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JEE Mathematics Solved Question Paper 2003

Multiple Choice Questions

If in $∆ ABC$ , r₁ < r₂ < r₃, then :

a < b < c
a > b > c
b > a < c
a < c < b

In a $∆ ABC$ , if 3a = b + c, then $\cot (\frac{B}{2}) \cot (\frac{C}{2})$ is equal to :

If in a triangle, if b = 20, c = 21 and $\sin (A) = \frac{3}{5}$ , then a is equal to

4.
A tower subtends angles $α, 2 α and 3 α$ respectively at points A, B and C, all lying on a horizontal line through the foot of the tower,then $\frac{AB}{BC}$ is equal to :

$\frac{\sin (3 α)}{\sin (2 α)}$

$1 + 2 \cos (2 α)$

$2 \cos (2 α)$

$\frac{\sin (2 α)}{\sin (α)}$

$1 + 2 \cos (2 α)$

$In ∆ ECD, \tan (3 α) = \frac{h}{CD}$

$\Rightarrow CD = hcot (3 α) . . . (i) In ∆ EBD, \tan (2 α) = \frac{h}{BD} \Rightarrow BD = hcot (2 α) . . . (ii) In ∆ EAD, \tan (α) = \frac{h}{AD} \Rightarrow AD = hcot (α) . . . (iii) From Eqs . (ii) and (iii), AD - BD = hcot (α) - hcot (2 α) AB = h (\cot (α) - \cot (2 α)) . . . (iv)$

$From Eqs . (i) and (ii), we get BD - CD = hcot (2 α) - hcot (3 α) BC = h (\cot (2 α) - \cot (3 α)) . . . (v) From Eqs . (iv) and (v), we get \frac{AB}{BC} = \frac{h (\cot (α) - \cot (2 α))}{2 (\cot (2 α) - \cot (3 α))} \Rightarrow \frac{AB}{BC} = \frac{\frac{\cos (α)}{\sin (α)} - \frac{\cos (2 α)}{\sin (2 α)}}{\frac{\cos (2 α)}{\sin (2 α)} - \frac{\cos (3 α)}{\sin (3 α)}} = \frac{\frac{\sin (2 α - α)}{\sin (α) \sin (2 α)}}{\frac{\sin (3 α - 2 α)}{\sin (2 α) \sin (3 α)}} = \frac{\sin (3 α)}{\sin (α)} = \frac{3 \sin (α) - 4 \sin^{3} (α)}{\sin (α)} = 3 - 4 \sin^{2} (α)$

$= - 3 - 2 (1 - \cos (2 α)) = 1 + 2 \cos (2 α)$

A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ballis drawn from it. Then, the probability for the ball chosen be white, is :

$\frac{2}{15}$
$\frac{7}{15}$
$\frac{8}{15}$
$\frac{14}{15}$

For a poisson variate X, if P(X = 2) = 3P(X = 3), then the mean of X is :

1
$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{4}$

A random variate X takes the values 0, 1, 2, 3 and its mean is 1.3. If P(X = 3) = 2P(X = 1) and P(X = 2) = 0.3, then P(X = 0) is equal to :

The co-ordinate axes are rotated through an angle 135°. If the co-ordinates of a point P inthe new system are known to be (4, - 3), then the co-ordinates of P in the original system are :