Maths

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

The probability, that in a randomly selected 3-digit number at least two digits are odd, is

Let the plane $P:\overrightarrow{r}.\overrightarrow{a}=d$ contains the line of intersection of two planes $\overrightarrow{r}.\left(\widehat{i}+3\widehat{j}-\widehat{k}\right)=6$ and $\overrightarrow{r}.\left(6\widehat{i}+5\widehat{j}-\widehat{k}\right)=7.$ If the plane P passes through the point $\left(2,3,\frac{1}{2}\right)$ , then the value of $\frac{{\left|13\overrightarrow{a}\right|}^2}{d^2}$ is equal to

The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

If two distinct points Q, R lie on the line of intersection of the planes x + 2y – z = 0 and 3x 5y + 2z = 0 and PQ = PR = $\sqrt[]{18}$ where the point P is (1, 2, 3), then the area of the triangle PQR is equal to

If the tangents drawn at the points O(0, 0) and $P\left(1+\sqrt[]{5},2\right)$  on the circle x² + y² – 2x – 4y = 0 intersect at the point Q, then the area of the triangle OPQ is equal to

Let the eccentricity of the hyperbola $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1be\sqrt[]{\frac{5}{2}}$ and length of its latus rectum be $6\sqrt[]{2},ify=2x+c$ is a tangent to the hyperbola H, then the value of c² is equal to

Let y = y(x) be the solution of the differential equation $x\left(1-x^2\right)\frac{dy}{dx}+\left(3x^{2}y-y-4x^3\right)=0,x>1,withy\left(2\right)=-2.$ Then y(3) is equal to$$