Directions for the Questions 1 to 3:Refer to the data given below and answer the questions.

Given  \( \dfrac{a}{b}=\dfrac{1}{2},\quad \dfrac{c}{d}=\dfrac{1}{3} \) and  \( z=\dfrac{a+c}{b+d} \), answer the questions below on limits of z.

Q. If y ≥ 0 and p ≥ 0 then the limits of ‘z’ are:

Directions for the Questions 1 to 3: Refer to the data given below and answer the questions. Given 

\( \dfrac{a}{b}=\dfrac{1}{2},\quad \dfrac{c}{d}=\dfrac{1}{3} \) and  \( z=\dfrac{a+c}{b+d} \) , answer the questions below on limits of z.

Q. c ≤ 0 and 1/3≤ z ≤1/2 only if:

Directions for the Questions 1 to 3: Refer to the data given below and answer the questions. Given

\( \dfrac{a}{b}=\dfrac{1}{2},\quad \dfrac{c}{d}=\dfrac{1}{3} \) and \( z=\dfrac{a+c}{b+d} \) , answer the questions below on limits of z.

Q. If a = –31, which of the following value of ‘d’ gives the highest value of ‘z’?

Find the integral solution of: 5y – 1 < (y + 1)² < (7y – 3)

If  \( f(a)=\dfrac{a-1}{a+1},\quad x\ge0 \) and if \( y=f\!\left(\dfrac{1}{a}\right) \), then

If f and g are real functions defined by f(a) = a + 2 and g(a) = 2a² + 5, then fog is equal to

If ‘p’ and ‘q’ are the roots of the equation x² – 10x + 16 = 0, the value of (1 – p) (1 – q) is

Given that ‘a’ and ‘b’ are positive real numbers such that a + b = 1, then what is the minimum value of

\( \sqrt{12+\dfrac{1}{a^2}}+\sqrt{12+\dfrac{1}{b^2}} \)

Let p, q and r be distinct positive integers satisfying p < q < r and p + q + r = k. What is the smallest value of k that does not determine p, q, r uniquely?

Given odd positive integers p, q and r which of the following is not necessarily true?