1
GATE CE 2024 Set 2
MCQ (Single Correct Answer)
+1
-0.33

Consider two Ordinary Differential Equations (ODEs):

P: $ \dfrac{dy}{dx} = \dfrac{x^4 + 3x^2 y^2 + 2y^4}{x^3 y} $

Q: $ \dfrac{dy}{dx} = -\dfrac{y^2}{x^2} $

Which one of the following options is CORRECT?

A

P is a homogeneous ODE and Q is an exact ODE.

B

P is a homogeneous ODE and Q is not an exact ODE.

C

P is a nonhomogeneous ODE and Q is an exact ODE.

D

P is a nonhomogeneous ODE and Q is not an exact ODE.

2
GATE CE 2024 Set 2
MCQ (Single Correct Answer)
+2
-0.833

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having a heart attack by 50%. Medicine X is taken by 50 of these 100 patients. The probability that a randomly selected patient, out of the 100 patients, takes medicine X and has a heart attack is

A

40%

B

60%

C

20%

D

30%

3
GATE CE 2024 Set 2
MCQ (More than One Correct Answer)
+2
-0.833

Three vectors $\overrightarrow{p}$, $\overrightarrow{q}$, and $\overrightarrow{r}$ are given as

$ \overrightarrow{p} = \hat{i} + \hat{j} + \hat{k}$

$ \overrightarrow{q} = \hat{i} + 2\hat{j} + 3\hat{k}$

$ \overrightarrow{r} = 2\hat{i} + 3\hat{j} + 4\hat{k}$

Which of the following is/are CORRECT?

A

$ \overrightarrow{p} \times (\overrightarrow{q} \times \overrightarrow{r}) + \overrightarrow{q} \times (\overrightarrow{r} \times \overrightarrow{p}) + \overrightarrow{r} \times (\overrightarrow{p} \times \overrightarrow{q}) = \overrightarrow{0}$

B

$ \overrightarrow{p} \times (\overrightarrow{q} \times \overrightarrow{r}) = (\overrightarrow{p} \cdot \overrightarrow{r}) \overrightarrow{q} - (\overrightarrow{p} \cdot \overrightarrow{q}) \overrightarrow{r}$

C

$ \overrightarrow{p} \times (\overrightarrow{q} \times \overrightarrow{r}) = (\overrightarrow{p} \times \overrightarrow{q}) \times \overrightarrow{r}$

D

$ \overrightarrow{r} \cdot (\overrightarrow{p} \times \overrightarrow{q}) = (\overrightarrow{q} \times \overrightarrow{p}) \cdot \overrightarrow{r}$

4
GATE CE 2024 Set 2
Numerical
+2
-0.833

The expression for computing the effective interest rate $(i_{eff})$ using continuous compounding for a nominal interest rate of 5% is

$i_{eff} = \lim\limits_{m \to \infty} \left(1 + \frac{0.05}{m}\right)^m - 1$

The effective interest rate (in percentage) is ___________ (rounded off to 2 decimal places).

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