Prove that for any positive integer $$k$$,
$${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]$$
Hence prove that $$\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} $$
Answer
(B)
The first part involves mathematical induction and trigonometric sum-to-product formulas, while the second part requires integration of trigonometric functions.
2
Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $$e=1.$$
This requires advanced calculus techniques, specifically dealing with trigonometric substitutions and properties of definite integrals.
4
Let $$A$$ and $$B$$ be two events such that $$P\,\,\left( A \right)\,\, = \,\,0.3$$ and $$P\left( {A \cup B} \right) = 0.8.$$ If $$A$$ and $$B$$ are independent events then $$P(B)=$$ ................
Answer
(E)
5/7
5
A is a set containing $$n$$ elements. $$A$$ subset $$P$$ of $$A$$ is chosen at random. The set $$A$$ is reconstructed by replacing the elements of $$P.$$ $$A$$ subset $$Q$$ of $$A$$ is again chosen at random. Find the probability that $$P$$ and $$Q$$ have no common elements.
Answer
(C)
$$\left( \frac{3}{4} \right)^n$$
6
Let $$\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k ,$$ and $$\overrightarrow C = 4\overrightarrow i - 3\overrightarrow j + 7\overrightarrow k .$$ Determine a vector $$\overrightarrow R .$$ Satisfying $$\overrightarrow R \times \overrightarrow B = \overrightarrow C \times \overrightarrow B $$ and $$\overrightarrow R \,.\,\overrightarrow A = 0$$
Answer
(A)
- \widehat i - 8\widehat j + 2\widehat k
7
Let $${z_1}$$ = 10 + 6i and $${z_2}$$ = 4 + 6i. If Z is any complex number such that the argument of $${{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \over 4}$$ , then prove that $$\left| {z - 7 - 9i} \right| = 3\sqrt 2 $$.
Answer
(A)
The argument condition geometrically represents a circle passing through z1 and z2. The center of the circle can be determined by considering the geometry and is 7+9i. Then using the formula of the radius the given equation can be proved.
The equation $$\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,$$
In the variable x, has real roots. Then p can take any value in the interval
Answer
(D)
$$\left( {0,\pi } \right)$$
10
If $$A,\,B$$ and $$C$$ are in arithmetic progression, determine the values of $$A,\,B$$ and $$C$$.
Answer
(B)
A = 45°, B = 60°, C = 75°
11
If $$\,x < 0,\,\,y < 0,\,\,x + y + {x \over y} = {1 \over 2}$$ and $$(x + y)\,{x \over y} = - {1 \over 2}$$, then x =..........and y =.........
Answer
(A)
x = -1/4, y = -1/4
12
The number of solutions of the equation sin$${(e)^x} = {5^x} + {5^{ - x}}$$ is
Answer
(A)
0
13
Prove that $${{{n^7}} \over 7} + {{{n^5}} \over 5} + {{2{n^3}} \over 3} - {n \over {105}}$$ is an integer for every positive integer $$n$$
Answer
(A)
The expression is always an integer due to Fermat's Little Theorem and modular arithmetic.
14
The number $${\log _2}\,7$$ is
Answer
(C)
an irrational number
15
If $${\log _3}\,2\,,\,\,{\log _3}\,({2^x} - 5)\,,\,and\,\,{\log _3}\,\left( {{2^x} - {7 \over 2}} \right)$$ are in arithmetic progression, determine the value of x.
Answer
3
16
Line $$L$$ has intercepts $$a$$ and $$b$$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line $$L$$ has intercepts $$p$$ and $$q$$, then
A line cuts the $$x$$-axis at $$A (7, 0)$$ and the $$y$$-axis at $$B (0, -5)$$. A variable line $$PQ$$ is drawn perpendicular to $$AB$$ cutting the $$x$$axis in $$P$$ and they $$Y$$-axis in $$Q$$. If $$AQ$$ and $$BP$$ intersect at $$R$$, find the locus of R.
Answer
(B)
x^2 + y^2 - 7x + 5y = 0
18
A circle touches the line y = x at a point P such that OP = $${4\sqrt 2 \,}$$, where O is the origin. The circle contains the point (- 10, 2) in its interior and the length of its chord on the line x + y = 0 is $${6\sqrt 2 \,}$$. Determine the equation of the circle.
Answer
(B)
x^2 + y^2 + 18x - 2y + 32 = 0
19
If $$f\left( x \right) = \left| {x - 2} \right|$$ and $$g\left( x \right) = f\left[ {f\left( x \right)} \right]$$, then $$g'\left( x \right) = ...............$$ for $$x > 20$$
Answer
(C)
1
20
Let $$f(x)$$ be a quadratic expression which is positive for all the real values of $$x$$. If $$g(x)=f(x)+f''(x)$$, then for any real $$x$$,
Answer
(B)
$$g(x)>0$$
21
Let $$f(x)$$ be a quadratic expression which is positive for all the real values of $$x$$. If $$g(x)=f(x)+f''(x)$$, then for any real $$x$$,
Answer
(B)
$$g(x)>0$$
22
In a triangle $$ABC$$, angle $$A$$ is greater than angle $$B$$. If the measures of angles $$A$$ and $$B$$ satify the equation $$3{\mathop{\rm sinx}\nolimits} - 4si{n^3}x - k = 0,$$ $$0 < k < 1$$, then the measure of angle $$C$$ is
Answer
(C)
$${2\pi \over 3}$$
23
A vertical tower $$PQ$$ stands at a point $$P$$. Points $$A$$ and $$B$$ are located to the South and East of $$P$$ respectively. $$M$$ is the mid point of $$AB$$. $$PAM$$ is an equilateral triangle; and $$N$$ is the foot of the perpendicular from $$P$$ and $$AB$$. Let $$AN$$$$=20$$ mrtres and the angle of elevation of the top of the tower at $$N$$ is $${\tan ^{ - 1}}\left( 2 \right)$$. Determine the height of the tower and the angles of elevation of the top of the tower at $$A$$ and $$B$$.
Answer
A
B
C
24
Show that $$2\sin x + \tan x \ge 3x$$ where $$0 \le x < {\pi \over 2}$$.
Answer
A
B
C
E
25
A point $$P$$ is given on the circumference of a circle of radius $$r$$. Chord $$QR$$ is parallel to the tangent at $$P$$. Determine the maximum possible area of the triangle $$PQR$$.
Answer
(D)
$$ rac{3sqrt{3}}{4}r^2$$
26
Let $$f:R \to R$$ and $$\,\,g:R \to R$$ be continuous functions. Then the value of the integral
$$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \right)} \right]dx} $$ is
