The equation of a common tangent to the parabolas \(y = x ^{2}\) and \(y =-( x -2)^{2}\) is.

If the projection of the vector \(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) on the sum of the two vectors \(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}\) and \(-\lambda \hat{i}+2 \hat{j}+3 \hat{k}\) is \(1,\) then \(\lambda\) is equal to ..... .

\(\smallint \frac{{{{\sin }^2}x{{\cos }^2}x}}{{{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}dx\)

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly \(7\) solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\) then \(\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}\) is equal to :

For \(k \in R\), let the solutions of the equation \(\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0\,<\,|x|<\,\frac{1}{\sqrt{2}}\) be \(\alpha\) and \(\beta\), where the inverse trigonometric functions take only principal values. If the solutions of the equation \(x ^{2}- bx -5=0\) are \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}\) and \(\frac{\alpha}{\beta}\), then \(\frac{b}{k^{2}}\) is equal to\(......\)

If \(f(a+b+1-x)=f(x),\) for all \(x,\) where \(a\) and \(b\) are fixed positive real numbers, then \(\frac{1}{a+b} \int\limits_{a}^{b} x(f(x)+f(x+1)) d x\) is equal to 

The total number of \(3\)-digit numbers, whose greatest common divisor with \(36\) is \(2\) , is

If \(\cos \,x\,\frac{{dy}}{{dx}} - y\,\sin \,x = 6x,\,\left( {0 < x < \frac{\pi }{2}} \right)\) and \(y\left( {\frac{\pi }{3}} \right) = 0\), then \(y\left( {\frac{\pi }{6}} \right)\) is equal to 

If \(y=y(x)\) is solution of the differential equation \(x\frac{{dy}}{{dx}} + 2y = {x^2}\) satisfying \(y(1)=1\) then \(y\left( {\frac{1}{2}} \right)\) is equal to

If \(A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i=\sqrt{-1}\), and \(\mathrm{Q}=\mathrm{A}^{\mathrm{T}} \mathrm{BA}\), then the inverse of the matrix \(\mathrm{A} \mathrm{Q}^{2021} \mathrm{~A}^{\mathrm{T}}\) is equal to :

Let \(A\) and \(B\) be two sets containing four and two elements respectively. Then the number of subsets of the set \(A \times B,\) each having at least three elements is :

 A hyperbola passes through the point \(P\left( {\sqrt 2 ,\sqrt 3 } \right)\) has foci at \(\left( { \pm 2,0} \right)\). Then the tangent to this hyperbola at  \(P\) also passes through the point

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 ...........