Let\(\vec{a}=\hat{i} - 2\hat{j} + 3\hat{k}, \vec{b}=\hat{i} - \hat{j} + \hat{k} \) and \(\vec{c}\)be a vector such that\(\vec{a} + (\vec{b}×\vec{c}) = \vec{0}\) and \(\vec{b}.\vec{c} = 5.\)Then the value of 3(\(\vec{c}.\vec{a}\)) is equal to
Let y = y(x), x > 1, be the solution of the differential equation\((x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1}\)with \(y(2) = \frac{1+e^4}{2e^4}\). If \(y(3) = \frac{e^α + 1}{βe^α}\) , then the value of α + β is equal to ____.
The number of solutions of the equation sin x = cos2 x in the interval (0, 10) is _____.
Let f and g be twice differentiable even functions on (–2, 2) such that\(ƒ(\frac{1}{4})=0, ƒ(\frac{1}{2})=0, ƒ(1) =1\) and \(g(\frac{3}{4}) = 0 , g(1)=2\).Then, the minimum number of solutions of f(x)g′′(x) + f′(x)g′(x) = 0 in (–2, 2) is equal to_____.
Let the coefficients of x–1 and x–3 in the expansion of\((2x^{\frac{1}{5}} - \frac{1}{x^{\frac{1}{5}}} )^{15} , x > 0\)be m and n respectively. If r is a positive integer such that\(mn² = ^{15}C_r.2^r\)then the value of r is equal to ______.
Let\(M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \\ \end{bmatrix}\)where α is a non-zero real number an\(N = \sum\limits_{k=1}^{49} M^{2k}. \) If \((I - M^2)N = -2I\)then the positive integral value of α is ____ .
Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 – 2x and g(f(x)) = 4x2 + 6x + 1, then the value of f(2) + g(2) is ____________ .
For real number a, b (a > b > 0), let\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)and \(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)Then the value of (a – b)2 is equal to _____.
Let\(f(x)=\frac{x−1}{x+1},x∈R− \left\{0,−1,1\right\}\)If ƒn+1(x) = ƒ(ƒn(x)) for all n∈N, then ƒ6(6) + ƒ7(7) is equal to :
Let\(A = \{z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1\}\)and\(B = \{z \in \mathbb{C} : \text{arg}(\frac{z-1}{z+1}) = \frac{2\pi}{3}\}\)Then \(A∩B\) is :