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1

JEE Advanced 2014 Paper 2 Offline

MCQ (Single Correct Answer)
The quadratic equation $$p(x)$$ $$ = 0$$ with real coefficients has purely imaginary roots. Then the equation $$p(p(x))=0$$ has
A
one purely imaginary root
B
all real roots
C
two real and two purely imaginary roots
D
neither real nor purely imaginary roots
2

IIT-JEE 2012 Paper 2 Offline

MCQ (Single Correct Answer)

Let $$\alpha$$(a) and $$\beta$$(a) be the roots of the equation $$(\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)x + (\root 6 \of {1 + a} - 1) = 0$$ where $$a > - 1$$. Then $$\mathop {\lim }\limits_{a \to {0^ + }} \alpha (a)$$ and $$\mathop {\lim }\limits_{a \to {0^ + }} \beta (a)$$ are

A
$$ - {5 \over 2}$$
B
$$ - {1 \over 2}$$
C
$$ - {7 \over 2}$$
D
$$ - {9 \over 2}$$

Explanation

Let a + 1 = t6. Thus, when a $$\to$$ 0, t $$\to$$ 1.

$$\therefore$$ $$({t^2} - 1){x^2} + ({t^3} - 1)x + (t - 1) = 0$$

$$ \Rightarrow (t - 1)\{ (t + 1){x^2} + ({t^2} + t + 1)x + 1\} = 0$$,

as t $$\to$$ 1

$$2{x^2} + 3x + 1 = 0$$

$$ \Rightarrow 2{x^2} + 2x + x + 1 = 0$$

$$ \Rightarrow (2x + 1)(x + 1) = 0$$

Thus, x = $$-$$1, $$-$$1/2

or, $$\mathop {\lim }\limits_{a \to {0^ + }} \alpha (a) = - {1 \over 2}$$

and $$\mathop {\lim }\limits_{a \to {0^ + }} \beta (a) = - 1$$

3

IIT-JEE 2011 Paper 2 Offline

MCQ (Single Correct Answer)
A value of $$b$$ for which the equations $$$\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr } $$$

have one root in common is

A
$$ - \sqrt 2 $$
B
$$ - i\sqrt 2 $$
C
$$i\sqrt 5 $$
D
$$\sqrt 2 $$
4

IIT-JEE 2011 Paper 1 Offline

MCQ (Single Correct Answer)
Let $$\left( {{x_0},{y_0}} \right)$$ be the solution of the following equations
$$\matrix{ {{{\left( {2x} \right)}^{\ell n2}}\, = {{\left( {3y} \right)}^{\ell n3}}} \cr {{3^{\ell nx}}\, = {2^{\ell ny}}} \cr } $$
Then $${x_0}$$ is
A
$${1 \over 6}$$
B
$${1 \over 3}$$
C
$${1 \over 2}$$
D
$$6$$

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