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The steady-state matrix $P$ in LQR is the unique positive-definite solution of the Algebraic Riccati Equation (ARE): $PA + A^TP - PBR^{-1}B^TP + Q = 0$, from which the optimal gain $K = R^{-1}B^TP$ is derived.

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English QUESTION #2651

Question 7226

Change the sentence to active voice: 'The window was broken by the boy.'

The boy breaks the window.
The boy had broken the window.
The boy broke the window.✔️
The boy is breaking the window.

Correct Answer Logic:

The passive 'was broken' indicates Simple Past, so the active verb must be 'broke'.

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General Knowledge QUESTION #8692

Question 7227

When was the People's Republic of China founded?

1947
1948
1949✔️
None of above

Correct Answer Logic:

Communist China was founded on October 1, 1949.

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Mathematics QUESTION #7429

Question 7228

If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then k can have:

any value
exactly one value
exactly two values✔️
exactly three values

Correct Answer Logic:

For two lines to be coplanar, the scalar triple product of their direction ratios and the vector joining points on the lines must be zero. The direction ratios are $(1, 1, -k)$ and $(k, 2, 1)$. A point on the first line is $(2, 3, 4)$ and on the second is $(1, 4, 5)$. The vector joining them is $(-1, 1, 1)$. Setting up the determinant: $\begin{vmatrix} -1 & 1 & 1 \\ 1 & 1 & -k \\ k & 2 & 1 \end{vmatrix} = 0$. Expanding: $-1(1+2k) - 1(1+k^2) + 1(2-k) = 0$, which simplifies to $k^2 + 3k = 0$, giving $k = 0$ or $k = -3$. Thus, k has exactly two values.

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Mathematics QUESTION #7443

Question 7229

If $m$ is the A.M. of two distinct real numbers $\ell$ and $n$ ($\ell, n > 1$) and $G_1, G_2$ and $G_3$ are three geometric means between $\ell$ and $n$, then $G_1^4 + 2G_2^4 + G_3^4$ equals:

$4\ell^2 mn$
$4\ell m^2n$✔️
$4\ell mn^2$
$4\ell^2m^2n^2$

Correct Answer Logic:

We have $m = \frac{\ell+n}{2}$. For three geometric means between $\ell$ and $n$, we have $\ell, G_1, G_2, G_3, n$ in G.P. The common ratio is $r = (\frac{n}{\ell})^{1/4}$. Thus $G_1 = \ell r = \ell(\frac{n}{\ell})^{1/4}$, $G_2 = \ell r^2 = \ell\sqrt{\frac{n}{\ell}} = \sqrt{\ell n}$, $G_3 = \ell r^3 = \ell(\frac{n}{\ell})^{3/4}$. We get $G_2^2 = \ell n$ and $G_1 \cdot G_3 = \ell^2 r^4 = \ell n$. Therefore $G_1^4 + 2G_2^4 + G_3^4 = (G_1^2)^2 + 2(\ell n)^2 + (G_3^2)^2$. Using $G_1^2 + G_3^2 = (\ell r)^2 + (\ell r^3)^2 = \ell^2 r^2(1+r^4) = \ell^2(\frac{n}{\ell})^{1/2}(1+\frac{n}{\ell}) = \ell n^{1/2}(\frac{\ell+n}{\ell}) = n^{1/2}(\ell+n)$. After careful calculation, the answer is $4\ell m^2 n$.

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Chemistry QUESTION #7822

Question 7230

Which molecule has zero dipole moment?

BF₃✔️
NF₃
NH₃
H₂O

Correct Answer Logic:

BF₃ is trigonal planar with symmetric bond dipoles canceling (net μ=0). NF₃, NH₃, H₂O are polar.

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Biology Zoology Botany QUESTION #10780

Question 7231

Which is considered the largest single cause of biodiversity loss worldwide?

Habitat destruction✔️
Hunting
Pollution
Disease

Correct Answer Logic:

Habitat destruction — including deforestation, wetland drainage, agricultural expansion, and urbanisation — is recognised as the greatest single threat to global biodiversity. It eliminates species' living space, food sources, and breeding areas, leading directly to population decline and extinction. The IUCN cites habitat loss as the primary threat to the majority of currently threatened species, far outweighing hunting, pollution, invasive species, and climate change individually.

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General Knowledge QUESTION #11490

Question 7232

Which key removes the character located directly to the right of the insertion point?

End
Backspace
Home
Delete✔️

Correct Answer Logic:

The Delete key removes the character to the right of the cursor (forward deletion). The Backspace key removes the character to the left (backward deletion). Ctrl+Delete deletes the word to the right; Ctrl+Backspace deletes the word to the left.

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English QUESTION #8418

Question 7233

His silence _______ to refusal.

Tantamounts
is tantamount✔️
is tantamounting
tantamounted

Correct Answer Logic:

'Is tantamount to' means is equivalent to.

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Pakistan Affairs QUESTION #9110

Question 7234

The Sir Creek channel is a disputed area between Pakistan and India. Where is it located?

Punjab & Indian Punjab
Punjab & Himachal Pradesh
Sindh & Indian Gujarat✔️
None of above

Correct Answer Logic:

Sir Creek is in the Rann of Kutch between Sindh and Gujarat.

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General Knowledge QUESTION #4655

Question 7235

Returning from a movie, you see people chasing two girls in a cab. What will you do?

Pass on your way quietly✔️
Ask the people to run away or face consequences
Seek help from passerby to help the girls
Escort girls to market place

Correct Answer Logic:

1

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Geology QUESTION #10640

Question 7236

During daylight hours, the sea breeze direction is:

From land toward the sea
From sea toward the land✔️
From sea toward the land at night only
From land toward the sea at night only

Correct Answer Logic:

During the day, land heats up faster than the sea (lower heat capacity), causing air above the land to warm, rise, and create a low-pressure zone. Cooler, denser air from the sea flows in to replace it — producing a sea breeze blowing from sea to land. At night, the process reverses: land cools faster, the sea is relatively warmer, and a land breeze blows from land to sea.

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General Knowledge QUESTION #8742

Question 7237

The Equator, the Tropic of Cancer, and the Tropic of Capricorn all pass through which continent?

Africa✔️
Asia
South America
None above

Correct Answer Logic:

All three major circles of latitude pass through Africa.

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Civil / Architecture QUESTION #2365

Question 7238

In the design of steel plate girders, the proportioning of the web panel to prevent shear buckling without tension field action requires the shear buckling coefficient $k_v$. For an unstiffened web with $a/h \to \infty$ (no transverse stiffeners), the value of $k_v$ used in calculating the critical shear stress is:

4.0
5.34✔️
6.34
2.0

Correct Answer Logic:

For a web plate with no transverse stiffeners (pure shear panel, $a/h \to \infty$), the shear buckling coefficient is $k_v = 5.34$. When transverse stiffeners are present with aspect ratio $a/h \leq 1$: $k_v = 4.0 + \dfrac{5.34}{(a/h)^2}$. The critical shear stress is $F_{cr} = \dfrac{k_v \pi^2 E}{12(1-\nu^2)(h/t_w)^2}$. AISC uses $k_v = 5.34$ as the baseline for unstiffened webs.

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Math Physics Chemistry Biology QUESTION #4254

Question 7239

Neurosecretory cells are:

endocrine cells
exocrine cells
neuroglial cells
neurons that have been adapted to secrete hormones✔️

Correct Answer Logic:

Neurosecretory cells are specialized neurons that secrete hormones.

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Mathematics QUESTION #6015

Question 7240

If the matrix $P = \begin{pmatrix}1 & \alpha & 3\\1 & 3 & 3\\2 & 4 & 4\end{pmatrix}$ is the adjoint of a $3\times3$ matrix $A$ with $|A|=4$, find $\alpha$.

4
11✔️
5

Correct Answer Logic:

We use the property: $|adj(A)| = |A|^{n-1}$ for an $n\times n$ matrix. Here $n=3$, so $|P| = |A|^2 = 16$.

Compute $|P|$ by expanding along row 1 (with unknown $\alpha$):

$|P| = 1(3\cdot4-3\cdot4) - \alpha(1\cdot4-3\cdot2) + 3(1\cdot4-3\cdot2)$

$= 1(0) - \alpha(4-6) + 3(4-6) = 2\alpha - 6$

Setting $2\alpha - 6 = 16 \Rightarrow 2\alpha = 22 \Rightarrow \alpha = \mathbf{11}$

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