$$P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$$ if and only if the relation between $$P(A)$$ and $$P(B)$$ is .............
Απάντηση
(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.
Απάντηση
(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 $$abc=$$ .......
Απάντηση
(C)
-1
4
If $$\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C} $$ are three non-coplannar vectors, then -
$${{\overrightarrow A .\overrightarrow B \times \overrightarrow C } \over {\overrightarrow C \times \overrightarrow A .\overrightarrow B }} + {{\overrightarrow B .\overrightarrow A \times \overrightarrow C } \over {\overrightarrow C .\overrightarrow A \times \overrightarrow B }} = $$ ................
Απάντηση
(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 $$\overrightarrow A .\overrightarrow B = \overrightarrow {3\,} $$ ..........
Απάντηση
(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 ........
Απάντηση
(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............................
Απάντηση
(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
Απάντηση
(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.
Απάντηση
(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.
Απάντηση
(B)
FALSE
13
If $${\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)$$, then x lies in the interval-
Απάντηση
(A)
$$(2,\infty )$$
14
The product of any r consecutive natural numbers is always divisible by r!
Απάντηση
(B)
FALSE
15
Use method of mathematical induction $${2.7^n} + {3.5^n} - 5$$ is divisible by $$24$$ for all $$n > 0$$
Απάντηση
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?
Απάντηση
(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 ________.
Απάντηση
(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 .............
Απάντηση
(C)
III
20
Three lines $$px + qy + r = 0$$, $$qx + ry + p = 0$$ and $$rx + py + q = 0$$ are concurrent if
Απάντηση
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.
Απάντηση
(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 $$A$$.
Απάντηση
A
C
23
If three complex numbers are in A.P. then they lie on a circle in the complex plane.
Απάντηση
(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.............
Απάντηση
(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 $$\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt 3 } \right),\,\,\left( {3,\sqrt 3 } \right)$$.
Απάντηση
(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 ...........
Απάντηση
(B)
$$0$$
27
If $$f\left( x \right) = {\log _x}\left( {In\,x} \right),$$ then $$f'\left( x \right)$$ at $$x=e$$ is ................
Απάντηση
(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 ...........
Απάντηση
(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.
Απάντηση
(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 $$a = b\tan {1 \over 2}\left( {\alpha + \beta } \right)$$
Απάντηση
(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 $$BC$$.
Απάντηση
(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 $$x+2y=0$$.
Απάντηση
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.
