Maths

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 $\sqrt[]{11}y-3x=\frac{5\sqrt[]{77}}{3}+11$  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 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, β)). Then the value of 7a + 3β) 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……….

The number of real solutions of the equation e^4x + 4e^3x – 58e^2x + 4e^x + 1 = 0, is……………

Let R₁ and R₂ be relations on the set {1,2,…., 50} such that

R₁ = $\left\{\left(p,p^n\right):pisaprimeandn\ge 0isanint\text{?}eger\right\}and$  

R₂ = $\left\{\left(p,p^n\right):pisaprimeandn=0or1\right\}.$  

Then, the number of elements in R₁ – R₂ is…………