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Question

# If [.] denotes the greatest function, then answer the following by appropriately matching the lists based ​​​​​​on the information given in Column I and Column II Column IColumn IIa. 1∫−1[x+[x+[x]]] dxp. 3b. 5∫2([x]+[−x]) dxq. 5c. 3∫−1sgn (x−[x]) dxr. 4d. 25π/4∫0((tan6(x−[x])+tan4(x−[x])) dxs. −3

A
(a)(s),(b)(q),(c)(r),(d)(s)
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B
(a)(s),(b)(r),(c)(r),(d)(q)
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C
(a)(s),(b)(s),(c)(r),(d)(q)
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D
(a)(q),(b)(s),(c)(r),(d)(p)
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Solution

## The correct option is C (a)→(s),(b)→(s),(c)→(r),(d)→(q)(a) I=1∫−1[x+[x+[x]]] dx We know that, [x+n]=[x]+n ; if n is an integer ⇒I=31∫−1[x] dx⇒I=30∫−1−1 dx+31∫00 dx⇒I=−3 (a)→(s) (b) I=5∫2([x]+[−x]) dx We know that, [x]+[−x]={−1, if x is not an integer 0, if x is an integer ⇒I=−5∫21 dx=−3 (b)→(s) (c) 3∫−1sgn (x−[x]) dx We know that, sgn (x−[x])={1, if x is not an integer0, if x is an integer Hence, 3∫−1sgn (x−[x]) dx=4 (c)→(r) (d) I=25π/4∫0((tan6(x−[x])+tan4(x−[x])) dx We know that, 0<x≤π4⇒[x]=0 ⇒I=25π/4∫0tan6x+tan4x dx⇒I=25π/4∫0tan4x(tan2x+1) dx⇒I=25π/4∫0tan4xsec2x dx Put tanx=t⇒sec2x dx=dt ⇒I=251∫0t4 dt⇒I=255=5 (d)→(q)

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