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If n is a positive integer such that $$(\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}>999$$, then the smallest value of n is
Correct Answer: 6
$$(\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}>999$$
$$(\sqrt[7]{10})^{1+2+...+n}>999$$
$$10^{\frac{1+2+...+n}{7}}>999$$
For minimum value of n,
$$\frac{1+2+...+n}{7}=3$$
1 + 2 + ... + n = 21
We can see that if n = 6, 1 + 2 + 3 + ... + 6 = 21.
The number of integer solutions of the equation $$\left(x^{2} - 10\right)^{\left(x^{2}- 3x- 10\right)} = 1$$ is
Correct Answer: 4
Case 1: When $$x^2-3x-10=0$$ and $$x^2-10\ne\ 0$$
$$x^2-3x-10=0\ $$or, $$(x-5)(x+2) = 0$$
or, x= 5 or -2
Case 2: $$x^2-10=1$$
$$x^2-11=0$$
No integer solutions
Case 3: $$x^2-10=-1\ and\ x^2-3x-10\ is\ even$$
$$x^2-9=0$$
or, (x+3)(x-3)=0
or, x= -3 and 3
for x= -3 and +3 $$x^2-3x-10$$ is even
In total 4 values of x satisfy the equations
The number of distinct integer values of n satisfying $$\frac{4-\log_{2}n}{3-\log_{4}n} < 0$$, is
Correct Answer: 47
Let $$\ \log_2n=y$$
$$\ \ \dfrac{\ 4-y}{3-\dfrac{y}{2}}<0$$
$$\ \ \left(4-y\right)\left(3-\dfrac{y}{2}\right)<0$$
$$\ \ \left(4-y\right)\left(6-y\right)<0$$
$$\ \ \left(y-4\right)\left(y-6\right)<0$$
$$4 < y < 6$$
$$4<\log_2n<6$$
$$2^4<n<2^6$$
$$16<n<64$$
n can take values from 17 to 63(inclusive).
The number of n values possible = 47
In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is
Given that the average marks of 4 girls and 6 boys is 24.
Let us assume 'b' is the marks scored by a boy and 'g' is the marks scored by a girl.
=> 4g + 6b = 10*24 = 240 ---(1)
Given that, $$b\le g\le2b$$
We need to find the distinct possible values of 2g + 6b = 2g + 240 - 4g = 240 - 2g.
From (1)
when b = g => 10g = 240 => g = 24
when b = g/2 => 7g = 240 => g = 240/7
=> 240 - 2g varies from 240 - 2*24 to 240 - 2*240/7
=> 171.42 to 192 => Integer values of 172 to 192 => 21 values.
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