Let A = \left( {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right). and $$10$$ B = \left( {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right). if $$B$$ is the inverse of matrix $$A$$, then $$\alpha$$ is

Let A = \left( {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right). The only correct statement about the matrix $$A$$ is

If $$a_1,a_2,a_3,.........,a_n,......$$ are in G.P., then the value of the determinant \left| {\matrix{ {\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr {\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr {\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {a_{n + 8}}} \cr } } \right|, is

The mean and the variance of a binomial distribution are $$4$$ and $$2$$ respectively. Then the probability of $$2$$ successes is :

The probability that $$A$$ speaks truth is {4 \over 5}, while the probability for $$B$$ is {3 \over 4}. The probability that they contradict each other when asked to speak on a fact is :

If $$x=e^{y+e^y+e^{y+.....\infty }}$$ , $$x>0,$$ then {{{dy} \over {dx}}} is

Solution of the differential equation $$ydx+\left(x+x^{2}y\right)dy=0$$ is

The differential equation for the family of circle $$x^2+y^2-2ay=0,$$ where a is an arbitrary constant is :

If the sum of the slopes of the lines given by $$x^2-2cxy-7y^2=0$$ is four times their product $$c$$ has the value :

If one of the lines given by $$6x^2-xy+4c{y}^2=0$$ is $$3x+4y=0,$$ then $$c$$ equals :

Let $$A\left(2,-3\right)$$ and $$B\left(-2,1\right)$$ be vertices of a triangle $$ABC$$. If the centroid of this triangle moves on the line $$2x+3y=1$$, then the locus of the vertex $$C$$ is the line :

The equation of the straight line passing through the point $$(4,3)$$ and making intercepts on the co-ordinate axes whose sum is $$-1$$ is :

Let $$S(K)$$ $$=1+3+\mathrm{5...}+\left(2K-1\right)=3+K^2.$$ Then which of the following is true

If $$\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$$, then the angle beetween A and B is

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $$60^{\circ }$$ and when he retires $$40$$ meters away from the tree the angle of elevation becomes $$30^{\circ }$$. The breadth of the river is :

The coefficient of the middle term in the binomial expansion in powers of $$x$$ of $${\left(1+\alpha x\right)}^4$$ and $${\left(1-\alpha x\right)}^6$$ is the same if $$\alpha$$ equals

The coefficient of $$x^n$$ in expansion of $$\left(1+x\right){\left(1-x\right)}^n$$ is

If {S_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}}} \,\,and\,\,{t_n} = \sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}},\,}then {{{t_{ n}}} \over {{S_n}}} is equal to

\int {{{dx} \over {\cos x - \sin x}}} is equal to

If \int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,} then value of $$(A,B)$$ is