If \(A\) is the area in the first quadrant enclosed by the curve \(C: 2 x^2-y+1=0\), the tangent to \(C\) at the point \((1,3)\) and the line \(x+y=1\), then the value of \(60 A\) is

If \(f\left( x \right) = {\log _e}\,\left( {\frac{{1 - x}}{{1 + x}}} \right)\), \(\left| x \right| < 1\), then \(f\left( {\frac{{2x}}{{1 + {x^2}}}} \right)\) is equal to

If the function \(f(x)=\left\{\begin{array}{c}\frac{\log _{e}\left(1-x+x^{2}\right)+\log _{e}\left(1+x+x^{2}\right)}{\sec x-\cos x}, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\} \\ k \end{array}\right.\) is continuous at \(x =0\), then \(k\) is equal to.

Let \(f\) and \(g\) be twice differentiable functions on \(R\) such that \(f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x\) \(f^{\prime}(1)=4 g^{\prime}(1)-3=9\) \(f(2)=3 g(2)=12\) Then which of the following is NOT true ?

Let \(y=y(x)\) be the solution of the differential equation \((x+1) y^{\prime}-y=e^{3 x}(x+1)^{2}\), with \(y(0)=\frac{1}{3}\). Then, the point \(x=-\frac{4}{3}\) for the curve \(y = y ( x )\) is

The distance of the point \(Q(0,2,-2)\) form the line passing through the point \(\mathrm{P}(5,-4,3)\) and perpendicular to the lines \(\overrightarrow{\mathrm{r}}=(-3 \hat{\mathrm{i}}+2 \hat{\mathrm{k}})\) \(\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}), \lambda \in \mathbb{R}\) and \( \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\) \(\mu(-\hat{i}+3 \hat{J}+2 \hat{K}), \mu \in \mathbb{R}\) is

Let \(a\) be the sum of all coefficients in the expansion of \(\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}\) and \(b=\lim _{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)\). If the equations \(\mathrm{cx}^2+\mathrm{dx}+\mathrm{e}=0\) and \(2 \mathrm{bx}^2+\mathrm{ax}+4=0\) have a common root, where \(c, d, e \in R\), then \(d: c: e\) equals

If some three consecutive in the binomial expansion of \({\left( {x + 1} \right)^n}\) in powers of \(x\) are in the ratio \(2 : 15 : 70\), then the average of these three coefficient is

If \(A\) is a \(3 \times 3\) matrix and \(| A |=2\), then \(\left|3 \operatorname{adj}\left(|3 A| A^2\right)\right|\) is equal to \(.........\).

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

The point of intersection of the normals to the parabola \(y^ 2\, = 4x\) at the ends of its latus rectum is 

If \(A = \left[ {\begin{array}{*{20}{c}}
1&{\sin \,\theta }&1\\
{ - \,\sin \,\theta }&1&{\sin \,\theta }\\
{ - 1}&{ - \,\sin \,\theta }&1
\end{array}} \right];\) then for all \(\theta \, \in \,\left( {\frac{{3\pi }}{4},\frac{{5\pi }}{4}} \right),\) det \((A)\) lies in the interval

The mean of the coefficients of \(x, x^2, \ldots \ldots x ^7\) in the binomial expansion of \((2+x)^9\) is \(...........\).