\( \vec{a}, \vec{b}, \vec{c} \) are three vectors such that \(|\vec{a}| = 3\), \(|\vec{b}| = 2\sqrt{2}\), \(|\vec{c}| = 5\), and \( \vec{c} \) is perpendicular to the plane of \( \vec{a} \) and \( \vec{b} \).
If the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{4} \), then
\[ |\vec{a} + \vec{b} + \vec{c}| = \ ? \]
Find \( \frac{dy}{dx} \) for the given function:
\[ y = \tan^{-1} \left( \frac{\sin^3(2x) - 3x^2 \sin(2x)}{3x \sin(2x) - x^3} \right). \]
The function \( y = 2x^3 - 8x^2 + 10x - 4 \) is defined on \([1,2]\). If the tangent drawn at a point \( (a,b) \) on the graph of this function is parallel to the X-axis and \( a \in (1,2) \), then \( a = \) ?
A, B, C, D are square matrices such that A + B is symmetric, A - B is skew-symmetric, and D is the transpose of C.
If
\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & 3 & -2 \\ 3 & -4 & 5 \end{bmatrix} \]
and
\[ C = \begin{bmatrix} 0 & 1 & -2 \\ 2 & -1 & 0 \\ 0 & 2 & 1 \end{bmatrix} \]
then the matrix \( B + D \) is:
For \( n \in \mathbb{N} \), the largest positive integer that divides \( 81^n + 20n - 1 \) is \( k \). If \( S \) is the sum of all positive divisors of \( k \), then find \( S - k \).
If A = \(\begin{bmatrix} 0 & 3\\ 0 & 0 \end{bmatrix}\)and f(x) = x+x2+x3+.....+x2023, then f(A)+I =
The quadratic equation whose roots are sin218° and cos2 36° is
If x = log (y +√y2 + 1 ) then y =
A bag contains four balls. Two balls are drawn randomly and found them to be white. The probability that all the balls in the bag are white is
The number of diagonals of a polygon is 35. If A, B are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having AB as one of its sides is:
The orthocenter of the triangle whose sides are given by x + y + 10 = 0, x - y - 2 = 0 and 2x + y - 7 = 0 is
For l ∈ R, the equation (2l - 3) x2 + 2lxy - y2 = 0 represents a pair of distinct lines
In a triangle BC, if the mid points of sides AB, BC, CA are (3,0,0), (0,4,0),(0,0,5) respectively, then AB2 + BC2 + CA2 =
If ∫ \(\frac{x^{49} Tan^{-1} (x^{50})}{(1+x^{100})}\)dx = k(Tan-1 (x50))2 + c, then k =
\(∫\frac{dx}{(x2+1) (x2+4)} =\)
\(∫\frac{dx}{(x-1)^{34} (x+2)^{\frac54}}=\)
If order and degree of the differential equation corresponding to the family of curves y2 = 4a(x+a)(a is parameter) are m and n respectively, then m+n2 =
The general solution of the differential equation (x2 + 2)dy +2xydx = ex(x2+2)dx is
if |a| = 4, |b| = 5, |a - b| = 3 and θ is the angle between the vectors a and b, then cot2 θ =