1
IIT-JEE 1981
Subjective
+4
-0
For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)} \right| < 1.$$ If $$f(0)=f(1)$$, then show that $$\left| {f\left( x \right)} \right| < 1$$ for all $$x$$ in $$\left[ {0,1} \right]$$.
2
IIT-JEE 1981
Subjective
+2
-0
Evaluate $$\int {\left( {{e^{\log x}} + \sin x} \right)\cos x\,\,dx.} $$
3
IIT-JEE 1981
MCQ (Single Correct Answer)
+2
-0.5
The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$
A
$$-1$$
B
$$2$$
C
$$1 + {e^{ - 1}}$$
D
none of these
4
IIT-JEE 1981
MCQ (Single Correct Answer)
+2
-0.5
Let $$a, b, c$$ be non-zero real numbers such that
$$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has
A
no root in $$(0, 2)$$
B
at least one root in $$(0, 2)$$
C
a double root in $$(0, 2)$$
D
two imaginary roots
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