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 $$\lambda .$$

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..........................

The expression $$2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]$$ is equal to

Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is $${1 \over 2}\,{\left| z \right|^2}$$ .

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

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

For $$a \le 0,$$ determine all real roots of the equation $$${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$$

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..........................

The solution of equation $${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is ........................

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?

The solution of the equation $$lo{g_7}$$ $$lo{g_5}$$ $$\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is .............

The points $$\left( {0,{8 \over 3}} \right),\,\,\left( {1,\,3} \right)$$ and $$\left( {82,\,30} \right)$$ are vertices of

All points lying inside the triangle formed by the points $$\left( {1,\,3} \right),\,\left( {5,\,0} \right)$$ and $$\left( { - 1,\,2} \right)$$ satisfy

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

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

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..............................

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.

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 ...............

There exists a triangle $$ABC$$ satisfying the conditions

If in a triangle $$ABC$$, $$\cos A\cos B + \sin A\sin B\sin C = 1,$$ Show that $$a:b:c = 1:1:\sqrt 2 $$

The principal value of $${\sin ^{ - 1}}\left( {\sin {{2\pi } \over 3}} \right)$$ is

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

If the line $$ax+by+c=0$$ is a normal to the curve $$xy=1$$, then

Evaluate : $$\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi } $$

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 ..............

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

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

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.