Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6: 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

The coefficient of $x^9$ in the expansion of $\left(1+x\right)\left(1+x^2\right)\left(1+x^3\right)\dots .\left(1+x^{100}\right)$ is

Suppose that $\vec{p},\vec{q}\text{and}\vec{r}$ are three non-coplanar vectors in $R^3$. Let the components of a vector $\vec{s}$ along $\vec{p},\vec{q}\text{and}\vec{r}$ be 4, 3 and 5, respectively. If the components of this vector $\vec{s}$ along $\left(-\vec{p}+\vec{q}+\vec{r}\right),\left(\vec{p}-\vec{q}+\vec{r}\right)\text{and}\left(-\vec{p}-\vec{q}+\vec{r}\right)$ are x, y and z, respectively, then the value of $2\vec{x}+\vec{y}+\vec{z}$ is

Suppose that $\vec{p},\vec{q}\text{and}\vec{r}$ are three non-coplanar vectors in $R^3$. Let the components of a vector $\vec{s}$ along $\vec{p},\vec{q}\text{and}\vec{r}$ be 4, 3 and 5, respectively. If the components of this vector $\vec{s}$ along $\left(-\vec{p}+\vec{q}+\vec{r}\right),\left(\vec{p}-\vec{q}+\vec{r}\right)\text{and}\left(-\vec{p}-\vec{q}+\vec{r}\right)$ are x, y and z, respectively, then the value of $2\vec{x}+\vec{y}+\vec{z}$ is

Let E1 and E2, be two ellipses whose centers are at the origin.The major axes of E1 and E2, lie along the x-axis and the y-axis, respectively. Let S be the circle $x^2+{\left(y-1\right)}^2=2$. The straight line x+ y =3 touches the curves S, E1 and E2 at P,Q and R, respectively. Suppose that $PQ=PR=\frac{2\sqrt{2}}{3}$.If e1 and e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is(are):

Let $f:R\vec{R}$be a continuous odd function, which vanishes exactly at one point and $f\left(1\right)=\frac{1}{2}\dot{}$Suppose that $F\left(x\right)={\int }_{-1}^{x}f\left(t\right)dtf\text{or}allx\in \left[-1,2\right]andG\left(x\right)={\int }_{-1}^{x}t\left|f\left(f\left(t\right)\right)\right|dtf\text{or}allx\in \left[-1,2\right]\dot{I}\frac{{f\left(\lim \right)}_{x\vec{1}}\left(F\left(x\right)\right)}{G\left(x\right)}=\frac{1}{14},$Then the value of $f\left(\frac{1}{2}\right)$is

If$\alpha ={\int }_0^1\left(e^{9}x+3{\tan }^{\left(-1\right)x}\right)\left(\frac{12+9x^2}{1+x^2}\right)dxwher\eta n^{-1}$takes only principal values, then the value of $\left({\left(\log \right)}_e\left|1+\alpha \right|-\frac{3\pi }{4}\right)is$

Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$are $\left(f_1,0\right)and\left(f_2,0\right)$where $f_1>0and{f}_2<0.$Let $P_{1}and{P}_2$be two parabolas with a common vertex at (0, 0) and with foci at $\left(f_1.0\right)and$(2f_2 , 0), respectively. Let$T_1$be a tangent to $P_1$which passes through $\left(2f_2,0\right)$and $T_2$be a tangents to $P_2$which passes through $\left(f_1,0\right)$. If $m_1$is the slope of $T_1$and $m_2$is the slope of $T_2,$then the value of $\left(\frac{1}{m12}+m22\right)$is

Consider the hyperbola $H:x^2-y^2=1$ and a circle S with centre $N\left(x_2,0\right)$ Suppose that H and S touch each other at a point $(P\left(x_1,y_1\right)$ with $x_1>1\text{and}{y}_1>0$ The common tangent to H and S at P intersects the x-axis at point M. If (l,m) is the centroid of the triangle $\Delta PMN$ then the correct expression is (A) $\frac{dl}{{dx}_1}=1-\frac{1}{3x_1^2}$ for $x_1>1$ (B) $\frac{dm}{{dx}_1}=\frac{x_{!}}{3\left({\sqrt{x}}_1^2-1\right)})f\text{or}{x}_1>1$ (C) $\frac{dl}{{dx}_1}=1+\frac{1}{3x_1^2}f\text{or}{x}_1>1$ (D) $\frac{dm}{{dy}_1}=\frac{1}{3}f\text{or}{y}_1>0$

Let $f\left(x\right)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$for all $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ . Then the correct expression (s) is (are) (a) ${\int }_0^{\frac{\pi }{4}}xf\left(x\right)dx=\frac{1}{12}$ (b)${\int }_0^{\frac{\pi }{4}}f\left(x\right)dx=0$(c)${\int }_0^{\frac{\pi }{4}}xf\left(x\right)=\frac{1}{6}$ (d) ${\int }_0^{\frac{\pi }{4}}f\left(x\right)dx=\frac{1}{12}$