1
IIT-JEE 2009 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
In a triangle $$ABC$$ with fixed base $$BC$$, the vertex $$A$$ moves such that $$$\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.$$$

If $$a, b$$ and $$c$$ denote the lengths of the sides of the triangle opposite to the angles $$A, B$$ and $$C$$, respectively, then

A
$$b+c=4a$$
B
$$b+c=2a$$
C
locus of point $$A$$ is an ellipse
D
locus of point $$A$$ is a pair of straight lines
2
IIT-JEE 2009 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-0

Match the conics in Column I with the statements/expressions in Column II :

Column I Column II
(A) Circle (P) The locus of the point ($$h,k$$) for which the line $$hx+ky=1$$ touches the circle $$x^2+y^2=4$$.
(B) Parabola (Q) Points z in the complex plane satisfying $$|z+2|-|z-2|=\pm3$$.
(C) Ellipse (R) Points of the conic have parametric representation $$x = \sqrt 3 \left( {{{1 - {t^2}} \over {1 + {t^2}}}} \right),y = {{2t} \over {1 + {t^2}}}$$
(D) Hyperbola (S) The eccentricity of the conic lies in the interval $$1 \le x \le \infty $$.
(T) Points z in the complex plane satisfying $${\mathop{\rm Re}\nolimits} {(z + 1)^2} = |z{|^2} + 1$$.

A
(A)$$\to$$(P); (B)$$\to$$(S), (T); (C)$$\to$$(R); (D)$$\to$$(R), (S)
B
(A)$$\to$$(P); (B)$$\to$$(S), (T); (C)$$\to$$(R); (D)$$\to$$(Q), (S)
C
(A)$$\to$$(P); (B)$$\to$$(S), (T); (C)$$\to$$(S); (D)$$\to$$(R), (S)
D
(A)$$\to$$(P); (B)$$\to$$(P), (T); (C)$$\to$$(R); (D)$$\to$$(Q), (S)
3
IIT-JEE 2009 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1

Let $$f$$ be a non-negative function defined on the interval $$[0,1]$$.

If $$\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,0 \le x \le 1} } $$, and $$f(0) = 0$$, then

A
$$f\left( {{1 \over 2}} \right) < {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) > {1 \over 3}$$
B
$$f\left( {{1 \over 2}} \right) > {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) > {1 \over 3}$$
C
$$f\left( {{1 \over 2}} \right) < {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
D
$$f\left( {{1 \over 2}} \right) > {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
4
IIT-JEE 2009 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is
A
$$e-1$$
B
$$\int\limits_1^e {\ln \left( {e + 1 - y} \right)dy} $$
C
$$e - \int\limits_0^1 {{e^x}dx} $$
D
$$\int\limits_1^e {\ln y\,dy} $$
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