1
IIT-JEE 2009 Paper 1 Offline
+3
-0

Match the statements/expressions in Column I with the open intervals in Column II :

Column I Column II
(A) Interval contained in the domain of definition of non-zero solutions of the differential equation $${(x - 3)^2}y' + y = 0$$ (P) $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$
(B) Interval containing the value of the integral $$\int\limits_1^5 {(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)dx}$$ (Q) $$\left( {0,{\pi \over 2}} \right)$$
(C) Interval in which at least one of the points of local maximum of $${\cos ^2}x + \sin x$$ lies (R) $$\left( {{\pi \over 8},{{5\pi } \over 4}} \right)$$
(D) Interval in which $${\tan ^{ - 1}}(\sin x + \cos x)$$ is increasing (S) $$\left( {0,{\pi \over 8}} \right)$$
(T) $$( - \pi ,\pi )$$

A
(A)$$\to$$(P), (Q), (S); (B)$$\to$$(P), (T), (S); (C)$$\to$$(P), (Q), (R), (T); (D)$$\to$$(S)
B
(A)$$\to$$(P), (Q), (S); (B)$$\to$$(P), (T), (R); (C)$$\to$$(P), (Q), (R), (T); (D)$$\to$$(R)
C
(A)$$\to$$(P), (Q), (S); (B)$$\to$$(P), (T), (S); (C)$$\to$$(S), (Q), (R), (T); (D)$$\to$$(S)
D
(A)$$\to$$(P), (T), (S); (B)$$\to$$(P), (T), (S); (C)$$\to$$(P), (Q), (R), (T); (D)$$\to$$(S)
2
IIT-JEE 2008 Paper 2 Offline
+3
-1
Let a solution $$y=y(x)$$ of the differential equation,

$$x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0$$ satify $$y\left( 2 \right) = {2 \over {\sqrt 3 }}.$$

STATEMENT-1 : $$y\left( x \right) = \sec \left( {{{\sec }^{ - 1}}x - {\pi \over 6}} \right)$$ and

STATEMENT-2 : $$y\left( x \right)$$ given by $${1 \over y} = {{2\sqrt 3 } \over x} - \sqrt {1 - {1 \over {{x^2}}}}$$

A
STATEMENT-1 is True, STATEMENT-2 is True;STATEMENT-2 is a correct explanation for STATEMENT-1
B
STATEMENT-1 is True, STATEMENT-2 is True;STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C
STATEMENT-1 is True, STATEMENT-2 is False
D
STATEMENT-1 is False , STATEMENT-2 is True
3
IIT-JEE 2008 Paper 2 Offline
+4
-1

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I Column II
(A) The minimum value of $${{{x^2} + 2x + 4} \over {x + 2}}$$ is (P) 0
(B) Let A and B be 3 $$\times$$ 3 matrices of real numbers, where A is symmetric, B is skew-symmetric and (A + B) (A $$-$$ B) = (A $$-$$ B) (A + B). If (AB)$$^t$$ = ($$-1$$)$$^k$$ AB, where (AB)$$^t$$ is the transpose of the matrix AB, then the possible values of k are (Q) 1
(C) Let $$a=\log_3\log_3 2$$. An integer k satisfying $$1 < {2^{( - k + 3 - a)}} < 2$$, must be less than (R) 2
(D) If $$\sin \theta = \cos \varphi$$, then the possible values of $${1 \over \pi }\left( {\theta + \varphi - {\pi \over 2}} \right)$$ are (S) 3

A
A - iii; B - ii, iv; C - iii, iv; D - i, iii
B
A - iii; B - ii; C - iii, iv; D - i, iii
C
A - ii; B - ii, iv; C - iii, iv; D - i
D
A - ii; B - ii, iv; C - iii, iv; D - i, iii
4
IIT-JEE 2008 Paper 1 Offline
+3
-1

Let a and b be non-zero real numbers. Then, the equation

$$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$ represents :

A
four straight lines, when c = 0 and a, b are of the same sign
B
two straight lines and a circle, when a = b, and c is of sign opposite to that of a
C
two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a
D
a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a
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