IIT JEE & NEET Question Bank with Solution: Mathematical Induction: Mathematics MCQ

Q.1. $I f space space n greater than 1 comma space a n d space left parenthesis 1 plus x right parenthesis to the power of n space end exponent equals C subscript 0 plus end subscript C subscript 1 x plus C subscript 2 x squared plus... C subscript n x to the power of n space t h e n space v a l u e space o f space C subscript 0 minus end subscript 2 C subscript 1 plus 3 C subscript 2 minus 4 C subscript 3 plus.... plus left parenthesis negative 1 right parenthesis space blank to the power of n left parenthesis n plus 1 right parenthesis C subscript n space i s$

A) -1

B) 0

C) 1

D) 2

Explanation-

$P u t t i n g space x equals negative 1 space i n space t h e space e q u a t i o n left parenthesis 1 right parenthesis space o f space a l t e r n a t i v e space s o l u t i o n space t o space E x a m p l e space 9 comma space w e space g e t C subscript 0 minus 2 C subscript 1 plus 3 C subscript 2 minus 4 C subscript 3 plus... plus open parentheses negative 1 to the power of n close parentheses open parentheses n plus 1 close parentheses C subscript n equals 0 plus 0 equals 0$

Q.2. $P u t t i n g space x equals negative 1 space i n space t h e space e q u a t i o n left parenthesis 1 right parenthesis space o f space a l t e r n a t i v e space s o l u t i o n space t o space E x a m p l e space 9 comma space w e space g e t C subscript 0 minus 2 C subscript 1 plus 3 C subscript 2 minus 4 C subscript 3 plus... plus open parentheses negative 1 to the power of n close parentheses open parentheses n plus 1 close parentheses C subscript n equals 0 plus 0 equals 0$

A) (35,45)

B) (20,45)

C) (35,20)

D) (45,35)

Explanation-

$open parentheses 1 minus y close parentheses to the power of m space open parentheses 1 plus y close parentheses to the power of n space equals open parentheses 1 minus m y plus blank to the power of m C subscript 2 y squared minus... close parentheses space open parentheses 1 plus n y plus blank to the power of n C subscript 2 y squared plus... close parentheses equals space 1 plus open parentheses n minus m close parentheses y plus open parentheses blank to the power of n C subscript 2 plus blank to the power of m C subscript 2 minus m n close parentheses space y squared plus... w e space a r e space g i v e n n minus m equals a subscript 1 space end subscript equals 10 space space space space space space space space space space to the power of n C subscript 2 plus blank to the power of m C subscript 2 minus m n space equals a subscript 2 equals 10. rightwards double arrow space space space space space space n equals m plus 10 space a n d space space space space space space space space space space space 1 half n open parentheses n minus 1 close parentheses plus 1 half m open parentheses m minus 1 close parentheses minus m n equals 10 rightwards double arrow space space space space space space open parentheses m plus 10 close parentheses space open parentheses m plus 9 close parentheses plus m open parentheses m minus 1 close parentheses minus 2 m open parentheses m plus 10 close parentheses equals 20 rightwards double arrow space space space space space space space m squared plus 19 m plus 90 plus m squared minus m minus 2 m squared minus 20 m space equals 20 rightwards double arrow space space space space space space minus 2 m equals space minus 70 space space rightwards double arrow space space m equals 35. therefore space space n equals 45$

Q.3. In the expansion of $open parentheses straight x cubed minus 1 over straight x squared close parentheses to the power of 15$ , the constant term equals

A) $blank to the power of 15 C subscript 9 to the power of blank end subscript$

B) 0

C) - $open parentheses blank to the power of 15 C subscript 9 close parentheses$

D) $blank to the power of 15 C subscript 11$

Explanation-

$T subscript r plus 1 comma space t h e space left parenthesis r plus 1 right parenthesis t h space t e r m space i n space t h e space e x p a n s i o n space o f space open parentheses x cubed minus 1 over x squared close parentheses to the power of 15 space i s end subscript space space space space space T subscript r plus 1 space equals end subscript to the power of 15 C subscript r space open parentheses x cubed close parentheses to the power of 15 minus r end exponent space open parentheses negative 1 over x squared close parentheses to the power of r space equals blank to the power of 15 C subscript r space left parenthesis negative 1 right parenthesis to the power of r space X to the power of 45 minus 5 r end exponent T o space o b t a i n space c o n s tan t space t e r m comma space w e space s e t space 45 minus 5 r space equals 0 space rightwards double arrow r space equals 9 therefore space C o e f f i c i e n t space o f space c o n s tan t space t e r m space i s space blank to the power of 15 C subscript 9 space left parenthesis negative 1 right parenthesis to the power of 9 space end exponent equals negative blank to the power of 15 C subscript 9$

Q.4. If $x to the power of 2 k space end exponent space o c c u r s space i n space t h e space e x p a n s i o n space o f space open parentheses x plus 1 over x squared close parentheses to the power of n minus 3 end exponent comma space t h e n$

A) n- 2k is multiple of 2

B) n- 2k is a multiple of 3

C) k=0

D) none of these

Explanation-

$T subscript r plus 1 space end subscript t h e space left parenthesis r plus 1 right parenthesis t h space t e r m space i n space t h e space e x p a n s i o n space o f space open parentheses x plus 1 over x squared close parentheses to the power of n minus 3 end exponent space i s space g i v e n n b y space space T subscript r plus 1 space equals space end subscript to the power of n minus 3 end exponent C subscript r space open parentheses x close parentheses to the power of n minus 3 minus r end exponent open parentheses 1 over x squared close parentheses to the power of r space equals blank to the power of n minus 3 end exponent C subscript r x to the power of n minus 3 minus 3 r end exponent A s space x to the power of 2 k end exponent space o c c u r s space i n space t h e space e x p a n s i o n space o f space open parentheses x plus 1 over x squared close parentheses to the power of n minus 3 end exponent comma space w e space m u s t space h a v e space n minus 3 minus 3 r equals 2 k f o r space s o m e space n o n minus n e g a t i v e space i n t e g e r space r. rightwards double arrow 3 left parenthesis 1 plus r right parenthesis equals n minus 2 k space space space space rightwards double arrow space n minus 2 k space i s space a space m u l t i p l e space o f space 3.$

Q.5. If the coefficients of $x to the power of 2 space end exponent a n d space x to the power of 8 space i n space open parentheses 2 plus x over 3 close parentheses to the power of n space end exponent$ are equal, then value of n is

A) 56

B) 55

C) 47

D) 19

Explanation-

$T subscript r space plus 1 comma space t h e space left parenthesis r plus 1 right parenthesis space t h space t e r m space i n space t h e space e x p a n s i o n space o f space open parentheses 2 plus x over 3 close parentheses space i s space g i v e n space b y end subscript T subscript r plus 1 space equals space scriptbase C subscript r to the power of 2 to the power of n minus r end exponent end exponent space open parentheses table row x row 3 end table close parentheses space equals blank to the power of n space end scriptbase presuperscript n C subscript r open parentheses table row cell 2 to the power of n minus r end exponent end cell row cell 3 to the power of r end cell end table close parentheses space x to the power of r end subscript A c c o r d i n g space t o space t h e space g i v e n space c o n d i t i o n blank to the power of n C subscript 7 open parentheses table row cell 2 to the power of n minus 7 end exponent end cell row cell 3 to the power of 7 end cell end table close parentheses space equals blank to the power of n C subscript 8 space open parentheses table row cell 2 to the power of n minus 8 end exponent end cell row cell 3 to the power of 8 end cell end table close parentheses space space rightwards double arrow open parentheses table row cell blank to the power of n C subscript 8 end cell row cell n to the power of blank C subscript 8 to the power of blank end subscript end cell end table close parentheses space equals open parentheses table row cell 2 to the power of n minus 8 end exponent end cell row cell 3 to the power of 8 end cell end table close parentheses. open parentheses table row cell 3 to the power of 7 end cell row cell 2 to the power of n minus 7 end exponent end cell end table close parentheses$ $rightwards double arrow fraction numerator n factorial over denominator 7 factorial space left parenthesis n minus 7 right parenthesis space factorial space end fraction space fraction numerator 8 factorial space left parenthesis n minus 8 right parenthesis space factorial over denominator n factorial space end fraction space equals 1 over 6 space rightwards double arrow space space fraction numerator 8 over denominator n minus 7 end fraction space equals 1 over 6 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 48 space equals n minus 7 space rightwards double arrow space space space n equals 55$

Q.6. $T h e space c o e f f i c i e n t space o f space x to the power of r space end exponent i n space t h e space e x p a n s i o n space o f space S equals space open parentheses x plus 3 to the power of blank close parentheses to the power of n minus 1 end exponent plus open parentheses x plus 3 close parentheses to the power of n minus 2 end exponent open parentheses x plus 2 close parentheses plus open parentheses x plus 3 close parentheses to the power of n minus 3 end exponent open parentheses x plus 2 close parentheses squared plus .... plus open parentheses x plus 2 close parentheses to the power of n minus 1 end exponent i s$

A) $3 to the power of n minus r end exponent minus 2 to the power of n minus r end exponent$

B) $blank to the power of n C subscript r open parentheses 3 to the power of r minus 2 to the power of r close parentheses$

C) $blank to the power of n C subscript r open parentheses 3 to the power of n minus r end exponent minus 2 to the power of n minus r end exponent close parentheses$

D) none of these

Explanation-

$W e space c a n space w r i t e space space space S equals fraction numerator open parentheses x plus 3 close parentheses to the power of n minus 1 end exponent open square brackets 1 minus open parentheses begin display style fraction numerator x plus 2 over denominator x plus 3 end fraction end style close parentheses to the power of n close square brackets over denominator 1 minus begin display style fraction numerator x plus 2 over denominator x plus 3 end fraction end style end fraction equals open parentheses 3 plus x close parentheses to the power of n minus open parentheses 2 plus x close parentheses to the power of n therefore space C o e f f i c e i n t space o f space x to the power of r equals blank to the power of n C subscript r open parentheses 3 to the power of n minus r end exponent minus 2 to the power of n minus r end exponent close parentheses$

Q.7. If $10 to the power of straight n space plus space 3.4 to the power of straight n plus 2 end exponent space plus space straight lambda$ is exactly divisible by 9 for all $straight n space element of space straight N comma$ then the least positive integral value of $straight lambda$ is

A) 5

B) 3

C) 7

D) 1

Q.8. If a,b are distinct rational numbers, then for all n $element of$ N the number $straight a to the power of straight n space minus space straight b to the power of straight n$ is divisible by

A) a - b

B) a + b

C) 2a - b

D) a - 2b

Q.9. For all n $element of$ N, $straight n to the power of 5 over 5 space plus space straight n cubed over 3 space plus space fraction numerator 7 over denominator 15 straight n end fraction$ is

A) an integer

B) a natural number

C) a positive fraction

D) none of these

Q.10. The sum of n terms of the series 1 + (1 + a) + (1 + a + $straight a squared$ ) + (1 + a + $straight a squared space plus space straight a cubed$ ) +... is

A) $fraction numerator straight n over denominator 1 minus straight a end fraction minus fraction numerator straight a left parenthesis 1 minus straight a to the power of straight n right parenthesis over denominator left parenthesis 1 minus straight a right parenthesis squared end fraction$

B) $fraction numerator straight n over denominator 1 minus straight a end fraction plus fraction numerator straight a left parenthesis 1 minus straight a to the power of straight n right parenthesis over denominator left parenthesis 1 minus straight a right parenthesis squared end fraction$

C) $fraction numerator straight n over denominator 1 minus straight a end fraction plus fraction numerator straight a left parenthesis 1 plus straight a to the power of straight n right parenthesis over denominator left parenthesis 1 minus straight a right parenthesis squared end fraction$

D) $negative fraction numerator straight n over denominator 1 minus straight a end fraction plus fraction numerator straight a left parenthesis 1 minus straight a to the power of straight n right parenthesis over denominator left parenthesis 1 minus straight a right parenthesis squared end fraction$

IIT JEE & NEET Question Bank with Solution

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Wednesday, August 25, 2021

Mathematical Induction: Mathematics MCQ

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