Find the derivative of $$\sin \left( {{x^2} + 1} \right)$$ with respect to $$x$$ first principle.
Відповідь
(C)
$$2x,cos left( {{x^2} + 1}
ight)$$
2
From a point $$O$$ inside a triangle $$ABC,$$ perpendiculars $$OD$$, $$OE, OF$$ are drawn to the sides $$BC, CA, AB$$ respectively. Prove that the perpendiculars from $$A, B, C$$ to the sides $$EF, FD, DE$$ are concurrent.
Відповідь
(C)
The perpendiculars from A, B, C to EF, FD, DE are concurrent at a point related to the isogonal conjugate of O with respect to triangle ABC.
3
Balls are drawn one-by-one without replacement from a box containing $$2$$ black, $$4$$ white and $$3$$ red balls till all the balls are drawn. Find the probability that the balls drawn are in the order $$2$$ black, $$4$$ white and $$3$$ red.
${1 \over 2}\log |\sin x - \cos x| + {x \over 2} + C
5
A triangle $$ABC$$ has sides $$AB=AC=5$$ cm and $$BC=6$$ cm Triangle $$A'B'C'$$ is the reflection of the triangle $$ABC$$ in a line parallel to $$AB$$ placed at a distance $$2$$ cm from $$AB$$, outside the triangle $$ABC$$. Triangle $$A''B''C''$$ is the reflection of the triangle $$A'B'C'$$ in a line parallel to $$BC$$ placed at a distance of $$2$$ cm from $$B'C'$$ outside the triangle $$A'B'C'$$. Find the distance between $$A$$ and $$A''$$.
Відповідь
(C)
8\sqrt{17/5} cm
6
If x = a + b, y = a$$\gamma $$ + b$$\beta $$ and z = a$$\beta $$ +b$$\gamma $$ where $$\gamma $$ and $$\beta $$ are the complex cube roots of unity, show that xyz = $${a^3} + {b^3}$$.
Відповідь
(B)
The statement is true; xyz = a^3 + b^3.
7
Find the equation of the circle whose radius is 5 and which touches the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$ at the point (5, 5).
Відповідь
(A)
$${x^2}, + ,{y^2}, - ,18x,, - 16y, + 120 = 0,$$
8
One side of rectangle lies along the line $$4x + 7y + 5 = 0.$$ Two of its vertices are $$(-3, 1)$$ and $$(1, 1).$$ Find the equations of the other three sides.
Відповідь
A
B
C
9
A straight line segment of length $$\ell $$ moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio $$1 : 2$$
Відповідь
(C)
$$9{x^2} + 36{y^2} = 4{\ell ^2}$$
10
Six X' s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.
Відповідь
(A)
26
11
Sketch the solution set of the following system of inequalities:
$$${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.$$$
Відповідь
(D)
The solution set is a bounded region.
12
Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$
Відповідь
(C)
x = 3
13
Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$
Відповідь
(C)
x = 3
14
Show that the square of $$\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}$$ is a rational number.
Відповідь
(C)
The square of the expression equals 1.
15
Solve the following equation for $$x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0$$
