Find the derivative of $$\sin \left( {{x^2} + 1} \right)$$ with respect to $$x$$ first principle.
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(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.
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(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 $$A''$$.
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(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 = $${a^3} + {b^3}$$.
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(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).
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(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.
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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 $$1 : 2$$
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(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.
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(A)
26
11
Sketch the solution set of the following system of inequalities:
$$${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.$$$
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(D)
The solution set is a bounded region.
12
Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$
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(C)
x = 3
13
Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$
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(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.
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(C)
The square of the expression equals 1.
15
Solve the following equation for $$x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0$$
