If P = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right], then P50 is :

Let A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]. If A$$-$$1 = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$\in$$ R, I is a 2 $$\times$$ 2 identity matrix then 4($$\alpha$$ $$-$$ $$\beta$$) is equal to :

Let \theta \in \left( {0,{\pi \over 2}} \right). If the system of linear equations $$(1+{\cos }^2\theta )x+{\sin }^2\theta y+4\sin 3\theta z=0$$ $${\cos }^2\theta x+(1+{\sin }^2\theta )y+4\sin 3\theta z=0$$ $${\cos }^2\theta x+{\sin }^2\theta y+(1+4\sin 3\theta )z=0$$ has a non-trivial solution, then the value of $$\theta$$ is :

If A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right), B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right), $$i=\sqrt{-1}$$, and Q = ATBA, then the inverse of the matrix A Q2021 AT is equal to :

Let A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right). Then A2025 $$-$$ A2020 is equal to :

If the matrix A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right) satisfies $$A(A^3+3I)=2I$$, then the value of K is :

Let A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right), where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval :

Let A(a, 0), B(b, 2b + 1) and C(0, b), b $$\ne$$ 0, |b| $$\ne$$ 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is :

Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [$$\lambda$$])z = [$$\lambda$$] has a solution is :

If the following system of linear equations 2x + y + z = 5 x $$-$$ y + z = 3 x + y + az = b has no solution, then :

If {a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}, r = 1, 2, 3, ....., i = $$\sqrt{-1}$$, then the determinant \left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right| is equal to :

If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations x + (cos $$\gamma$$)y + (cos $$\beta$$)z = 0 (cos $$\gamma$$)x + y + (cos $$\alpha$$)z = 0 (cos $$\beta$$)x + (cos $$\alpha$$)y + z = 0 has :

Consider the system of linear equations $$-$$x + y + 2z = 0 3x $$-$$ ay + 5z = 1 2x $$-$$ 2y $$-$$ az = 7 Let S1 be the set of all a$$\in$$R for which the system is inconsistent and S2 be the set of all a$$\in$$R for which the system has infinitely many solutions. If n(S1) and n(S2) denote the number of elements in S1 and S2 respectively, then

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :

The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :

When a missile is fired from a ship, the probability that it is intercepted is {1 \over 3} and the probability that the missile hits the target, given that it is not intercepted, is {3 \over 4}. If three missiles are fired independently from the ship, then the probability that all three hit the target, is :

The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :

Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is :