Revision Plan-II (Dpp # 4)_mathematics_english

8/16/2019 Revision Plan-II (Dpp # 4)_mathematics_english

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DATE : 20.04.2016 PART TEST-02 (PT-02)

Syllabus : Sequence & Series and Binomial Theorem

TARGET : JEE (Advanced) 2016TEST INFORMATION

Course : VIJETA(ADP) & VIJAY(ADR) Date:17-04-2016

DPP

NO.

04

MMAATTHHEEMMAATTIICCSS

DDPPPP DDAAIILLYY PPRRAACCTTIICCEE PPRROOBBLLEEMMSS

TTEESSTT IINNFFOORRMMAATTIIOONN

REVISION DPP OFSEQUENCE & SERIES AND BINOMIAL THEOREM

Total Marks : 135 Max. Time : 120 min.Single choice Objective ('–1' negative marking) Q.1 to Q.17 (3 marks 3 min.) [51, 51]Multiple choice objective ('–1' negative marking) Q.18 to Q.32 (4 marks 3 min.) [60, 45] Subjective Questions ('–1' negative marking) Q.33 to Q.37 (3 marks 3 min.) [15, 15] Comprehension ('–1' negative marking) Q.38 to Q.40 (3 marks 3 min.) [9, 9]

1. The sum3 4

1! 2! 3! 2! 3! 4!

+ . . . . +

2016

2014! 2015! 2016! is equal to

(A)1 1

–2 2014!

(B)1 1 –

2 2016! (C)

1

2016!– 2018! (D)

1 1 –

2017! 2018!

2. Let A,G,H are respectively the A.M., G.M. and H.M. between two positive numbers. If xA = yG = zHwhere 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 + 2x

2)743

(2 + 3x – 4x2)

744 is

(A) 2974 (B) 1487 (C) 1 (D) 0

4. If ai, i = 1, 2, 3, 4 be four real numbers of same sign then the minimum value ofi

j

a

a where i, j {1,2 3, 4} and i j is(A) 6 (B) 8 (C) 12 (D) 24

5. The value of1

13

21

13

41

13

81

13

.........to is

(A) 3 (B) 6/5 (C) 3/2 (D) 2

6. The remainder, when 1523 + 2323 is divided by 38, is

(A) 4 (B) 17 (C) 23 (D) 0

7. The value of 20

220r

r 0

r 20 – r C is equal to

(A) 400 . 39C20

(B) 400 . 40C19

(C) 400 . 39C19

(D) 400 . 38C20

8. The term independent from ‘x’ in the expansion of

301

1 xx – 1

is

(A) 30C20

(B) 0 (C) 30C10

(D) 30C5

9. If cos(x – y), cos y, cos(x + y) are in H.P. , then the value of |cos y . secx

2| is equal to (x 2n)

(A) 2 (B) 1 (C) 2 (D) None of these


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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

(A) c3n2(n + 1)2 (B) 2c3n2(n + 1)2 (C)32c

3 n2(n+1)(2n+1) (D)

2

3c3n2(n–1)(2n– 1)

11_. Sum to n terms of the series

tan sec2 + tan2.sec22 + ........ + tan2n–1.sec2n.(A) tan2 – tan2n–1 (B) tan2n – tan (C) tan – tan2n (D) tan2n–1 – tan2

12_. Let f(n) =n n

kr

r 0 k r

C

. The total number of divisors of f(9) is :

(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 colouredred and the annular regions are coloured alternately green & red, such that no two adjacent regions areof 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 + 20x 2) –1 is given by(A) 5n – 4n (B) 5n + 1 – 4n + 1 (C) 5n + 1 – 4n – 1 (D) 5n – 1 – 4n + 1

15. If ninth term in the expansion ofx– 1

3

x– 13

111

log (9 7)31

log (3 1)8

13

3

is 660, then the value of x is

(A) 4 (B) 1 or 2 (C) 0 or 1 (D) 3

16. The value of50 5050 50

0 501 2C CC C – .........3 4 5 53

is equal to

(A) 1

503

0

x 1– x dx (B) 1

50

0

x 1– x dx (C)1

2652 (D)

1

70278

17_. The value ofn

C0. cosn +n

C1. cos(n – 2) +n

C2. cos(n – 4) + ........ +n

Cn. 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) t

50 = 502 – 1 (B) S

20 = 11460 (C) t

50 = 4.502 – 1 (D) S

20 = 11640

19. If a =20

20r

r 0

C

, b =

920

r

r 0

C

, c =

2020

r

r 11

C , then

(A) a = b + c (B) b = 219

–1

2 20

C10

(C) c = 219

+1

2 20

C10 (D) a – 2c = 102 1.3.5.....19

10!

20_. Consider the series2016.n + 2015.(n – 1) + 2014.(n – 2) + 2013.(n – 3) + ........ , where Sn is the sum of first n terns of theseries. Which of the following is/are true

(A) Sn =n(n 1)(6049 n)

6

(B) Sn =

n(n 1)(3025 n)

3

(C) S20 = 422030 (D) S20 = 420700

21. The natural numbers are written as a sequence of digits 123456789101112 . . . , then in thesequence(A) 190

th digit is 1 (B) 201

st digit is 3

(C) 2014

th

digit is 8 (D) 2013

th

digit is same as 2014

th

digit


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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

(A) |10 –9| =1

32 (B) 0 – 1, 1 – 2, 2 – 3, . . . are in G.P.

(C) 0 – 2, 2(1 – 2), 1 – 3 are in H.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)a m

b n – m 1

(B)

a n

b n – m 1

(C)

an

b (D)

am

b

25. If a, b, c are any three terms of an A.P. such that a b then b – ca – b

may be equal to

(A) 0 (B) 3 (C) 1 (D) 2

26. If Sn =1 5 11 19 29 41

........3! 4! 5! 6! 7! 8!

is sum of n terms of sequence then

(A) t100 =10099

102! (B) S2016 =

1 1 –

2 2018 2016!

(C) S2016 =

1 1 –

4 2018 2016! (D)

n

n

1lim S

2

27. If a1, a2, a3, . . . . . , are in A.P. with common difference d and bK = aK + aK+1 + . . . + aK+n–1 for K N then

(A)n

2K n

K 1

b n a

(B) n

2

K n

K 1

b n 1 a

(C) bK =n

2[an + a1 + 2d(K – 1)] (D)

n

K

K 1

b

= n(n + 1)an

28. If100

C6 + 4(100

C7) + 6(100

C8) + 4(100

C9) +100

C10 has valuexCy then x + y can take value

(A) 112 (B) 114 (C) 196 (D) 198

29. (2 – 3x + 2x2 + 3x3)20 = a0 + a1x + . . . + a60x60

, then

(A)30

2r–1

r 1

a 0

(B)30

40 202r

r 1

a 2 2

(C) a0 = 2 (D) a59 = 40(319)

30. If f(m) =m

203030–r m–r

r 0

C C

, then (if n < k then take nCk = 0)

(A) Maximum value of f(m) is 50C25

(B) f(0) + f(1) + f(2) + . . . . + f(25) = 249 +1

2. 50C

25

(C) f(33) is divisible by 37 (D) 50

2

m 0

f(m)

= 100C50


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31. If n

8 3 7 = 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 aG.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 +4 9 16

3 9 27 + . . . . . . . , then find the value of [S] (where [.] is G.I.F.)

34. The value ofnLim

n r 1n r

r tnr 1 t 0

1C C 3

5

is equal to

35. If , are roots of the quadratic equation ax2 + bx + c = 0 and , are roots of the quadratic equationcx2 + bx + a = 0. Such that , , , is an A.P. of distinct terms, then find the value of a + c.

36. If only 4

th

term in the expansion of

103x

2 8

has greatest numerical value, then find the number ofintegral values of x.

37. If 25C0 25C

2 + 2 . 25C

1 25C

3 + 3 . 25C

2 . 25C

4 + . . . . + 24 . 25C

23 . 25C

25 = 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 n

th 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.n

f(n)lim

g(n) =

(A) 1 (B) 2 (C) 3 (D) does not exist

39.n

F(n)lim

G(n) =

(A) 0 (B) 1 (C) 2 (D) does not exist

40.

nn

n n

F(n) f(n)lim – lim

G(n) g(n)

=

(A)e – 1

e 2 (B)

e 1

e e

(C)

1– e

e e (D)

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) 35. (AB) 36. (ACD)

37. B 38. C 39. ABC 40. ABC

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Revision Plan-II (Dpp # 4)_mathematics_english