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JEE Mathematics Solved Question Paper 2007

Multiple Choice Questions

11.

If H is the harmonic mean between P and Q, then the value of $\frac{H}{P} + \frac{H}{Q}$ is

2
$\frac{PQ}{P + Q}$
$\frac{1}{2}$
$\frac{P + Q}{PQ}$

12.

If $\cos (x) + \cos^{2} (x) = 1$ , then the value of $\sin^{12} (x) + 3 \sin^{10} (x) + 3 \sin^{8} (x) + \sin^{6} (x) - 1$ , is equal to :

13.

The product of all values of $(\cos (α) + isin (α))$ ^3/5 is :

1
$\cos (α) + isin (α)$
$\cos (3 α) + isin (3 α)$
$\cos (5 α) + isin (5 α)$

14.

The imaginary part of $\frac{{(1 + i)}^{2}}{i (2 i - 1)}$ is :

$\frac{4}{5}$
0
$\frac{2}{5}$
$- \frac{4}{5}$

15.

The equation of a directrix of the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1$ is :

3y = 5
y = 5
3y = 25
y = 3

16.

If the normal at (ap, 2ap) on the parabola y² = 4ax, meets the parabola again at (aq² , 2aq), then

p² + pq + 2 = 0
p² - pq + 2 = 0
q² + pq + 2 = 0
p² + pq + 1

17.

The length of the straight line x - 3y = 1 intercepted by the hyperbola x² - 4y² = 1 is :

$\sqrt{10}$
$\frac{6}{5}$
$\frac{1}{\sqrt{10}}$
$\frac{6}{5} \sqrt{10}$

18.

The curve described parametrically by x = t² + 2t - 1, y = 3t + 5 represents :

an ellipse
a hyperbola
a parabola
a circle

19.

The function f (x) = x² e^{- 2x}, x > 0. Then the maximum value of f (x) is :

$\frac{1}{e}$
$\frac{1}{2 e}$
$\frac{1}{e^{2}}$
$\frac{4}{e^{4}}$

20.
If (x + y) $\sin (u) = x^{2} y^{2}, then x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ is equal to

$\frac{1}{e}$

$\frac{1}{2 e}$

$\frac{1}{e^{2}}$

$\frac{4}{e^{4}}$

$\frac{4}{e^{4}}$

$(x + y) \sin (u) = x^{2} y^{2}$ ...(i)
On differentiating partially w.r.t. x of Eq. (i),

we get

$(1 + 0) \sin (u) + (x + y) \cos (u) \frac{\partial u}{\partial x} = 2 {xy}^{2} \Rightarrow xsin (u) + (x^{2} + xy) \cos (u) \frac{\partial u}{\partial x} = 2 x^{2} y^{2} . . . (ii) On differentiating partially w . r . t . y of Eq . (i), we get (0 + 1) \sin (u) + (x + y) \cos (u) \frac{\partial u}{\partial y} = 2 x^{2} y \Rightarrow ysin (u) + (xy + y^{2}) \cos (u) \frac{\partial u}{\partial y} = 2 x^{2} y^{2} . . . (iii) On adding Eqs . (ii) and (iii), we get (x + y) \sin (u) + (x + y) \{x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}\} \cos (u) = 4 x^{2} y^{2} \Rightarrow x^{2} y^{2} + (x + y) \{x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}\} \cos (u) = 4 x^{2} y^{2} \Rightarrow (x + y) (x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}) \cos (u) = 3 x^{2} y^{2} \Rightarrow (x + y) (x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}) \cos (u) = 3 (x + y) \sin (u)$