Engineering Maths MCQ with Answers Set-2

Let $C_{1},…, C_{n}$ be scalars, not all zero, such that  $\sum_{i=1}^{n}c_{i}a_{i} = 0$where $a_{i} $ are column vectors in $R^{n}$.

Consider the set of linear equations

Ax=b

where A = $[a_{1},…, a_{n}]$ and b = $\sum_{i=1}^{n}a_{i} = 0$, the set equation is:

Compute $\lim_{x \to 3}\frac{x^{4}-81}{2x^{2}-5x-3}$

Assume that multiplying a matrix $G_{1}$ of dimension p x q with another matrix $G_{2}$  of dimension q x r requires pqr scalar multiplications. Computing the product of n matrices $G_{1}G_{2}G_{3}$ ................ $G_{n}$ can be done by parenthesizing in different ways. Define $G_{i}G_{i+1}$ as an explicitly computed pair for a given paranthesization  if they are directly multiplied. For example, in the matrix multiplication chain $G_{1}G_{2}G_{3}G_{4}G_{5}G_{6}$  using paranthesization $(G_{1}(G_{2}G_{3}))(G_{4}(G_{5}G_{6}))$, $G_{2}G_{3}$ and $G_{5}G_{6}$ are the only explicitly computed pairs.

Consider a matrix multiplication chain $F_{1}F_{2}F_{3}F_{4}F_{5}$, where matrices $F_{1},F_{2},F_{3},F_{4}$ and $F_{5}$ are of dimensions 2 x 25, 25 x 3 , 3 x 16, 16 x 1 and 1 x 1000, respectively. In the paranthesization of $F_{1}F_{2}F_{3}F_{4}F_{5}$ that minimizes the total number of scalar multiplications, the explicitly computed pairs is/are

Let p,q, and r be propositions and the expression $(p\rightarrow q)\rightarrow r$ be a contradiction. Then, the expression $(r\rightarrow p)\rightarrow q$ is

Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium(M) and low(L). Let $H_{G}$ denote the probability that Guwahati has high temperature. Similarly, P($M_{G})$ and P($L_{G}$) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P ($H_{D}$), P($M_{D}$) and P($L_{D}$) for Delhi. The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature.

Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature ($H_{G}$) then the probability of Delhi also having a high temperature ($H_{D}$) is 0.40; i.e., P($H_{D}|H_{G}$) = 0.40. Similarly, the next two entries are P($M_{D}|H_{G}$) = 0.48 and P($L_{D}|H_{G}$) = 0.12. Similarly for the other rows.

If it is known that , P ($H_{G}$) = 0.2, P ($M_{G}$) = 0.5, and P ($L_{G}$) = 0.3 then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _________.

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial $3x^{2}+6xY+3Y+6$ has only real roots is (rounded off to 1 decimal place)________.