In a certain town 25% families own a phone and 15% own a car, 65% families own neither a phone nor a car. 2000 families own both a car and a phone. Consider the following statements in this regard:
If \[\Delta (x)=\left| \begin{matrix} 1 & \cos x & 1-\cos x \\ 1+\sin x & \cos x & 1+\sin x-\cos x \\ \sin x & \sin x & 1 \\ \end{matrix} \right|,\] then \[\int_{0}^{\pi /2}{\Delta (x)\,\,dx}\] equals
Let a, b, c be positive real numbers. The following system of equations in x, y and z \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}-\frac{{{z}^{2}}}{{{c}^{2}}}=1,\] \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}+\frac{{{z}^{2}}}{{{c}^{2}}}=1\] and \[\frac{-{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}+\frac{{{z}^{2}}}{{{c}^{2}}}=1,\] has
If \[A=\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right]\] and \[I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right],\] then which one of the following holds for all \[n\ge 1,\] by the principle of mathematical induction.
The value of f(0) so that the function \[f\,\,(x)=\frac{{{\cos }^{-1}}(1-{{\{x\}}^{2}})\,\,{{\sin }^{-1}}\,\,(1-\{x\})}{\{x\}-{{\{x\}}^{3}}},\]\[x\ne 0\]({x} denotes fractional part of x) becomes continuous at x = 0 is
Let a, b, c be non-zero real number such that \[\int_{0}^{1}{(1+{{\cos }^{8}}x)\,\,(a{{x}^{2}}+bx+c)\,\,dx}\]\[=\int_{0}^{2}{(1+{{\cos }^{8}}x)\,\,(a{{x}^{2}}+bx+c)\,\,dx.}\]Then the quadratic equation \[a{{x}^{2}}+bx+c=0\] has
If the lines \[\frac{x-a+d}{\alpha -\delta }=\frac{y-a}{\alpha }=\frac{z-a-d}{\alpha +\delta }\] and \[\frac{x-b+c}{\beta -\gamma }=\frac{y-b}{\beta }=\frac{z-b-c}{\beta +\gamma }\] are coplanar, then equation to the plane in which they lie, is
If the function \[f(x)=\left\{ \begin{matrix} x+{{a}^{2}}\sqrt{2}\sin x, & 0\le x\le \pi /4 \\ x\cot x+b, & \pi /4\le x<\pi /2 \\ b\sin 2x-a\cos 2x, & \pi /2\le x\le \pi \\ \end{matrix} \right.\] is continuous in the interval \[[0,\pi ],\] then the values of (a, b) are
A bag contains 'a' number of white and 'b' number of black balls. Two players A and B alternately draw a ball from the bag, replacing the ball each time after the draw till one of them draws a white ball and wins the game. If A begins the game and the probability of A winning the game is three times that of B, then find the value of a:b.
The number of distinct solutions of the equation \[\frac{5}{4}{{\cos }^{2}}2x+{{\cos }^{4}}x+{{\sin }^{4}}x+{{\cos }^{6}}x+{{\sin }^{6}}x=2\] in the interval \[[0,2\pi ]\] is
The equation of the curve satisfying the differential equation \[y\left( x+{{y}^{3}} \right)\,\,dx=x\left( {{y}^{3}}-x \right)\,\,dy\]and passing through the point (1,1) is:
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. What is the probability that out of 5 such bulbs none will fuse after 150 days of use
Let \[f\,\,(x)=\sin \,\,\left( \frac{\pi }{6}\,\,\sin \,\,(\frac{\pi }{2}\sin \,\,x) \right)\] for all \[x\in R\] and \[g(x)=\frac{\pi }{2}\sin x\] for all \[x\in R.\] Let (fog) (x) denote \[f\,\,(g\,\,(x))\] and \[(gof)\,\,(x)\] denote \[g\,\,(f\,\,(x)).\]Then which of the following is false?
If \[{{x}^{a}}{{y}^{b}}={{e}^{m}},\] \[{{x}^{c}}{{y}^{d}}={{e}^{n}},\] \[{{\Delta }_{1}}=\left| \begin{matrix} m & b \\ n & d \\ \end{matrix} \right|,\] \[{{\Delta }_{2}}=\left| \begin{matrix} a & m \\ c & n \\ \end{matrix} \right|\] and \[{{\Delta }_{3}}=\left| \begin{matrix} a & b \\ c & d \\ \end{matrix} \right|,\] then find values of x and y are respectively.
The distance of the point \[\left( -1,-5,-10 \right)\]from the point of intersection of the line \[\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}\] and the plane \[x-y+z=5,\] is
The numbers P, Q and R for which the function \[f\left( x \right)=P{{e}^{2x}}+Q{{e}^{x}}+Rx\] satisfies the conditions \[f\left( 0 \right)=-1,\] \[f'\,\,(log\,\,2)=31\] and \[\int_{0}^{\log 4}{[f(x)-Rx]\,\,dx=\frac{39}{2}}\] are given by
The value of \[{{\sin }^{-1}}\frac{1}{\sqrt{2}}+{{\sin }^{-1}}\frac{\sqrt{2}-1}{\sqrt{6}}+{{\sin }^{-1}}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}.....\infty \] is
Consider two events A and B such that \[P(A)=\frac{1}{4},\] \[P\left( \frac{B}{A} \right)=\frac{1}{2},\] \[P\left( \frac{A}{B} \right)=\frac{1}{4}\]. For each of the following statements, which is true
I. \[P({{A}^{C}}\text{/}{{B}^{C}})=\frac{3}{4}\]
II. The events A and B are mutually exclusive
III. \[P\,\,(A\text{/}B)+P\,\,(A\text{/}{{B}^{C}})=1\]