The position vectors of the points $$A, B, C$$ and $$D$$ are $$3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\widehat j - 4\widehat k,\, - \widehat i + \widehat j + 2\widehat k$$ and $$4\widehat i + 5\widehat j + \lambda \widehat k,$$
respectively. If the points $$A, B, C$$ and $$D$$ lie on a plane, find the value of $$\lambda .$$
คำตอบ
(C)
{{146} \over {17}}
2
Let $$\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$$ and $$\overrightarrow c = {c_1}i + {c_2}j + {c_3}k$$ be three non-zero vectors such that $$\overrightarrow c $$ is a unit vector perpendicular to both the vectors $$\overrightarrow a $$ and $$\overrightarrow b .$$ If the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is $${\pi \over 6},$$ then
$${\left| {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{b_1}} & {{b_2}} & {{b_3}} \cr
{{c_1}} & {{c_2}} & {{c_3}} \cr
} } \right|^2}$$ is equal to
From the point A(0, 3) on the circle $${x^2} + 4x + {(y - 3)^2} = 0$$, a chord AB is drawn and extended to a point M such that AM = 2AB. The equation of the locus of M is..........................
คำตอบ
(A)
x^2 + y^2 + 8x - 6y + 9 = 0
4
The expression $$2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]$$ is equal to
คำตอบ
(B)
1
5
Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is $${1 \over 2}\,{\left| z \right|^2}$$ .
คำตอบ
(D)
The area of the triangle is $${1 over 2},{left| z
ight|^2}$$.
6
If $$S$$ is the set of all real $$x$$ such that $${{2x - 1} \over {2{x^3} + 3{x^2} + x}}$$ is positive, then $$S$$ contains
คำตอบ
D
A
7
If $$a,\,b$$ and $$c$$ are distinct positive numbers, then the expression
$$\left( {b + c - a} \right)\left( {c + a - b} \right)\left( {a + b - c} \right) - abc$$ is
คำตอบ
(B)
negative
8
For $$a \le 0,$$ determine all real roots of the equation $$${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$$
คำตอบ
A
B
E
9
If the quadratic equations $${x^2} + ax + b = 0$$ and $${x^2} + bx + a = 0$$ $$(a \ne b)$$ have a common root, then the numerical value of a + b is..........................
คำตอบ
(A)
-1
10
The solution of equation $${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is ........................
คำตอบ
(D)
4
11
If $${C_r}$$ stands for $${}^n{C_r},$$ then the sum of the series
$${{2\left( {{n \over 2}} \right){\mkern 1mu} !{\mkern 1mu} \left( {{n \over 2}} \right){\mkern 1mu} !} \over {n!}}\left[ {C_0^2 - 2C_1^2 + 3C_2^2 - } \right......... + {\left( { - 1} \right)^n}\left( {n + 1} \right)C_n^2\mathop ]\limits^ \sim \,,$$
where $$n$$ is an even positive integer, is equal to
A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?
คำตอบ
(B)
64
13
The solution of the equation $$lo{g_7}$$ $$lo{g_5}$$ $$\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is .............
คำตอบ
(C)
4
14
The points $$\left( {0,{8 \over 3}} \right),\,\,\left( {1,\,3} \right)$$ and $$\left( {82,\,30} \right)$$ are vertices of
คำตอบ
(D)
none of these
15
All points lying inside the triangle formed by the points $$\left( {1,\,3} \right),\,\left( {5,\,0} \right)$$ and $$\left( { - 1,\,2} \right)$$ satisfy
คำตอบ
A
C
16
A vector $$\overline a $$ has components $$2p$$ and $$1$$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, $$\overline a $$ has components $$p + 1$$ and $$1$$, then
คำตอบ
(B)
$$p = 1$$ or $$p = - {1 \over 3}$$
17
Let $${z_1}$$ and $${z_2}$$ be complex numbers such that $${z_1}$$ $$ \ne $$ $${z_2}$$ and $$\left| {{z_1}} \right| =\,\left| {{z_2}} \right|$$. If $${z_1}$$ has positive real and $${z_2}$$ has negative imaginary part, then $${{{z_1}\, + \,{z_2}} \over {{z_1}\, - \,{z_2}}}$$ may be
คำตอบ
A
D
18
The equation of the line passing through the points of intersection of the circles $$3{x^2} + 3{y^2} - 2x + 12y - 9 = 0$$ and $${x^2} + {y^2} - 6x + 2y - 15 = 0$$ is..............................
คำตอบ
(C)
10x - 3y - 18 = 0
19
Lines 5x + 12y - 10 = 0 and 5x - 12y - 40 = 0 touch a circle $$C_1$$ of diameter 6. If the centre of $$C_1$$ lies in the first quadrant, find the equation of the circle $$C_2$$ which is concentric with $$C_1$$ and cuts intercepts of length 8 on these lines.
คำตอบ
(A)
x^2 + y^2 - 10x - 4y + 4 = 0
20
The derivative of $${\sec ^{ - 1}}\left( {{1 \over {2{x^2} - 1}}} \right)$$ with respect to $$\sqrt {1 - {x^2}} $$ at $$x = {1 \over 2}$$ is ...............
คำตอบ
(B)
4
21
There exists a triangle $$ABC$$ satisfying the conditions
คำตอบ
A
D
22
If in a triangle $$ABC$$, $$\cos A\cos B + \sin A\sin B\sin C = 1,$$ Show that $$a:b:c = 1:1:\sqrt 2 $$
คำตอบ
A
B
E
23
The principal value of $${\sin ^{ - 1}}\left( {\sin {{2\pi } \over 3}} \right)$$ is
คำตอบ
(D)
none
24
Let $$P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}$$ be a polynomial in a real variable $$x$$ with
$$0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.$$ The function $$P(x)$$ has
คำตอบ
(C)
only one minimum
25
If the line $$ax+by+c=0$$ is a normal to the curve $$xy=1$$, then
If $${{1 + 3p} \over 3},\,\,\,{{1 - p} \over 4}$$ and $$\,{{1 - 2p} \over 2}$$ are the probabilities of three mutually exclusive events, then the set of all values of $$p$$ is ..............
คำตอบ
(C)
$$1/3 \le p \le 1/2$$
28
A student appears for tests, $$I$$, $$II$$ and $$III$$. The student is successful if he passes either in tests $$I$$ and $$II$$ or tests $$I$$ and $$III$$. The probabilities of student passing in tests $$I$$, $$II$$ and $$III$$ are $$p, q$$ and $${1 \over 2}$$ respectively. If the probability that the student is successful is $${1 \over 2}$$, then
คำตอบ
(C)
$$p=1,$$ $$q=0$$
29
The probability that at least one of the events $$A$$ and $$B$$ occurs is $$0.6$$. If $$A$$ and $$B$$ occur simultaneously with probability $$0.2,$$ then $$P\left( {\overline A } \right) + P\left( {\overline B } \right)$$ is
คำตอบ
(C)
$$1.2$$
30
A lot contains $$20$$ articles. The probability that the lot contains exactly $$2$$ defective articles is $$0.4$$ and the probability that the lot contains exactly $$3$$ defective articles is $$0.6$$. Articles are drawn from the lot at random one by one without replacement and are tested till all defective articles are found. What is the probability that the testing procedure ends at the twelth testing.
