MAT 2003 Question Paper

For an equation $$ax^2 + bx + c = 0$$, 

the roots are : $$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Now, if $$b^2>4ac$$, roots are real and different.

If $$b^2=4ac$$, roots are equal.

If $$b^2<4ac$$, roots are imaginary.

Thus, only first statement is true.

=> Ans - (A)

$$ax^3 + bx^2 + cx + d = 0,$$ if its roots are $$\alpha, \beta and \gamma$$, then

$$\alpha+\beta+\gamma=\frac{-b}{a}$$

$$(\alpha \beta)+(\beta \gamma)+(\gamma \alpha)=\frac{c}{a}$$

$$\alpha \beta \gamma=\frac{-d}{a}$$

Thus, neither of the statements is true.

=> Ans - (D)

I. $$\frac{d}{dx}(sin^2(3x))=2 sin(3x)\times3=6 sin(3x)$$

Hence, first statement is not true.

II. $$\frac{d}{dx} (a^u) = a^u (\log a) \frac{du}{dx}$$

Hence, second statement is true.

=> Ans - (B)

$$sin(2\theta)=\frac{2 tan\theta}{1+tan^2\theta}$$

and $$cos(2\theta)=\frac{1- tan^2\theta}{1+tan^2\theta}$$

Thus, if $$y=2x$$, then only first statement is true.

=> Ans - (A)

It is given that $$z = x + iy$$, where $$i = (-1)$$

If $$x=0$$, then $$z=iy$$

Hence, first statement is not true.

II : $$If a + bi = c + di, then a = c, b = d$$

If an equation contains both real and imaginary numbers, then the real numbers are equal, and the coefficient of imaginary numbers are also equal.

Hence, second statement is true.

=> Ans - (B)

$$A \cap (B \cap C)$$ = $$(A \cap B)$$ $$\cup (A \cap C)$$

Thus, we need to know the value of both $$(A \cap B)$$ and $$(A \cap C)$$ to find the solution, which is given only in statement II.

Thus, the question can be answered by using one statement alone, but not by using other statement alone.

=> Ans - (A)

We have been given an expression of displacement in terms of time. 
Thus, differentiating this equation with respect to time once will give us an expression for velocity, and differentiating once again with respect to time will give us an expression for acceleration. 

Hence,
V = $$3t^2-8t+16$$ ..... (A) and 
A = $$6t-8$$ ............. (B)

While we have obtained the expressions for velocity and acceleration, we cannot determine their absolute values at a particular time simply from the given statement 1. However, if we take into account statement 2, we have:
$$20=3t^2-8t+16$$ which when solved gives t = 4+2(sqrt 7) 

When this value of t is put in (B), we get value of acceleration and hence, the answer can be found by using both the statements together but not by using either of them alone. 

Sum of $$'n'$$ terms of G.P. = $$\frac{a(r^n-1)}{r-1}$$

Now, using the first statement, if we know the $$n^{th}$$ term, then we can find the value of $$n$$ and hence the sum of $$n$$ terms.

Now, the second statement states that the next term after the $$N^{th}$$ term is thrice of it, which only means that the common ratio $$r=3$$, and thus we cannot find the sum using this statement.

Hence, the question can be answered using only one statement alone.

=> Ans - (A)

Using statement I, we know that $$\log_{96} 70=k$$ (say)

Now, $$\log_{70} 96=$$ $$\frac{1}{\log_{96} 70}=\frac{1}{k}$$

Thus, I statement is sufficient alone.

But, we cannot get the solution using second statement alone.

Thus, the question can be answered by using only one statement alone, but not by using another statement alone.

=> Ans - (A)