LetA=\(\begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}\)If B = I – 5C1(adjA) + 5C2(adjA)2 – …. – 5C5(adjA)5, then the sum of all elements of the matrix B is
Let \(\stackrel{→}{a} = \hat{i} + \hat{j} + \hat{2k}, \stackrel{→}{b} = \hat{2i} - \hat{3j} + \hat{k}\)and \(\stackrel{→}{c}= \hat{i} - \hat{j} + \hat{k}\)be three given vectors.Let \(\stackrel{→}{v}\) be a vector in the plane of \(\stackrel{→}{a}\) and \(\stackrel{→}{b}\) whose projection on \(\stackrel{→}{c}\) is \(\frac{2}{\sqrt3}\).If \(\stackrel{→}{v}.\hat{j}\) = 7 , then \(\stackrel{→}{v}.(\hat{i}+\hat{k})\) is equal to :
If the two lines \(l1:\frac{(x−2)}{3}=\frac{(y+1)}{−2},z=2 \)and\( l2:\frac{(x−1)}{1}=\frac{(2y+3)}{α}=\frac{(z+5)}{2} \)are perpendicular, then an angle between the lines l2 and \(l3:\frac{(1−x)}{3}=\frac{(2y−1)}{−4}=\frac{z}{4} \)is
Let the image of the point P(1, 2, 3) in the line \(L:\frac{x−6}{3}=\frac{y−1}{2}=\frac{z−2}{3} \)be Q. Let R (α, β, γ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22(α + β + γ) is equal to ________.
Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tanx(cosx – y). If the curve passes through the point (π/4, 0) then the value of \(\int_{0}^{\frac{\pi}{2}} y \,dx\)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 ____.
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 ____________ .
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 ____ .
The sum of the infinite series\(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} +\frac{51}{6^5} + \frac{70}{6^6}+…..\)is equal to
Let\(f(x) = \begin{vmatrix} a & -1 & 0\\ ax & a & -1\\ ax^2 & ax & a \end{vmatrix}\)a ∈ R. Then the sum of the square of all the values of a, for which 2f′(10) –f′(5) + 100 = 0, is
Let ƒ R ⇒ R be a function defined by
where [t] is the greatest integer less than or equal to t. Let m be the number of points where ƒ is not differentiable and
Then the ordered pair (m, I) is equal to :
\(\tan(2\tan^{−1}\frac{1}{5}+\sec^{−1}\frac{\sqrt5}{2}+2\tan^{−1}\frac{1}{8})\)is equal to :
The sum of all the elements of the set {α ∈ {1, 2, …, 100} : HCF(α, 24) = 1} is
