Simplify: \[\frac{\sqrt{7\sqrt{5}+17\sqrt{(21+8\sqrt{5})}-\sqrt{5}}}{4}\]
A)
\[\frac{7\sqrt{5}}{4}\]
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B)
\[\sqrt{5}\]
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C)
\[\frac{21}{4}\]
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D)
1
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E)
None of these
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If \[\frac{8-3\sqrt{7}}{8+3\sqrt{7}}=a-b\sqrt{7},\] then which one of the following represents the value of a and b?
A)
a = 164 and b = 22
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B)
a = 64 and b = 63
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C)
a = 127 and b= 48
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D)
a = 33 and b = 22
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E)
None of these
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If \[x=\frac{3+\sqrt{7}}{3-\sqrt{7}},\] then find the value of \[\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)\].
A)
127
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B)
254
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C)
\[96\,\sqrt{7}\]
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D)
\[48\,\sqrt{7}\]
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E)
None of these
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Factorise: \[25{{y}^{4}}+65{{y}^{2}}+42\]
A)
\[(5{{y}^{2}}+6)\,\,(5{{y}^{2}}+7)\]
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B)
\[(5y+1)\,\,(3{{y}^{3}}+1)\]
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C)
\[(5{{y}^{2}}+7)\,\,(7{{y}^{2}}+6)\]
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D)
\[(5{{y}^{2}}+6)\,\,(5+y)\]
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E)
None of these
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If \[\alpha \] and \[\beta \] are the zeroes of the polynomial \[4{{z}^{2}}+kz-12\] such that \[{{\alpha }^{2}}+{{\beta }^{2}}=16,\] then find the value of k.
A)
\[\sqrt{10}\]
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B)
4
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C)
\[5\,\sqrt{7}\]
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D)
\[4\,\sqrt{10}\]
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E)
None of these
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If \[\alpha ,\beta ,\gamma \] are the roots of the polynomial \[h(n)={{n}^{3}}-16{{n}^{2}}+18n-12,\] then find the value of\[\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }\].
A)
\[-\,\,\frac{1}{2}\]
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B)
\[-\,\,\frac{10}{9}\]
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C)
\[\frac{3}{2}\]
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D)
\[\frac{3}{8}\]
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E)
None of these
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\[({{\log }_{5}}5)\,\,({{\log }_{4}}2)\,\,({{\log }_{3}}9)\]is equal to:
A)
1
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B)
0
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C)
\[-1\]
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D)
2
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E)
None of these
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If \[{{\log }_{8}}x+{{\log }_{8}}\frac{1}{6}=\frac{1}{3},\] then the value of x is:
A)
12
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B)
16
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C)
18
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D)
24
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E)
None of these
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If \[\log 2=0.3010\]and\[\log 3=0.4771\], then log 4.5 is equal to:
A)
0.523
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B)
0.6532
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C)
0.4231
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D)
\[\frac{2}{9}\]
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E)
None of these
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Find the value of m for which the system of equation \[mx-2y=8\] and \[3x+5y=13\] have infinitely many solutions.
A)
\[\left( m\ne -\frac{6}{5}\And m=\frac{24}{13} \right)\]
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B)
\[\left( m=-\frac{6}{5}\And m=\frac{24}{13} \right)\]
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C)
\[\left( m\ne -\frac{6}{5}\And m\ne \frac{24}{13} \right)\]
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D)
\[\left( m=-\frac{6}{5}\And m\ne \frac{24}{13} \right)\]
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E)
None of these
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An aeroplane leaves an airport and flies due north at a speed of 1000 km/h. At same time, another plane flies due west at a speed of 1200 km/h from the same place. Distance between the two planes after 1.5 hours will be:
A)
2400 km
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B)
2520 km
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C)
2343 km
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D)
2434 km
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E)
None of these
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Seven times a two digit number is equal to four times the number obtained by reversing the digits. If the difference between the digits is 3, then find the number.
A)
36
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B)
89
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C)
58
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D)
74
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E)
None of these
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A vertical stick 20 cm long casts a 10 cm long shadow on the ground. At the same time a vertical pole casts a 60 cm long shadow on the ground. Height of the pole is:
A)
120 cm
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B)
70 cm
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C)
125 cm
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D)
30 cm
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E)
None of these
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Find the length of EC in the figure given below, if\[DE\parallel BC\].
A)
2.6 cm
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B)
2.3 cm
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C)
6.2 cm
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D)
2.4 cm
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E)
None of these
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In triangle PQR, PT is the angle bisector of\[\angle \,QPR.\] If PQ = 8 cm, QT = 4 cm and TR = 2 cm. Find PR.
A)
4 cm
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B)
5 cm
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C)
6 cm
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D)
7 cm
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E)
None of these
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Find the perimeter of a rhombus whose diagonals are 16 cm and 30 cm long.
A)
17 cm
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B)
69 cm
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C)
63 cm
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D)
68 cm
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E)
None of these
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Two opposite angles of a parallelogram are \[(3x+20){}^\circ \] and \[(50-x){}^\circ \]. Find the measure of x.
A)
\[4.5{}^\circ \]
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B)
\[15{}^\circ \]
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C)
\[7.5{}^\circ \]
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D)
\[3.5{}^\circ \]
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E)
None of these
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In a parallelogram ABCD, the bisectors of \[\angle A\]and \[\angle \,B\]meet at O. The measure of \[\angle \,AOB\] is :
A)
\[30{}^\circ \]
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B)
\[60{}^\circ \]
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C)
\[90{}^\circ \]
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D)
\[120{}^\circ \]
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E)
None of these
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If all sides of a parallelogram touches a circle, then the parallelogram will be a :
A)
Rectangle
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B)
Square
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C)
Rhombus
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D)
Trapezium
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E)
None of these
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If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, then the quadrilateral will be a :
A)
Square
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B)
Rectangle
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C)
Trapezium
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D)
Rhombus
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E)
None of these
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Find the ratio of the areas of the incircle and circumcircle of an equilateral triangle.
A)
2 : 3
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B)
3 : 4
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C)
1 : 4
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D)
4 : 5
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E)
None of these
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If D is the mid-point of the straight line joining \[A\,(-\,2,\,9)\] and B (4, 3), then the coordinates of D is:
A)
(4, 5)
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B)
\[(10,-3)\]
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C)
\[(-10,-4)\]
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D)
(1, 6)
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E)
None of these
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Two vertices of a \[\Delta \,ABC\] are given by A (2, 3) and \[B\,\,(-\,2,\,1)\] and its centroid is \[G\,\,\left( 1,\,\,\frac{2}{3} \right)\]. Find the coordinates of the third vertex C of the\[\Delta \,ABC.\]
A)
(2, 4)
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B)
\[(-\,1,-\,2)\]
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C)
\[(3,-\,2)\]
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D)
\[(-2,-3)\]
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E)
None of these
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The area of the triangle formed by \[(a,\,b+c),\] \[(b,\,c+a)\] and \[(c,\,a+b)\] is:
A)
0
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B)
\[abc\]
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C)
\[(a+b+c)\]
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D)
\[{{(a+b+c)}^{2}}\]
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E)
None of these
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If \[2\,\,\sin 2\theta =\sqrt{3},\] then the value of\[\theta \]is;
A)
\[\frac{\pi }{3}\]
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B)
\[\frac{\pi }{4}\]
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C)
\[\frac{\pi }{6}\]
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D)
\[\frac{\pi }{2}\]
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E)
None of these
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\[\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\operatorname{cotA}},\] can be reduced to:
A)
\[\sin A\]
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B)
\[\cos A\]
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C)
\[\sin \,A+cos\,A\]
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D)
\[\sin A-\cos A\]
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E)
None of these
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From a place which is 20 m above the ground, the angle of elevation and depression of the top and foot of a building are \[30{}^\circ \] and \[45{}^\circ \] respectively. The height of building is:
A)
\[\left( 20+\frac{20}{\sqrt{3}} \right)\,\,\,m\]
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B)
\[\left( 20-\frac{20}{\sqrt{3}} \right)\,\,\,m\]
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C)
\[\left( 20+20\,\sqrt{3} \right)\,\,\,m\]
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D)
\[\left( 20-20\,\sqrt{3} \right)\,\,\,m\]
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E)
None of these
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In what ratio are the volumes of a cylinder, a cone and a sphere, if each has the same radius and the same height?
A)
1 : 3 : 2
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B)
2 : 3 : 1
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C)
3 : 1 : 4
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D)
3 : 2 : 1
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E)
None of these
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A solid is made up of a cube and a hemisphere attached on its top. Each edge of the cube measures 5 cm and the hemisphere has a diametre of 4.2 cm. Find the total area to be painted.
A)
\[81.43\,\,c{{m}^{2}}\]
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B)
\[103.56\,\,c{{m}^{2}}\]
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C)
\[163.86\,\,c{{m}^{2}}\]
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D)
\[212.86\,\,c{{m}^{2}}\]
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E)
None of these
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A shuttlecock used for playing badminton has the shape of a frustum of a cone mounted on a hemisphere. The external diametres of the frustum are 5 cm and 2 cm and the height of the entire shuttlecock is 7 cm. Find its external surface area.
A)
\[18.39\,c{{m}^{2}}\]
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B)
\[37.13\,c{{m}^{2}}\]
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C)
\[74.27\,c{{m}^{2}}\]
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D)
\[82.56\,c{{m}^{2}}\]
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E)
None of these
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Find the mean of the following data.
Height (m) 15 16 17 18 19 20 21 No. of plants 10 5 7 12 4 6 8
A)
25.6
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B)
17.87
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C)
23.3
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D)
15.24
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E)
None of these
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Find m, if mean of the given distribution is 6.5.
A)
4.6
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B)
5
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C)
6
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D)
4
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E)
None of these
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Out of forty students, 14 are taking Mathematics and 26 are taking Environmental Studies. What is the probability that a randomly chosen student from this group is taking only the Environmental Studies?
A)
\[\frac{13}{15}\]
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B)
\[\frac{13}{20}\]
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C)
\[\frac{29}{40}\]
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D)
\[\frac{7}{5}\]
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E)
None of these
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The probability that the three friends have same birthday, if they were born in 1997 is:
A)
\[\frac{3}{365}\]
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B)
\[\frac{1}{365}\]
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C)
\[\frac{2}{365}\]
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D)
\[\frac{6}{365}\]
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E)
None of these
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Robert tosses a coin three times. The probability that he gets atleast two heads is:
A)
\[\frac{1}{2}\]
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B)
\[\frac{1}{3}\]
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C)
\[\frac{1}{4}\]
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D)
\[\frac{1}{5}\]
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E)
None of these
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Choose the group of letters which is different from the other.
A)
ABD
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B)
FGI
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C)
LMO
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D)
STU
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E)
None of these
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In a tennis tournament of singles, six players will play every other player exactly once. How many matches will be played during the tournament?
A)
6
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B)
12
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C)
15
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D)
30
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E)
None of these
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Out of four persons, B is taller than C, A is taller than D but not as tall as C. Who among them is the tallest?
A)
A
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B)
B
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C)
C
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D)
D
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E)
None of these
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If P denotes \[\div ,\] Q denotes x, R denotes + 1 and S denotes \[-,\] then 18 Q 12 P 4 R 5 S 6 = ?
A)
36
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B)
53
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C)
59
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D)
65
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E)
None of these
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How many numbers from 11 to 50 are there which are exactly divisible by 7 but not divisible by 3?
A)
2
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B)
4
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C)
5
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D)
6
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E)
None of these
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