$$P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$$ if and only if the relation between $$P(A)$$ and $$P(B)$$ is .............
პასუხი
(A)
P(A) = P(B)
2
In a multiple-choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks the correct answers. The candidate decides to tick the answers at random, If he is allowed upto three chances to answer the questions, find the probability that he will get marks in the questions.
პასუხი
(A)
1/5
3
If $$\left| {\matrix{
a & {{a^2}} & {1 + {a^3}} \cr
b & {{b^2}} & {1 + {b^3}} \cr
c & {{c^2}} & {1 + {c^3}} \cr
} } \right| = 0$$ and the vectors
$$\overrightarrow A = \left( {1,a,{a^2}} \right),\,\,\overrightarrow B = \left( {1,b,{b^2}} \right),\,\,\overrightarrow C = \left( {1,c,{c^2}} \right),$$ are non-coplannar, then the product $$abc=$$ .......
პასუხი
(C)
-1
4
If $$\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C} $$ are three non-coplannar vectors, then -
$${{\overrightarrow A .\overrightarrow B \times \overrightarrow C } \over {\overrightarrow C \times \overrightarrow A .\overrightarrow B }} + {{\overrightarrow B .\overrightarrow A \times \overrightarrow C } \over {\overrightarrow C .\overrightarrow A \times \overrightarrow B }} = $$ ................
პასუხი
(A)
0
5
If $$\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)$$ are given vectors, then a vector $$B$$ satifying the equations $$\overrightarrow A \times \overrightarrow B = \overrightarrow {\,C} $$ and $$\overrightarrow A .\overrightarrow B = \overrightarrow {3\,} $$ ..........
პასუხი
(A)
$$\frac{5}{3}\widehat i + \frac{2}{3}\widehat j + \frac{2}{3}\widehat k$$
6
A box contains $$100$$ tickets numbered $$1, 2, ....., 100.$$ Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than $$10.$$ The minimum number on them is $$5$$ with probability ........
პასუხი
(A)
1/9
7
Let $${x^2} + {y^2} - 4x - 2y - 11 = 0$$ be a circle. A pair of tangentas from the point (4, 5) with a pair of radi from a quadrilateral of area............................
პასუხი
(C)
8 sq units
8
If $$a,\,b,\,c$$ and $$u,\,v,\,w$$ are complex numbers representing the vertics of two triangles such that $$c = \left( {1 - r} \right)a + rb$$ and $$w = \left( {1 - r} \right)u + rv,$$ where $$w = \left( {1 - r} \right)u + rv,$$ is a complex number, then the two triangles
პასუხი
(B)
are similar
9
If $${z_1}$$ = a + ib and $${z_2}$$ = c + id are complex numbers such that $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1$$ and $${\mathop{\rm Re}\nolimits} ({z_1}\,{\overline z _2}) = 0$$, then the pair of complex numbers $${w_1}$$ = a + ic and $${w_2}$$ = b+ id satisfies -
If $${n_1}$$, $${n_2}$$,.......$${n_p}$$ are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.
პასუხი
(B)
FALSE
12
If $$P(x) = a{x^2} + bx + c\,\,and\,\,Q(x) = - a{x^2} + dx + c$$, where $$ac \ne \,0$$, then P(x) Q(x) = 0 has at least two real roots.
პასუხი
(B)
FALSE
13
If $${\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)$$, then x lies in the interval-
პასუხი
(A)
$$(2,\infty )$$
14
The product of any r consecutive natural numbers is always divisible by r!
პასუხი
(B)
FALSE
15
Use method of mathematical induction $${2.7^n} + {3.5^n} - 5$$ is divisible by $$24$$ for all $$n > 0$$
პასუხი
A
B
C
16
7 relatives of a man comprises 4 ladies and 3 gentlemen ; his wife has also 7 relatives ; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relatives and 3 of the wife's relatives?
პასუხი
(A)
485
17
If $$a,\,b,\,c$$ are in GP., then the equations $$\,\,\alpha {x^2} + 2bx + c = 0$$ and $$d{x^2} + 2ex + f = 0$$ have a common root if $${d \over a},\,{e \over b},{f \over c}$$ are in ________.
პასუხი
(A)
A.P.
18
Find the sum of the series :
$$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms} \right]} $$$
The orthocentre of the triangle formed by the lines $$x + y = 1,\,2x + 3y = 6$$ and $$4x - y + 4 = 0$$ lies in quadrant number .............
პასუხი
(C)
III
20
Three lines $$px + qy + r = 0$$, $$qx + ry + p = 0$$ and $$rx + py + q = 0$$ are concurrent if
პასუხი
A
B
C
21
One of the diameters of the circle circumscribing the rectangle $$ABCD$$ is $$4y = x + 7$$. If $$A$$ and $$B$$ are the points $$(-3, 4)$$ and $$(5, 4)$$ respectively, then find the area of rectangle.
პასუხი
(C)
32 sq. units
22
Two sides of rhombus $$ABCD$$ are parallel to the lines $$y = x + 2$$ and $$y = 7x + 3$$. If the diagonals of the rhombus intersect at the point $$(1, 2)$$ and the vertex $$A$$ is on the $$y$$-axis, find possible co-ordinates of $$A$$.
პასუხი
A
C
23
If three complex numbers are in A.P. then they lie on a circle in the complex plane.
პასუხი
(B)
FALSE
24
From the origin chords are drawn to the circle $${(x - 1)^2} + {y^2} = 1$$. The equation of the locus of the mid-points of these chords is.............
პასუხი
(A)
x^2 + y^2 - x = 0
25
No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices $$\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt 3 } \right),\,\,\left( {3,\sqrt 3 } \right)$$.
პასუხი
(B)
FALSE
26
If $${f_r}\left( x \right),{g_r}\left( x \right),{h_r}\left( x \right),r = 1,2,3$$ are polynomials in $$x$$ such that $${f_r}\left( a \right) = {g_r}\left( a \right) = {h_r}\left( a \right),r = 1,2,3$$
and $$F\left( x \right) = \left| {\matrix{
{{f_1}\left( x \right)} & {{f_2}\left( x \right)} & {{f_3}\left( x \right)} \cr
{{g_1}\left( x \right)} & {{g_2}\left( x \right)} & {{g_3}\left( x \right)} \cr
{{h_1}\left( x \right)} & {{h_2}\left( x \right)} & {{h_3}\left( x \right)} \cr
} } \right|$$ then $$F'\left( x \right)$$ at $$x = a$$ is ...........
პასუხი
(B)
$$0$$
27
If $$f\left( x \right) = {\log _x}\left( {In\,x} \right),$$ then $$f'\left( x \right)$$ at $$x=e$$ is ................
პასუხი
(A)
1/e
28
The set of all real numbers $$a$$ such that $${a^2} + 2a,2a + 3$$ and $${a^2} + 3a + 8$$ are the sides of a triangle is ...........
პასუხი
(B)
(5, ∞)
29
In a triangle $$ABC$$, if cot $$A$$, cot $$B$$, cot $$C$$ are in A.P., then $${a^2},{b^2},{c^2}$$, are in ............... progression.
პასუხი
(A)
Arithmetic
30
A ladder rests against a wall at an angle $$\alpha $$ to the horizintal. Its foot is pulled away from the wall through a distance $$a$$, so that it slides $$a$$ distance $$b$$ down the wall making an angle $$\beta $$ with the horizontal. Show that $$a = b\tan {1 \over 2}\left( {\alpha + \beta } \right)$$
პასუხი
(E)
There is no explanation provided.
31
In a triangle $$ABC$$, the median to the side $$BC$$ is of length
$$${1 \over {\sqrt {11 - 6\sqrt 3 } }}$$$ and it divides the angle $$A$$ into angles $${30^ \circ }$$ and $${45^ \circ }$$. Find the length of the side $$BC$$.
პასუხი
(C)
$$2$$
32
Find all the tangents to the curve
$$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$ that are parallel to the line $$x+2y=0$$.
პასუხი
A
B
33
Let $$f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.$$ Find the intervals in which $$\lambda $$ should lie in order that $$f(x)$$ has exactly one minimum and exactly one maximum.
