Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function such that \(f(0)=0, f(1)=1, f(2)=-1, f(3)=2\) and \(f(4)=-2\). Then, the minimum number of zeros of \(\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)\) is ...........

Let \(\quad S =\left\{ M =\left[ a _{ ij }\right], a _{ ij } \in\{0,1,2\}, 1 \leq i , j \leq 2\right\}\) be a sample space and \(A=\{M \in S: M\) is invertible \(\}\) be an event. Then \(P ( A )\) is equal to

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius \( = \sqrt 3 \) is

Let \(A =\{ x \in R :| x +1|<2\}\) and \(B=\{x \in R:|x-1| \geq 2\}\). Then which one of the following statements is NOT true ?

Let \(A B C D\) be a tetrahedron such that the edges \(\mathrm{AB}, \mathrm{AC}\) and AD are mutually perpendicular. Let the areas of the triangles \(\mathrm{ABC}, \mathrm{ACD}\) and ADB be 5,6 and 7 square units respectively. Then the area (in square units) of the \(\triangle \mathrm{BCD}\) is equal to :

Let \(a>0, b>0\). Let \(e\) and \(\ell\) respectively be the eccentricity and length of the latus rectum of the hyperbola \(\frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1\). Let \(e ^{\prime}\) and \(\ell^{\prime}\) respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If \(e ^{2}=\frac{11}{14} \ell\) and \(\left( e ^{\prime}\right)^{2}=\frac{11}{8} \ell^{\prime}\), then the value of \(77 a+44 b\) is equal to

Let, \(\alpha, \beta\) be the distinct roots of the equation \(\mathrm{x}^2-\left(\mathrm{t}^2-5 \mathrm{t}+6\right) \mathrm{x}+1=0, \mathrm{t} \in \mathrm{R}\) and \(\mathrm{a}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\). Then the minimum value of \(\frac{\mathrm{a}_{2023}+\mathrm{a}_{2025}}{\mathrm{a}_{2024}}\) is

Let \(x\) and \(y\) be real numbers such that \(50\left(\dfrac{2x}{1+3i} - \dfrac{y}{1-2i}\right) = 31 + 17i\), \(i = \sqrt{-1}\). Then the value of \(10(x - 3y)\) is :

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

Let \(Q\) be the mirror image of the point \(P (1,0,1)\) with respect to the plane \(S : x + y + z =5\). If a line \(L\) passing through \((1,-1,-1)\), parallel to the line \(PQ\) meets the plane \(S\) at \(R\), then \(QR ^{2}\) is equal to

\(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x =1\) is equal to

\(\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{\left( {1 - cos2x} \right)\left( {3 + \cos x} \right)}}{{x\;tan4x}}\) =

The area of the region   \(A=\left\{(x, y):|\cos x-\sin x| \leq y \leq \sin x, 0 \leq x \leq \frac{\pi}{2}\right\}\)