then the least value of (1 a1 a12 )(1 a2 a 2 ).....(1 an a 2 ) 1 2n 12 n 2 n is - (A) 3n (B) n3n (C) 33n (D) data inadequate 1 3 . Let a1, a2, a3,........ and b1, b2, b3,........ be arithmetic progression such that a1 = 25, b1 = 75 and a100 + b100 = 100, then - (A) The common difference in progression 'ai' is equal but opposite in sign to the common difference in progression 'bj'. (B) an + bn = 100 for any n. (C) (a1 + b1), (a2 + b2), (a3 + b3), ....... are in A.P. 100 (D) (ar br ) 104 r 1 1 4 . If the AM of two positive numbers be three times their geometric mean then the ratio of the numbers is - (A) 3 2 2 (B) 2 1 (C) 17 12 2 2 (D) 3 2 2 1 5 . If first and (2n – 1)th terms of an A.P., G.P. and H.P. are equal and their nth terms are a, b, c respectively, then - (A) a + c = 2b (B) a b c (C) a + c = b (D) b2 = ac 1 6 . Let a, x, b be in A.P. ; a, y, b be in G.P. and a, z, b be in H.P. If x = y + 2 and a = 5z then - (A) y2 = xz (B) x > y > z (C) a = 9, b = 1 (D) a = 9 , b = 1 44 1 7 . The pth term Tp of H.P. is q(q + p) and qth term Tq is p(p + q) when p > 1, q > 1, then - (A) Tp + q = pq (B) Tpq = p + q (C) Tp + q > Tpq (D) Tpq > Tp+q 1 8 . a, b, c are three distinct real numbers, which are in G.P. and a + b + c = xb, then - NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 (A) x < –1 (B) –1 < x < 2 (C) 2 < x < 3 (D) x > 3 1 9 . Let a , a , ......., a be in A.P. & h , h , .......h be in H.P. . If a = h = 2 & a = h = 3 then a h is - 12 10 12 10 11 10 10 47 (A) 2 (B) 3 (C) 5 (D) 6 BRAIN TEASERS ANSWER KEY EXERCISE-2 Que. 1 234 567 8 9 10 Ans. A A A,C B,D C A A,D B CA Que. 11 12 13 14 15 16 17 18 19 Ans. C A A,B,C,D C,D B,D A,B,C A,B,C A,D D E 21
JEE-Mathematics EXERCISE - 03 MISCELLANEOUS TYPE QUESTIONS FILL IN THE BLANKS 1 . The sum of n terms of two A.P.’s are in the ratio of (n + 7) : (3n + 11). The ratio of their 9th term is _________. 2 . The sum of the first nineteen terms of an A.P. a , a , a ................. if it is known that a + a + a + a = 224, 123 4 8 12 16 is ______________. 3 . If x R and the numbers (51+x + 51–x), a/2, (25x + 25–x) form an A.P. then ‘a’ must lie in the interval _________. 111 2 1 1 1 4. If 12 + 22 + 32 + .................. upto = , then + 32 + + ............ = ____________. 6 12 52 5 . When 9th term of an A.P. is divided by its 2nd term the quotient is 5 & when 13th term is divided by the 6th term, the quotient is 2 and remainder is 5. The first term and the common difference of the A.P. are _____________ & ________ respectively. 11 1 6 . The sum to infinity of the series + + + ......... is equal to _______________. 1 12 123 7 . If sin (x – y), sin x and sin (x + y) are in H.P., then sin x. sec y = ______________. 2 MATCH THE COLUMN Following questions contains statements given in two columns, which have to be matched. The statements in Column-I are labelled as A, B, C and D while the statements in Column-II are labelled as p, q, r and s. Any given s t a t e m e n t i n C o l u m n - I c a n h a v e c o r r e c t m a t c h i n g w i t h O N E s t a t e m e n t i n C o l u m n - I I . 1 . Column-I Column-II (A) If ai's are in A.P. and a1 + a3 + a4 + a5 + a7 = 20, a4 (p) 21 is equal to (B) Sum of an infinite G.P. is 6 and it's first term is 3. (q) 4 then harmonic mean of first and third terms of G.P. is (C) If roots of the equation x3 – ax2 + bx + 27= 0, are in G.P. (r) 2 4 with common ratio 2, then a + b is equal to (s) 6/5 (D) If the roots of x4 – 8x3 + ax2 + bx + 16 = 0 are positive real numbers then a is NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 2 . Column-I Column-II (A) nth term of the series 4, 11, 22, 37, 56, 79,....... (p) 2n2 + n (B) |12 – 22 + 32 – 42.......... 2n terms| is equal to (q) 2n2 + n + 1 (C) sum to n terms of the series 3, 7, 11, 15,....... is (r) – (n2 + n) (s) 1 (n2 n) (D) coefficient of xn in 2x(x – 1)(x – 2) ......... (x – n) is 2 ASSERTION & REASON These questions contains, Statement-I (assertion) and Statement-II (reason). (A) Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I. (B) Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for Statement-I. (C) Statement-I is true, Statement-II is false. (D) Statement-I is false, Statement-II is true. 22 E
JEE-Mathematics 1. are three in ab cb > 4 Statement-I : If a, b, c distinct positive number H.P., then + 2a b 2c b Because Statement-II : Sum of any number and it's reciprocal is always greater than or equal to 2. (A) A (B) B (C) C (D) D 2 . Statement-I : If x2y3 = 6(x, y > 0), then the least value of 3x + 4y is 10 Because m1a1 m2a2 1 m1 m2 Statement-II : If m1, m2 N, a1, a2 > 0 then (a1m1 a2m2 ) m1 m2 and equality holds when a1 = a2. (A) A (B) B (C) C (D) D 3 . Statement-I : For n N, 2n > 1 + n (2n 1 ) Because Statement-II : G.M. > H.M. and (AM) (HM) = (GM)2 (A) A (B) B (C) C (D) D a b c 3abc 3 abc 2 4 . Statement-I : If a, b, c are three positive numbers in G.P., then 3 . ab bc ca = Because Statement-II : (A.M.) (H.M.) = (G.M.)2 is true for any set of positive numbers. (A) A (B) B (C) C (D) D 5 . Statement-I : nth term (Tn) of the sequence (1, 6, 18, 40, 75, 126,....) is an3 + bn2 + cn + d, and 6a + 2b – d is = 4. Because Statement-II If the second successive differences (Differences of the differences) of a series are in A.P., then Tn is a cubic polynomial in n. (A) A (B) B (C) C (D) D 6 . Statement-I : The format of nth term (Tn) of the sequence (n2, n4, n32, n1024.......) is an2 + bn + c. Because Statement-II : If the second successive differences between the consecutive terms of the given sequence are in G.P., then Tn = a + bn + crn–1, where a, b, c are constants and r is common ratio of G.P. (A) A (B) B (C) C (D) D NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 COMPREHENSION BASED QUESTIONS Comprehension # 1 There are 4n + 1 terms in a sequence of which first 2n + 1 are in Arithmetic Progression and last 2n + 1 are in Geometric Progression the common difference of Arithmetic Progression is 2 and common ratio of Geometric Progression is 1/2. The middle term of the Arithmetic Progression is equal to middle term of Geometric Progression. Let middle term of the sequence is Tm and Tm is the sum of infinite Geometric Progression whose F I5 2 9 GH KJsum of first two terms is 4 n and ratio of these terms is 16 . On the basis of above information, answer the following questions : 1 . Number of terms in the given sequence is equal to - (A) 9 (B) 17 (C) 13 (D) none 2 . Middle term of the given sequence, i.e. Tm is equal to - (A) 16/7 (B) 32/7 (C) 48/7 (D) 16/9 3 . First term of given sequence is equal to - (A) –8/7, –20/7 (B) –36/7 (C) 36/7 (D) 48/7 E 23
JEE-Mathematics 4 . Middle term of given A. P. is equal to - (A) 6/7 (B) 10/7 (C) 78/7 (D) 11 (C) 3 (D) 6 5 . Sum of the terms of given A. P. is equal to - (A) 6/7 (B) 7 Comprehension # 2 : If ai > 0, i = 1, 2, 3, ..... n and m1, m2, m3, ....., mn be positive rational numbers, then m1a1 m2a2 ..... mn an a1m1 a m 2 ..... a m n 1 /(m1 m2 .....mn ) (m1 m2 .... mn) m1 m2 .... mn 2 n m1 m2 ... m n a1 a2 an is called weighted mean theorem where A* = m1a1 m2a2 .... m nan = Weighted arithmetic mean m1 m2 .... mn G* = a1m1 a m 2 .... a m n 1 /(m1 m2 .... mn ) 2 n = Weighted geometric mean and H* = m1 m2 ..... m n = Weighted harmonic mean m1 m2 .... mn a1 a2 an i.e., A* G* H* Now, let a + b + c = 5(a, b, c > 0) and x2y3 = 243(x > 0, y > 0) On the basis of above information, answer the following questions : 1 . The greatest value of ab3c is - (A) 3 (B) 9 (C) 27 (D) 81 2 . Which statement is correct - 11 11 11 11 (A) (B) (C) 5 (D) 25 1 6 1 5 1 3 1 25 1 9 1 191 abc abc abc abc (D) less than 15 3 . The least value of x2 + 3y + 1 is - (A) 15 (B) greater than 15 (C) 3 4 . Which statement is correct - (A) 2x 3y 3 5 2x 3y 5 xy NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 5 32 (B) 5 3 3x 2y xy (C) 2x 3y 3 5xy (D) 2x 3y 3 5 xy 5 3x 4y 5 2x 3y MISCELLANEOUS TYPE QUESTION ANSWER KEY EXERCISE-3 Fill in the Blanks 1. 12 : 31 2 . 1064 3 . [12, ] 4 . 2/8 5 . a = 3 d = 4 6 . 2 7. ± 2 Match the Column 1. (A) (q), (B) (s), (C) (p), (D) (r) 2. (A) (q), (B) (p), (C) (p), (D) (r) Assertion & Reason 1. C 2. A 3. C 4. C 5. A 6. B Comprehension Based Questions Comprehension # 1 : 1 . C 2. C 3. B 4. A 5. D Comprehension # 2 : 1 . C 2. C 3. B 4. B 24 E
EXERCISE - 04 [A] JEE-Mathematics CONCEPTUAL SUBJECTIVE EXERCISE 1 . Given that ax = by = cz = du & a , b , c , d are in GP, show that x , y , z , u are in HP . 2 . There are n AM’s between 1 & 31 such that 7th mean : (n 1)th mean = 5 : 9, then find the value of n. 3 . Find the sum of the series , 7 + 77 + 777 + ..... to n terms. qr 4 . If the pth, qth & rth terms of an AP are in GP. Show that the common ratio of the GP is p q . 5 . Express the recurring decimal 0.1 576 as a rational number using concept of infinite geometric series . 6 . If one AM ‘a’ & two GM’s p & q be inserted between any two given numbers then show that p3+ q3 = 2 apq. 7 . Find three numbers a , b , c between 2 & 18 which satisfy following conditions : (i) their sum is 25 (ii) the numbers 2, a, b are consecutive terms of an AP & (iii) the numbers b , c , 18 are consecutive terms of a GP. 1 1 1 1 2 1 1 3 n n n 8. Find the sum of the first n terms of the series : 1 2 3 4 ......... 9 . Let a1, a2, a3 ...... an be an AP . Prove that : 1 1 1 .......... 1 2 1 1 1 .......... 1 a1 an a n 1 an 2 an a1 = a2 a3 a2 a3 a1 an a1 an 1 0 . The harmonic mean of two numbers is 4 . The arithmetic mean A & the geometric mean G satisfy the relation 2 A + G² = 27 . Find the two numbers . 1 1 . Prove that : (ab + xy)(ax + by) 4abxy where a, b, x, y R+ 1 2 . If a, b, c R+ & a + b + c = 1; then show that (1 – a)(1 – b)(1 – c) 8abc 1 3 . If a, b, c are sides of a scalene triangle then show that (a + b + c)3 > 27 (a + b – c)(b + c – a)(c + a – b) bc ac ab 1 4 . For positive number a, b, c show that a b c abc 1 5 . The odd positive numbers are written in the form of a triangle NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 1 35 7 9 11 13 15 17 19 ............................ .................................. find the sum of terms in nth row. CONCEPTUAL SUBJECTIVE EXERCISE ANSWER KEY EXERCISE-4(A) 2. 14 3 . S = (7/81)(10n+1 – 9n – 10) 5. 35/222 7 . a = 5, b = 8, c = 12 8. n2 1 0 . 6, 3 15. n3 E 25
JEE-Mathematics EXERCISE - 04 [B] BRAIN STORMING SUBJECTIVE EXERCISE 1 . In a A.P. & an H.P. have the same first term, the same last term & the same number of terms; prove that the product of the rth term from the beginning in one series & the rth term from the end in the other is independent of r. 2 . Sum the following series to n terms and to infinity : 11 1 n n 1 (a) 1.4 .7 4 .7.10 7.10.13 ......... (b) r (r + 1) (r + 2) (r + 3) (c) r 1 4 r2 1 r 1 nn 3 . Find the value of the sum rs 2r 3s where rs is zero if r s & rs is one if r = s. r 1s 1 ni j 4 . Find the sum 1 . i 1 j1 k 1 5 . If there be 'm' A.P’s beginning with unity whose common difference is 1 , 2 , 3 .... m. Show that the sum of their nth terms is (m/2) (mn m + n + 1). 6 . If a , a , a .... a are in H.P., then prove that a a + a a + ..... + a a = (n – 1) a a . 123 n 12 23 n-1 n 1n 7 . If a, b, c are in H.P., b, c, d are in G.P. & c, d, e are in A.P., then Show that e = ab²/(2a b)². 8 . The value of x + y + z is 15, if a , x , y , z , b are in A.P. while the value of ; (1/x)+(1/y)+(1/z) is 5/3 if a , x , y , z , b are in H.P. Find a & b. 9. Prove that the sum of the infinite series 1.3 3.5 5.7 7.9 .......... 23 . 2 22 23 24 1 0 . If a , b, c be in G.P. & log a, log c, log b be in A.P., then show that the common difference of the A.P. must c b a be 3/2. 1 2x 3 x2 1 1 . Find the sum to n terms : (a) x 1 (x 1) (x 2) (x 1) (x 2) (x 3) ....... (b) a1 a2 a3 ....... 1 a1 1 a1 1 a2 1 a1 1 a2 1 a3 1 2 . In a G.P., the ratio of the sum of the first eleven terms to the sum of the last eleven terms is 1/8 and the ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is 2 . Find the number of terms in the G.P. 1 3 . Prove that the number 4 4 4 ............4 8 8 8 ..............8 9 is a per fect square of the number n digits (n-1) digits NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 6 6 6 ..............6 7. (n-1) digits 1 4 . Find the nth term and the sum to 'n' terms of the series : (a) 1 + 5 + 13 + 29 + 61 + ...... (b) 6 + 13 + 22 + 33 + ....... a b c3 1 5 . If a, b, c are three positive real number then prove that : bc ac ab 2 abc 1 6 . If a, b, c are the sides of a triangle and s = 2 , then prove that 8(s – a)(s – b)(s – c) abc. BRAIN STORMING SUBJECTIVE EXERCISE ANSWER KEY EXERCISE-4(B) 2 . (a) 1 1 1 n(n 1)(n 2)(n 3)(n 4) n1 24 1)(3n , 24 (b) 5 (c) 2n 1 , 2 6(3n 4) 3 . 6 (6n 1) 4 . [n(n+1)(n+2)]/6 8 . a = 1, b = 9 or b = 1, a = 9 5 1 1 . (a) 1 xn (b) 1 1 (x 1)(x 2)...........(x n) (1 a1 )(1 a2 )...........(1 an ) 1 2 . n = 38 1 1 4 . (a) 2n+1 – 3; 2n+2 – 4 – 3n (b) n2 + 4n +1 ; n(n 1)(2n 13) n 6 26 E
EXERCISE - 05 [A] JEE-Mathematics JEE-[MAIN] : PREVIOUS YEAR QUESTIONS 1 . If 1, log3 31x 2 , log3(4.3x – 1) are in A.P. then x equals. [AIEEE 2002] (1) log3 4 (2) 1 – log3 4 (3) 1 – log4 3 (4) log4 3 2 . Sum of infinite number of terms in G.P. is 20 and sum of their square is 100. The common ratio of G.P. is- [AIEEE 2002] (1) 5 (2) 3/5 (3) 8/5 (4) 1/5 3 . Fifth term of a G.P. is 2, then the product of its 9 terms is- [AIEEE 2002] (1) 256 (2) 512 (3) 1024 (4) None of these [AIEEE 2002] 4 . The sum of the series 13 – 23 + 33 – ..... + 93 = (1) 300 (2) 125 (3) 425 (4) 0 5 . Let Tr be the rth term of an A.P. whose first term is a and common difference is d. If for some positive 11 [AIEEE 2004] integers m,n, m n , Tm = n and Tn = m , then a – d equals (1) 0 (2) 1 1 11 (3) (4) mn mn 6 . If AM and GM of two roots of a quadratic equation are 9 and 4 respectively, then this quadratic equation is- [AIEEE 2004] (1) x2 – 18x + 16 = 0 (2) x2 + 18x –16 = 0 (3) x2 + 18x + 16 = 0 (4) x2 – 18x – 16 = 0 log an log an1 log an2 7 . If a1 , a2, a3 , ...... an , ..... are in G.P. then the value of the determinant log an3 log an4 log an5 , is- log an6 log an7 log an8 (1) 0 (2) 1 (3) 2 [AIEEE 04, 05] (4) –2 an , y = bn , z = 8 . If x = Cn where a, b, c are i n A.P. and |a| < 1, |b| < 1, n 0 n 0 n 0 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 |c| < 1 then x, y, z are in- [AIEEE 2005] (1) HP (2) Arithmetic - Geometric Progression (3) AP (4) GP 9. Let a1, a2, a3,....... be terms of an A.P. If a1 a2 .... ap p2 , p q then a6 equals -[AIEEE-2006] a1 a2 ...... aq q2 a21 2 11 41 7 (1) 7 (2) (3) (4) 41 11 2 1 0 . If a1, a2 ,....., an are in H.P., then the expression a1a2 + a2a3 + .......+ an–1an is equal to-[AIEEE-2006] (1) na1an (2) (n – 1)a1an (3) n(a1 – an) (4) (n – 1)(a1 – an) 1 1 . In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals- [AIEEE-2007] 1 (2) 5 1 5 1) 1 (1) 5 (3) ( (4) 2 (1 5 ) 2 2 E 27
JEE-Mathematics 1 2 . The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is [AIEEE 2008] (1) –4 (2) –12 (3) 12 (4) 4 13. The sum to infinity of the series 1 2 6 10 14 .... is :- [AIEEE-2009] 3 32 33 34 (1) 4 (2) 6 (3) 2 (4) 3 1 4 . A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = ... = a10 = 150 and a10, a11, .... are in an AP with common difference –2, then the time taken by him to count all notes is :- [AIEEE-2010] (1) 24 minutes (2) 34 minutes (3) 125 minutes (4) 135 minutes 1 5 . A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after :- [AIEEE-2011] (1) 20 months (2) 21 months (3) 18 months (4) 19 months 100 100 1 6 . Let an be the nth term of an A.P. If a2r and a2r1 , then the common difference of the A.P. is: r1 r1 [AIEEE-2011] (2) – (4) – (1) 200 (3) 100 1 7 . Statement–1 : The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + .... + (361 + 380 + 400) is 8000. n [AIEEE-2012] Statement–2 : k3 (k 1)3 n3 , for any natural number n. k 1 (1) Statement–1 is true, Statement–2 is false. (2) Statement–1 is false, Statement–2 is true. NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 (3) Statement–1 is true, Statement–2 is true ; Statement–2 is a correct explanation for Statement–1. (4) Statement–1 is true, Statement–2 is true ; Statement–2 is not a correct explanation for Statement–1. 1 8 . If 100 t ime s the 100th term of an A.P. w ith non-zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is : [AIEEE-2012] (1) zero (2) –150 (3) 150 times its 50th term (4) 150 1 9 . The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, ......, is : [JEE-MAIN 2013] (1) 7 (179 1020 ) (2) 7 (99 10 20 ) (3) 7 (179 10 20 ) (4) 7 (99 10 20 ) 81 9 81 9 PREVIOUS YEARS QUESTIONS ANSWER KEY EXERCISE-5 [A] 1. 2 2. 2 3. 2 4. 3 5. 1 6. 1 7. 1 8. 1 9. 2 10. 2 11. 3 12. 2 13. 4 14. 2 15. 2 16. 3 17. 3 18. 1 19. 3 E 28
EXERCISE - 05 [B] JEE-Mathematics JEE-[ADVANCED] : PREVIOUS YEAR QUESTIONS 1 . ( a ) Consider an infinite geometric series with first term ‘a’ and common ratio r. If the sum is 4 and the second term is 3/4, then - [JEE 2000, Screening, 1+1M out of 35] (A) a 7 , r 3 3 31 1 4 7 (B) a = 2, r (C) a , r (D) a = 3, r 8 22 4 ( b ) If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation - (A) 0 M 1 (B) 1 M 2 (C) 2 M 3 (D) 3 M 4 ( c ) The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an i n teg e r. [JEE 2000, Mains, 4M out of 100] 2 . ( a ) Let , be the roots of x2 - x + p = 0 and , be the roots of x2 - 4x + q = 0. If , , , are in G.P., then the integer values of p and q respectively, are - [JEE 2001 Screening 1+1+1M out of 35] (A) –2, –32 (B) –2, 3 (C) –6, 3 (D) –6, –32 ( b ) If the sum of the first 2n terms of the A.P. 2, 5, 8 ............. is equal to the sum of the first n terms of the A.P. 57, 59, 61, .............. then n equals - (A) 10 (B) 12 (C) 11 (D) 13 ( c ) Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are (A) not in A.P./G.P./H.P. (B) in A.P. (C) in G.P. (D) in H.P. ( d ) Let a , a ........... be positive real numbers in G.P.. For each n, let A , G , H , be respectively, the 12 nnn arithmetic mean, geometric mean and harmonic mean of a , a , a ,........... a . Find an expression for the 123 n G.M. of G , G ,.............G in terms of A , A , .................A , H , H ,..........H [JEE 2001 (Mains) ; 5M] 12 n 12 n 12 n NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 3 3 . ( a ) Suppose a, b, c are in A.P. and a2, b2, c2 are in G.P. If a < b < c and a b c , then the value of a is - 2 1 1 11 11 (A) (B) (C) 2 3 (D) 2 2 22 23 [JEE 2002 ( Screening), 3M] ( b ) Let a, b be positive real numbers. If a, A , A , b are in A.P. ; a, G , G , b are in G.P. and a, H , H , b are 12 12 12 in H.P., show that G1G2 A1 A 2 (2a b)(a 2b) . [JEE 2002, Mains, 5M out of 60] H1H2 H1 H2 9ab c 4 . If a, b, c are in A.P., a2, b2, c2 are in H.P., then prove that either a = b = c or a, b, form a G.P. 2 [JEE 2003, Mains, 4M out of 60] 5 . If a, b, c are positive real numbers, then prove that [(1 + a)(1 + b)(1 + c)]7 > 77 a4 b4 c4. [JEE 2004, 4M] 6 . The first term of an infinite geometric progression is x and its sum is 5. Then - [JEE 2004] (A) 0 x 10 (B) 0 x 10 (C) –10 < x < 0 (D) x > 10 E 29
JEE-Mathematics FGH KIJ7 . n1 (2n+1 – n – 2) where n > 1, and the runs scored in the kth If total number of runs scored in n matches is 4 match are given by k. 2n+1–k, where 1 k n. Find n. [JEE-05, Mains-2M out of 60] 8 . In quadratic equation ax2 + bx + c = 0, if are roots of equation, = b2 – 4ac and +, 2 +2, 3 + 3 are in G.P. then [JEE 2005 (screening)] (A) 0 (B) b = 0 (C) c = 0 (D) = 0 HGF JKI GFH JKI HGF IKJ9 .2 3 3 3 3 3 n If a n 4 4 4 4 .... (1) n 1 and bn = 1 – an then find the minimum natural number n0 such that b > a n n [JEE 2006, 6M out of 184] n n 0 Comprehension Based Question Comprehension # 1 Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let T = V – V – 2 and Q = T – T for r = 1,2,........ r r + 1 r r r + 1 r 1 0 . The sum V + V + ... + V is : [JEE 2007, 4M] 12 n 1 1 (A) n(n + 1) (3n2 – n + 1) (B) n(n + 1) (3n2 + n + 2) 12 12 1 1 (C) n(2n2 – n + 1) (D) (2n3 – 2n + 3) 2 3 1 1 . T is always : [JEE 2007, 4M] r (A) an odd number (B) an even number (C) a prime number (D) a composite number 1 2 . Which one of the following is a correct statement ? [JEE 2007, 4M] (A) Q1,Q2,Q3,...are in A.P. with common difference 5 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 (B) Q ,Q ,Q ,...are in A.P. with common difference 6 123 (C) Q ,Q ,Q ,...are in A.P. with common difference 11 123 (D) Q = Q = Q = ... 1 2 3 Comprehension # 2 Let A , G , H denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive 111 numbers. For n 2, let A and H has arithmetic, geometric and harmonic means as A , G , H re spectively : n–1 n–1 n n n 1 3 . Which one of the following statements is correct ? [JEE 2007, 4M] (A) G > G > G > ... 123 (B) G < G < G < ... 123 (C) G1 = G2 = G3 = ... (D) G < G < G < ... and G > G > G > ... 123 456 1 4 . Which one of the following statements is correct ? [JEE 2007, 4M] (A) A > A > A > ... (B) A < A < A < ... 123 123 (C) A > A > A > ... and A < A < A < ... (D) A < A < A < ... and A > A > A > ... 135 246 135 246 1 5 . Which one of the following statements is correct ? [JEE 2007, 4M] (A) H > H > H > ... (B) H < H < H < ... 123 123 (C) H > H > H > ... and H < H < H > ... (D) H < H < H < ... and H > H > H > ... 135 246 135 2 4 6 E 30
JEE-Mathematics 1 6 . Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1 + a2, b3 = b2 + a3 and b4 = b3 + a4. Statement -I : The numbers b1, b2, b3, b4 are neither in A.P. nor in G.P. and Statement -II : The number s b1, b2, b3, b4 are in H.P. [JEE 2008, 3M, –1M] (A) Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I. (B) Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I. (C) Statement-I is true, Statement-II is false. (D) Statement-I is false, Statement-II is true. 1 7 . If the sum of first n terms of an A.P. is cn2, then the sum of squares of these n terms is [JEE 2009, 3M, –1M] n 4n2 1 c2 n 4n2 1 c2 n 4n2 1 c2 n 4n2 1 c2 (A) (B) (C) (D) 6 3 3 6 18. Let S , k = 1,2,.......,100, denote the sum of the infinite geometric series whose first term is k 1 k k ! and the 1 1002 100 k 100! k 1 common ratio is . Then the value of k2 3k 1 Sk is [JEE 10, 3M] 1 9 . Let a1,a2,a3,.........,a11 be real numbers satisfying a1 = 15, 27 – 2a2 > 0 and ak = 2ak–1 – ak–2 for k = 3,4,........,11. If a12 a 2 .... a 2 90 , then the value of a1 a2 ... a11 is equal to [JEE 10, 3M] 2 11 11 11 2 0 . The minimum value of the sum of real numbers a–5, a–4, 3a–3, 1, a8 and a10 with a > 0 is [JEE 2011,4] p 2 1 . Sp ai ,1 p 100 . For any integer n Let a ,a ,a ,.........,a be an arithmetic progression with a = 3 and 123 100 1 i 1 with 1 < n < 20, let m = 5n. If Sm does not depend on n, then a is [JEE 2011, 4] 2 Sn 2 2 . Let a , a , a , ..... be in harmonic progression with a = 5 and a = 25. The least positive integer n for which 123 1 20 a < 0 is [JEE 2012, 3 (–1)] n (A) 22 (B) 23 (C) 24 (D) 25 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#07\\Eng\\01(b)-Sequence-series (Exercise).p65 4 n k ( k 1) 2 3 . Let S n (1) 2 k2 . Then Sn can take value(s) [JEE-Advanced 2013, 4, (–1)] k 1 (A) 1056 (B) 1088 (C) 1120 (D) 1332 2 4 . A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller to the numbers on the removed cards is k, then k – 20 = [JEE-Advanced 2013, 4, (–1)] PREVIOUS YEARS QUESTIONS ANSWER KEY EXERCISE-5 [B] 1 . (a) D, (b) A 1 2 . (a) A, (b) C, (c) D, (d) A1, A 2 ,..........A n H1, H2,.........Hn 2n 3 . (a) D 6. B 7 . n = 7 8. C 9. 6 10. B 11. D 12. B 13. C 14. A 17. C 18. 3 19. 0 20. 8 2 1 . 9 or 3 15. B 16. C 24. 5 22. D 23. A,D E 31
JEE-Mathematics CHECK YOUR GRASP EXERCISE - 01 SELECT THE CORRECT ALTERNATIVE (ONLY ONE CORRECT ANSWER) 1 . If ABCDEF is a regular hexagon and if AB AC AD AE AF AD , then is - (A) 0 (B) 1 (C) 2 (D) 3 2 . If a b is along the angle bisector of a & b then - (A) a & b are perpendicular (B) a b (C) angle between a & b is 60° (D) a b 3 . Given the points A (2, 3, 4) , B (3, 2, 5) , C (1, 1, 2) & D (3, 2, 4) . The projection of the vector AB on the vector CD is - 22 (B) 21 (C) 47 (D) –47 (A) 7 4 3 4. The vectors 3ˆi 2ˆj 2kˆ and ˆi 2kˆ are the adjacent sides of a parallelogram ABCD then the AB BC angle between the diagonals is - 1 49 (C) co s 1 1 3 85 85 10 (A) cos1 (B) – cos1 2 2 (D) cos1 5 . The values of a, for which the points A, B, C with position vectors 2ˆi – ˆj + kˆ, ˆi – 3ˆj – 5kˆ and aˆi 3ˆj kˆ respectively are the vertices of a right angled triangle with are - C 2 (A) –2 and 1 (B) 2 and –1 (C) 2 and 1 (D) –2 and –1 a . a a . b a . c 6. If ˆi ˆj kˆ, b ˆi ˆj kˆ, ˆi 2ˆj kˆ , then the value of b.a b .b b.c a c c.a c.b c.c (A) 2 (B) 4 (C) 16 (D) 64 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 7 . The area of the triangle whose vertices are A (1, –1, 2) ; B (2, 1, –1) ; C (3, –1, 2) is - (A) 13 (B) 2 13 (C) 13 (D) none 8. Let ˆi ˆj & & is - a b 2 ˆi kˆ . The point of intersection of the lines rxa bxa rxb axb (A) ˆi ˆj kˆ (B) 3 ˆi ˆj kˆ (C) 3 ˆi ˆj kˆ (D) ˆi ˆj kˆ 9. If are non-coplanar vectors and is a real number then ( a for - a, b, c a b) 2 b c bc b (A) exactly two values of (B) exactly three values of (C) no value of (D) exactly one value of 1 0 . Volume of the tetrahedron whose vertices are represented by the position vectors , A (0, 1, 2) ; B (3, 0, 1) ; C (4, 3, 6) & D (2, 3, 2) is - (A) 3 (B) 6 (C) 36 (D) none 30 E
JEE-Mathematics 1 1 . The sine of angle formed by the lateral face ADC and plane of the base ABC of the tetrahedron ABCD where A (3, –2, 1) ; B (3, 1, 5); C (4, 0, 3) and D (1, 0, 0) is - 2 5 33 2 (A) 29 (B) 29 (C) 29 (D) 29 1 2 . Given the vertices A (2, 3, 1), B (4, 1, –2), C (6, 3, 7) & D (–5, –4, 8) of a tetrahedron. The length of the altitude drawn from the vertex D is - (A) 7 (B) 9 (C) 11 (D) none 13. Let and be non-zero vectors such that and are non-collinear & satisfies 1| . a,b c a b (a b) c 3 b|| c|a If is the angle between the vectors b and c then sin equals - 2 2 1 22 (A) 3 (B) (C) 3 (D) 3 3 14. The value of ˆi ( ˆi ) ˆj ( ˆj ) kˆ ( kˆ) is - r r r (A) r (B) 2 r (C) 3r (D) 4 r 1 5 . A, B, C, D be four points in a space and if, | AB CD BC AD CA BD| = (area of triangle ABC) then the value of is - (B) 2 (C) 1 (D) none of these (A) 4 1 6 . If the volume of the parallelopiped whose conterminous edges are represented by –12ˆi kˆ, 3ˆj – kˆ, 2ˆi ˆj – 15kˆ is 546, then equals- (A) 3 (B) 2 (C) –3 (D) –2 17. Let = 2 ˆi + 3 ˆj – kˆ and = ˆi – 2 ˆj + 3 kˆ . Then the value of for which the vector a b b , is- = ˆi + ˆj + (2 – 1) kˆ is parallel to the plane containing and c a (A) 1 (B) 0 (C) –1 (D) 2 18. If a + 5b = c and a – 7 b = 2 c , then- NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 (A) a and c are like but and c are unlike vectors b (B) a and b are unlike vectors and so also a and c (C) b and c are like but a and b are unlike vectors (D) a and c are unlike vectors and so also b and c 19. If a , are three non-coplanar and , q , are reciprocal vectors to a , and respectively, then b, c p r b c ( a + n ).( + m + , m, n are scalars) + mb c p q n r ) is equal to : (where 20. (A) 2 + m2 + n2 two (B) m + mn + n and (C) 0 represent the (D) none of these If are non collinear vectors a, b, c sides of a ABC satisfying x &y (a b)x (b c)y (c a )(x y ) 0 then ABC is - (A) an acute angle triangle (B) an obtuse angle triangle (C) a right angle triangle (D) a scalene triangle 21. If A , B and C are three non-coplanar vectors then ( A + B + C ).[( A + B ) × ( A + C )] equals - E (A) 0 (B) [ A B C ] (C) 2[ A B C ] (D) –[ A B C ] 31
JEE-Mathematics SELECT THE CORRECT ALTERNATIVES (ONE OR MORE THAN ONE CORRECT ANSWERS) 2 2 . ABCD is a parallelogram. E and F be the middle points of the sides AB and BC, then - (A) DE trisect AC (B) DF trisect AC (C) DE divide AC in ratio 2 : 3 (D) DF divide AC in ratio 3 : 2 2 3 . a , b , c are mutually perpendicular vectors of equal magnitude then angle between a b c and a is - F I1 GF JI(B) cos 1 1 FG JI(C) – cos 1 1 (D) tan1 2 HG JK(A) cos 1 3 H K3 H K3 24. If , where and are any three vectors such that then and (a b) c a (b c) a, b c a . b 0, b.c 0 a c are - (A) perpendicular (B) parallel (C) non collinear (D) linearly dependent bc e j2 5 . If a , b & c are non coplanar unit vectors such that a b c = 2 , then the angle between - 3 (C) & is 3 (D) & is (A) a & b is 4 (B) a & b is 4 a c 4 a c 4 2 6 . If a, b, c, d, e, f are position vectors of 6 points A, B, C, D, E & F respectively such that , then - 3a 4b 6c d 4e 3f x (A) AB is parallel to CD (B) line AB, CD and EF are concurrent (C) x is position vector of the point dividing CD in ratio 1 : 6 7 (D) A, B, C, D, E & F are coplanar 2 7 . Read the following statement carefully and identify the true statement - (a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two lines parallel to a plane are parallel. (d) Two lines perpendicular to a plane are parallel. NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 (e) Two lines either intersect or are parallel. (A) a & b (B) a & d (C) d & e (D) a 2 8 . The vector 1 (2ˆi 2ˆj kˆ) is - 3 (A) unit vector (B) makes an angle /3 with vector 2ˆi 4ˆj 3kˆ (C) parallel to the vector ˆi ˆj (1 / 2)kˆ (D) perpendicular to the vector 3ˆi 2ˆj 2kˆ 29. If a vector of magnitude 3 6 is collinear with the bisector of the angle between the vectors 7 i 4 j 4 k r a & 2 i j 2 k , then b r= (A) i 7 j 2 k (B) i 7 j 2 k 13ˆi ˆj 10kˆ (D) i 7 j 2 k (C) E 5 32
JEE-Mathematics 3 0 . A parallelopiped is formed by planes drawn through the points (1, 2, 3) and (9, 8, 5) parallel to the coordinate planes then which of the following is the length of an edge of this rectangular parallelopiped - (A) 2 (B) 4 (C) 6 (D) 8 3 1 . If A (a ) ; B (b) ; C (c ) and D (d ) are four points such that a = –2 ˆi + 4 ˆj + 3 kˆ ; b = 2 ˆi – 8 ˆj ; c = ˆi – 3 ˆj + 5 kˆ ; d = 4 ˆi + ˆj – 7 kˆ , d is the shortest distance between the lines AB and CD, then (A) d = 0, hence AB and CD intersect (B) d = [ABCDBD] AB CD 23 (C) AB and CD are skew lines and d = 13 (D) d = [ABCDAC] AB CD NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 CHECK YOUR GRASP ANSWER KEY EXERCISE-1 Que. 1 2 3 456 7 8 9 10 Ans. D B C DCC A C CB Que. 11 12 13 14 15 16 17 18 19 20 Ans. B C D BAC B A AA Que. 21 22 23 24 25 26 27 28 29 30 Ans. D A,B B,D B,D A,D B,C B,D A,C,D A,C A,C,D Que. 31 Ans. B,C,D E 33
JEE-Mathematics BRAIN TEASERS EXERCISE - 02 SELECT THE CORRECT ALTERNATIVES (ONE OR MORE THAN ONE CORRECT ANSWERS) 1. Let 2ˆi ˆj kˆ, ˆi 2ˆj kˆ and ˆi ˆj 2kˆ be three vectors. A vector in the plane of and whose a b c b c projection on is magnitude 2/3 is - a (A) 2ˆi 3ˆj 3kˆ (B) 2ˆi 3ˆj 3kˆ (C) 2ˆi 5ˆj kˆ (D) 2ˆi ˆj 5kˆ 2 . Let a, b, c are three non-coplanar vectors such that r1 a b c , r2 b c a , r3 c a b , 3b . If r 2a 4c r 1 r1 2 r2 3 r3 , then - (A) 1 7 (B) 1 3 3 (C) 1 2 3 4 (D) 3 2 2 3. of a triangle ABC, a point M such that 1 . A point N is taken on the side such Taken on side AC AM AC CB 3 that BN CB then, for the point of intersection X of AB & MN which of the following holds good ? (A) 1 (B) 1 (C) 3 XB AB AX AB XN MN (D) XM 3XN 2 3 4 4 . Vector A has components A , A , A along the three axes. If the co-ordinates system is rotated by 90° 123 about z-axis, then the new components along the axes are - (A) A, A,A (B) A, A, A (C) A, A, A (D) A, A1, A 1 2 3 1 2 3 2 1 3 2 3 5 . Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation p ((x q ) p) q ((x r ) q ) r ((x p) r ) 0 . Then x is given by - (A) 1 2 (B) 1 (C) 1 (D) 1 (2 (p q r) (p q r) (p q r) p q r) 2 3 2 3 6 . A vector which makes equal angles with the vectors 1 (ˆi 2ˆj 2kˆ), 1 (4ˆi 3kˆ), ˆj is - 35 (A) 5ˆi ˆj 5kˆ (B) 5ˆi ˆj 5kˆ (C) 5ˆi ˆj 5kˆ (D) 5ˆi ˆj 5kˆ · a d a b cd 7. to The triple product simplifies - (D) none NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 (A) (b. d)[d a c] (B) (b.c)[a b d] (C) (b.a)[a b d] 8. If the vectors are non-coplanar and ,m,n are distinct real numbers, then a, b, c (a m ) (c ) = 0 implies - b nc) (b mc na ma nb (A) m + mn+ n = 0 (B) + m + n = 0 (C) 2 + m2+ n2 = 0 (D) 3 + m3 + n3 = 0 9. If unit vectors ˆi & ˆj are at right angles to each other and 3 ˆi 4 ˆj , 5 ˆi , and p q 4r p q 2 s p q , then - (A) r k s = r k s for all real k (B) r is perpendicular to s (C) r s is perpendicular to r s (D) r s p q 1 0 . The three vectors ˆi ˆj, ˆj kˆ, kˆ ˆi taken two at a time form three planes, The three unit vectors drawn perpendicular to these planes form a parallelopiped of volume : 1 (B) 4 33 4 (A) 3 (C) (D) 3 3 4 E 34
JEE-Mathematics 1 1 . If a, b, c are different real numbers and a ˆi b ˆj c kˆ ; b ˆi c ˆj a kˆ & c ˆi a ˆj b kˆ are position vectors of three non-collinear points A, B & C then - abc 3 (A) centroid of triangle ABC is ˆi ˆj kˆ (B) ˆi ˆj kˆ is equally inclined to the three vectors (C) perpendicular from the origin to the plane of triangle ABC meet at centroid (D) triangle ABC is an equilateral triangle. 1 2 . Identify the statement (s) which is/are incorrect ? a a2 a a b ab (A) (B) If a, b, c are non coplanar vectors and v.a v.b v.c 0 then must be a null vector v a b c d a b c d =0 (C) If and lie in a plane normal to the plane containing the vectors and then (D) If a, b, c and a ', b ', c ' are reciprocal system of vectors then a . b ' b.c ' c .a ' 3 13. Given parallelogram OACB. a, b & c respectively. a The lengths of the vectors OA , OB & AB are The scalar product of the vectors OC & OB is - a2 3b2 c2 3a2 b2 c2 3a2 b2 c2 a2 3b2 c2 (A) (B) (C) (D) 2 2 2 2 14. Consider ABC with A B and C = If b . = b.b +a.c ; = 3; = 4, then the (a) , (b) (c) . a c b–a c–b angle between the medians AM and BD is - (A) – cos–1 5 1 (B) – cos–1 1 1 5 (C) cos–1 5 1 (D) cos–1 1 1 5 13 3 13 3 1 5 . If the non zero vectors a & b are perpendicular to each other then the solution of the equation, r a b is - 1 1 r xa ab r xb ab (A) (B) (C) r x a b (D) none of these a.a b.b 1 6 . a , b , c be three non coplanar vectors and r be any arbitrary vector, then NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 ( a × b ) × ( r × c ) + ( b × c ) × ( r × a ) + ( c × a ) × ( r × b ) is equal to- (D) none of these (A) [ a b c ] r (B) 2 [ a b c ] r (C) 3[ a b c ] r 1 7 . a and b are mutually perpendicular unit vectors. r is a vector satisfying r . a = 0, r . b = 1 and [ r a b ] = 1, then r is - (A) a + ( a × b ) (B) b + ( a × b ) (C) a + b ( a × b ) (D) a – b + ( a × b )’ BRAIN TEASERS ANSWER KEY EXERCISE - 02 Que. 1 2 3 456 7 8 9 10 Ans. A,C B,C B,C C C B,C A B A,B,C D Que. 11 12 13 14 15 16 17 Ans. A,B,C,D A,C,D D AAB B E 35
JEE-Mathematics MISCELLANEOUS TYPE QUESTIONS EXERCISE - 03 TRUE / FALSE 1 . There exists infinitely many vectors of given magnitude which are perpendicular to a given plane. 2 . There exists infinitely many vectors of given magnitude which are perpendicular to a given line. 3. Given that a c = =0 & = then b b c d r a b r .a r . b r.c r.d 0 . 4. The point (1, 2, 3) lies on the line (2ˆi 3ˆj 4 kˆ) (ˆi ˆj kˆ) . r 5 . The area of a parallelogram whose two adjacent edges are two diagonals of a given parallelogram is double the area of given parallelogram. 6. If are three non-coplanar vectors, then [JEE 1985] A, B, C A.(B C) B.(A C ) 0 (C A ).B C.(A B) 7 . [2a 3b 3a 4b 4a 5b] 0 MATCH THE COLUMN Following question contains statements given in two columns, which have to be matched. The statements in Column-I are labelled as A, B, C and D while the statements in Column-II are labelled as p, q, r and s. Any given statement in Column-I can have correct matching with ONE statement in Column-II. 1 . Column-I Column-II (A) ABC is a triangle. If P is a point inside the ABC (p) centroid NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 (q) orthocentre such that areas of the triangle PBC, PCA and PAB, (r) incentre all are equal, then with respect to the ABC, P is its (s) circumcentre (B) If a, b, c are the position vectors of the three non-collinear points A, B and C respectively such that the vector V PA PB PC is a null vector, then with respect to the ABC, P is its (C) If P is a point inside the ABC such that the vector R (BC)(PA ) (CA )(PB) (AB)(PC) is a null vector, then with respect to the ABC, P is its (D) If P is a point in the plane of the triangle ABC such that the scalar product PA .CB and PB.AC vanishes, then with respect to the ABC, P is its 2 . Let a, b, c be vectors then - Column-I Column-II [a a] |2 b] (A) + b , b + c, c + (p) |b [a c (B) [( a × b )× (a × c )]. b (q) ( a . b )[ a b c] (C) [a × b, b × c, c × a ] (r) 2[ a b (s) b c] [a (D) b . {( a × b ) × ( c × b )} c ]2 36 E
JEE-Mathematics 3 . Column-I Column-II (A) Let ˆi ˆj & 2ˆi kˆ . If the point of intersection of the lines (p) 0 a b ra ba & is 'P', then 2 (OP) (where O is the origin) is rb ab (B) If ˆi 2ˆj 3kˆ, b 2ˆi ˆj kˆ and 3ˆi 2ˆj kˆ and a (b c) is equal to (q) 5 a c , then x + y + z is equal to xa yb zc (C) The number of values of x for which the angle between the vectors (r) 7 x9ˆi (x3 1)ˆj 2kˆ & (x3 1)ˆi xˆj 1 kˆ is obtuse a b 2 (D) Let P 2x – y + z = 7 & P x + y + z = 2. If P be a point that lies on (s) 1 1 1 2 P1, P2 and XOY plane, Q be the point that lies on P1, P2 and YOZ plane and R be the point that lies on P1, P2 & XOZ plane, then [Area of triangle PQR] (where [.] is greatest integer function) ASSERTION & REASON These questions contains, Statement I (assertion) and Statement II (reason). (A) Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I. (B) Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I (C) Statement-I is true, Statement-II is false (D) Statement-I is false, Statement-II is true 1 . Statement-I : The volume of a parallelopiped whose co-terminous edges are the three face diagonals of a given parallelopiped is double the volume of given parallelopied. Because Statement-II : For any vectors we have a, b, c [a b b c c a] 2[a b c] (A) A (B) B (C) C (D) D 2 . Statement-I : Let A(a) & B(b) be two points in space. Let P(r ) be a variable point which moves in space NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 such that PA.PB 0 , such a variable point traces a three-dimensional figure whose volume is given by a b2 2a.b .ab 2 6 Because Statement-II : Diameter of sphere subtends acute angle at any point inside the sphere & its volume is given by 4 r3 , where 'r' is the radius of sphere. 3 (A) A (B) B (C) C (D) D 3. Statement-I : Let a, b, c be there non-coplanar vectors. Let p1 be perpendicular to plane of a & b, p2 perpendicular to plane b & c, p3 perpendicular to plane of c & a then p 1, p2 & p3 are non-coplanar. Because Statement-II : c]2 [a b b c c a] [a b (A) A (B) B (C) C (D) D E 37
JEE-Mathematics 4. Statement-I : If be two lines such that & where , µ t r a b & r p µd b td a p sb & s be non-zero scalars then the two lines have unique point of intersection. Because Statement-II : Two non-parallel coplanar lines have unique point of intersection. (A) A (B) B (C) C (D) D b a, b and c 5. Statement-I : If ˆi, ˆj and ˆi ˆj , then a and b are linearly independent but are linearly a c dependent. Because a and b a, b and c Statement-II : If are linearly dependent and is any vector, then are linearly dependent. c (A) A (B) B (C) C (D) D COMPREHENSION BASED QUESTIONS Comprehension # 1 : G F' Three forces ƒ1, ƒ2 & ƒ 3 of magnitude 2, 4 and 6 units respectively act O F along three face diagonals of a cube as shown in figure. Let P1 be a ƒ1 D' parallelopiped whose three co-terminus edges be three vectors ƒ1, ƒ2 & ƒ 3 . Let the joining of mid-points of each pair of opposite edges of parallelopiped D E ƒ2 C ƒ3 P1 meet in point X. A B B' On the basis of above information, answer the following questions : 1 . The magnitude of the resultant of the three forces is - (A) 5 (B) 10 (C) 15 (D) none of these (D) 50 2 2 . The volume of the parallelopiped P1 is - (D) 2.5 (A) 48 2 (B) 96 2 (C) 24 2 3 . (OX ) is equal to - (A) 5 (B) 1.5 (C) 2 Comprehension # 2 : Consider three vectors ˆi ˆj kˆ , 2ˆi 4ˆj kˆ and r ˆi ˆj 3kˆ and let be a unit vector. NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 p q s On the basis of above information, answer the following questions : 1. , and are - p q r (A) linearly dependent (B) can form the sides of a possible triangle (C) such that the vector q r is orthogonal to p (D) such that each one of these can be expressed as a linear combination of the other two 2. If = (u+v+w) equals to - p q r up vq wr , then (A) 8 (B) 2 (C) –2 (D) 4 3. The magnitude of the vector p.s r + q.s + p is - q r p r.s q (A) 4 (B) 8 (C) –2 (D) 2 38 E
JEE-Mathematics Comprehension # 3 : Three points A(1, 1, 4), B(0, 0, 5) & C(2, –1, 0) forms a plane. P is a point lying on the line ˆi 3ˆj (ˆi ˆj kˆ) . r The perpendicular distance of point P from plane ABC is 2 6 . 3 'Q' is a point inside the tetrahedron PABC such that resultant of vectors AQ , BQ , CQ & PQ is a null vector. On the basis of above information, answer the following questions : 1 . Co-ordinates of point 'P' is - (A) (2, 4, 1) (B) (1, 3, 0) (C) (4, 6, 3) (D) (7, 9, 6) 2 . Volume of tetrahedron PABC is - 4 81 2 81 81 6 81 (A) (B) (C) (D) 9 9 9 9 3 . Co-ordinates of point 'Q' is - (A) 5 , 1, 5 (B) (5, 1, 5) (C) 5 , 1, 5 (D) 5 , 5, 5 4 2 2 4 4 2 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 MISCELLANEOUS TYPE QUESTION ANSWER KEY EXERCISE -3 True / False 5. T 6. T 7. T 1. F 2. T 3. F 4. T Match the Column 2. (A) (r), (B) (q), (C) (s), (D) (p) 1. (A) (p), (B) (p), (C) (r), (D) (q) 3. (A) (s), (B) (r), (C) (p), (D) (p) Assertion & Reason 1. A 2. C 3. A 4. D 5. B Comprehension Based Questions Comprehension # 1 : 1. B 2 . C 3 . A Comprehension # 2 : 1. C 2 . B 3 . A Comprehension # 3 : 1. A 2 . B 3 . A E 39
JEE-Mathematics CONCEPTUAL SUBJECTIVE EXERCISE EXERCISE - 04 [A] 1 . The sides of parallelogram are 2ˆi 4ˆj 5kˆ and ˆi 2ˆj 3kˆ . Find the unit vectors, parallel to their diagonals. 2 . If G is the centroid of a triangle ABC, then prove that GA GB GC 0 3 . Find out whether the following pairs of lines are parallel, non-parallel & intersecting, or non-parallel & non-intersecting. ˆi ˆj 2kˆ (3ˆi 2ˆj 4kˆ) ˆi ˆj 3kˆ (ˆi ˆj kˆ) r1 r1 (a) 2ˆi ˆj 3kˆ µ(6ˆi 4ˆj 8kˆ) (b) r2 2ˆi 4ˆj 6kˆ µ(2ˆi ˆj 3kˆ) r2 ˆi kˆ (ˆi 3ˆj 4kˆ) r1 (c) r2 2ˆi 3ˆj µ(4ˆi ˆj kˆ) 4 . (a) Show that the points a 2b 3c;2a 3b 4c & 7b 10c are collinear. (b) Prove that the points A = (1, 2, 3), B(3, 4, 7), C(–3, –2, –5) are collinear & find the ratio in which B divides AC. 5 . Points X & Y are taken on the sides QR & RS, respectively of a parallelogram PQRS, so that QX 4XR & The line XY cuts the line PR at Z. Prove that 21 . RY 4YS . PZ 25 PR 6 . Using vectors prove that the altitudes of a triangle are concurrent. 7 . Using vectors show that the mid-point of the hypotenuse of a right angled triangle is equidistant from its vertices. 8 . Using vectors show that a parallelogram whose diagonals are equal is a rectangle. 9 . Using vectors show that a quadrilateral whose diagonals bisect each other at right angles is a rhombus. 1 0 . Two medians of a triangle are equal, then using vector show that the triangle is isosceles. 1 1 . 'O' is the origin of vectors and A is a fixed point on the circle of radius 'a' with centre O. The vector OA is denoted by variable point 'P' lies on the tangent at A & Show that | a |2 . Hence if P (x, y) a. A OP r . a.r & A (x , y) deduce the equation of tangent at A to this circle. 1 1 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 12. be a vector on rectangular coordinate system with sloping angle 60°. Suppose that ˆi is geometric Let u u mean of and 2ˆi where ˆi is the unit vector along x-axis then has the value equal to a b where u u u a, b N . Find the value (a + b)3 + (a – b)3. 1 3 . a, b,c and d are the position vectors of the points A (x, y, z); B (y, –2z, 3x) ; C (2z, 3x, –y) and D (1, –1, 2) respectively. If 3 ; ^ ^ ; a ^ and ^ ˆj is obtuse, then find x, y, z. a 2 a b a c d a 2 1 4 . If r and s are nonzero constant vectors and the scalar b is chosen such that r bs is minimum, then show that the value of 2 b 2 is equal to 2 . bs r s r 1 5 . (a) Find a unit vector â which makes an angle (/4) with axis of z & is such that â+i+j is a unit vector. 2 2 aa2 ab (b) Prove that b a b b2 40 E
JEE-Mathematics 16. Given four non zero vectors a , and a , and are coplanar but not collinear pair by b, c d . The vectors b c ^ ^ ^ ^ = 3, pair and vectord a , b is not coplanar with vectors and and = = , = then c ab bc da db ^ prove that = cos–1(cos – cos) dc 1 7 . Given three points on the xy plane O(0, 0), A(1, 0) and B(–1, 0). Point P is moving on the plane satisfying the condition . + 3 OA . = 0. If the maximum and minimum values of are M and m PA PB OB PA PB respectively then find the values of M2 + m2. 18. If O is origin of reference, point A( a ); D( a + ); F( + a ); G( a + + ) where B( b ); C( c ); b E( b + c ); c b c a = a1 ˆi + a2 ˆj + a3 kˆ ; = b1 ˆi + b2 ˆj + b3 kˆ and = c1 ˆi + c2 ˆj + c3 kˆ , then prove that these points are b c vertices of a cube having length of its edge equal to unity provided the matrix. a1 a2 a3 b1 b2 b 3 is orthogonal. Also find the length XY such that X is the point of intersection of CM and GP; c1 c2 c3 Y is the point of intersection of OQ and DN where P, Q, M, N are respectively the midpoint of sides CF, BD, GF and OB 1 9 . Let A = 2 i + k , B = i + j + k , and C = 4 i – 3 j + 7 k Determine a vector R , satisfying R × B = C × B and R · A = 0 2 0 . If a,b,c,d are position vectors of the vertices of a cyclic quadrilateral ABCD prove that : a x b b x d d x a b x c c x d d x b (b a) . (d a) (b c) . (d c) 0 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 21. Let a = 3 i – j and 1 ˆi + 3 ˆj and = a = – p a = 2 b x + (q2 – 3) b , y + q b . If x y , then express p 2 as a function of q, say p = f(q), (p 0 and q 0) and find the intervals of monotonicity of f(q). 22. If ˆi ˆj kˆ, ˆi 2ˆj 2kˆ & ˆi 2ˆj kˆ , find a unit vectors normal to the vectors and . a b c ab b c [REE 2000] 2 3 . . a Prove that a b = b a b 2 4 . If a, b, c are non-coplanar vectors and d is a unit vector, then find the value of, | (a . d)(b c) (b. d)(c a) (c . d)(a b) | independent of d . [REE 99] 25. Find the vector which is perpendicular to ˆi 2ˆj 5 kˆ and 2ˆi 3ˆj kˆ and r.(2ˆi ˆj kˆ) 8 0 . r a b E 41
JEE-Mathematics 2 6 . Two vertices of a triangle are at ˆi 3ˆj and 2ˆi 5ˆj and its orthocentre is at ˆi 2ˆj . Find the position vector of third vertex. [REE 2001] 2 7 . Find the point R in which the line AB cuts the plane CDE where = i + 2j + k , = 4j + 4k , d = 2i 2j + 2k & a b = 2i + j + 2k c e = 4i + j + 2k. 28. Solve for : + ( . ) = and are non zero non collinear and 0 x x× a x b a c , where a c a.b CONCEPTUAL SUBJECTIVE EXERCISE ANSWER KEY EXERCISE - 4(A) NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 1 . 3 ˆi 6 ˆj 2 kˆ, 1 ˆi 2 ˆj 8 kˆ 3. (a) parallel (b) the lines intersect at the point p.v. 2 ˆi 2 ˆj 7 7 7 69 69 69 (c) lines are skew 4 . (b) Externally in ratio 1 : 3 1 1 . xx1 + yy1 = a2 12. 28 1 3 . x = 2, y = –2, z = –2 1 5 . (a) 1 i 1 j 1 k 1 7 . 3 4 18. 11 19. – ˆi – 8 ˆj + 2 kˆ 22 2 3 21. p = q(q2 3) ; decreasing in q (–1, 1), q 0 22. ˆi 2 4 . 25. 13ˆi 11ˆj 7kˆ = 3i + 3k [a b c] r 4 R= r 1 ( a.c a.b b c) a (b a) 2 6 . 5 ˆi 17 ˆj kˆ where R 28. 77 a 27. p.v. of 2 42 E
EXERCISE - 04 [B] JEE-Mathematics BRAIN STORMING SUBJECTIVE EXERCISE 1 . The position vectors of the points A, B, C are respectively (1, 1, 1) ; (1, 1, 2) ; (0, 2, 1). Find a unit vector parallel to the plane determined by ABC & perpendicular to the vector (1, 0, 1). 2. If a = a1 ˆi + a2 ˆj + a3 kˆ ; = b1 ˆi + b2 ˆj + b3 kˆ and = c1 ˆi + c2 ˆj + c3 kˆ then show that the value of the b c .ˆi .ˆj .kˆ a a a scalar triple product n n is (n3 + 1) .ˆi .ˆj .kˆ a b nbc c a b b b c .ˆi .ˆj .kˆ c c 3 . 2 Given that are four vectors such that a + = µ , . q =0& b b b = 1, where µ is a scalar a, b, p, q p then prove that = a.q p p.q a p.q 4 . ABCD is a tetrahedron with pv's of its angular point as A(–5, 22, 5); B(1, 2, 3); C(4, 3, 2) and D(–1, 2, –3). If the area of the triangle AEF where the quadrilaterals ABDE and ABCF are parallelograms is S then find the values of S. 5 . Given four points P1, P2, P3 and P4 on the coordinate plane with origin O which satisfy the condition 3 OPn 1 + OPn 1 = 2 OPn , n = 2, 3 (a) If P1, P2 lie on the curve xy = 1, then prove that P3 does not lie on this curve. (b) If P1, P2, P3 lie on the circle x2 + y2 = 1, then prove that P4 lies on this circle. 6. Find a vector which is coplanar with the vectors ˆi + ˆj – 2 kˆ and ˆi – 2 ˆj + kˆ and is orthogonal to the vector v NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 2ˆi ˆj kˆ . It is given that the projection of along the vector ˆi ˆj kˆ is equal to 6 3. v 7. If ; (p 0) prove that p2 . px (x x a) b x b (b.a)a a2p) (b x a) p (p2 Solve the following equation for the vector p ; pxa p.b c bxc a,b,c 8 . where are non zero non coplanar vectors and is neither perpendicular to nor hence show that p x abc c is a b to c , a a.c perpendicular to b c . 9. Solve the simultaneous vector equations for the vectors x and y . x c y a and yc xb where c is a non zero vector. E 43
JEE-Mathematics 10 . Let (a1 a)2 (a1 b)2 (a1 c)2 = 0 and if the vectors = ˆi + a ˆj + a2 kˆ ; = ˆi + b ˆj + b2 kˆ ; (b1 a)2 (b1 b)2 (b1 c)2 (c1 a)2 (c1 b)2 (c1 c)2 ˆi + c ˆj + c2 kˆ are non coplanar, show that the vectors 1 = ˆi + a1 ˆj + a12 kˆ ; ˆi + b1 ˆj + b12 kˆ = 1 = ˆi + c1 ˆj c12 kˆ and = + are coplanar. 1 1 1 . The vector OP = ˆi + 2 ˆj + 2 kˆ turns through a right angle, passing through the positive x-axis on the way. Find the vector in its new position. 12. If x y a, y z b, x.b , x.y 1 and y.z 1 , then find x, y & z in terms of a, b & . [REE 98] 1 3 . Find the value of such that a, b, c are all non-zero and [REE 2001] (4ˆi 5ˆj)a (3ˆi 3ˆj kˆ)b (ˆi ˆj 3kˆ)c (aˆi bˆj ckˆ) BRAIN STORMING SUBJECTIVE EXERCISE ANSWER KEY EXERCISE-4(B) NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 1. 1 (i 5 j k) 4. 110 5 . 9 6 . 9(ˆj kˆ) 33 a . cab c a.b b.c b b. b c a c.1a) cc2 b c y b (c1.b) cc2 a c 8.p 9. ( , a c x b x a . b a . b b b a b a a b ab a b a 11. 4 ˆi 1 ˆj 1 kˆ 12. x ; y ; z 13. 2 29 222 b 2 b 2 a a 44 E
EXERCISE - 05 [A] JEE-Mathematics JEE-[MAIN] : PREVIOUS YEAR QUESTIONS 1 . If a, b, c are three non zero vectors out of which two are not collinear. If a + 2b and c ; b + 3c and a are collinear then a + 2b + 6c is- [AIEEE-2002] (1) Parallel to c (2) Parallel to a (3) Parallel to b (4) 0 2 . If [ a b c ] = 4 then [ a × b b × c c × a ] = [AIEEE-2002] (1) 4 (2) 2 (3) 8 (4) 16 3 . If c = 2 (a × b) + 3µ(b × a) ; a × b 0, c.(a × b)=0 then- [AIEEE-2002] (1) = 3µ (2) 2 = 3µ (3) + µ = 0 (4) None of these 4. If = 2 ˆi + ˆj + 2 kˆ , = 5 ˆi – 3 ˆj + kˆ , then orthogonal projection of on is- [AIEEE-2002] a b a b (1) 3 ˆi – 3 ˆj + kˆ 9(5ˆi 3ˆj kˆ) (5ˆi 3ˆj kˆ) (4) 9(5 ˆi – 3 ˆj + kˆ ) (2) (3) 35 35 5. A unit vector perpendicular to the plane of = 2 ˆi – 6 ˆj – 3 kˆ , b = 4 ˆi + 3 ˆj – kˆ is- [AIEEE-2002] a 4ˆi 3ˆj kˆ 2ˆi – 6ˆj 3kˆ 3ˆi – 2ˆj 6kˆ 2ˆi – 3ˆj – 6kˆ (1) (2) (3) (4) 26 7 7 7 6. Let = ˆi + ˆj , = ˆi – ˆj and = ˆi + 2 ˆj + 3 kˆ . If nˆ is a unit vector such that . nˆ = 0 and . nˆ =0, then u v w u v |w . nˆ | is equal to- [AIEEE-2003] (1) 3 (2) 0 (3) 1 (4) 2 7 . A particle acted on by constant forces 4 ˆi + ˆj – 3 kˆ and 3 ˆi + ˆj – kˆ is displaced from the point ˆi + 2 ˆj + 3 kˆ to the point 5 ˆi + 4 ˆj + kˆ . The total work done by the forces is- [AIEEE-2003] (1) 50 units (2) 20 units (3) 30 units (4) 40 units 8. The vectors = 3 ˆi + 4 kˆ and =5 ˆi – 2 ˆj + 4 kˆ are the sides of a triangle ABC. The length of the median AB AC through A is- [AIEEE-2003] (1) 288 (2) 18 (3) 72 (4) 33 9. a, b, c are three vectors, such that a + b + c = 0 , | a | = 1, | b | = 2, | c | = 3, then a . b + b . c + c . a is equal to- NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 [AIEEE-2003] (1) 1 (2) 0 (3) –7 (4) 7 1 0 . Consider point A, B, C and D with postion vectors 7 ˆi – 4 ˆj + 7 kˆ , ˆi – 6 ˆj + 10 kˆ , – ˆi – 3 ˆj + 4 kˆ and 5 ˆi – ˆj + 5 kˆ respectively. Then ABCD is a- [AIEEE-2003] (1) parallelogram but not a rhombus (2) square (3) rhombus (4) None of these 11. If u, v and w are three non-coplanar vectors, then ( u +v – w ).( u –v) × (v – w ) equals- [AIEEE-2003] (1) 3 .( (2) 0 (3) .( × (4) .( × u v× w) u v w) u w v) 1 2 . Let a , b and c be three non-zero vectors such that no t wo of the se are colli near. If the vector a + 2 b is collinear with c and b + 3 c is collinear with a ( being some non-zero scalar) then a + 2 b + 6 c equals- [AIEEE-2004] (2) b (3) c (4) 0 (1) a 1 3 . A particle is acted upon by constant forces 4 ˆi + ˆj – 3 kˆ and 3 ˆi + ˆj – kˆ which displace it from a point ˆi + 2 ˆj + 3 kˆ to the point 5 ˆi + 4 ˆj + kˆ . Then work done in standard units by the forces is given by-[AIEEE-2004] (1) 40 (2) 30 (3) 25 (4) 15 E 45
JEE-Mathematics 14. If a , b, c are non-coplanar vectors and is a real number, then the vectors a + 2 b + 3 c , b + 4 c and (2 – 1) c are non-coplanar for- [AIEEE-2004] (1) all values of (2) all except one value of (3) all except two values of (4) no value of 1 5 . Let u , v , w be such that | u | = 1, | v | = 2, |w | = 3. If the projection of v along u is equal to projection of w along u and v and w are perpendicular to each other then | u – v + w | equals- [AIEEE-2004] (1) 2 (2) 7 (3) 14 (4) 14 16. = 1 | || | . If is the acute angle between Let a , b and c be non-zero vectors such that ( a × b ) × c b c a 3 the vectors b and c , then sin equals- [AIEEE-2004] 1 2 2 22 (1) 3 (2) (3) 3 (4) 3 3 1 7 . If C is the mid point of AB and P is any point outside AB, then- [AIEEE-2005] [AIEEE-2005] (1) PA + PB = 2 PC (2) PA + PB = PC (3) PA + PB + 2 PC = 0 (4) PA + PB + PC = 0 18. For any vector × ˆi )2 + × ˆj )2 + × kˆ )2 is equal to- a , the value of ( a (a (a (1) 3 a 2 (2) a 2 (3) 2 a 2 (4) 4 a 2 1 9 . Let a, b and c be distinct non-negative numbers. If the vectors a ˆi + a ˆj + c kˆ , ˆi + kˆ and c ˆi + c ˆj + b kˆ lie in a plane, then c is- [AIEEE-2005] (1) the Geometric Mean of a and b (2) the Arithmetic Mean of a and b (3) equal to zero (4) the Harmonic Mean of a and b 2 0 . If a , b , c are non-coplanar vectors and is a real number then [( a + b ) 2 b c ] = [ a b + c d ] for- [AIEEE-2005] (1) exactly one value of (2) no value of (3) exactly three values of (4) exactly two values of 21. Let = ˆi – kˆ , = x ˆi + ˆj + (1 – x) kˆ and = y ˆi + x ˆj + (1 + x – y) kˆ . Then , a b c [a b, c ] depends on- [AIEEE-2005] (1) only y (2) only x (3) both x and y (4) neither x nor y NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 2 2 . If ( a × b ) × c = a × ( b × c ), where a , b and c are any three vectors such that a . b 0, b . c 0, then a and c are- [AIEEE-2006] (1) inclined at an angle of /6 between them (2) perpendicular (3) parallel (4) inclined at an angle of /3 between them 1 2 3 . ABC is a triangle, right angled at A. The resultant of the forces acting along AB , AC with magnitudes AB 1 and AC respectively is the force along AD , where D is the foot of the perpendicular from A onto BC. the magnitude of the resultant is- [AIEEE-2006] (AB) (AC) 11 1 AB2 AC2 (1) AB AC (2) AB + AC (3) (4) (AB)2 (AC )2 AD 2 4 . The values of a, for which the points A, B, C with position vectors 2 ˆi – ˆj + kˆ , ˆi – 3 ˆj – 5 kˆ and a ˆi – 3 ˆj + kˆ are- [AIEEE-2006] respectively are the vertices of a right-angled triangle with C = 2 (1) –2 and –1 (2) –2 and 1 (3) 2 and –1 (4) 2 and 1 46 E
JEE-Mathematics 2 5 . If uˆ and vˆ are unit vectors and is the acute angle between them, then 2 uˆ × 3 vˆ is a unit vector for- [AIEEE-2007] (1) Exactly two values of (2) More than two values of (3) No value of (4) Exactly one value of 2 6 . Let a = ˆi + ˆj + kˆ , b = ˆi – ˆj + 2 kˆ and c = x ˆi + (x – 2) ˆj – kˆ . If the vector c lies in the plane of a and b , then x equals- [AIEEE-2007] (1) 0 (2) 1 (3) –4 (4) –2 b 27. The vector = ˆi + 2 ˆj + kˆ , lies in the plane the vectors = ˆi + ˆj and = ˆj + kˆ and bisect the angle a c between b and c . Then which one of the following gives possible values of and ? [AIEEE-2008] (1) = 2, = 2 (2) = 1, = 2 (3 ) = 2, = –1 (4) = 1, = 1 2 8 . The non-zero vectors a , b and c are related a = 8 b and c = –7 b . Then the angle between a and c is- [AIEEE-2008] (1) 0 (2) /4 (3) /2 (4) 2 9 . If u, v, w are non-coplanar vectors and p, q are real numbers, then the equality – – = 0 holds for :- [AIEEE-2009] [3u pv pw ] [pv w qu] [2w qv qu] (1) More than two but not all values of (p, q) (2) All values of (p, q) (3) Exactly one value of (p, q) (4) Exactly two values of (p, q) 30. Let ˆj kˆ and ˆi ˆj kˆ . Then the vector satisfying abc 0 and a .b3 is : [AIEEE-2010] a c b (1) ˆi ˆj 2kˆ (2) 2ˆi ˆj 2kˆ (3) ˆi ˆj 2kˆ (4) ˆi ˆj 2kˆ 31. If the vectors ˆi ˆj 2kˆ , 2ˆi 4ˆj kˆ and ˆi ˆj kˆ are mutually orthogonal, then a b c (, ) = [AIEEE-2010] (1) (–3, 2) (2) (2, –3) (3) (–2, 3) (4) (3, –2) 32. The vectors a and b are not perpendicular and c and d are two vectors satisfying : bc bd and a.d 0 . Then the vector d is equal to :- [AIEEE-2011] (1) (2) (3) (4) b b.c c c a.c b b b.c c c a.c b a.b a.b a.b a.b 1 3ˆi kˆ 1 a b If a b a b a 2b 10 7 3 3 . 2 . NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 and 2ˆi 3ˆj 6kˆ , then the value of is :- [AIEEE-2011] (1) 5 (2) 3 (3) – 5 (4) – 3 3 4 . If the vectors pˆi ˆj kˆ,ˆi qˆj kˆ and ˆi ˆj rkˆ (p q r 1) are coplanar, then the value of pqr – (p + q + r) is :- [AIEEE-2011] (1) –2 (2) 2 (3) 0 (4) –1 35. Let be three non-zero vectors which are pairwise non-collinear. If is collinear with and a, b, c a 3b c b 2c is colliner with a , then a 3b 6c is : [AIEEE-2011] (1) a c (2) a (3) c (4) 0 36. Let aˆ and bˆ be two unit vectors. If the vectors aˆ 2 bˆ and 5 aˆ 4 bˆ are perpendicular to each other, c d then the angle between aˆ and bˆ is : [AIEEE-2012] (1) (2) 6 (3) (4) 3 4 2 E 47
JEE-Mathematics 3 7 . Let ABCD be a parallelogram such that AB q, AD p and BAD be an acute angle. If r is the vector that coincides with the altitude directed from the vertex B to the side AD, then r is given by : [AIEEE-2012] (1) 3 p (2) 3 p r 3q .q p r 3q .q p (p . p) (p . p) . . r q p . q p r q p . q p (3) p p (4) p p 38. If the vectors 3ˆi 4 kˆ and 5ˆi 2ˆj 4kˆ are the sides of a triangle ABC, then the length of the median AB AC through A is : [JEE (Main)-2013] (1) 18 (2) 72 (3) 33 (4) 45 PREVIOUS YEARS QUESTIONS ANSWER KEY EXERCISE-5 [A] NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans 4 4 2 2 3 1 4 4 3 4 3 4 1 3 3 Que. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Ans 4 1 3 1 2 4 3 3 4 4 4 4 4 3 1 Que. 31 32 33 34 35 36 37 38 Ans 1 2 3 1 4 4 3 3 E 48
JEE-Mathematics EXERCISE - 05 [B] JEE-[ADVANCED] : PREVIOUS YEAR QUESTIONS 1. Select the correct alternative : (a) If the vectors a, b & c form the sides BC, CA & AB respectively of a triangle ABC, then 2. 3. (A) a . b b .c c .a 0 (B) a b b c c a 4. 5. (C) a . b b .c c .a (D) a b b c c a 0 6. 7. ( b ) Let the vectors a, b, c & d be such that a b c d 0 . Let P & P be planes determined 8. 12 9. by the pairs of vectors a, b & c, d respectively. Then the angle between P and P is : E 12 (A) 0 (B) /4 (C) /3 (D) /2 b b 2a (c) If a, & c are unit coplanar vectors, then the scalar triple product 2b c 2c a (A) 0 (B) 1 (C) 3 (D) 3 [JEE 2000 (Screening) 1+1+1M out of 35] Let ABC and PQR be any two triangles in the same plane. Assume that the perpendicular from the points A, B, C to the sides QR, RP, PQ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from P, Q, R to BC, CA, AB respectively are also concurrent. [JEE 2000 ( Mains) 10M out of 100] (a) If aˆ, bˆ and cˆ are unit vectors, then aˆ bˆ 2 bˆ cˆ 2 cˆ aˆ 2 does not exceed (A) 4 (B) 9 (C) 8 (D) 6 (b) Let ˆi kˆ , b xˆi ˆj (1 x)k and yˆi xˆj (1 x y )kˆ . Then b, depends on a c [a, c] (A) only x (B) only y (C) neither x nor y (D) both x and y [JEE ‘2001 (Screening) 1+1M out of 35] Show by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices. [JEE 2001 (Mains) 5M out of 100] Find 3-dimensional vectors v1, v2, v3 satisfying v1 . v1 4 , v1 . v2 2 , v1 . v 3 6 , v2 . v2 2 , v2 . v3 5 , v3 . v3 29 . [JEE 2001 (Mains) 5M out of 100] Let A (t) f1 (t)ˆi f2 (t)ˆj and B(t) g1 (t)ˆi g2 (t)ˆj , t [0,1] , where f, f, g, g are continuous functions. If 1 2 1 2 A (t) and B(t) are non-zero vectors for all t and A (0) 2ˆi 3ˆj , A(1) 6ˆi 2ˆj , B(0) 3ˆi 2ˆj and B(1) 2ˆi 6ˆj , then show that A (t) and B(t) b are parallel for some t. NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 ( a ) If and are two unit vectors such that and [JEE 2001 (Mains) 5M out of 100] a b a 2b 5a 4b are perpendicular to each other then the angle between a and b is - [JEE 2002 (Screening), 3M] (A) 45° (B) 60° (C) co s 1 1 (D) co s 1 2 3 7 ( b ) Let V 2 i j k and W i 3k . If U is a unit vector, then the maximum value of the scalar triple product [U V W ] is - [JEE 2002 (Screening), 3M] (A) –1 (B) 10 6 (C) 59 (D) 60 Let v be the volume of the parallelopiped formed by the vectors a a1ˆi a 2 ˆj a 3 kˆ, b2ˆj r = 1, 2, b b1ˆi b3 kˆ , c1ˆi c2ˆj c 3 kˆ. If a , b, c, where 3, are non-negative real c r r r 3 [JEE 2002 (Mains)] numbers and ar br cr 3L , show that V L3. r 1 The value of a for which the volume of parallelopiped formed by the vectors ˆi aˆj kˆ and ˆj akˆ and aˆi kˆ as coterminous edge is minimum is - [JEE 03 (Screening), 3M] (A) –3 (B) 3 (C) 1 / 3 (D) none of these 49
JEE-Mathematics 1 0 . If u , v , w are three non-coplanar unit vectors and , , are the angles between u and v , v and w , w and u respectively and x , y , z are unit vectors along the bisectors of the angles , , respectively. 1 ]2 sec2 sec2 sec2 [JEE 03 (Mains) 4M] Prove that [x y y z z x] [u v w 16 2 2 2 11. (a) If for vectors and ˆj kˆ, ˆi ˆj kˆ then vector is – a b, a . b 1, a b a b (A) ˆi ˆj kˆ (B) 2 ˆj kˆ (C) ˆi (D) 2 ˆi ( b ) A given unit vector is orthogonal to 5ˆi 2ˆj 6kˆ and coplanar with ˆi ˆj kˆ and 2ˆi ˆj kˆ then the vector is - 3ˆj kˆ 6ˆi 5kˆ 2ˆi 5kˆ 2ˆi ˆj 2kˆ (A) (B) (C) (D) 10 61 29 3 [JEE 04 (screening) 3+3M] 12. and are four distinct vectors satisfying the conditions & a c b d , then prove a, b, c d ab cd that a . b c . d a .c b. d [JEE 04 (Mains) 2M] a,b,c b .a b .a 13. If are three non-zero, non-coplanar vectors and b1 b |a|2 a , b2 b |a|2 a , c1 c |ca|.a2 a + |bc|.2c b1 , c2 c |ca|.a2 a b1 . c b1 c3 c |ca|. 2a a b 1c|.2c b1 , c4 c |ca.|a2 a b. c b1 | b1|2 , + | – | b|2 then the set of orthogonal vectors is - (A) (a , b1 , c3 ) (B) (a , b1 , c2 ) (C) (a , b1 , c1 ) (D) (a , b2 , c2 ) [JEE 05 (screening) 3M] 1 4 . Incident ray is along the unit vector v and the reflected ray is along the unit vector w . The normal is along unit vector â outwards. Express w in terms of â and v . [JEE 05 (Mains) 4M out of 60] 15. (a) Let ˆi 2ˆj kˆ, ˆi ˆj kˆ and ˆi ˆj kˆ . A vector in the plane of and whose projection on a b c a b c 1 has the magnitude equal to 3 is - (A) 4ˆi ˆj 4kˆ (B) 3ˆi ˆj 3kˆ (C) 2ˆi ˆj 2kˆ (D) 4ˆi ˆj 4kˆ NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 [JEE 06, 3M] ( b ) Let A be vector parallel to line of intersection of planes P1 and P2 through origin. P1 is parallel to the vectors 2ˆj 3kˆ and 4ˆj 3kˆ and P2 is parallel to ˆj kˆ and 3ˆi 3ˆj , then the angle between vector A and 2ˆi ˆj 2kˆ is - 3 (A) (B) (C) 6 (D) 2 4 4 [JEE 06, 5M] 1 6 . The number of distinct real values of , for which the vectors 2ˆi ˆj kˆ , ˆi 2ˆj kˆ and ˆi ˆj 2kˆ are coplanar, is :- [JEE 07, 3M] (A) zero (B) one (C) two (D) three 17. Let a, b, c be unit vectors such that abc 0 . Which one of the following is correct? [JEE 07, 3M] (A) a b b c c a 0 (B) a b b c c a 0 (C) (D) a b, b c, c a are mutually perpendicular ab bc ac 0 50 E
JEE-Mathematics 1 8 . Let the vectors PQ , QR , RS , ST , TU and UP represent the sides of a regular hexagon. Statement-1 : PQ (RS ST) 0 . because Statement-2 : PQ RS 0 and PQ ST 0 . [JEE 07, 3M] (A) Statement-1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1. (C) Statement-1 is True, Statement-2 is False. (D) Statement-1 is False, Statement-2 is True. 1 9 . The edges of a parallelopiped are of unit length and are parallel to non–coplanar unit vectors aˆ , bˆ , cˆ such that aˆ . bˆ = bˆ . cˆ = cˆ . aˆ = 1 Then, the volume of the parallelopiped is :- [JEE 08, 3M, –1M] . 2 1 1 3 1 (A) (B) (C) (D) 2 22 2 3 20. Let two non-collinear unit vectors aˆ and bˆ form an acute angle. A point P moves so that at any time t the position origin) is given by aˆ cost + bˆ sint. When P is farthest from origin O, let vector (where O is the OP M be the length of OP and uˆ be the unit vector along OP . Then - [JEE 08, 3M, –1M] (A) uˆ aˆ bˆ and M = (1 + aˆ . bˆ )1/2 (B) uˆ aˆ bˆ and M = (1 + aˆ . bˆ )1/2 aˆ bˆ aˆ bˆ (C) uˆ aˆ bˆ and M = (1 + 2 aˆ . bˆ )1/2 (D) uˆ aˆ bˆ and M = (1 + 2 aˆ . bˆ )1/2 aˆ bˆ aˆ bˆ 1 a, b, c d ab . cd a.c , then :- 2 1 . 1 and = [JEE 2009, 3M, –1M] If and are unit vectors such that 2 (A) are non-coplanar (B) d are non-coplanar a, b, c (D) b, c, (C) b, d are non-parallel d are parallel and b, are parallel a, c 2 2 . Match the statements / expressions given in Column I with the values given in Column II [JEE 2009, 8M] Column–I Column–II (A) Root(s) of the equation 2sin2 + sin22=2 (P) 6 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 (B) Points of discontinuity of the function f(x) = 6x cos 3x (Q) 4 where [y] denotes the largest integer less than or equal to y (R) 3 (C) Volume of the parallelepiped with its edges represented by the vectors ˆi ˆj , ˆi 2ˆj and ˆi ˆj kˆ (S) (D) Angle between vectors and where and are unit 2 a b a, b c (T) vectors satisfying ab 3c 0 23. Let P, Q, R and S be the points on the plane with position vectors – 2ˆi ˆj, 4ˆi,3ˆi 3ˆj and 3ˆi 2ˆj respectively. E The quadrilateral PQRS must be a [JEE 10, 3M, –1M] (A) parallelogram, which is neither a rhombus nor a rectangle (B) square (C) rectangle, but not a square (D) rhombus, but not a square 51
JEE-Mathematics 24. and are vectors in space given by ˆi 2ˆj and 2ˆi ˆj 3 kˆ , then the value of If a b a b 5 14 ab a 2b . is [JEE 10, 3M, –1M] 2a b 2 5 . Two adjacent sides of a parallelogram ABCD are given by AB 2ˆi 10ˆj 11kˆ and AD ˆi 2ˆj 2kˆ The side AD is rotated by an acute angle in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle is given by - [JEE 10, 5M, –1M] 8 17 1 45 (A) 9 (B) (C) 9 (D) 9 9 26. (a) Let ˆi ˆj kˆ, ˆi ˆj kˆ and ˆi ˆj kˆ be three vectors. A vector in the plane of a b c v 1 a and b , whose projection on c is , is given by 3 (A) ˆi 3ˆj 3kˆ (B) 3ˆi 3ˆj kˆ (C) 3ˆi ˆj 3kˆ (D) ˆi 3ˆj 3kˆ ( b ) The vector(s) which is/are coplanar with vectors ˆi ˆj 2kˆ and ˆi 2ˆj kˆ and perpendicular to the vector ˆi ˆj kˆ is/are (A) ˆj kˆ (B) ˆi ˆj (C) ˆi ˆj (D) ˆj kˆ (c) Let ˆi kˆ, ˆi ˆj and ˆi 2ˆj 3kˆ be three given vectors. If is a vector such that a b c r r b c b and r.a 0 , then the value of r.b is [JEE 2011, 3+4+4] 27. are unit vectors satisfying | | c|2 | |2 9 , then ( a ) If a, b and c a b|2 b c a |2a 5b 5c| is NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 (b) 29 and (2ˆi 3ˆj 4 kˆ) (2ˆi 3ˆj 4 kˆ) , then a possible If a and b are vectors such that | a b| a b value of 2ˆj 3kˆ) is (a b).(7ˆi (A) 0 (B) 3 (C) 4 (D) 8 [JEE 2012, 4+3] 28. Let 3ˆi ˆj 2kˆ and ˆi 3ˆj 4 kˆ determine diagonals of a parallelogram PQRS and ˆi 2ˆj 3kˆ PR SQ PT be another vector. Then the volume of the parallelepiped determined by the vectors PT, PQ and PS is [JEE-Advanced 2013, 2M] (A) 5 (B) 20 (C) 10 (D) 30 2 9 . Consider the set of eight vectors V aˆi bˆj ckˆ : a, b, c {1,1} . Three non-coplanar vectors can be chosen from V in 2p ways. Then p is [JEE-Advanced 2013, 4, (–1)] 52 E
JEE-Mathematics 3 0 . Match List-I with List-II and select the correct answer using the code given below the lists. List-I List-II 1. 100 P. Volume of parallelepiped determined by vectors a, b and 2. 30 c is 2. Then the volume of the parallelepiped determined by and c is vectors 2 a b ,3 b c a Q. Volume of parallelepiped determined by vectors and a, b c is 5. Then the volume of the parallelepiped determined by and 2 is vectors 3 a b , b c c a R. Area of a triangle with adjacent sides determined by vectors 3. 24 a and b is 20. Then the area of the triangle with adjacent a b 2a 3b sides determined by vectors and is S. Area of a parallelogram with adjacent sides determined by 4. 60 vectors a and b is 30. Then the area of the parallelogram ab with adjacent sides determined by vectors and a is Codes : P QR S (A) 4 2 3 1 (B) 2 3 1 4 (C) 3 4 1 2 (D) 1 4 3 2 [JEE-Advanced 2013, 3, (–1)] NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\02-VECTOR(EXERCISES).p65 PREVIOUS YEARS QUESTIONS ANSWER KEY EXERCISE-5 [B] 1 . (a) B ; (b) A ; (c) A 3. (a) B (b) C 5. 2ˆi , ˆi j , 3ˆi 2ˆj 4kˆ 1 2 . (a) B ; (b) C v1 v2 v3 9 . D 11. (a) C (b) A 1 3 . B 14. w = v –2 (â. v )â 1 5 . (a) A ; (b) B,D 16. C 17. B 18. C 19. A 20. A 21. C 23. A 24. 5 2 2 . (A) (Q, S), (B) (P, R, S, T), (C) (T), (D) (R) 28. C 29. 5 25. B 30. C 2 6 . (a) C; (b) A,D; (c) 9 27. (a) 3; (b) C E 53
JEE-Mathematics 3D-COORDINATE GEOMETRY POINT 1. INTRODUCTION : In earlier classes we have learnt about points, lines, circles and conic section in two dimensional geometry. In two dimensions a point represented by an ordered pair (x, y) (where x & y are both real numbers) In space, each body has length, breadth and height i.e. each body exist in three dimensional space. Therefore three independent quantities are essential to represent any point in space. Three axes are required to represent these three quantities. 2 . RECTANGULAR CO-ORDINATE SYSTEM : In cartesian system of the three lines are mutually perpendicular, such a system is called rectangular cartesian co-ordinate system. Co-ordinate axes and co-ordinate planes : z x' When three mutually perpendicular planes intersect at a point, then mutually perpendicular lines are obtained and these lines also pass through that point. If y' Oy we assume the point of intersection as origin, then the three planes are known as co-ordinate planes and the three lines are known as co-ordinate axes. Octants : Every plane bisects the space. Hence three co-ordinate plane divide the space x z' in eight parts. These parts are known as octants. 3 . COORDINATES OF A POINT IN SPACE : Z Let O be a fixed point, known as origin and let OX, OY and OZ be three CE mutually perpendicular lines, taken as x-axis, y-axis and z-axis respectively, in k such a way that they form a right handed system. F x' P(x, y, z) The plane s XOY, YOZ and ZOX are know n as xy-plane, yz-plane and y' BY i Oj zx-plane respectively. Let P be a point in space and distances of P from yz, zx and xy planes be X A D z' x, y, z respectively (with proper signs) then we say that coordinates of P are (x, y, z). Also OA = |x|, OB = |y|, OC = |z| NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 4 . DISTANCE FORMULA : The distance between two points A (x1, y1, z1) and B (x2, y2, z2) is given by AB [(x2 x1 )2 (y2 y1 )2 (z2 z1 )2 ] ( a ) Distance from Origin : Let O be the origin and P (x, y, z) be any point, then OP (x2 y2 z2 ) ( b ) Distance of a point from coordinate axes : Let P(x, y, z) be any point in the space. Let PA, PB and PC be the perpendiculars drawn from P to the axes OX, OY and OZ respectively. Then PA (y2 z2 ) ; PB (z2 x2 ) ; PC (x2 y2 ) Illustration 1 : Prove by using distance formula that the points P (1, 2, 3), Q (–1, –1, –1) and R (3, 5, 7) are c o l li nea r. 54 E
JEE-Mathematics Solution : We have PQ = (1 1)2 (1 2)2 (1 3)2 = 4 9 16 29 Q R = (3 1)2 (5 1)2 (7 1)2 = 16 36 64 116 2 29 and PR = (3 1)2 (5 2)2 (7 3)2 4 9 16 29 Ans. Since QR = PQ + PR. Therefore the given points are collinear. Illustration 2 : Find the locus of a point the sum of whose distances from (1, 0, 0) and (–1, 0, 0) is equal to Solution : 10. Let the points A(1,0,0), B (–1,0,0) and P(x,y,z) Given : PA + PB =10 (x 1)2 (y 0)2 (z 0)2 + (x 1)2 (y 0)2 (z 0)2 = 10 (x 1)2 y2 z2 = 10 – (x 1)2 y2 z2 Squaring both sides, we get ; (x – 1)2 + y2 + z2 = 100 + (x + 1)2 + y2 + z2 – 20 (x 1)2 y2 z2 – 4x –100 = – 20 (x 1)2 y2 z2 x + 25 = 5 (x 1)2 y2 z2 Ans. Again squaring both sides we get x2 + 50x + 625 = 25 {(x2 +2x +1) + y2 +z2} 24x2 + 25y2 + 25z2 – 600 = 0 i.e. required equation of locus 5 . SECTION FORMULAE : Let P(x1, y1, z1) and Q(x2, y2, z2) be two points and let R (x, y, z) divide PQ in the ratio m1 : m2. Then co-ordinates of R(x, y, z) = m1 x2 m2 x1 , m1y2 m2y1 , m1z2 m2 z1 m1 m2 m1 m2 m1 m2 If (m1/m2) is positive, R divides PQ internally and if (m1/m2) is negative, then externally. Mid-Point : Mid point of PQ is given by x1 x2 , y1 y2 , z1 z2 2 2 2 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 Illustration 3 : Find the ratio in which the plane x – 2y + 3z = 17 divides the line joining the points (–2, 4, 7) and (3, –5, 8). Solution : Let the required ratio be k : 1 The co-ordinates of the point which divides the join of (–2, 4, 7) and (3, –5, 8) in the ratio k : 1 are 3k 2 , 5 k 4 , 8k 7 k 1 k 1 k 1 Since this point lies on the plane x – 2y + 3z – 17 = 0 3k 2 2 5k 4 3 8k 7 17 0 k 1 k 1 k 1 (3k – 2) –2 (–5k + 4) +3 (8k + 7) = 17 k + 17 3k + 10k + 24k – 17k = 17 + 2 + 8 – 21 37k – 17k = 6 20k = 6 ; k = 63 20 10 3 Ans. Hence the required ratio = k : 1 = 10 : 1 = 3 : 10 E 55
JEE-Mathematics Do yourself 1: ( i ) Find the distance between the points P(3, 4, 5) and Q(–1, 2, –3). ( i i ) Show that the points A(0, 7, 10), B(–1, 6, 6) and C(–4, 9, 6) are vertices of an isosceles right angled triangle. ( i i i ) Find the locus of a point such that the difference of the square of its distance from the points A(3, 4, 5) and B(–1, 3, –7) is equal to 2k2. ( i v ) Find the co-ordinates of points which trisects the line joining the points A(–3, 2, 4) and B(0, 4, 7) ( v ) Find the ratio in which the planes (a) xy (b) yz divide the line joining the points P(–2, 4, 7) and Q(3, –5, 8). 6 . CENTROID OF A TRIANGLE : Let A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) be the vertices of a triangle ABC. Then its centroid G is given by G x1 x2 x3 , y1 y2 y3 , z1 z2 z3 3 3 3 Illustration 4 : If the centroid of a tetrahedron OABC where A, B, C, are given by (a, 2, 3), (1, b, 2) and (2, 1, c) respectively be (1, 2, – 1), then distance of P (a, b, c) from origin is - (A) 107 (B) 14 (C) 107 14 (D) none of these F I1 1 1 GH KJCentroid is Solution : x, y, z (1, 2, 1) 444 a 1 2 0 1 2 b 1 0 2 3 2 c 0 1 a = 1, b = 5, c = – 9 4 44 OP a2 b2 c2 107 Ans. (A) 7 . DIRECTION COSINES OF LINE : z If be the angles made by a line with x-axis, y-axis & z-axis respectively then y cos, cos & cos are called direction cosines of a line, denoted by , m & n E respectively. Note : x NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 (i) If line makes angles with x, y & z axis respectively then is another set of angle that line makes with principle axes. Hence if , m & n are direction cosines of line then –, –m & –n are also direction cosines of the same line. (ii) Since parallel lines have same direction. So, in case of lines, which do not pass through the origin. We can draw a parallel line passing through the origin and direction cosines of that line can be found. Important points : ( i ) Direction cosines of a line : Take a vector A aˆi bˆj ckˆ parallel to a line whose D.C’s are to be found out. A.ˆi a | A| cos a ai + bj + ck cos a similarly, cos b ; cos c | A| | A| | A| cos2 cos2 cos2 1 2 m2 n2 1 56
JEE-Mathematics (ii) Direction cosine of axes : Since the positive x-axes makes angle 0°, 90°, 90° with axes of x, y and z respectively, D.C.’s of x axes are 1, 0, 0. D.C.’s of y-axis are 0, 1, 0 D.C.’s of z-axis are 0, 0, 1 8. DIRECTION R ATIOS : 9. Any three numbers a, b, c proportional to direction cosines , m, n are called direction ratios of the line. 7. mn E i.e. abc There can be infinitely many sets of direction ratios for a given line. Direction ratios and Direction cosines of the line joining two points : Let A(x1, y1, z1) and B(x2, y2, z2) be two points, then d.r.’s of AB are x2 – x1, y2 – y1, z2 – z1 and the d.c.’s of AB 11 1 are r (x2 – x1), r (y2 – y1), r (z2 – z1) where r [(x2 x1 )2 ] REL ATION BETWEEN D.C’S & D.R’S : mn abc 2 m2 n2 2 m2 n2 a2 b2 c2 a2 b2 c2 a ; b c a2 b2 c2 m= ; n= a2 b2 c2 a2 b2 c2 Important point : Direction cosines of a line are unique but Dr's of a line in no way unique but can be infinite. NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 PROJECTIONS : z P(x,y,z) ( a ) Projection of line segment OP on co-ordinate axes : y O Let line segment make angle with x-axis A Thus, the projections of line segment OP on axes are the absolute values x of the co-ordinates of P. i.e. Projection of OP on x-axis = |x| Projection of OP on y-axis = |y| Projection of OP on z-axis = |z| Now, in OAP, angle A is a right angle and OA = x OP = x2 y2 z2 cos xx x2 y2 z2 | OP| if |OP| = r, then x = |OP|cos = r Similarly y = |OP|cos = mr, z = nr, where , m, n are DC’s of line ( b ) Projection of a line segment AB on coordinate axes : Projection of the point A(x , y , z ) on x-axis is E(x , 0, 0). Projection of point B(x , y , z ) on x-axis is 1 11 1 222 F(x , 0, 0). 2 Hence projection of AB on x-axis is EF = |x – x |. 21 57
JEE-Mathematics Similarly, projection of AB on y and z-axis are |y – y |, |z – z | respectively. 21 21 Note : Projection is only a length therefore it is always taken as positive. ( c ) Projection of line segment AB on a line having direction cosines , m, n : B(x2,y2,z2) Let A(x , y , z ) and B(x , y , z ). z A(x1,y1,z1) 111 222 Now projection of AB on EF = CD = AB cos (x2 x1 ) (y2 y1 )m (z2 z1 )n EC DF (x2 x1 )2 (y2 y1 )2 (z2 z1 )2 y = (x2 x1 )2 (y2 y1 )2 (z2 z1 )2 × x = (x2 x1 ) (y2 y1 )m (z2 z1 )n Illustration 5 : A line OP makes with the x-axis an angle of measure 120° and with y-axis an angle of measure 60°. Find the angle made by the line with the z-axis. Solution : = 120° and = 60° 11 cos = cos 120° = – and cos = cos 60° = but cos2 + cos2 + cos2 = 1 22 1 2 12 + cos2 = 1 2 2 11 1 cos = ± 1 cos2 = 1 – 135° 2 44 2 Ans. = 45° or Illustration 6 : Find the projection of the line segment joining the points (–1, 0, 3) and (2, 5, 1) on the line whose direction ratios are 6, 2, 3. m n 2 m 2 n2 1 1 Solution : The direction cosines , m, n of the line are given by 6 2 3 62 22 32 49 7 6, m 2,n 3 7 77 The required projection is given by = |(x2 – x1) + m(y2 – y1) + n(z2 – z1)| = 62 3 [2 (1)] (5 0) (1 3) NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 77 7 = 6 3 2 5 3 2 = 18 10 6 18 10 6 22 Ans. 777 = 77 7 77 Do yourself - 2 : ( i ) Find the projections of the line segment joining the origin O to the point P(3, 2, –5) on the axes. ( i i ) Find the projections of the line joining the points P(3, 2, 5) and Q(0, –2, 8) on the axes. ( i i i ) Find the direction ratios & direction cosines of the line joining the points O(0, 0, 0) and P(2, 3, 4). 1 1 . ANGLE BETWEEN TWO LINES : Let be the angle between the lines with d.c.’s 1, m1, n1 and 2, m2, n2 then cos = 1 2 + m1m2 + n1n2. If a1, b1, c1 and a2, b2, c2 be D.R.’s of two lines then angle between them is given by cos (a1a2 b1b2 c1c2 ) (a 2 b12 c 2 ) ( a 2 b 2 c 2 ) 1 1 2 2 2 58 E
Illustration 7 : If a line makes angles , , with four diagonals of a cube, JEE-Mathematics then cos2 + cos2 + cos2 + cos2 equals - (D) 3/4 (A) 3 (B) 4 (C) 4/3 Solution : Let OA, OB, OC be coterminous edges of a cube and OA = OB = OC = a, then co-ordinates of its vertices are O(0, 0, 0), A(a, 0, 0), B(0, a, 0), C(0, 0, a), L(0, a, a), M(a, 0, a), N(a, a, 0) and P (a, a, a) Direction ratio of diagonal AL, BM, CN and OP are Z 1, 1, 1 , 1 , 1, 1 , 1, 1 , 1 , 1, 1, 1 CL 3 3 3 3 3 3 3 3 3 3 3 3 M Let , m, n be the direction cosines of the given line, then O P BY cos = 1 m 1 n 1 m n A 3 3 3 3 X N Similarly cos = m n , cos m n and cos m n 33 3 4 Ans. (C) cos2 + cos2 + cos2 + cos2 = 3 Illustration 8 : (a) Find the acute angle between two lines whose direction ratios are 2, 3, 6 and 1, 2, 2 respectively. (b) Find the measure of the angle between the lines whose direction ratios are 1, –2, 7 and Solution : (a) 3, –2, –1. a1 = 2, b1 = 3, c1 = 6; a2 = 1, b2 = 2, c2 = 2. If be the angle between two lines whose d.r’s are given, then a1a2 b1b2 c1c2 2 1 3 2 6 2 2 6 12 20 cos 2 b12 2 2 2 2 = 22 32 62 12 22 22 = 73 21 1 1 2 2 2 a c a b c NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 co s 1 20 21 (b) 12 (2)2 72 54 32 (2)2 (1)2 14 The actual direction cosines of the lines are 1 , 2 , 7 and 3 , 2 , 1 54 54 54 14 14 14 If is the angle between the lines, then cos 1 3 2 2 7 1 14 54 14 54 54 14 3 4 7 90 Ans. = 54. 14 = 0 E 59
JEE-Mathematics 1 2 . PERPENDICULAR AND PAR ALLEL LINES : Let the two lines have their d.c.’s given by 1, m1, n1 and 2, m2, n2 respectively then they are perpendicular if = 90° i.e. cos = 0, i.e. 1 2 + m1m2 + n1n2 = 0. Also the two lines are parallel if = 0 i.e. sin = 0, i.e. 1 m1 n1 2 m2 n2 N o t e : If instead of d.c.’s, d.r.’s a1, b1, c1 and a2, b2, c2 are given, then the lines are per pendicular if a1a2 + b1b2 + c1c2 = 0 and parallel if a1 b1 c1 . a2 b2 c2 Illustration 9 : If the lines whose direction cosines are given by a + bm + cn = 0 and fmn + gn + hm = 0 are perpendicular, then f gh equals - abc (A) 0 (B) –1 (C) 1 (D) none of these Solution : Eliminating n between the given relations, we find that (fm + g) a bm + hm = 0 c or ag 2 (af bg ch ) bf 0 ........(i) m m Let 1 and 2 , are roots of (i), then 1 2 bf m1 m2 m1 m2 ag 12 m1m2 ........(ii) f/a g/b Similarly m1m2 n1n2 ........(iii) g/b h/c From (ii) and (iii), we get 12 m1m2 n1n2 NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 f/a g/b h/c = .f/a ; mm = .g/b ; nn = .h/c 12 12 12 + m1m2 + n1n2 = f g h 12 a b c f gh { + m m + n n = 0} =0 Ans. (A) abc 12 12 12 Do yourself - 3 : ( i ) Find the angle between the lines whose direction ratios are 1, –2, 1 and 4, 3, 2. ( i i ) If a line makes and angle with axes, then prove that sin2 + sin2 + sin2 = 2. ( i i i ) Find the direction cosines of the line which is perpendicular to the lines with direction cosines proportional to (1, –2, –2) & (0, 2, 1). 60 E
JEE-Mathematics PLANE 13 . DEFINITION : A geometrical locus is a plane, such that if P and Q are any two points on the locus, then every point on the line PQ is also a point on the locus. 1 4 . EQUATIONS OF A PLANE : The equation of every plane is of the first degree i.e. of the form ax + by + cz + d = 0, in which a, b, c are constants, not all zero simultaneously. ( a ) Equation of plane passing through a fixed point : Vector form : If a be the position vector of a point on the plane and n be a vector normal to the plane then it’s vectorial equation is given by . 0 = d , where r a n r.n d a .n constant. Cartesian form : If and aˆi bˆj ckˆ , then cartesian equation of plane will be a(x1, y1, z1 ) n a(x – x1) + b(y – y1) + c(z – z1) = 0 ( b ) Plane Parallel to the Coordinate Planes : (i) Equation of yz plane is x = 0. (ii) Equation of zx plane is y = 0. (iii) Equation of xy plane is z = 0. (iv) Equation of the plane parallel to xy plane at a distance c is z = c or z = –c. (v) Equation of the plane parallel to yz plane at a distance c is x = c or x = –c (vi) Equation of the plane parallel to zx plane at a distance c is y = c or y = –c. (c) Equations of Planes Parallel to the Axes : If a = 0, the plane is parallel to x-axis i.e. equation of the plane parallel to x-axis is by + cz + d = 0. Similarly, equations of planes parallel to y-axis and parallel to z-axis are ax + cz + d = 0 and ax + by + d = 0, respectively. NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 ( d ) Equation of a Plane in Intercept Form : Equation of the plane which cuts off intercepts a, b, c from the axes x, y, z respectively is x + y + z = 1 . abc ( e ) Equation of a Plane in Normal Form : O n Vector form : If nˆ is a unit vector normal to the plane from the origin and d be the perpendicular distance of plane from origin then its r vector equation is r . nˆ = d . Cartesian form : If the length of the perpendicular distance of the plane from the origin is p and direction cosines of this perpendicular are (, m, n), then the equation of the plane is x + my + nz = p. ( f ) Equation of a Plane through three points : Vector form : If A, B, C are three points having P.V.'s a, b, c respectively, then vector equation of the plane is [ r a b ] [ r b c ] [ r c a ] [ a b c ] . E 61
JEE-Mathematics Cartesian form :The equation of the plane through three non-collinear points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is x – x1 y – y1 z – z1 x2 – x1 y2 – y1 z2 – z1 = 0 x3 – x1 y3 – y1 z3 – z1 Illustration 10 : Find the equation of the plane through the points A(2, 2, –1), B(3, 4, 2) and Solution : C(7, 0, 6). The general equation of a plane passing through (2, 2, –1) is a (x – 2) + b (y – 2) + c (z + 1) = 0 ........(i) It will pass through B (3, 4, 2) and C (7, 0, 6) if a (3 – 2) + b (4 – 2) + c (2 + 1) = 0 or a + 2b + 3c = 0 ........(ii) and a (7 – 2) + b (0 – 2) + c (6 + 1) = 0 or 5a – 2b + 7c = 0 ........(iii) Solving (ii) and (iii) by cross-multiplication, we have a b c or a b c (say) 14 6 15 7 2 10 5 2 3 a = 5, b = 2 and c = –3 Substituting the values of a, b and c in (i), we get 5 (x –2) + 2 (y – 2) – 3 (z + 1) = 0 or 5(x – 2) +2 (y – 2) –3 (z + 1) = 0 5x + 2y – 3z = 17, which is the required equation of the plane Ans. Illustration 11 : A plane meets the co-ordinates axis in A,B,C such that the centroid of the ABC is the point Solution : xyz (p,q,r) show that the equation of the plane is pqr =3 Let the required equation of plane be : xyz ......(i) =1 abc Then, the co-ordinates of A, B and C are A(a, 0, 0), B(0, b, 0), C(0, 0, c) respectively a b c NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 So the centroid of the triangle ABC is 3 , 3 , 3 But the co-ordinate of the centroid are (p,q,r) abc = p, = q, = r 333 xyz Putting the values of a, b and c in (i), we get the required plane as 3p 3q 3r =1 xyz pqr =3 Ans. Do yourself - 4 : E ( i ) Equation of a plane is 3x + 4y + 5z = 7. (a) Find the direction cosines of its normal (b) Find the points where it intersects the axes. (c) Find its intercept form. (d) Find its equation in normal form (in cartesian as well as in vector form) ( i i ) Find the equation of the plane passing through the points (2, 3, 1), (3, 0, 2) and (–1, 2, 3). 62
JEE-Mathematics 1 5 . ANGLE BETWEEN TWO PLANES : Vector form : If r .n1 d1 and r .n2 d2 be two planes, then angle between these planes is the angle between their normals c o s n 1 . n2 | n 1 || n 2 | Planes are perpendicular if n 1 . n2 0 and they are parallel if n 1 n2 Cartesian form : Consider two planes ax + by + cz + d = 0 and a' x + b' y + c' z + d' = 0. Angle between these planes is the angle between their normals. cos aa ' bb ' cc ' a 2 b 2 c2 a '2 b '2 c '2 Planes are perpendicular if aa' + bb' + cc' = 0 and they are parallel if a b c . a' b' c' Planes parallel to a given Plane : Equation of a plane parallel to the plane ax + by + cz + d = 0 is ax + by + cz + d' = 0. d' is to be found by other given condition. Illustration 12 : Find the angle between the planes x + y + 2z = 9 and 2x – y + z = 15 Solution : We know that the angle between the planes a x + b y + c z + d = 0 and 1 111 ax + by + cz + d = 0 is given by cos = a1a2 b1b2 c1c2 2 2 2 2 a 2 b12 c12 a 2 b 2 c 2 1 2 2 2 Therefore, angle between x + y + 2z = 9 and 2x – y + z = 15 is given by cos = (1)(2) (1)(1) (2)(1) 1 = Ans. 12 12 22 22 (1)2 12 2 3 Illustration 13 : Find the equation of the plane through the point (1, 4, –2) and parallel to the plane –2x + y – 3z = 7. NODE6\\E\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Maths\\Unit#10\\ENG\\03-3D-COORDINATE GEO.p65 Solution : Let the equation of a plane parallel to the plane –2x + y – 3z = 7 be –2x + y – 3z + k = 0 This passes through (1, 4, –2), therefore (–2) (1) + 4 – 3 (–2) + k = 0 –2 + 4 + 6 + k = 0 k = –8 Putting k = –8 in (i), we obtain –2x + y – 3z – 8 = 0 or –2x + y – 3z = 8 Ans. This is the equation of the required plane. Do yourself - 5 : ( i ) Prove that the planes 3x – 2y + z + 17 = 0 and 4x + 3y – 6z – 25 = 0 are perpendicular. ( i i ) Find the angle between the planes 3x + 4y + z + 7 = 0 and –x + y – 2z = 5 1 6 . A PLANE THROUGH THE LINE OF INTERSECTION OF TWO GIVEN PLANES : Consider two planes u ax + by + cz + d = 0 and v a' x + b' y + c' z + d' = 0. The equation u + v = 0, a real parameter, represents the plane passing through the line of intersection of given planes and if planes are parallel, this represents a plane parallel to them. E 63
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