Maths

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……….

The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking it was found that, in the observations, 20 was misread as 5. Then, the correct variance 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…………

Let p, q, r be three logical statements. Consider the compound statements

$S_1:\left(\left(\sim p\right)\vee q\right)\vee \left(\left(\sim p\right)\vee r\right)and$

$S_2:p\to \left(q\vee r\right)$

Then, which of the following is NOT true?

Let AB and PQ be two vertical poles, 160m apart from each other. Let C be the middle points of B and Q, which are feet of these two poles. Let $\frac{\pi }{8}$  and q be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan²q is equal to