JEE Maths Quiz on Binomial Theorem and Mathematical Induction
JEE Maths Quiz on Binomial Theorem and Mathematical Induction : In this article you will get to Online test for JEE Main, JEE Advanced, UPSEE, WBJEE and other engineering entrance examinations that will help the students in their preparation. These tests are free of cost and will useful in performance and inculcating knowledge. In this post we are providing you quiz on Binomial Theorem and Mathematical Induction
Quiz on Binomial Theorem and Mathematical Induction
Q1. In the polynomial (x − 1) (x − 2) (x − 3) . . . . . (x − 100), what is the coefficient of x^{99}?
a) 5050
b) – 5050
c) 4040
d) – 4040
b) – 5050
Solution:
(x − 1) (x − 2) (x − 3) . . . . . (x − 100)
Number of terms = 100
Coefficient of x^{99} = (x − 1) (x − 2) (x − 3) . . . . . (x − 100)
= (−1 −2 −3 −. . . . . . −100)
= − (1 + 2 + . . . . . +100)
= − [100 * 101 / 2
= – 5050
Q2. If (1 + ax)^{n }= 1 + 8x + 24x^{2} +…., then the value of a and n are ________.
a) a = 2 and n = 4
b) a = 2 and n = 4
c) a = 4 and n = 6
d) a = 8 and n = 2
b) a = 2 and n = 4
Q3. If the coefficient of x^{7} in (ax^{2 }+ [1/bx])^{11} is equal to the coefficient of x^{−7 }in (ax − 1/bx^{2})^{11}, then ab = __________.
a) 1
b) 1
c) 2
d) 2
b) 1
Q4. In the expansion of (x + a)^{n}, the sum of odd terms is P and sum of even terms is Q, then the value of (P^{2 }− Q^{2}) will be _________.
a) (x^{3} − a^{2})^{n}
b) (x^{2} − a^{2})^{2}
c) (x^{2} − a^{2})^{n}
d) (x^{n} − a^{n})^{2}
c) (x^{2} − a^{2})^{n}
Solution:
(x + a)^{n} = x^{n} +^{ n}C_{1}x^{n−1 }a + . . . . . = (x^{n }+ ^{n}C_{2}x^{n−2}a^{2 }+ . . . . . . . + (^{n}C_{1}x^{n−1}a + ^{n}C_{3}x^{n−3}a^{3 }+ . . . . .) = P+Q
(x − a)^{n }= P − Q
As the terms are altered,
P^{2 }− Q^{2 }= (P + Q) (P − Q) = (x + a)^{n }(x − a)^{n}
P^{2 }− Q^{2 }= (x^{2} − a^{2})^{n}
Q5. (1 + x)^{n }− nx − 1 is divisible by (where n∈N)
a) 2x
b) x^{2}
c) 2x^{3}
d) All of these
b) x^{2}
(1 + x)^{n }= 1 + nx + ([n (n − 1)] / [2!]) * x^{2 }+ ([n (n − 1) (n − 2)] / [3!]) * x^{3 }+ . . . . .
(1 + x)^{n }− nx − 1 = x^{2} [([n (n − 1)] / [2!]) + ([n (n − 1) (n − 3)] / [3!]) * x + . . . . .]
From above it is clear that (1 + x)^{n }− nx − 1 is divisible by x^{2}.
Trick: (1 + x)^{n }− nx − 1, put n = 2 and x = 3;
Then 4^{2 }− 2 * 3 − 1 = 9 is not divisible by 6, 54 but divisible by 9, which is given by option (b) i.e., x^{2 }= 9.
Q6. The approximate value of (1.0002) 3000 is __________.
a) 1.8
b) 1.6
c) 1.9
d) 1.7
b) 1.6
Solution:
(1.0002) 3000 = (1 + 0.0002) 3000
= 1 + (3000) (0.0002) + ([(3000) (2999)] / [1.2]) * (0.0002)^{2}
= 1 + (3000) (0.0002)
= 1.6
Q7. The digit in the unit place of the number (183!) + 3^{183} is _____.
a) 5
b) 7
c) 9
d) 11
b) 7
We know that n! terminates in 0 for n³ at 5 and 3^{4n} terminator in 1, (because 3^{4 }= 81)
Therefore, 3^{180 }= (3^{4})^{45} terminates in 1.
Also 3^{3 }= 27 terminates in 7
3^{183 }= 3^{180 }* 3^{3} terminates in 7.
183! + 3^{183} terminates in 7 i.e. the digit in the unit place = 7
Q8. If the three consecutive coefficients in the expansion of (1 + x)^{n} are 28, 56 and 70, then the value of n is ______.
a) 2
b) 4
c) 6
d) 8
d) 8
Solution:
Let the three consecutive coefficients be ^{n}C_{r−1 }= 28, ^{n}C_{r }= 56 and ^{n}C_{r+1 }= 70, so that
^{n}C_{r }/ ^{n}C_{r−1 }= [n − r + 1] / [r] = 56 / 28 = 2 and ^{n}C_{r+1} / ^{n}C_{r }= [n − r] / [r + 1] = 70 / 56 = 5 / 4
This gives n + 1 = 3r and 4n − 5 = 9r
4n − 5 / n + 1 = 3
⇒ n = 8
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