The probability that an event $$A$$ happens in one trial of an experiment is $$0.4.$$ Three independent trials of the experiment are performed. The probability that the event $$A$$ happens at least once is
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(B)
$$0.784$$
2
Two events $$A$$ and $$B$$ have probabilities $$0.25$$ and $$0.50$$ respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is $$0.14$$. Then the probability that neither $$A$$ nor $$B$$ occurs is
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(A)
$$0.39$$
3
The equation $$\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 < x \le {\pi \over 2}$$ has
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(A)
no real solution
4
$$ABC$$ is a triangle with $$AB=AC$$. $$D$$ is any point on the side $$BC$$. $$E$$ and $$F$$ are points on the side $$AB$$ and $$AC$$, respectively, such that $$DE$$ is parallel to $$AC$$, and $$DF$$ is parallel to $$AB$$. Prove that
$$$DF + FA + AE + ED = AB + AC$$$
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(A)
The statement is always true.
5
Given $$A = {\sin ^2}\theta + {\cos ^4}\theta $$ then for all real values of $$\theta $$
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(B)
$${3 \over 4} \le A \le 1$$
6
The smallest positive integer n for which $${\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1$$ is
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(D)
none of these
7
Find the real values of x and y for which the following equation is satisfied $$\,{{(1 + i)x - 2i} \over {3 + i}} + {{(2 + 3i)y + i} \over {3 - i}} = i$$
For all $$\theta $$ in $$\left[ {0,\,\pi /2} \right]$$ show that, $$\cos \left( {\sin \theta } \right) \ge \,\sin \,\left( {\cos \theta } \right).$$
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(D)
The inequality holds because \(\cos(x) \ge \sin(x)\) for all \(x \in [0, \pi/4]\) and because of the properties of sine and cosine functions in the interval \([0, \pi/2]\).
11
Find all the real values of $$x,$$ for which $$y$$ takes real values.
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(E)
$${[-1, 2) \cup [3, \infty)}$$
12
Given $${n^4} < {10^n}$$ for a fixed positive integer $$n \ge 2,$$ prove that $${\left( {n + 1} \right)^4} < {10^{n + 1}}.$$
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(C)
We must show that $\frac{(n+1)^4}{n^4} < 10$.
13
has solution satisfying the conditions $$x > 0,\,y > 0.$$
Find the solution set of the system
$$$\matrix{
{x + 2y + z = 1;} \cr
{2x - 3y - w = 2;} \cr
{x \ge 0;\,y \ge 0;\,z \ge 0;\,w \ge 0.} \cr
} $$$
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(A)
x = 1, y = 0, z = 0, w = 0
15
Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always
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(B)
real
16
The least value of the expression $$2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)$$ for x > 1, is
Trả lời
(B)
2
17
If $$\,({x^2} + px + 1)\,$$ is a factor of $$(a{x^3} + bx + c)$$, then
Trả lời
(C)
$${a^2} - {c^2} = ab$$
18
The interior angles of a polygon are in arithmetic progression. The smallest angle is $${120^ \circ }$$, and the common difference is $${5^ \circ }$$, Find the number of sides of the polygon.
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(A)
9
19
The point $$\,\left( {4,\,1} \right)$$ undergoes the following three transformations successively.
Reflection about the line $$y=x$$.
Translation through a distance 2 units along the positive direction of x-axis.
Rotation through an angle $$p/4$$ about the origin in the counter clockwise direction.
Then the final position of the point is given by the coordinates.
A straight line $$L$$ is perpendicular to the line $$5x - y = 1.$$ The area of the triangle formed by the line $$L$$ and the coordinate axes is $$5$$. Find the equation of the Line $$L$$.
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A
B
21
A square is inscribed in the circle $${x^2} + {y^2} - 2x + 4y + 3 = 0$$. Its sides are parallel to the coordinate axes. The one vertex of the square is
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(D)
none of these
22
Two circles $${x^2} + {y^2} = 6$$ and $${x^2} + {y^2} - 6x + 8 = 0$$ are given. Then the equation of the circle through their points of intersection and the point (1, 1) is
$$\frac{dy}{dx} = \frac{5}{3} \cdot \frac{1}{(1-x)^2} - 2\sin(4x+2), x < 1$$ and $$\frac{dy}{dx} = -\frac{5}{3} \cdot \frac{1}{(x-1)^2} - 2\sin(4x+2), x > 1$$
24
$$ABC$$ is a triangle, $$P$$ is a point on $$AB$$, and $$Q$$ is point on $$AC$$ such that $$\angle AQP = \angle ABC$$. Complete the relation
$$${{area\,\,of\,\,\Delta APQ} \over {area\,\,of\,\,\Delta ABC}} = {{\left( {...} \right)} \over {A{C^2}}}$$$
Trả lời
(D)
AP²
25
$$ABC$$ is a triangle with $$\angle B$$ greater than $$\angle C.\,D$$ and $$E$$ are points on $$BC$$ such that $$AD$$ is perpendicular to $$BC$$ and $$AE$$ is the bisector of angle $$A$$. Complete the relation
$$$\angle DAE = {1 \over 2}\left[ {\left( {} \right) - \angle C} \right]$$$
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(B)
\angle B
26
In a $$\Delta ABC,\,\angle A = {90^ \circ }$$ and $$AD$$ is an altitude. Complete the relation $${{BD} \over {BA}} = {{AB} \over {\left( {....} \right)}}$$.
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(B)
BC
27
$$ABC$$ is a triangle. $$D$$ is the middle point of $$BC$$. If $$AD$$ is perpendicular to $$AC$$, then prove that
$$$\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}$$$
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(C)
Applying the Law of Cosines and the properties of medians to express \(\cos A\) and \(\cos C\), and using the given perpendicularity to simplify to the desired form. Considering Stewart's theorem.
28
(ii) $$AB$$ is vertical pole. The end $$A$$ is on the level ground. $$C$$ is the middle point of $$AB$$. $$P$$ is a point on the level ground. The portion $$CB$$ subtends an angle $$\beta $$ at $$P$$. If $$AP = n\,AB,$$ then show that tan$$\beta $$ $$ = {n \over {2{n^2} + 1}}$$
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(A)
The height of the tower is abc tan(θ) / 4Δ and tan(β) = n / (2n^2 + 1)
