Consider the system of equations \(x-2 y+3 z=-1, x-3 y+4 z=1\) and \(-x+y-2 z=k\)
Statement 1 The system of equations has no solution for \(k \neq 3\).
Statement 2 The determinant \(\left|\begin{array}{ccc}1 & 3 & -1 \\ -1 & -2 & k \\ 1 & 4 & 1\end{array}\right| \neq 0\), for \(k \neq 3\).

A line $y=mx+1$ intersects the circle ${\left(x-3\right)}^2+{\left(y+2\right)}^2=25$ at the points $P$ and $Q.$ If the midpoint of the line segment $PQ$ has $x-$coordinate $-\frac{3}{5},$ then which one of the following options is correct?

Two fair dice, each with faces numbered $1,2,3,4,5$ and $6$, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14p$ is_____

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function with $f\left(0\right)=0$. If $y=f\left(x\right)$ satisfies the differential equation $\frac{dy}{dx}=\left(2+5y\right)\left(5y-2\right)$, then the value of  $\underset{x\to -\infty }{\lim }f\left(x\right)$ is 

Let \((x, y)\) be such that \(\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}\). Match the statements in Column I with the values in Column II.

Let \(\omega \neq 1\) be a cube root of unity and \(S\) be the set of all non-singular matrices of the form \(\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\), where each of \(a, b\) and \(c\) is either \(\omega\) or \(\omega^2\). Then, the number of distinct matrices in the set \(S\) is

Let \(f(\theta)=\sin \left[\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right]\), where \(-\frac{\pi}{4} < \theta < \frac{\pi}{4}\). Then, the value of \(\frac{d}{d(\tan \theta)}(f(\theta))\) is

The acidic hydrolysis of ether (X) shown below is fastest when

Let $f :\left[-\frac{1}{2}, 2\right]\to \mathbb{R}$ and $g:\left[-\frac{1}{2},2\right]\to \mathbb{R}$ be functions defined by $f\left(x\right)=\left[x^2-3\right]$ and $g\left(x\right)=\left|x\right|f\left(x\right)+\left|4x-7\right|f\left(x\right)$ , where $\left[y\right]$ denotes the greatest integer less than or equal to $y$ for $y\in \mathbb{R}$ . Then

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \mathrm{mm}$. The circular scale has $100$ divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	$0$ division	$4$ divisions
Attempt-$1$: With wire	$4$ divisions	$20$ divisions
Attempt-$2$: With wire	$4$ divisions	$16$ divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

A factory has a total of three manufacturing units, \(M_1, M_2\), and \(M_3\), which produce bulbs independent of each other. The units \(M_1, M_2\), and \(M_3\) produce bulbs in the proportions of 2:2:1, respectively. It is known that \(20 \%\) of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by \(M_1, 15 \%\) are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by \(M_2\) is \(\frac{2}{5}\).
If a bulb is chosen randomly from the bulbs produced by \(M_3\), then the probability that it is defective is ____.

Match the four starting materials (P, Q, R, S) given in List I with the corresponding reaction schemes (I, II, III, IV) provided in List II and select the correct answer using the code given below the lists.

	List – I		List – II
A.		P.
B.		Q.
C.		R.
D.		S.

Let the curve C be the mirror image of the parabola ${\mathrm{y}}^{\mathrm{2 }}$$=4\mathrm{x}$ with respect to the line $\mathrm{x}+\mathrm{y}+4=0.$ If A and B are the points of intersection of C with the line $\mathrm{y}=\mathrm{ }-5,$ then the distance between A and B is