1. If \[f\left(x\right)=x\left(x-2\right)\left(x-4\right),1\leq x\leq4\]
then a number
satisfying the conditions of the mean value theorem
is
a) 1
b) 2
c) 5/2
d) 7/2
Explanation:
2. The sum of the intercepts of a tangent to \[\sqrt{x}+\sqrt{y}=\sqrt{a},a>0\]
upon the coordinate axes is
a) 2a
b) a
c) a/2
d) \[\sqrt{a}\]
Explanation:
3. Let x and y be two real numbers such that x > 0 and
xy = 1. The minimum value of x + y is
a) 1
b) 1/2
c) 2
d) 1/4
Explanation:
4. The maximum value of \[\frac{x^{2}-x+1}{x^{2}+x+1}\] for all real values
of x is
a) 1/2
b) 1
c) 2
d) 3
Explanation:
5. If \[y=2x+\cot^{-1}x+\log\left(\sqrt{1+x^{2}}-x\right)\]
then y
a) decreases on \[\left(-\infty ,\infty\right)\]
b) decreases on \[\left[0 ,\infty\right)\]
c) neither decreases nor increases on \[\left[0 ,\infty\right)\]
d) increases on \[\left(-\infty ,\infty\right)\]
Explanation:
6. Let \[g\left(x\right)=(\log(1+x) )^{-1}-x^{-1},x>0\]
then
a) 1 < g(x) < 2
b) -1 < g(x) < 0
c) 0 < g(x) < 1
d) \[\frac{1}{2} < g(x)< 1\]
Explanation:
7. Given n real numbers \[a_{1},a_{2},....a_{n}\] , the value of x for
which sum of the square of all the deviations is least
is
a) \[a_{1}+a_{2}+....+a_{n}\]
b) \[2\left(a_{1}+a_{2}+....+a_{n}\right)\]
c) \[a_1^2+a_2^2+....+a_n^2\]
d) \[\frac{a_{1}+a_{2}+....+a_{n}}{n}\]
Explanation:
8.The number of solutions of the equation \[a^{f\left(x\right)}+g\left(x\right)=0\] , where \[a>0,g(x)\neq0\] and g(x) has minimum
value 1/4, is
a) one
b) two
c) infinitely many
d) zero
Explanation:
9. Suppose that \[\frac{4}{\sin x}+\frac{1}{1-\sin x}=a\]
has at least one solution on the interval \[\left(0,\pi/2\right)\]
Then a has minimum value of x =
a) \[\sin^{-1}2/3\]
b) \[\sin^{-1}1/4\]
c) \[\cos^{-1}4/5\]
d) 1
Explanation: \[\sin^{-1}2/3\]
10. If the tangent at (1, 1) on \[y^{2}=x\left(2-x\right)^{2}\] meets the
curve again at P, then P is
a) (4, 4)
b) (-1, 2)
c) (9/4, 3/8)
d) (3/4, 7/4)
Explanation:
