MATHS
ASSIGNMENTS Exercise - 01
OBJECTIVE 1.
2.
If a1, a2, a3,.............are in AP then a p, aq, ar are in AP if p, q, r are in in (A) AP (B) GP (C) HP
(D) none of these
The product of n positive numbers num bers is unity. unity. Then their sum is (A) a positive integer
(B) divisible by n
(C) (C) eq equal to to n +
1 n
(D) never less than n
3.
If p, q, r , s N and they are four consecutive terms of an AP then the p th, q th, r th, s th terms of a GP in (A) AP (B) GP (C) HP (D) none of these
4.
If in a progression a 1, a2, a3,........, etc., (ar – ar + 1 ) bears a constant ratio with a r .ar + 1 then the terms of the progression are in (A) AP (B) GP (C) HP (D) none of these
5.
If
a 2a 3 a 1a 4
a a 3 2 3 then a1, a2, a3, a4 are in a1 a 4 a1 a 4
a 2 a3
(A) AP 6.
(B) GP
(C) HP
(D) none of these
Let x, y, y, z be three positive prime prim e numbers. The progression in which whi ch terms ( not necessarly consecutive ) is (A) AP (B) GP
(C) HP
x,
y,
z can be three
(D) none of these
7.
Let f(x) = 2x + 1. Then the number of real number of real values of x for which the three unequal numbers f(x), f(2x), f(4x) are in GP is (A) 1 (B) 2 (C) 0 (D) none of these
8.
If a r 0, r N , a1, a2, a3,.........., a2n are in AP then a1 a 2n a1
a2
a 2 a 2 n 1 a2
a3
(A) (n - 1) 9.
(B)
a 3 a 2n 2
a4
a3
.........
n (a 1 a 2 n ) a1
a n 1
If a1, a2, a3,........, a2n + 1 are in AP then
(A) 10.
n (n 1) a 2 a1 . 2 a n 1
(B)
n ( n 1) 2
a n 1 is equal to a n a n 1 an
n 1 (C) a 2 n 1 a 1 a 2 n 1 a 1
a1
a n 1
a 2n
(D) none of these
a2 a a ........... n 2 n is equal to a 2n a 2 a n 2 a n
(C) (n + 1) (a2 - a1)
(D) none of these
Let a1, a2, a3,........ be in AP and a p, aq, ar be in G.P. .P. Then aq : a p is equal to (A)
r p q p
(B)
q p r q
(C)
r q q p
(D) none of these 28
MATHS 11.
If in an AP, AP, t 1 = log10 a, tn + 1 = log10 b and t2n + 1 = log10 c then a, b, c are in (A) AP (B) GP (C) HP (D) none of these
12.
If n !, 3 n ! and (n + 1) ! are in GP then th en n !, 5 n ! and (n + 1) ! are in (A) AP (B) GP (C) HP (D) none of these
13.
In an AP, AP, the pth term t erm is q and the t he (p + q)th term is 0. Then the qth term is (A) - p (B) p (C) p + q (D) p - q
14.
In a sequence of (4n + 1) terms the first (2n + 1) terms as in i n AP whose common difference is 2, and the last (2n + 1) terms te rms are in GP whose common ratio is 0.5. If the middle terms of the AP and GP are equal then the middle term of the sequence is (A)
15.
n.2 2
n
n 1
(B)
1
2 2 2 If x 9 y 25z
(A) AP
n.2 2
2n
n 1
1
(C) n . 2n
(D) none of these
15 5 3 xyz then x, y, z in x y z (B) GP
(C) HP
(D) none of these
16.
If a, b, c, d and p are distinct real numbers such that (a2 + b2 + c2)p2 - 2(ab + bc + cd)p + (b 2 + c2 + d2) 0 then a, b, c, d are in (A) AP (B) GP (C) HP (D) none of these
17.
The largest term common to the sequences 1, 11, 21, 31,.........to 31,......... to 100 terms and 31, 36, 41, 46,....to 46,.. ..to 100 terms is (A) 381 (B) 471 (C) 281 (D) none of these
18.
The interior angles of a convex polygon are in AP, AP, the common difference being 5 0. If the smallest angle is (A) 9
2 3
then the number of sides is (B) 16
(C) 7
(D) none of these
19.
The minimum number of terms of 1 + 3 + 5 + 7 + ........that add up to a number exceeding 1357 is (A) 15 (B) 37 (C) 35 (D) 17
20.
In the value of 100! the number of zeros zero s at the end is (A) 11 (B) 22 (C) 23
(D) 24
21.
If ( 2n r ) r , n N, r N is expressed as the sum of k consecutive odd od d natural numbers then k is equal to (A) r (B) n (C) r + 1 (D) n + 1
22.
The sum of all odd proper divisiors divis iors of 360 is (A) 77 (B) 78 (C) 81 (D) none of these In the squence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,. ....... . .........., ...., where n consecutive terms te rms have the value n, the 150th term is (A) 17 (B) 16 (C) 18 (D) none of these
23.
29
MATHS 24.
In the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8,..............., where k consecutive terms have the value k (k = 1, 2, 4, 8,...), the 1025 th term is (A) 29 (B) 210 (C) 211 (D) 28
25.
Let {tn} be a sequence of integers in GP in which t 4 : t6 = 1 : 4 and t 2 + t5 = 216. Then t 1 is (A) 12 (B) 14 (C) 16 (D) none of these
26.
If log
5c , log 3 b a and log are in AP, where a, b, c are in G.P., then a, b, c are the a 5c 3 b
lengths of sides of (A) an isosceles triangle (C) a scalene triangle 27.
If x, 2y, 3z are in AP, where the distinct numbers x, y, z are in GP, then the common ratio of the GP is (A) 3
28.
(B) an equilateral triangle (D) none of these
(B)
1 3
(C) 2
If x > 0 and a is known positive number then the least values of ax + (A) a2
(B) a
(C) 2a
(D) a x
1 2
is (D) none of these
29.
If a, a 1, a 2, a3, ............a2n - 1 , b are in AP, a, b 1, b 2, b 3,......., b2n - 1 , b are in GP and a, c1, c2, c3,......., c2n - 1, b are in HP, where a, b are positive, then the equation a nx2 - bnx + cn = 0 has its roots (A) real and unequal (B) real and equal (C) imaginary (D) none of these
30.
If a, x, b are in AP, a, y, b are in GP and a, z, b are in HP such that x = 9z and a > 0, b > 0 then (A) | y | = 3z (B) x = 3 | y | (C) 2y = x + z (D) none of these
30
MATHS
Exercise - 02
SUBJECTIVE 1.
Find the sum of first 24 terms of the A.P. a 1 , a 2 , a 3 , ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
2.
The interior angles of a polygon are in arithmetic progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.
3.
The ratio between the sum of n terms of two A.P.’s is 7n + 1 : 4n + 27. Find the ratio between their n th terms.
4.
The r th , s th and t th terms of a certain G..P. are R, S and T respectively. Prove that R s–t . St–r . Tr–s = 1.
5.
The sum of three numbers in G.P. is 42. If the first two numbers are increased by 2 and third is decreased by 4, the resulting numbers form an A.P. Find the numbers of G.P.
6.
If one G.M. G and two arithmetic means p and q be inserted between any given numbers, then show that G2 = (2p – q) (2q – p).
7.
The sum of an infinite geometric series is 162 and the sum of its first n terms is 160. If the reciprocal of its common ratio is an integer, find all possible values of the common ratio, n and the first term of the series.
8.
Evaluate: 1 + 2.2 + 3.2 2 + 4.23 + ... + 100.2 99 .
9.
If pth, q th , r th terms of an A.P. be a, b, c respectively, then prove that p(b – c) + q (c – a) + r (a – b) = 0.
10.
Find S of the G.P. whose first term is 28 and the fourth term is
11.
If H be the H.M. between a & b, then show that (H – 2a) (H – 2b) = H 2
12.
Find the sum of n terms of the series, the rth term of which is (2r + 1) 2 r .
13.
Let x = 1 + 3a + 6a 2 + 10a3 + ..., |a| < 1; y = 1 + 4b + 10b 2 + 20b3 + ..., |b| < 1. Find S = 1 + 3 (ab) + 5 (ab)2 ... in terms of x and y.
14 .
After striking the floor a certain ball rebounds 4/5 th of the height from which it has fallen. Find the total distance that it travels before coming to rest if it gently dropped from a height of 120 meter.
15.
Show that the number 1111.....1 is a composite number..
4 49
.
91digits
31
MATHS
Exercise - 03
OBJECTIVE a
1.
2.
3.
If three numbers are in HP then the numbers obtained by subtracting half of the middle number from each of them are in (A) AP (B) GP (C) HP (D) none of these a, b, c, d, e are five numbers in which the first three are in AP and the last three are in HP. If the three numbers in the middle are in GP then the numbers in the odd places are in (A) AP (B) GP (C) HP (D) none of these If a, b, c are in AP then a
6.
3
(B)
2
(A)
9.
10.
ab
are in (C) HP
(D) none of these
2
(C)
3
1
(D) none of these
2
a1 a 2 n g1g 2 n
a 2 a 2 n 1 g 2g 2 n 1
..........
an
a n 1
g n g n 1
is equal to
(B) 2nh (C) nh (D) none of these h Let a1 = 0 and a1, a2, a3,.........., an be real numbers such that | a i | = | ai - 1 + 1| for all i then the AM of the numbers a 1, a2, a3,.........., an has the value A where 1
(B) A < -1
(C) A
1
(D) A
1
2 2 2 Let there be a GP whose first term is a and the common ratio is r. If A and H are the arithmetic mean and the harmonic mean respectively for the first n terms of the GP, A. H is equal to (A) a2 r n - 1 (B) a r n (C) a2 r n (D) none of these If the first and the (2n - 1) th terms of an AP, a GP and an HP are equal and their n th terms are a, b and c respectively then (A) a = b = c (B) a b c (C) a + c = b (D) ac - b2 = 0
a n b n a n 1 b n 1 (A) 0
11.
1
2n
(A) A 8.
ca
, c
Let a, b be two positive numbers, where a > b and 4 GM = 5 HM for the numbers. Then a is 1 (A) 4b (B) b (C) 2b (D) b 4 If a, a1, a2, a3,........, a2n, b are in AP and a, g 1, g2, g3, ............., g2n, b are in GP and h is the HM of a and b then
7.
1
The AM of two given positive numbers is 2. If the larger number is increased by 1, the GM of the numbers becomes equl to the AM of the given numbers. Then the HM of the given numbers is (A)
5.
, b
bc (B) GP
(A) AP 4.
1
is the HM between a and b if n is (B)
1 2
(C) -
1 2
(D) 1
1 1 is equal to P Q
If the harmonic mean between P and Q be H then H
(A) 2
(B)
PQ PQ
(C)
PQ PQ
(D)
1 2 32
MATHS 12.
If p, q, r be three positive real numbers, then the value of (p + q) (q + r) (r + p) is (A) > 8 pqr (B) < 8 pqr (C) 8 pqr (D) none of these
13.
Let t r r.(r!) . Then
15
t r is equal to
r 1
(A) 15 ! - 1
(B) 15 ! + 1
(C) 16 ! - 1
(D) none of these
14.
Four numbers are in arithmetic progression. The sum of first and last terms is 8 and the product of both middle terms is 15. The least number of the series is (A) 4 (B) 3 (C) 2 (D) 1
15.
If n arithmatic mean are inserted between 2 and 38 , then the sum of the resulting series is obtained as 200 , then the value of n is (A) 6 (B) 8 (C)9 (D)10
16.
The sum of three consecutive terms in a geometric progression is 14. If 1 is added to the first and the second term and 1 is subtracted from the third term. The resulting new terms are in arithmetic progression. Then the lowest of the original terms is (A) 1 (B) 2 (C) 4 (D) 8
17.
The angles of a triangle are in A. P. and the ratio of the greatest to the smallest angle is 3 : 1. Then the smallest angle is (A)
(B)
6
(C)
3
4
(D) none of these
18.
Let S be the sum, P be the product and R be the sum of the reciprocals of n terms of a GP. Then P2R n: Sn is equal to (A) 1 : 1 (B) (common ratio)n : 1 (C) (first term)2 : (common ratio) n (D) none of these
19.
If a2, b2, c2 are in A.P., then
b c (B) G.P.
(A) A.P. 20.
21.
22.
If
a
,
b ca
,
c a b (C) H.P.
(D) none of these
a n 1 b n 1
is the A.M. between a and b, then n is equal to a n b n (A) -1 (B) -2 (C) 0 (D) 1 The length of the side of a square is ‘a’ meter. A second square is formed by joining the middle points of the sides of the square. Then a third square is formed by joining the middle points of the sides of the second square and so on.Then the sum of the area of squares which carried up to infinity is (A) a (B) 2a2 (C) 3a2 (D) 4a2 If Sn
nP
difference is (A) P + Q
n 2
(n 1)Q , where Sn denotes the sum of the first n terms of an A.P., then common
(B) 2P + 3Q
(C) 2Q
(D) Q
33
MATHS 23.
The three sides of a right angled triangle are in G. P. The tangents of the two acute angles are (A)
(C)
24.
25.
5 1
5 1
and
2
2
1
5 and
5 1
(B)
2
y log z log x z log x log y is (D) 16
a, b, c are three positive numbers and abc 2 has the greatest value 1 2
, c
(C) a b c
2
(D) none of the foregoing pairs of numbers
5
If x, y, z are positive then the minimum value of x log y log z (A) 3 (B) 1 (C) 9
(A) a b
5 1
and
1
(B) a b
4
1
1 4
1 64
, c
. Then
1 2
(D) none of then
3
26.
The sum of all the numbers between 200 and 400 which are divisible by 7 is (A) 9872 (B) 7289 (C) 8729 (D) 8279
27.
Observe that1 3 = 1,23 = 3 + 5,33 = 7 + 9 + 11, 4 3 = 13 + 15 + 17 + 19.Then n 3 as a similar series is
n (n 1) n (n 1) n (n 1) 1 2 1 1 ........ 2 1 2n 3 2 2 2
(A) 2
(B) (n2 + n + 1) + (n2 + n + 3) + (n2 + n + 5) + ......+ (n2 +3n – 1) (C) (n2 – n + 1) + (n2 – n + 3) + (n2 – n + 5) + ......+ (n2 + n –1) (D) none of these 10
28.
Let t r 2 (A)
29.
30.
r / 2
2 21 1 10
2
2
20
r / 2
. Then
(B)
t
2 r
r 1
is equal to
2 21 1 10
2
19
(C)
2 21 1 2
20
1
The sum of the series : 1 2 – 22 + 32 – 42 + 52 – 62 + . . . – 1002 is (A) –10100 (B) –5050 (C) –2525
Suppose that F(n + 1) = (A) 50
2 F( n ) 1
2 (B) 52
(D) none of these
(D) –5500
for n = 1, 2, 3, . . . and F(1) = 2. Then F(101) equals (C) 54
(D) none of these
34
MATHS
Exercise - 04
SUBJECTIVE 1.
Let Sn denote the sum of first n terms of an A.P. If S 2n = 3Sn , then show that the ratio S 3n/Sn is equal to 6.
2.
If the roots of the equation x 3 – 12x2 + 39x – 28 = 0 are in A.P., then find the common difference.
3.
If x = 1 + a + a2 + a3 + ... to
(|a| < 1) and y = 1 + b + b 2 + b3 + ... to (|b| < 1), then prove that
1 + ab + a 2 b2 + a3 b3 + ... to
=
4.
x y 1
.
(i)
33 ...16 terms. Evaluate: 1 1 3 13 5
(ii)
Sum to n terms the series 1 2 – 22 + 32 – 42 + 52 – 62 + ...
13
13 23
xy
13 23
5.
If the A.M. of a and b is twice as great as their G.M., then show that a : b = (2 3):(2 3) .
6.
a, b, c are the first three terms of a geometric series, If the harmonic mean of a and b is 12 and that of b and c is 36, find the first five terms of the series.
7.
An AP and an HP have the same first term, the same last term and the same number of terms ; prove that the product of the r th term from the beginning in one series and the r th term from the end in the other is independent of r.
8.
The sum of first ten terms of an A.P. is equal to 155, and the sum of the first two terms of a G.P. is 9, find these progressions, if the first term of A.P. is equal to common ratio of G.P. and the first term of G.P. is equal to common difference of A.P.
9.
The series of natural numbers is divided into groups (1); (2, 3, 4); (5, 6, 7, 8, 9); . .. and so on. Show that the sum of the numbers in the nth group is (n – 1) 3 + n3.
10.
Suppose x and y are two real numbers such that the rth mean between x and 2y is equal to the rth mean between 2x and y when n arithmetic means are inserted between them in both the cases. Show that n 1 r
11.
y
1. x
An A.P. and a G.P. with positive terms have the same number of terms and their first terms as well as last terms are equal. Show that the sum of the A.P. is greater than or equal to the sum of the G.P.
35
MATHS 12.
Solve the following equations for x and y,
R|log S |T
10
x log10 x1 / 2 log10 x1/ 4 .......... 1 3 5...........(2 y 1)
y
4 7 10 ..........( 3y 1)
20
.
7 log10 x
13.
Given that ax = by = cz = du and a, b, c, d are in GP, show that x, y, z, u in HP.
14.
The first and last terms of an A.P. are a and b. There are altogether (2n + 1) terms. A new series is formed by multiplying each of the first 2n terms by the next term. Show that the sum of new series is
15.
(4 n
2
1)(a 2 b 2 ) (4n 2 2)ab 6n
.
Sum the following series to n terms and to infinity : (i)
1 13 . .5
1 35 . .7
1 5.7.9
...........
(ii)
n
(iii)
r( r 1)( r 2)( r 3)
(v)
If A = 1
B
=
2
(iv)
1
1
1
3
n
n 1
...........
n2 2
14 . .7
1 4.7 .1 0
1 71013 . .
...........
n
r 1
1
1
4r 1 1 r 1
2
and
R 1 2 3 ( n 1) U ......... V, 3.2 W T ( n 1)n n(n 1) ( n 1)( n 2)
S
then
show
that A = B.
36
MATHS
Inequalities 1.
(a) If xi > 0, (i = 1, 2, ... n), then prove that (x 1 + x2 + ... + x n)
1 1 1 2 ... x x n . x n 1 2
(b) If a1, a2, ......an are n non-zero real numbers, prove that a 1
2.
(i)
If
a 1,
a 2,
na 1a 2 .......a n (ii)
.......,
a n are
n
positive
2
........ a n 2
real
n2 a1
2
numbers,
....... a n show
2
.
that
a1n a 2 n ......... a n n .
If a, b, c are three distinct positive real numbers. Prove that
b c a
ca b
a b c
6 or,,
bc (b + c) + ca (c + a) + ab (a + b) > 6abc.
3.
(iii)
If a, b, c are three distinct positive a2 (1 + b2) + b2 (1 + c2) + c2 (1 + a2) > 6abc.
(iv)
If a, b, c, d are distinct positive real number, prove that a8(1 + b8) + b8 (1 + c8) + c8(1 + d8) + d8(1 + a8) > 8a3 b3 c3 d3.
(v)
Show that, if a, b, c, d be four positive unequal quantities and s = a + b + c + d, then (s - a) (s - b) (s - c) (s - d) > 81 abcd.
(vi)
If a, b, c, d are distinct positive real numbers such that 3s = a + b + c + d, then prove that abcd > 81(s - a) (s - b) (s - c) (s - d).
numbers,
prove
that
If ai < 0 for all i = 1, 2, .........., n prove that (i)
1 1 1 a1 a 2 ......... a n ....... n2 . an a1 a 2
(ii)
(1 - a1 + a12 ) ( 1 - a2 + a22 )..........(1 - an + an2) > 3n (a1 a2....an) (where n is even). ab
Prove that
a 2 b2 ab
5.
Prove that
x 2 y2 z2 xyz
6.
If none of b 1, b2, .........,bn is zero, prove that
4.
real
a a bb x y z
x
x
y
y z
z
x y z 3
x yz
2
a1 a ....... n (a12 ........ a n 2 ) ( b12 ....... b n 2 ) . b n b1 7.
Show that 1 2 ........ n
n
n 1 2
(n 1)3 / 2 .
37
MATHS 8.
By considering the sequence 1, a 2, a 4 ,..........a 2n ,........, where 0 < a < 1, prove that (i) (ii) 1 - a2n > nan - 1 (1 - a2) 1 - a2n < n(1 - a2) .
9.
If x, y, z are postive and x + y + z = 1, prove that
10.
If n5 < 5n for a fixed positive integer n 6 , show that (n + 1)5 < 5n + 1 .
11.
(i)
If a, b, c are the sides of a triangle , then prove that a 2 + b2 + c2 > ab + bc + ca.
(ii)
In a triangle ABC prove that
(iii)
If a, b, c be the length of the sides of a scalene triangle, prove that (a + b + c)3 > 27 (a + b - c) (b + c - a) (c + a - b) .
(iv)
If a, b, c are positive real numbers representing the sides of a scalene triangle, prove that
3 2
a bc
1 1 1 1 1 1 8 x y z
b c a
c ab
2.
b 2 c 2 ab + bc + ca < a + b + c < 2 (ab + bc + ca) or , 1 2 and hence prove ab bc ca (a b c) 2 2 4. that 3(ab + bc + ca ) < ( a + b + c ) < 4(ab + bc + ca) or 3 ab bc ca 2
12.
13.
2
a
2
If A, B and C are the angles of a triangle, prove that :
A sin B sin C 1 2 2 2 8
A cos B cos C 3 3 2 2 2 8
(i)
sin
(iii)
cos A + cos B + cos C
(i)
1 /( n 1) If n is a positive integer, prove that {( n 1)!}
3 2
(ii)
.
(iv)
cos
A tan 2 B tan 2 C 1 2 2 . 2
tan 2
1
(ii) (iii)
n 1
( n!)1/ n .
1 1 . If n is a positive integer, show that 1 1 n n 1 For every positive real number a 1 and for every positive integer n prove that
1 na 1 n
n 1
an .
By assigning weights 1 and n to the numbers 1 and 1 + (x/n) respectively, prove that if x x > – n, then 1 n 1
15.
n n 1
n
14.
2
Prove that
1 2n 1
n 1
n
x 1 . n
1.3.5..........(2n 1) 2.4.6..........2n
n 1 2n 1
nI.
38
MATHS
Exercise - 05 IIT NEW PATTERN QUESTIONS Section I
Fill in the blanks
1.
The sum of three numbers in A.P. is 15 whereas sum of their squares is 83. The numbers are...
2.
Four numbers in A.P. whose sum is 20 and sum of their squares is 120, are... .....
3.
The sum of all two–digit numbers which when divided by 4, yield unity as remainder is.........
4.
Let x, | x + 1 | and | x – 1 | are the I st three terms of an A.P., its sum of I st upto 20 terms is..........
5.
If a, b, c are the positive reals numbers and (a – c) 2 = 4(b2 – ac) then. a, b, c are in ...... ....
Section II More than one correct : 1.
2.
3. 4.
If the first two terms of a progression are 8 and 4 respectively, then correct statement(s) is/are: (A) if third term is 2, then the term are in G. P. (B) if third term is 0, then the terms are in A. P. 8 (C) if third term is , then the terms are in H. P.. 3 (D) if the third term is 3, then terms are in A. P. The next term of G. P. x, x 2 + 3, x3 + 15 can be 1029 (A) 16 (B) 32 (C) (D) 64 16 If 40 +
155
150
(C)
4
G1G 2 H1H 2
H1H 2 A1
A2
4
+
145
+ . . . + n terms = 625 then n can be 4 (A) 24 (B) 25 (C) 40 (D) 37 Let a and b be two positive real numbers. Suppose A 1, A2 are two Arithmetic means, G 1, G2 and H1, H2 are Geometric and Harmonic Means between a and b, then (A)
+
A1 A 2
(B)
H1 H2
9ab
(D)
(2a b) (a 2b)
G1G 2 H1H 2
–
2 a b 9 9b a
H1
G1G 2 H1H 2
5
H2 A1 A 2
Section IIIAssertion/Reason (A)
1.
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 Statement - 1 is False, Statement - 2 is True (D) Let a, b, c and d be distinct positive real numbers in H.P. Statement–1(A): a+d>b+c 1 1 1 1 Statement–2(R) : a
2.
Statement-1(A) :
d
b
c
P is a point (a, b, c). Let A, B, C be images of P in yz, zx and xy plane
respectively, then equation of plane must be Statement-2(R) :
x
y
z
1.
a b c The direction ratio of the line joining origin and point (x, y, z) must be x, y, z. 39
MATHS 3.
Statement-1(A): 11 11 …… 1 (up to 91 terms) is a prime number. Statement-2(R): If
b c a c a b a b c , , a b c
1 1 1 , , are also in A.P.. a b c Statement-1(A): If a(b – c) x 2 + b (c – a) x + c(a – b) = 0 has equal roots, then a, b, c are in H.P. Statement-2(R): Sum of the roots and product of the root are equal Are in A.P., then
4.
Section IV Comprehensions Write Up I The sum of n terms of a series each term of which is composed of r factors in arithmetical progression, the first factors of the several terms being in the same arithmetical progression. Let the series be denoted by u 1 + u2 + u3 + ........+ un, where
un = (a nb) (a n 1 b . ) (a n 2 b . )......(a n r 1 b . ) . Replacing n by n – 1, we have un – 1 = (a n 1 b . ) (a nb) (a n 1 b . )........(a n r 2 b . ) (a n 1 b . ) un
a n r 1 b . u n 1 v n , say
Replacing n by n + 1 we have (a n r b . ) un
v n 1
Therefore by subtraction ; (r + 1) b . un = vn + 1 – vn. Similarly, (r + 1) b . u n – 1 = v n – vn – 1 , ......................................... (r + 1) b . u 2 = v3 – v2, (r +1) b . u 1 = v2 – v1. By addition, (r + 1) b . S n = vn + 1 – v1 ; that is,
1.
2.
v n 1 v1 (r 1) b
=
(a n r b . )u n (r 1) b
C , say ;
where C is a quantity independent of n, which may be founded by ascribing to n some particular value. The above result gives us the following convenient rule : Write down the nth term, affic the next factor at the end divide by the number of factors thus increased and by the common difference. The sum of n terms of series 1 . 3 . 5 + 3 . 5 . 7 + 5 . 7 . 9 + ... ........ is (A) n(2n 3 + 8n2 + 7n – 2) (B) (2n 3 + 8n2 – 7n – 2) (C) n(2n 3 – 8n 2 + 7n – 2) (D) none of these The sum of n terms of series 1 . 5 . 9 + 2 . 6 . 10 + 3 . 7 . 11 + ......... (A)
n 4
(n 1) ( n 8) ( n 9)
(B)
n 4
( n 1) (n 8) ( n 9)
n
(n 1) (n 8) (n 9) (D) none of these 4 The sum of n terms of series 1 . 2 . 3 . 4 + 2 . 3 . 4 . 5 + 3 . 4 . 5 . 6 + ......... (C)
3.
S n =
(A)
(C)
n 5
n 5
n
n ( n 1) (n 2) ( n 3) ( n 4)
(B)
n (n 1) (n 2) (n 3) (n 4)
(D) none of these
5
n ( n 1) ( n 2) ( n 3) (n 4)
40
MATHS Write Up II Suppose a series of n terms is given by S n = t1 + t2 + t3 + ...+ tn.
Then S n–1 = t1 + t2 + t3 + ...+ tn–1, n > 1. Subtracting we get, S n – Sn–1 = t n , n 2 . Further if we
1.
2.
put n = 1 in the first sum then S 1 = t1. Thus we can write t n = Sn – Sn–1 , n 2 and t 1 = S1. The above results can be used to find the terms of any kind of series, independent of its nature, provided the sum to first n terms is given. If sum to n terms of a series is of the form an 2 + bn, where a and b are constants, then the fourth term of the series is (A) 5a + b (B) 7a + b (C) 9a + 3b (D) 16a + 4b The sum of n terms of a series is a . 2 n – b, where a and b are constants then the series is (A) A.P. (B) G.P. (C) A.G.P. (D) G.P. from second term onwards
3.
n
If the sum of n terms of a series is a . 2 – b, then the sum
1
t r 2
(A) a
a
(B)
(C)
2
1 a
Section VI Match the Column Match the series in Column I with their sum in Column II 1. Column I
(A)
(B)
(C)
(D)
1 – 1 1 – 1 1 – 1 22 32 42 ..... to infinity equals 23 –1 33 –1
23 1 33 1 1.2.3
1 1.2.3.4
1
1 2.3.4
43 – 1 43
3.4.5
1 2.3.4.5
(D)
(p)
(q)
+..... to inifinity equals
(r)
1 3.4.5.6
+....to
2 a
Column II
....to inifinity equals
1 1
r
(s)
1 4
1 2 2 3 1 18
inifnity equals 2.
3 4 a b Consider the matrices A = and B = 0 1 and let P be any orthogonal matrix and 1 1 Q = PAPT and R = P TQK P also S = PBP T and T = PTSK P Column I (A) If we vary K from 1 to n then the first row first column elements at R will form (B) If we vary K from 1 to n then the 2nd row 2nd column elements at R will form (C) If we vary K from 1 to n then the first row first column elements of T will form (D) If we vary K from 3 to n then the first row 2nd column elements of T will represent the sum of
Column II (p) G.P. with common ratio a
(q) A.P. with common differece 2 (r) G.P. with common ratio b (s) A.P.with common difference-2
41
MATHS
Exercise - 06
AIEEE FLASH BACK 1.
If 1, log9 (31 – x + 2), log3 (4.3x – 1) are in A.P. then x equals (A) log3 4
2.
(B) 1 – log3 4
(C) 1 – log4 3
(D) log4 3
Fifth term of an GP is 2, then the product of its 9 terms is (A) 256
3.
[AIEEE-2002]
(B) 512
[AIEEE-2002]
(C) 1024
(D) none of these
Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is
[AIEEE-2002]
(A) 5 4.
(B) 3/5
[AIEEE-2002]
(B) –425
(C) 475
(D) –475
The sum of integers from 1 to 100 that are divisible by 2 or 5 is (A) 3000
6.
(D) 1/5
13 – 23 + 33 – 43 +....+ 93 = (A) 425
5.
(C) 8/5
(B) 3050
[AIEEE-2002]
(C) 3600
(D) 3250
Let f(x) be a polynomail function of second degree. If f(1) = f(–1) and a, b, c are in A.P. then f ' (A), f ' (B), f ' (C) are in
7.
(A) Arithmetic - Geometric Progression
(B) A.P.
(C) G.P.
(D) H.P.
The sum of the series 1 1.2
–
1 2.3
(A) loge 8.
[AIEEE-2003]
1 3.4
[AIEEE-2003]
...... up to is equal to
4 e
(B) 2 loge 2
(C) loge 2 – 1
(D) loge 2
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equtation
[AIEEE-2004]
(A) x2 – 18x – 16 = 0 (B) x2 – 18x + 16 = 0 (C) x2 + 18x – 16 = 0 (D) x2 + 18x + 16 = 0 9.
Let Tr be the rth term of an A.P. whose first term is a and common difference is d. If for some positive 1 1 [AIEEE-2004] integers m, n, m n, Tm = and Tn = , then a – d equals n m (A)
10.
1 m
1 n
(B) 1
(C)
2
2
The sum of the first n terms of the series 1 + 2.2
1
(D) 0
mn
2
2
2
2
+ 3 + 2.4 + 5 + 2.6 ... is
n is even. when n is odd the sum is
n(n 1)2 (A) 2
(B)
n 2 (n 1) 2
n(n 1) 2 2
when
[AIEEE-2004]
(C)
n(n 1) 2 4
(D)
3n(n 1) 2
42
MATHS
11.
The sum of series
(A)
2!
a
If x =
4!
,y
1
6!
c
n
n 0
n
[AIEEE-2004]
(C)
2e
b , z
n 0
+..... is
(e –1) 2
(B)
e n
1
(e 2 – 2)
12.
1
(e2 –1) 2e
(D)
(e2 –1) 2
where a, b, c are in A.P. and |a| < 1, |b|, < 1, |c| < 1 then x, y, z
n 0
are in
13.
[AIEEE-2005]
(A) G.P.
(B) A.P.
(C) Arithmetic - Geometric Progression
(D) H.P.
If in a ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P, then sin A, sin B and sin C are in
14.
(A) G.P.
(B) A.P.
(C) Arithmetic - Geometric Progression
(D) H.P.
The sum of the series 1+
(A)
15.
1 4.2!
1 16.4!
[AIEEE-2005]
1 64.6!
e –1
e 1
(B)
e
(C)
e
e –1 2 e
....a p p 2 2 , p q , then Let a1, a2, a3 ... be terms on A.P. If a1 a 2 ....a q q 41
7
(B)
11
(C)
2
2 7
(D)
a6 a 21
e 1 2 e
equals [AIEEE-2006]
(D)
11 41
If a1, a2,...an are in H.P., then the expression a 1a2 + a2a3 +...+an–1an is equal to (A) n(a1 – an)
17.
...... ad inf. is
a1 a 2
(A) 16.
[AIEEE-2005]
The sum of series
(A)
–
e
(B) (n – 1) (a1 – an)
2!
–
1 3!
1 2
(B)
e
1 4!
(C) na1an
[AIEEE-2006]
(D) (n – 1) a 1an
– .... up to infinity is
[AIEEE-2007]
1 2
(C) e –2
(D) e –1
43
MATHS
Exercise - 07
IIT FLASH BACK (OBJECTIVE ) (A)
Fill in the blanks
1.
The sum of integers from 1 to 100 that are divisible by 2 or 5 is.................
2.
The solution of the equation log7 log5
3.
The sum of the first n terms of the series 1 2 + 2.22 + 32 + 2.42 + 52 + 2.62 + ..........is n (n + 1)2/2, when n is even. When n is odd, the sum is........... [IIT - 88]
4.
Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the [IIT - 92] two number are in the ratio............
5.
Let n be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of (1 + x) n [IIT - 94] are in A.P., then the value of n is............
6.
For any odd integer n 1 , n3 - (n - 1) 3 + ..........+ (-1)n-1 13 = ..............
7.
x = 1 + 3a + 6a 2 + 10a3 + ..........| a | < 1 [REE-96] 2 3 2 y = 1 + 4b + 10b + 20b + ....... | b | < 1, find S = 1 + 3ab + 5(ab) + ..........in terms of x and y.
8.
Let p and q be roots of the equation x 2 - 2x + A = 0, and let r and s be the roots of the equation x2 - 18x + B = 0. If p < q < r < s are in arithmetic progression, then A =............................., and B =.................. [IIT - 97]
9.
Let x be the arithmetic mean and y, z be the two geometric means between any two positive numbers. Then
y3 z 3 xyz
x 5 x
0 is ...............
[IIT - 84] [IIT - 86]
= ............
[IIT - 96]
[IIT - 97]
(B)
Multiple choice questions with one or more than one correct answer :
1.
If the first and the (2n – 1) st terms of an A.P., a G.P. and an H.P. are equal and their n th terms are a, b and c respectively, then [IIT - 88] (A) a = b = c (B) a b c (C) a + c = b (D) ac – b2 = 0
2.
Indicate the correct alternative(s), for 0 / 2 , if : x
cos n0
2n
, y sin , z cos2 n sin2 n then :
(A) xyz = xz + y
2n
n 0
[IIT - 93]
n0
(B) xyz = xy + z
(C) xyz = x + y + z
(D) xyz = yz + x
44
MATHS 3.
Let Tr be the r th term of an AP, for r = 1, 2, 3...........If for some positive integers m, n we have Tm = (A)
4.
1 n
and Tn =
1 m
1
(B)
mn
1 m
[IIT - 98]
1
(C) 1
n
If x > 1, y > 1, z > 1 are in GP, then (A) AP
5.
, then Tmn equals :
1
1
,
1
,
1 n x 1 n y 1 n z
(B) HP
Let n be an odd integer. If sin n
(D) 0 are in :
(C) GP n
b sin r
r
[IIT - 98]
(D) none of these
, for every value of , then
[IIT - 98]
r 0
(A) b0 = 1, b1 = 3 (C) b0 = –1, b1 = n
6.
(B) b0 = 0, b1 = n (D) b0 = 0, b1 = n2 + 3n + 3
For a positive integer n, let a(n) = 1
1 2
1
1
1
4
(2 ) 1
........ 3
n
, then
(A) a (100) 100
(B) a (100) 100
(C) a ( 200) 100
(D) a(200) > 100
(C)
Multiple choice questions with one correct answer :
1.
The third term of geometric progression is 4. The product of the first five terms is (A) 43 (B) 45 (C) 44 (D) none of these
2.
[IIT - 82]
The rational number, which equals the number 2.357 with recurring decimal is (A)
3.
[IIT - 99]
2355 1001
(B)
2379
(C)
997
2355 999
(D) none of these
[IIT - 83]
If a, b, c are in G.P., then the equations ax 2 + 2bx + c = 0 and dx 2 + 2ex + f = 0 have a common d e f , , are in a b c (A) A.P. (B) G.P.
root if
4.
)
(B) (1, 2)
Sum of the first n terms of the series (A) 2n – n – 1
6.
(D) none of these
[IIT - 85]
(D) none of these
[IIT - 85]
If log0.3(x – 1) < log0.09(x – 1), then x lies in the interval (A) ( 2,
5.
(C) H.P.
The number log2 7 is (A) an integer (C) an irrational number
(B) 1 – 2 –n
(C) (–2, –1) 1 2
3
7
4
8
15
........ is equal to 16 (C) n + 2n – 1 (D) 2n + 1
[IIT - 88]
(B) a rational number (D) a prime number
[IIT - 90] 45
MATHS
7.
If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the [IIT - 98] relation : (A) 0 < M 1 (B) 1 M 2 (C) 2 M 3 (D) 3 M 4
8.
The harmonic mean of the roots of the equation d5 2 i x (A) 2
(B) 4
2
d4 5i x 8 2 5 0 is
(C) 6
[IIT - 99]
(D) 8
9.
Let a1, a2, ........a10, be in A.P. and h1, h2, ........, h10 be in H.P. If a1 = h1 = 2 and a 10 = h10 = 3 then a 4 [IIT - 99] h7 is : (A) 2 (B) 3 (C) 5 (D) 6
10.
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 : [IIT - 2000] (A) a =
11.
12.
13.
14.
15.
Let
7 4
,r=
3 7
(B) a = 2, r =
3 8
(C) a =
3 2
,r=
1 2
(D) a = 3, r =
P., then the integral values of p and q respectively, are [IIT - 2000] (A) -2, - 32 (B) -2, 3 (C) -6, 3 (D) -6, -32 If the sum of the first 2n terms of the A. P. 2, 5, 8, .............is equal to the sum of the first n [IIT - 2001] terms of the A.P. 57, 59, 61,..........., the n equals (A) 10 (B) 12 (C) 11 (D)13 [IIT –2001] Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd and bcd are (A) Not in A.P./G.P./H.P. (B) in A.P. (C) in G.P. (D) H.P. The number of solutions of log 4(x – 1) = log2(x – 3) is [IIT - 2001] (A) 3 (B) 1 (C) 2 (D) 0 Suppose a, b, c are in A.P. a 2, b2, c2 are in G.P. If a < b < c and a + b + c =
(A)
17.
4
, be the roots of x 2 - x + p = 0 and , be the roots of x 2 - 4x + q = 0. If , , , are in G..
3 2
, then the value of
a is
16.
1
[IIT - 2002]
1
(B)
1
(C)
1
1
2 2 2 2 3 3 An infinite G.P. has first term ‘x’ and sum ‘5’, then x belongs to (A) x < – 10 (B) –10 < x < 0 (C) 0 < x < 10 In the quadratic equation ax 2 + bx + c = 0,
(D)
1 2
1 2 [IIT - 2004]
(D) x > 10
b 2 4ac and and , 2 2 , 3 3 , are in
, are the root of ax 2 + bx + c = 0, then (A) 0 (B) b 0 (C) c 0 G.P. where
[IIT - 2005]
(D)
0
(D)
Assertion & Reason
1.
Suppose four distinct positive numbers a 1 , a2 , a3 , a4 are in G.P. Let b1 = a 1 , b 2 = b 1 + a 2 , b 3 = b 2 + a 2 and b 4 = b 3 + a 4 . The numbers b 1 , b2 , b3 , b4 are neither in A.P. nor in G.P. Statement - 1 : Statement - 2 : The numbers b 1 , b2 , b3 , b4 are in H.P.
46
MATHS
Exercise - 08
IIT FLASH BACK (SUBJECTIVE) 1.
If a 1, a 2 ,........, a n are in arithmetic progression, where a i > 0 for all i, show that 1 a1
a2
1 a2
a3
.........
1 a n 1
an
n 1 a1
an
.
[IIT - 82]
2.
Does there exist a geometric progression containing 27, 8 and 12 as three of its terms ? If it exits, how many such progressions are possible ? [IIT - 83]
3.
Find three numbers a, b, c between 2 and 18 such that (i) their sum is 25 (ii) the numbers 2, a, b sare consecutive terms of an A.P. and (iii) the numbers b, c, 18 are consecutive terms of a G.P. [IIT - 83]
4.
If 1, a1, a2,...,an – 1 are the n roots of unity, then show that (1 – a 1) (1 – a 2) (1 – a 3)....(1 – an–1) = n [IIT - 84]
5.
If a > 0, b > 0 and c > 0, prove that (a b c)
6.
. 2 2 p . 3 3 ......... p k k and p1, p2,....., pk are distinct primes, If n is a natural number such that n p1 1 p
1 1 1 9 . a b c
then show that n n k n 2 .
[IIT - 84]
n
7.
Find the sum of series :
(1)
r n
r 0
8.
1 3r 7 r 15r C r r 2 r 3r 4 r .......up to m terms 2 2 2 2
[IIT - 85]
The sum of the squres of three distinct real numbers, which are in G.P., is S 2. If their sum is a S, show
1 2 that a , 1 1, 3 . 3 9.
[IIT - 84]
[IIT - 86]
Solve for x the following equation : log(2x + 3) (6x2 + 23 x + 21) = 4 – log(3x + 7) (4x2 + 12x + 9).
[IIT - 87]
7 are in arithmetic progression, determine the value of x. 2
10.
x If log32 . log3(2x – 5) and log 3 2
11.
[IIT - 90] If p be the first of n arithmetic means between two numbers and q be the first of n harmonic means 2
n 1 p. between the same two numbers, prove that the value of q cannot be between p and n 1 [IIT - 91] 12.
If S1, S2, S3, ... Sn are the sums of infinite geometric series whose first terms are 1, 2,3, ..., n and whose common ratios are S12 S22 S32 ... S2n 12 .
1 1 1 1 , , ,..., respectively, then find the value of 2 3 4 n 1 [IIT - 91] 47
MATHS 13.
14. 15. 16. 17.
18.
The sum of the first ten terms of an AP is 155 and the sum of first two terms of a GP is 9. The first term of the AP is equal to the common ratio of the GP and the first term of the G P is equal to the [REE-93] common difference of the AP. Find the two progressions. th th th If the (m + 1) , (n + 1) and (r + 1) terms of an AP are in GP and m, n, r are in HP. Find that the [REE-94] ratio of the common difference to the first term of the AP. The real numbers x 1, x2, x3 satisfying the equation x 3 x 2 x 0 are in A.P. Find the [IIT - 96] intervals in which and lie. a, b, c are the first three terms of a geometric series. If the harmonic mean of a and b is 12 and that of b and c is 36, find the first five terms of the series. [REE-98] The sum of an infinite geometric series is 162 and the sum of its n terms is 160. If the inverse of its common ratio is an integer, find all possible values of the common ratio, n and the first terms of the series. [REE-99] [IIT - 99] Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations u + 2v + 3w = 6, 4u + 5v + 6w = 12, 6u + 9v = 4, then show that the roots of the equation
1 1 1 x 2 [( b c) 2 (c a ) 2 (d b) 2 ]x u v w 0 are reciprocals of each other.. u v w 19.
20. 21. 22.
The fourth power of the common difference of an arithmetic progression with integer entries added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of and [IIT - 2000] integer. 2 Given that , are roots of the equation, Ax - 4x + 1 = 0 and , the roots of the equation, Bx2 - 6x + 1= 0 , find values of A and B, such that , , and are in H.P.. [REE-2000] 2 The sum of roots of the equation ax + bx + c = 0 is equal to the sum of squares of their reciprocals. Find whether bc 2, ca2 and ab2 in A.P., G.P. or H.P. ? [REE-2000] Solve the following equations for x and y log2 x + log4 x + log16 x +................ = y
23.
24.
1 3 5.......... (2 y 1)
= 4 log4 x.
[REE-2001]
Let a1, a2,..............be positive real numbers in G.P. for each n, let A n, Gn, Hn , be respectively, the arithmetic mean, geometric mean and harmonic mean of a 1, a2, a3,..............an. Find an expression [IIT - 2001] for the G.M. of G1, G2, .........Gn in terms of A1, A2...........An, H1, H2, ..........Hn. Let a, b be positive real numbers. If a, A 1, A2, b are in arithmetic progression a, G 1, G2, b are geometric progression and a, H 1 , H 2 , b are in harmonic progression, show that G1G 2 H 1H 2
25.
5 9 13.......... (4 y 1)
A1 A 2 H1 H 2
(2a b) (a 2 b) 9ab
.
[IIT - 2002]
If a,b,c are in A.P. a 2,b2,c2 are in H.P. Then prove that either a = b = c or a , b, -c/2 form a G..P. [IIT - 2003]
26.
Prove that (a + 1) 7 (b + 1)7 (c + 1)7 > 77 a4 b4 c4, where a , b, c R .
[IIT - 2004]
27.
An infinite G.P has first term x and sum 5, then find the exhaustive range of x ?
[IIT - 2004]
28.
For n = 1, 2, 3, . . . , let 2
3
n
3 3 3 An = – + – . . . + (–1) n–1 , and Bn = 1 – An. 4 2 4 n Find the smallest natural number n 0 such that Bn > An for all n n0. 3
[IIT - 2006] 48
MATHS
ANSWER SHEET OBJECTIVE
Exercise - 01 1.
A
2.
D
3.
B
4.
C
5.
C
6.
D
7.
C
8.
B
9.
A
10.
C
11.
B
12.
A
13.
B
14.
A
15.
C
16.
B
17.
D
18.
A
19.
B
20.
D
21.
A
22.
A
23.
A
24.
B
25.
A
26.
D
27.
B
28.
C
29.
C
30.
B
SUBJECTIVE
Exercise - 02 900
1.
14n 6
3.
8n 23
2.
9
5.
6, 12, 24 OR 24, 12, 6
7.
1 1 1 r , or n = 4, 2 or 1 and a = 108, 144 or 160 3 9 81
8.
99.2100 + 1
12.
n+2
n2
n+1
–2
+2
10.
7/6
13.
S
1 ab (1 ab) 2
where a = 1 – x –1/3 & b = 1 – y –1/4
14. 1080 m
OBJECTIVE
Exercise - 03 1.
B
2.
B
3.
A
4.
A
5.
A
6.
A
7.
C
8.
A
9.
D
10.
A
11.
A
12.
A
13.
C
14.
D
15.
B
16.
B
17.
A
18.
A
19.
A
20.
D
21.
B
22.
D
23.
B
24.
A
25.
B
26.
C
27.
C
28.
B
29.
B
30.
B
49
MATHS
SUBJECTIVE
Exercise - 04 2. d 3
n(n 1)
, when n be even &
n(n 1)
4.(i)
446
6.
8, 24, 72, 216, 648
8.
A.P. is 2 + 5 + 8 + 11 + & G.P is 3 + 6 + 12 + 24 + or A.P. is & G.P. is
(ii)
2 3
25 3
2
625 6
2
, when n be odd
25 2
79 6
83 6
...
12.
x = 105, y = 10
15.
(i) Sn = (1/12) - [1/{4(2n + 1) (2n + 3)}] ; S = 1/12 (ii) Sn = (1/24) - [1/{6(3n + 1) (3n + 4) }] ; S = 1/24 (iii) (1/5)n (n + 1) (n + 2) (n + 3) (n + 4) (iv) n/(2n + 1)
IIT NEW PATTERN
Exercise - 05
Section I Fill in the blanks 1. 2. (3, 5, 7) (2, 4, 6, 8) A.P. 5.
3.
1210
4.
350 or 180
Section II More than one correct (A) (B)(C) (C) (D) 1. 2.
3.
(B) (C)
4.
(A) (B) (C)
Section III Assertion/Reason 1. 2. B B
3.
D
4.
C
Section IV Comprehensions Write Up I
1.
(A)
2.
(C)
3.
(C)
2.
(D)
3.
(C)
Write Up II
1.
(B)
Section VI Match the Column A - q, B -r, C - p, D - s 1. A-q, B-s, C-p, D-p 2.
50
MATHS
AIEEE FLASH BACK
Exercise - 06 1.
(B)
2.
(B)
3.
(B)
4.
(A)
5.
(B)
6.
(B)
7.
(A)
8.
(B)
9.
(D)
10.
(B)
11.
(B)
12.
(D)
13.
(B)
14.
(D)
15.
(D)
16.
(D)
17.
(D)
IIT JEE FLASH BACK (OBJECTIVE)
Exercise - 07 (A)
1.
3050
2.
4
3.
4.
4:1&1:4
5.
7
6.
7. S =
(B) 1. 4. (C) 1. 5. 9. 13. 17. (D) 1.
1 ab
Where a = 1 - x -1/3 and b = 1 - y-1/4
(1 ab) 2
n 1 2
n2 1 4
(2 n 1) ( n 1) 2
8.
-3, 77
BD B
2. 5.
BC B
3. 6.
C AD
B C D D C
2. 6. 10. 14.
C C D B
3. 7. 11. 15.
A A A D
9.
2
A B C B
4. 8. 12. 16.
C
IIT JEE FLASH BACK (SUBJECTIVE)
Exercise - 08 2.
Yes, infinite
3.
10.
3
12.
5, 8 , 12
7.
2mn 1 2 mn (2n 1)
1 1 , , , 3 27
17.
r 1 / 9 ; n = 2 ; a = 144
OR r 1 / 3 ; n = 4 ; a = 36
20.
A=3;B=8
A .P.
23.
bA , A
n
1
2
21.
1 4
3
14.
15.
n (2n 1) (4n 1) 3
13. (3 + 6 + 12 + .........) ; (2/3 + 25/3 + 625/6 + ...............) ; (2, 5, 8, ..........);
2
9.
,.......... A n gb H1 , H 2 ,........... H n g
22.
16.
F G 25 , 79 ,..............J I H 2 6 K
8, 24, 72, 216, 648
OR r = 1/81 ; n = 1 ; a = 160
x = 2 2 and y = 3
1 2n
28. least value of n 0 = 2 51
