Maths

Let $f$ be a twice differentiable function on $\left(\mathrm{1,6}\right)$ . If $f\left(2\right)=8,f^{\mathrm{\text{'}}}\left(2\right)=5,f^{\mathrm{\text{'}}}\left(x\right)\ge 1$ and $f$ " $\left(x\right)\ge 4$ , for all $x\in \left(\mathrm{1,6}\right)$ , then:

Let $f\left(x\right)=|x-2|$ and $g\left(x\right)=f\left(f\right(x\left)\right),x\in \left[\mathrm{0,4}\right]$ . Then ${\int }_0^3?\left(g\right(x)-f(x\left)\right)dx$ is equal to:

Let $\alpha$ and $\beta$ be roots of $x^2-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6x+q=0$ . If $\alpha ,\beta$ , $\gamma ,\delta$ , form a geometric progression. Then ratio $(2q+p):(2q-p)$ is:

Let the lines y + 2x = $\sqrt[]{11}+7\sqrt[]{7}$ and $2y+x=2\sqrt[]{11}+6\sqrt[]{7}$ be normal to a circle C : ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2.$ If the line $\sqrt[]{11}y-3x=\frac{5\sqrt[]{77}}{3}+11$ is tangent to the circle C, then the value of ${\left(5h-8k\right)}^2+5r^2$ is equal to

The number of elements in the set $\left\{z=a+ib\in C:a,b\in Zand1<\left|z-3+2i\right|<4\right\}$ is

The number of positive integers k such that the constant term in the binomial expansion of ${\left(2x^3+\frac{3}{x^k}\right)}^{12},x\ne 0is{2}^8.\mathcal{l},where\mathcal{l}$  is an odd integer, is……….

Let A = $\left\{1,a_1,a_2....a_{18},77\right\}$  be a set of integers with 1 < a₁ < a₂ < < a₁₈ < 77. Let the set A + A = $\left\{x+y:x,y\in A\right\}$  contains exactly 39 elements. Then, the value of a₁ + a₂ +…. + a₁₈ is equal to

Let $\mathcal{l}$  be a line which is normal to the curve y = 2x² + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line $\mathcal{l}$  and O is origin, then the area of the triangle OPQ is equal to

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ration 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (a, b). Then the value of 7a + 3b is equal to……….

If $\overrightarrow{a}=2\widehat{i}+\widehat{j}+3\widehat{k},\overrightarrow{b}=3\widehat{i}+3\widehat{j}+\widehat{k}and\overrightarrow{c}=c_1\widehat{i}+c_2\widehat{j}+c_3\widehat{k}$ are coplanar vectors and $\overrightarrow{a}.\overrightarrow{c}=5,\overrightarrow{b}\perp \overrightarrow{c},then122\left(c_1+c_2+c_3\right)$ is equal to……….