For the following questions answer them individually
Let $$\overrightarrow{a}=-\widehat{i}+2\widehat{j}+2\widehat{k},\overrightarrow{b}=8\widehat{i}+7\widehat{j}-3\widehat{k} \text { and } \overrightarrow{c}$$ be a vector such that $$\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{b}$$. If $$\overrightarrow{c}\cdot(\widehat{i}+\widehat{j}+\widehat{k})=4$$, then $$\mid\overrightarrow{a}+\overrightarrow{c}\mid^{2}$$ is equal to :
Let PQ and MN be two straight lines touching the circle $$x^{2}+y^{2}-4x-6y-3=0$$ at the points A and B respectively. Let O be the centre of the circle and $$\angle AOB=\pi/3$$. Then the locus of the point of intersection of the lines PQ and MN is:
The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to:
If $$x^{2}+x+1=0$$, then the value of $$\left(x+\frac{1}{x}\right)^{4}+\left(x^{2}+\frac{1}{x^{2}}\right)^{4}+\left(x^{3}+\frac{1}{x^{3}}\right)^{4}+...+\left(x^{25}+\frac{1}{x^{25}}\right)^{4}$$ is:
Let $$f: RÂ \rightarrow (0, \infty)$$ be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f" (3) = 4. Then $$\lim_{x \rightarrow 1}\left(\log_{a}\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^{2}}}\right)$$ ls equal to :
The value of cosec10° - $$\sqrt{3}$$ secl0° is equal to :
If the coefficient of x in the expansion of $$(ax^{2}+bx+c)(1-2x)^{26}$$. is - 56 and the coefficients of $$x^{2}\text{ and }x^{3}$$ are both zero, then a + b + c is equal to:
Let a point A lie between the parallel lines $$L_{1}\text{ and }L_{2}$$ such that its distances from $$L_{1}\text{ and }L_{2}$$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines $$L_{1}\text{ and }L_{2}$$, respectively, is:
Let O be the vertex of the parabola $$x^{2}=4y$$ and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2: 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is:
Let $$\overrightarrow{c} \text{ and } \overrightarrow{d}$$ be vectors such that $$\mid\overrightarrow{c}+\overrightarrow{d}\mid=\sqrt{29}$$ and $$\overrightarrow{c}\times( 2\widehat{i}+3\widehat{j}+4\widehat{k})=(2\widehat{i}+3\widehat{j}+4\widehat{k})\times\overrightarrow{d}$$. If $$\lambda_{1}, \lambda_{2}( \lambda_{1}> \lambda_{2})$$ are the possible values of $$(\overrightarrow{c}+\overrightarrow{d})\cdot(-7\widehat{i}+2\widehat{j}+3\overrightarrow{k})$$, then the equation $$K^{2}x^{2}+(K^{2}-5K+\lambda_{1})xy+\left(3K+\frac{\lambda_{2}}{2} \right)y^{2}-8x+12y+\lambda_{2}=0$$ represents a circle, for K equal to :
