Consider a line \(\vec{r}=(\hat{i}+4 \hat{j})+\lambda(2 \hat{i}-2 \hat{j}+3 \hat{k})\), then which of the following statements are correct?
(A) It passes through point \((9,-4,12)\)
(B) It passes through point \((1,4,-1)\)
(C) Its direction cosine's are \(\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}\)
(D) Its Cartesian equation is \(\frac{x-1}{2}=\frac{y-4}{-2}=\frac{z}{3}\)
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If A and B are two events such that \(P(A) \neq 0\) and \(P(A / B)=1\) then :

Which of the following functions are increasing on \(x \in\left(0, \frac{\pi}{2}\right)\) ?
(A) \(f(x)=\sin x\)
(B) \(f(x)=\cos x\)
(C) \(f(x)=\tan x\)
(D) \(f(x)=\cos 3 x\)
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Which of the following statements are correct about the "Central Limit Theorem"?
(A) The sampling distribution of the sample mean approaches the normal distribution as the sample size gets larger.
(B) A sample size of 30 or more is considered to be sufficient to hold the "Central Limit Theorem".
(C) As the sample size becomes larger, the prediction of characteristics of the population becomes more accurate.
(D) The sampling distribution of the sample mean approaches bell shaped curveas the sample size gets larger.
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The feasible region of the constraints \(x \geq 0, x+y \leq 1\) and \(x-y \leq 1\) is situated in:

Mr. Jain wishes to purchase a shop for ₹25,00,000 with down payment of ₹2, 00, 000. If he can amortize the balance at 9% per annum compounded monthly for 25 years, what will be his monthly installment if it is given that \(\left(a_{\frac{300}{0.0075}}=119.1616\right)\)?

Let \(|\vec{a}|=5,|\vec{b}|=2\) and \(\vec{a} \cdot \vec{b}=6\), then the value of \(|\vec{a} \times \vec{b}|\) is equal to

In reference to Inferential Statistics, if \(\bar{x}\) is a sample mean of random data \(\left\{x_i\right\}_{i=1}^n\) and n is the sample size, then the formula \(\frac{1}{n-1} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2\) represents

Inlet pipe A can fill a tank in 30 minutes and outlet pipes B and C can empty the tank in 2 hours each. If all three pipes operate together, the tank will be filled in

Read the passage carefully and answer the questions based on the passage:
Pollen pistil interaction involves all events from the landing of pollen grains on the stigma until the pollen tube enters the embryo sac (when the pollen is compatible). When a pollen tube grows through the style and enters into the ovules, it finally discharges two male gametes in one of the synergids. Syngamy and triple fusion are two fusion events occur in angiosperms. Thus, angiosperms exhibit double fertilization. The products of these fusions are the diploid zygote and triploid primary endosperm nucleus. Zygote develops into embryo and primary endosperm cell forms the endosperm tissue. The developing embryo passes through different stages before maturation.
Which of the following is not a stage of growing embryo in dicotyledon plants?

For any events \(A\) and \(B\) of a sample space \(S\), which of the following statements are TRUE?
(A) \(P(S \mid B)=1\)
(B) \(P(A \cap B)=P(A)+P(B)+P(A \cup B)\)
(C) \(P(\bar{A} \mid B)=1-P(A \mid B)\)
(D) \(P(A \mid B)=\frac{P(A \cap B)}{P(B)}, P(B) \neq 0\)
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The spermatids are transformed into spermatozoa by the process called as

A coil \(C_1\) is connected to a galvanometer G. Second coil \(C_2\) is connected to a battery through a tapping key. The steady current in the coil \(C_2\) produces a steady magnetic field. As coil \(C_2\) is moved towards the coil \(C_1\), the galvanometer shows a deflection. To obtain a large deflection, one or more of the following steps can be taken:
(A) Use a rod made of soft iron inside the coil \(C_2\)
(B) Connect the coil \(C_2\) to a more powerful battery
(C) Move the arrangement of coil \(C_2\) rapidly towards the coil \(C_1\)
(D) hold the key pressed continuously
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