1
IIT-JEE 2011 Paper 1 Offline
+3
-1

Let a, b and c be three real numbers satisfying

$$[\matrix{ a & b & c \cr } ]\left[ {\matrix{ 1 & 9 & 7 \cr 8 & 2 & 7 \cr 7 & 3 & 7 \cr } } \right] = [\matrix{ 0 & 0 & 0 \cr } ]$$ .......(E)

If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

A
0
B
12
C
7
D
6
2
IIT-JEE 2011 Paper 1 Offline
+3
-1

Let a, b and c be three real numbers satisfying

$$[\matrix{ a & b & c \cr } ]\left[ {\matrix{ 1 & 9 & 7 \cr 8 & 2 & 7 \cr 7 & 3 & 7 \cr } } \right] = [\matrix{ 0 & 0 & 0 \cr } ]$$ ........(E)

Let $$\omega$$ be a solution of $${x^3} - 1 = 0$$ with $${\mathop{\rm Im}\nolimits} (\omega ) > 0$$. If a = 2 with b and c satisfying (E), then the value of $${3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}$$ is equal to

A
$$-$$2
B
2
C
3
D
$$-$$3
3
IIT-JEE 2011 Paper 1 Offline
+3
-1

Let a, b and c be three real numbers satisfying

$$[\matrix{ a & b & c \cr } ]\left[ {\matrix{ 1 & 9 & 7 \cr 8 & 2 & 7 \cr 7 & 3 & 7 \cr } } \right] = [\matrix{ 0 & 0 & 0 \cr } ]$$ ........ (E)

Let b = 6, with a and c satisfying (E). If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation ax2 + bx + c = 0, then $$\sum\limits_{n = 0}^\infty {{{\left( {{1 \over \alpha } + {1 \over \beta }} \right)}^n}}$$ is

A
6
B
7
C
$${6 \over 7}$$
D
$$\infty$$
4
IIT-JEE 2011 Paper 1 Offline
Numerical
+3
-1

Let $$f:[1,\infty ) \to [2,\infty )$$ be a differentiable function such that $$f(1) = 2$$. If $$6\int\limits_1^x {f(t)dt = 3xf(x) - {x^3} - 5}$$ for all $$x \ge 1$$, then the value of f(2) is ___________.

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