MATHEMATICS DPP
DPP
TARGET : JEE (Advanced) 2016 T EST INFORM ATION
Course : VIJETA(ADP) & VIJAY(ADR)
Date:17-04-2016
NO.
04
DAILY PRACTICE PROBLEMS
TEST INFORMATION DATE : 20.04.2016
PART TEST-02 (PT-02)
Syllabus : Sequence & Series and Binomial Theorem
REVISION DPP OF SEQUENCE & SERIES AND BINOMIAL THEOREM Total Marks : 135 Single choice Objective ('–1' negative marking) Q.1 to Q.17 (3 marks 3 min.) Multiple choice objective ('–1' negative marking) Q.18 to Q.32 Subjective Questions ('–1' negative marking) Q.33 to Q.37 Comprehension ('–1' negative marking) Q.38 to Q.40
Max. Time : 120 min. [51, 51]
(4 marks 3 min.) (3 marks 3 min.) (3 marks 3 min.)
[60, 45] [15, 15] [9, 9]
3 4 2016 +....+ is equal to 1! 2! 3! 2! 3! 4! 2014! 2015! 2016! 1 1 1 1 1 1 1 (A) – (B) – (C) (D) – 2 2014! 2 2016! 2016!– 2018! 2017! 2018!
1.
The sum
2.
Let A,G,H are respectively the A.M., G.M. and H.M. between two positive numbers. If xA = yG = zH where x, y, z are non-zero quantities then x, y, z are in (A) A.P. (B) G.P. (C) H.P. (D) A.G.P.
3.
The sum of the coefficients of the polynomial obtained by collection of like terms after the expansion of (1 – 2x + 2x2)743(2 + 3x – 4x2) 744 is (A) 2974 (B) 1487 (C) 1 (D) 0 ai If ai, i = 1, 2, 3, 4 be four real numbers of same sign then the minimum value of aj
4.
where i, j {1,2 3, 4} and i j is (A) 6 (B) 8 5.
6.
(D) 24
1 1 1 1 The value of 1 1 2 1 4 1 8 .........to is 3 3 3 3 (A) 3 (B) 6/5 (C) 3/2
(D) 2
The remainder, when 1523 + 2323 is divided by 38, is (A) 4 (B) 17 (C) 23
(D) 0
20
7.
(C) 12
The value of
r 20 – r
20
Cr
2
is equal to
r 0
(A) 400 . 39C20
8.
9.
(B) 400 . 40C19
(C) 400 . 39C19
The term independent from ‘x’ in the expansion of 1 x (A) 30C20 (B) 0 (C) 30C10
x – 1 1
(D) 400 . 38C20 –30
If cos(x – y), cos y, cos(x + y) are in H.P. , then the value of |cos y . sec (A) 2
(B) 1
(C)
2
is (D) 30C5 x | is equal to (x 2n) 2 (D) None of these
Corporate Office : CG Tower, A-46 & 52, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) - 324005 Website : www.resonance.ac.in | E-mail :
[email protected] Toll Free : 1800 200 2244 | 1800 258 5555 | CIN: U80302RJ2007PLC024029
PAGE NO.-1
10.
If the sum of n terms of an A.P. is cn(n + 1), where c 0, then sum of cubes of these terms is 2 2c 3 2 (A) c3n2(n + 1)2 (B) 2c3n2(n + 1)2 (C) n (n+1)(2n+1) (D) c3n2(n–1)(2n– 1) 3 3
11_.
Sum to n terms of the series tansec2 + tan2.sec22 + ........ + tan2n–1.sec2n. n–1 n n (A) tan2 – tan2 (B) tan2 – tan (C) tan – tan2 n
12_.
Let f(n) =
n–1
(D) tan2
– tan2
n
k
Cr . The total number of divisors of f(9) is :
r 0 k r
(A) 7
(B) 8
(C) 9
(D) 6
13.
Concentric circles of radii 1, 2, 3, ......100 cm are drawn. The interior of the smallest circle is coloured red and the annular regions are coloured alternately green & red, such that no two adjacent regions are of the same color. Then the total area of green regions is Xgiven by (A) 1000 sq. cm (B) 5050 sq. cm (C) 4950 sq. cm (D) 5151 sq. cm
14.
The coefficient of xn in the expansion of (1 – 9x + 20x2)–1 is given by (A) 5n – 4n (B) 5n + 1 – 4n + 1 (C) 5n + 1 – 4n – 1
15.
1 x –1 1 log3 (9 7) If ninth term in the expansion of 3 3 1 x –1 log 1) 3 (3 38 (A) 4 (B) 1 or 2 (C) 0 or 1 50
16.
C0 – 3
The value of
50
C1 4
1
(A)
50
C2 ......... 5
x
1– x
50
(B)
dx
0
x 1– x
50
11
is 660, then the value of x is (D) 3
50
C50 is equal to 53
1 3
(D) 5n – 1 – 4n + 1
dx
0
(C)
1 2652
(D)
1 70278
17_.
The value of n C0. cosn + nC1. cos(n – 2) + nC2. cos(n – 4) + ........ + nCn. cos(n – 2n) is : (A) 2ncosn (B) 2nsinn (C) 2n+1cosn (D) 2n+1sinn
18.
If tn denotes the nth term and Sn denotes sum to first n terms of the series 3 + 15 + 35 + 63 + . . . . . ., then (A) t50 = 502 – 1 (B) S20 = 11460 (C) t50 = 4.502 – 1 (D) S20 = 11640
19.
If a =
20
20
9
20
Cr , b =
r 0
20
Cr , c =
r 0
19
20_.
Cr , then
r 11
(B) b = 2 – 1 2
20
C10
(D) a – 2c =
1 20 C10 2 210 1.3.5.....19 10!
Consider the series 2016.n + 2015.(n – 1) + 2014.(n – 2) + 2013.(n – 3) + ........ , where Sn is the sum of first n terns of the series. Which of the following is/are true n(n 1)(6049 n) 6 (C) S20 = 422030
(A) Sn =
21.
20
19
(A) a = b + c (C) c = 2 +
n(n 1)(3025 n) 3 (D) S20 = 420700
(B) Sn =
The natural numbers are written as a sequence of digits 123456789101112 . . . , then in the sequence (A) 190th digit is 1 (B) 201st digit is 3 th (C) 2014 digit is 8 (D) 2013th digit is same as 2014th digit Corporate Office : CG Tower, A-46 & 52, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) - 324005 Website : www.resonance.ac.in | E-mail :
[email protected] Toll Free : 1800 200 2244 | 1800 258 5555 | CIN: U80302RJ2007PLC024029
PAGE NO.-2
22.
If N = 72014, then (A) sum of last four digits of N is 23 (B) Number of divisors of N are 2014 (C) Number of composite divisors of N are 2013 (D) If number of prime divisors of N are p then number of ways to express a non-zero vector coplanar with two given non-collinear vectors as a linear combination of the two vectors is p + 1.
23.
Consider the sequence of numbers 0, 1, . . . . , n where 0 = 17.23, 1 = 33.23 and r+2 =
r r 1 . 2
Then 1 32 (C) 0 – 2, 2(1 – 2), 1 – 3 are in H.P.
(A) |10 –9| =
(B) 0 – 1, 1 – 2, 2 – 3, . . . are in G.P. (D) |10 – 9| = |8 – 7|
24.
Given 'n' arithmetic means are inserted between each of the two sets of numbers a, 2b and 2a, b where a, b R. If mth mean of the two sets of numbers is same then a m a n a a (A) (B) (C) n (D) m b n –m 1 b n –m 1 b b
25.
If a, b, c are any three terms of an A.P. such that a b then (A) 0
26.
(B)
(C) 1
3
(D) 2
1 5 11 19 29 41 ........ is sum of n terms of sequence then 3! 4! 5! 6! 7! 8! 10099 1 1 (A) t100 = (B) S2016 = – 102! 2 2018 2016!
If Sn =
(C) S2016 =
27.
b–c may be equal to a–b
1 1 – 4 2018 2016!
(D) lim Sn n
1 2
If a1, a2, a3, . . . . . , are in A.P. with common difference d and bK = aK + aK+1 + . . . + aK+n–1 for K N then n
(A)
n
bK n2 an
(B)
K 1
2
b
K
n 1 an
K 1 n
n [an + a1 + 2d(K – 1)] 2
(C) bK =
(D)
b
K
= n(n + 1)an
K 1
28.
If 100C6 + 4(100C7) + 6(100C8) + 4(100C9) + 100C10 has value xCy then x + y can take value (A) 112 (B) 114 (C) 196 (D) 198
29.
(2 – 3x + 2x2 + 3x3)20 = a0 + a1x + . . . + a60x , then
60
30
(A)
30
a
2r –1
(B)
0
r 1
If f(m) =
2r
240 220
19
(C) a0 = 2
(D) a59 = 40(3 )
r 1
m
30.
a
30
20
C30–r Cm–r , then (if n < k then take nCk = 0)
r 0
(A) Maximum value of f(m) is 50C25
(B) f(0) + f(1) + f(2) + . . . . + f(25) = 249 +
(C) f(33) is divisible by 37
(D)
50
f(m)
2
1 50 . C25 2
= 100C50
m 0
Corporate Office : CG Tower, A-46 & 52, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) - 324005 Website : www.resonance.ac.in | E-mail :
[email protected] Toll Free : 1800 200 2244 | 1800 258 5555 | CIN: U80302RJ2007PLC024029
PAGE NO.-3
31.
If 8 3 7
n
= I + f, where 'I' is an integer, n N and 0 < f < 1, then
(A) is an odd integer
(B) is an even integer (C) ( + f) (1 – f) = 1
(D) ( + f) (1 – f) = 2n
32.
Given four positive numbers in A.P. If 5, 6, 9 and 15 are added respectively to these numbers, we get a G.P. , then (A) Common ratio of G.P. is 3/2 (B) Common ratio of G.P. is 2/3 (C) Common difference of A.P. is 3 (D) First term and common difference of AP are equal
33.
If S = 1 +
34.
The value of Lim
4 9 16 + . . . . . . . , then find the value of [S] (where [.] is G.I.F.) 3 9 27 n
n
35.
r 1 1 n r t C C 3 is equal to r t n r 1 t 0
5
If , are roots of the quadratic equation ax2 + bx + c = 0 and , are roots of the quadratic equation cx2 + bx + a = 0. Such that , , , is an A.P. of distinct terms, then find the value of a + c. 10
36.
37.
3x If only 4th term in the expansion of 2 8 integral values of x.
has greatest numerical value, then find the number of
If 25C0 25C2 + 2 . 25C1 25C3 + 3 . 25C2 . 25C4 + . . . . + 24 . 25C23 . 25C25 = k . 49C + 50C , then find the value of 2k – – . (where , < 25)
Comprehension (Q. No. 38 to 40) Let f(n) denotes the nth term of the sequence 2, 5, 10, 17, 26, . . . . . and g(n) denotes the nth term of the sequence 2, 6, 12, 20,30, . . . . Let F(n) and G(n) denote respectively the sum of n terms of the above sequences. 38.
39.
f(n) = n g(n) (A) 1
(B) 2
(C) 3
(D) does not exist
F(n) = n G(n) (A) 0
(B) 1
(C) 2
(D) does not exist
lim
lim
n
n
40.
f(n) F(n) lim – nlim = n G(n) g(n)
(A)
e –1
e 1
(B)
e 2
(C)
e e
1– e
(D)
e e
e e 1 e
ANSWERKEY OF DPP # 03
1.
(D)
2.
(C)
3.
(A)
4.
(A)
5.
(B)
6.
(C)
7.
(C)
8.
(C)
9.
(B)
10.
(D)
11.
(C)
12.
(C)
13.
(C)
14.
(D)
15.
(C)
16.
(A)
17.
(C)
18.
(ABD)
19.
(AC)
20.
(ABCD) 21. (ABCD)
22.
(ABC) 23.
(ABC) 24.
(ACD)
25.
(CD)
26.
(AB)
27.
(AC)
28.
(BCD) 29.
(BD)
30.
(CD)
31.
(AC)
32.
(AD)
33.
(ACD)
34.
(BC)
(AB)
36.
(ACD)
37.
(B)
38.
(C)
39.
(ABC) 40.
35.
(ABC)
Corporate Office : CG Tower, A-46 & 52, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) - 324005 Website : www.resonance.ac.in | E-mail :
[email protected] Toll Free : 1800 200 2244 | 1800 258 5555 | CIN: U80302RJ2007PLC024029
PAGE NO.-4
