Following are my answers:
Q1. In the figure shown above,the measure of angle PRS is how many degrees greater than the measure of angle PQR?
(1) The measure of angle QPR is 30
(2) The sum of the measures of angles PQR and PRQ is 150
We have to find the difference between <PRS and <PQR.
1.
<PQR + <QPR = <PRS ( Exterior angle theorem : Sum of two interior angles = opposite exterior angle)
<PQR + 30 = <PRS
Hence, <PRS - <PQR = 30 degrees.
So, 1 is sufficient.
2. <PQR + <PRQ = 150
<PQR + <PRQ +<QPR = 180 Hence <QPR = 30;
If we have <QPR, we have already proved in 1 that we can find <PRS - <PQR.
Hence Sufficient.
As each statement alone is sufficient, the answer is D.
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Q2. The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A and inversely proportional to the concentration of chemical B present.If the concentration of chemical B is increased by 100%,which one of the following is the closest to the percentage change in the concentration of chemical A required to keep the reaction rate unchanged?
a)100% decrease
b)50% decrease
c)40% decrease
d)40% increase
e)50% increase
Let the rate of the reaction be R
Let concentration of chemical A be A
Let concentration of chemical B be B
Then R is proportional to A²
R is also proportional to 1/B
Hence, R is proportional to A²/B
If C is a constant, R=C*(A²/B)
If the concentration of B is increased 100% B becomes 2B ( B+(100/100)*B = 2B)
Let A2 be the new concentration of chemical A for the rate to be constant
Then R=C*(A²/B) = C*(A2²/(2*B))
Hence, A² = A2²/(2) So A² = A2/√2
A2 = √2 * A = 1.41 * A
Hence A becomes 1.41 * A If the concentration of B is increased 100%
So, there is a 41% increase in A.
Answer is D
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Q3.For every integer k from 1 to 10,inclusive,the kth term of a certain sequence is given by (-1)^k+1 * (1/2^k).If T is the sum of the 1st 10 terms in the sequence then T is,
a)Greater than 2
b)between 1 and 2
c)between ½ and 1
d)between ¼ and ½
e)less than ¼
Kth term of a sequence is =
Rk = (-1)^k+1 * (1/2^k)
R1 = (-1)^2 * (1/2)^1 = 1/2
R2 = (-1)^3 * (1/2)^2 = -1/4
similarly, R3= 1/8, R4 = -1/16 etc
So sum of 1st 10 terms in the sequence =
S = 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 +1/128 etc upto the 10th term ( there will be total of 11 terms as S is inclusive of 1st and 10 term)
S = 1/2 +(- 1/4 + 1/8) + (- 1/16 + 1/32) etc ( there will be a total of 5 pairs like this as the sequence has 11 terms. )
S = 1/2 -1/8 -1/32 etc
If we sum up the negative terms
1/8 + 1/32 +.... we can see that sum is greater than 1/8 but less than 1/4 ( as 1/32+1/128 etc <1/8)
So the sum is greater than 1/2-1/4 =1/4 but less than 1/2 as we are subtracting 1/8 + 1/32 from 1/2
Hence the sum is between 1/2 and 1/4
Answer is D
