The value of \(k\) for which the following system of equations does not possess a unique solution is:
\(\begin{array}{l}k x+3 y-z=1 \\ x+2 y+z=2 \\ -k x+y+2 z=-1\end{array}\)

If \(\vec{a}, \vec{b}, \vec{c}\) are vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\) and \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=5\), then the expression \(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) equals

A letter is known to have come either from KOLKATA or TATANAGAR. On the envelope just two consecutive letters TA are visible. The probability that letter has come from TATANAGAR is

Let \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\), where \(0 \leq \theta \leq 2 \pi\), then which of the following are true?
(A) \(|A|=2+2 \sin ^2 \theta\)
(B) \(|A|=2+\sin ^2 \theta\)
(C) minimum value of \(|A|\) is 1
(D) maximum value of \(|A|\) is 4.
Choose the correct answer from the options given below:

If \(A\) is any event associated with sample space and If \(E_1, E_2, E_3\) are mutually exclusive and exhaustive events. Then which of the following are true?
(A) \(P(A)=P\left(E_1\right) P\left(E_1 \mid A\right)+P\left(E_2\right) P\left(E_2 \mid A\right)+P\left(E_3\right) P\left(E_3 \mid A\right)\)
(B) \(P(A)=P\left(A \mid E_1\right) P\left(E_1\right)+P\left(A \mid E_2\right) P\left(E_2\right)+P\left(A \mid E_3\right) P\left(E_3\right)\)
(C) \(P\left(E_i \mid A\right)=\frac{P\left(A \mid E_i\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(A \mid E_i\right) P\left(E_i\right)}, i=1,2,3\)
(D) \(P\left(A \mid E_i\right)=\frac{P\left(E_i \mid A\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(E_i \mid A\right) P\left(E_i\right)}, i=1,2,3\)
Choose the correct answer from the options given below:

If \(\left|\begin{array}{ccc}-a^2 & a b & a c \\ a b & -b^2 & b c \\ a c & b c & -c^2\end{array}\right|=\lambda a^2 b^2 c^2\), then the value of \(\lambda\) is:

If \(|\vec{a}|=1,|\vec{b}|=2,|2 \vec{a}+\vec{b}|=2 \sqrt{3}\) then \(|\vec{a}-\vec{b}|\) is :

Match List I with List - II.

List-I	List-II
(A) Radio waves	(I) 1 mm to 700 nm
(B) Infrared waves	(II) 700 nm to 400 nm
(C) Light waves	(III) 1 nm to \(10^{-3} nm\)
(D) X-rays	(IV) > 0.1 m

Choose the correct answer from the options given below :

Match List-I with List-II

List-I (Phenomenon)	List-II (Characteristics)
(A) Reflection	(I) Used for reducing glare
(B) Refraction	(II) Change in path when a light is incident on opaque medium
(C) Interference	(III) Change in path when a light is incident on transparent medium
(D) Polarisation	(IV) Light added to light to produce darkness

Choose the correct answer from the options given below:

Why do the actinoids exhibit a higher number of oxidation states than lanthanoids?

The capacitances of three capacitors are in the ratio 2:3:4. Their equivalent capacitance in parallel combination is more than the equivalent capacitance in series combination by (21/26) pF. The capacitances of the capacitors are, respectively:

A current carrying solenoid of cross-sectional area \( 2 \, cm^2 \) and 2000 turns has a magnetic moment of \( 0.80 \, Am^2 \). The current flowing through the solenoid is

Which of the following is not a female contraceptive?