If a<0 and n is any positive even integer, then `(a)^(1/n)` to the set of real numbers. (Does not belong/belongs)
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If `n` is an even positive integer, then find the value of `x` if the greatest term in the expansion of `1+x` ^n may have the greatest coefficient als
Correct Answer - `x gt (n)/(n+2)` and `x lt (n+2)/(n)` If n is even, the greatest coefficient is `.^(n)c_(n//2)`. Therefore, the greatest term is `.^(n)C_(n//2)x^(n//2)`. `:. .^(n)C_(n//2)x^(n//2)gt.^(n)C_((n//2)-1)x^((n-2)//2)` and `.^(n)C_(n//2)x^(n//2) gt .^(n)C_((n//2)+1)x^((n//2)+1)`...
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If n is a positive integer, then `(sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n)` is (1) an irrational number (2) an odd positive integer (3) an even positive inte
Correct Answer - A `(3sqrt(3) + 1)^(2n)- (sqrt(3) - 1)^(2n)` `=2[.^(2n)C_(1)(sqrt(3))^(2n-1)+.^(2n)C_(3)(sqrt(3))^(2n-3)+.^(2n)C_(5)(sqrt(3))^(2n-5) + "….."]` = an irrational number
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A positive integer `n` is of the form `n=2^(alpha)3^(beta)`, where `alpha ge 1`, `beta ge 1`. If `n` has `12` positive divisors and `2n` has `15` posi
Correct Answer - B `(b)` `n=2^(alpha)*3^(beta)` No. of divisors `=(alpha+1)(beta+1)=12` `2n=2^(alpha+1)3^(beta)` No.of divisors `=(alpha+2)(beta+1)=15` `implies(alpha+2)/(alpha+1)=(5)/(4)` `implies4alpha+8=5alpha+5impliesalpha=3` `impliesbeta=2implies3n=2^(3)3^(3)` No. of divisors `=(3+1)(3+1)=16`
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Two distinct numbers `a` and `b` are chosen randomly from the set `{2,2^(2),2^(3),….2^(25)}`. Then the probability that `log_(a)b` is an integer is
Correct Answer - B `(b)` `log_(a)b=(logb)/(loga)` Let `b=2^(m)` and `a=2^(n)` where `m` and `n` denote the exponents on the base `2` in the given set then `log_(a)b=(m)/(n)` Therefore, `log_(a)b` is an...
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La belongs to group 3 while Hg belongs to group 12 and both belong to period 6 of the periodic table.
i. La and Hg both belongs to period 6. Therefore, n = 6. ii. Elements of group 3 to group 12 belong to the d-block of the modem periodic table. iii. The general outer...
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3. Suppose set \( A \) consists of first 250 natural numbers that are multiples of 3 and set \( B \) consists of first 200 even natural numbers. How many elements does \( A \cup B \) have? (a) 324 (b) 364 (c) 384 (d) 400
Given: n(A)=250 n(B)=200 A={3,6,9,12,.......750} B=2,4,6,8,10.......400} Therefore,A⋂B={6,12,18....396) =>n(A⋂B)=396/6=66 n(A∪B)=n(A)+n(B)-n(A⋂B) =250+200-66 ...
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Which of the following sets in an finite/infinite set ? `(i)` Set of divisors of `24` `(ii)` Set of all real number which lie between `1` and `2` `(ii
Correct Answer - `(i)` Finite `(ii)` Infinite `(iii)` Finite `(iv)` Finite
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Suggest a universal set for the set given below. A = {Even numbers less than 100} B={Odd natural numbers less than 100 in natural numbers }
Correct Answer - `mu = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12---------, 99}`
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Which of the following are examples of the null set ? (i) Set of odd numbers divisible by 2. (ii) Set of even prime numbers. (iii) `A={x:x in N,1 lt x
Correct Answer - (i),(iv),(v),(vi)
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If an element X belongs to II group and another element Y belongs to VII group then the compound formed between these elements has fomula :
Correct option is C) X Y2
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