The midpoint of the line joining the points \[\left( -\,10,8 \right)\] and \[\left( -\,6,12 \right)\] divides the line joining the points \[\left( 4,-\,2 \right)\] and \[\left( -\,2,4 \right)\] in the ratio of:
If \[\omega \] is an imaginary cube root of unity. then value of the expression \[2\,(1+\omega )\,\,(1+{{\omega }^{2}})+3\,(2+\omega )\,\,(2+{{\omega }^{2}})\]\[+.....+(n\,+1)\,\,(n\,+\omega )\,\,(n\,+{{\omega }^{2}})\]
A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of eleven steps he is one step away from the starting point.
A, B, C be three exhaustive and mutually exclusive events associated with a random experiment. If \[P(B)=\frac{3}{2}P(A)\] and \[P(C)=\frac{1}{2}P(B)\] the P(A) is equal to:
The domain of the derivative of the function \[f(x)=\left\{ \begin{matrix} {{\tan }^{-1}}x, & if\,\,x||\,\le \,1 \\ \frac{1}{2}(|x|-1), & if\,\,|x|\,\le \,1 \\ \end{matrix} \right.\] is
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
Let \[-\frac{\pi }{6}<\theta <-\frac{\pi }{12}\] Suppose \[{{\alpha }_{1}}\] and \[{{\beta }_{1}}\] are the roots of the equation \[{{x}^{2}}-2x\sec \theta +1=0\] and \[{{\alpha }_{2}}\] and \[{{\beta }_{2}}\] are the roots of the equation\[{{x}^{2}}+2x\tan \theta -1=0\]. If \[{{\alpha }_{1}}>{{\beta }_{1}}\] and \[{{\alpha }_{2}}>{{\beta }_{2}},\] then \[{{\alpha }_{1}}+{{\beta }_{2}}\] equals
A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is \[30{}^\circ \]. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is \[60{}^\circ \]. Then the time taken (in minutes) by him, from B to reach the pillar is:
Let \[{{b}_{i}}>1\] for i =1, 2 ... 101. Suppose\[\log {}_{e}{{b}_{1}}\], \[\log {}_{e}{{b}_{2}},.....,\,\,log{}_{e}{{b}_{101}}\]are in Arithmetic progression (A.P) with the common difference \[\log {}_{e}\] 2. Suppose \[{{a}_{1}},\,{{a}_{2}},\,.....,\,{{a}_{101}}\] are in A.P. such that \[{{a}_{1}}={{b}_{1}}\]and\[{{a}_{51}}={{b}_{51}}\]. If \[t={{b}_{1}}+{{b}_{2}}+.......+{{b}_{51}}\] and \[s={{a}_{1}}+{{a}_{2}}+.....+{{a}_{51}}\] then
Two sides of a rhombus are along the lines, \[x-y+1=0\] and \[7x-y-5=\text{0}\]. If its diagonals intersect at \[\left( -\,1,-\,2 \right),\] then which one of the following is a vertex of this rhombus?
The mean square deviations about \[-\,1\] and 1 of a set of observations are 7 and 3 respectively. Find the standard deviation of this set of observations.
Directions: Study the following information to answer the given questions. Five Women Madhu, Kanchan, Chandni, Sheela and Rekha are married to Doctor, Naval Officer, Lawyer, Sales Manager and Engineer. The ladies are Accountant, Teacher and Doctor by Profession, while two are Housewives.
(i) One husband and Wife have the same profession.
(ii) Madhu and Kanchan are neither Housewives nor they are married to Doctor or Lawyer.
(iii) Sheela and Rekha are neither Teacher nor Accountant and their Husbands are neither the Engineer nor are in Navy.
(iv) The Sales Manager is not Madhu's or Chandni's husband. His Wife is an Accountant.
(v) Rekha is not a Doctor
(vi) Chandni is not a Teacher and the Teacher's husband is naval officer
Who among the following is engaged in Doctor's Profession with her husband?
Directions: Study the following information to answer the given questions. Five Women Madhu, Kanchan, Chandni, Sheela and Rekha are married to Doctor, Naval Officer, Lawyer, Sales Manager and Engineer. The ladies are Accountant, Teacher and Doctor by Profession, while two are Housewives.
(i) One husband and Wife have the same profession.
(ii) Madhu and Kanchan are neither Housewives nor they are married to Doctor or Lawyer.
(iii) Sheela and Rekha are neither Teacher nor Accountant and their Husbands are neither the Engineer nor are in Navy.
(iv) The Sales Manager is not Madhu's or Chandni's husband. His Wife is an Accountant.
(v) Rekha is not a Doctor
(vi) Chandni is not a Teacher and the Teacher's husband is naval officer
Which of the following pair is the correct match of husband - wife?
Directions: Study the following information to answer the given questions. Five Women Madhu, Kanchan, Chandni, Sheela and Rekha are married to Doctor, Naval Officer, Lawyer, Sales Manager and Engineer. The ladies are Accountant, Teacher and Doctor by Profession, while two are Housewives.
(i) One husband and Wife have the same profession.
(ii) Madhu and Kanchan are neither Housewives nor they are married to Doctor or Lawyer.
(iii) Sheela and Rekha are neither Teacher nor Accountant and their Husbands are neither the Engineer nor are in Navy.
(iv) The Sales Manager is not Madhu's or Chandni's husband. His Wife is an Accountant.
(v) Rekha is not a Doctor
(vi) Chandni is not a Teacher and the Teacher's husband is naval officer
Directions: Some statements are given followed by some conclusions. You have to consider the statements to be true even if they seem to be at variance from commonly known facts. You have to decide which of the following conclusions if any follow from the given statements:
Statements:
All boys are girls. Some girls are men. No men is a women
Directions: Some statements are given followed by some conclusions. You have to consider the statements to be true even if they seem to be at variance from commonly known facts. You have to decide which of the following conclusions if any follow from the given statements:
Statements
Some birds are animals. Some animals are idiots,
All idiots are humans. No idiot is bat. Some bats are animals.
Directions: Some statements are given followed by some conclusions. You have to consider the statements to be true even if they seem to be at variance from commonly known facts. You have to decide which of the following conclusions if any follow from the given statements:
Statements:
All boys are girls. Some girls are not men. No men is a women. Some women are girls. No woman is human.
Change in positions of beads in the four figures above follows a sequence. Following the same sequence, which of the figures below should appear as the fifth figure above?
Let P be the point on the parabola \[{{y}^{2}}=4x\] which is at the shortest distance from the center S of the circle \[{{x}^{2}}+{{y}^{2}}-4x-16y+64=0\]. Let Q be the point on the circle dividing the line segment SP internally. Then
A)
\[SP=2\sqrt{5}\]
doneclear
B)
\[SQ:QP=(\sqrt{5}+1):4\]
doneclear
C)
The slope of the tangent to the circle at Q is \[\frac{1}{2}\]
In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and \[2s=x+y+z\]. If \[\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}\] and area of in circle of the triangle XYZ is \[\frac{8\pi }{3},\] then
A)
Area of the triangle XYZ is \[6\sqrt{6}\]
doneclear
B)
The radius of circumcircle of the triangle XYZ is \[\frac{35}{4}\sqrt{6}\]
The value of \[\sum\limits_{k=1}^{13}{\frac{1}{\sin \,\left( \frac{\pi }{4}+\frac{(k-1)\pi }{6} \right)\,\,\sin \,\,\left( \frac{\pi }{4}+\frac{k\pi }{6} \right)}}\] is equal to
The circle \[{{C}_{1}}:{{x}^{2}}+{{y}^{2}}=3,\] with center at O, intersects the parabola \[{{x}^{2}}=2y\] at the point P in the first quadrant. Let the tangent to the circle \[{{C}_{1}}\]at P touches other two circles \[{{C}_{2}}\] and \[{{C}_{3}},\] at \[{{R}_{2}}\]and \[{{R}_{3}},\] respectively. Suppose \[{{C}_{2}}\], and \[{{C}_{3}},\]have equal radii \[2\sqrt{3}\]and centers \[{{Q}_{2}}\] and \[{{Q}_{3}}\], respectively. If \[{{Q}_{2}}\] and \[{{Q}_{3}},\] lie on the y-axis, then
A)
\[{{R}_{2}}{{R}_{3}}=4\sqrt{6}\]
doneclear
B)
Area of the triangle \[O{{R}_{2}}{{R}_{3}}\] is \[6\sqrt{2}\]
doneclear
C)
Area of the triangle \[P{{Q}_{2}}{{Q}_{3}}\] is \[6\sqrt{2}\]
\[Let\,\,f(x)\,\,\,=\left[ \begin{matrix} {{\{1+|\sin x|\}}^{a/|\sin x|}}\,\,\,\,\,, & \frac{\pi }{6}<x<0 \\ b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, & x=0 \\ {{e}^{\tan 2x/\tan 3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}}, & 0<x<\frac{\pi }{6} \\ \end{matrix} \right.\]. Determine a and b such that f(x) is continuous at x = 0.
Area of the triangle with vertices (a, b),\[({{x}_{1}},{{y}_{1}})\] and \[({{x}_{2}},{{y}_{2}}),\] where \[a,\,{{x}_{1}}\] and \[{{x}_{2}},\] are in G.P. with common ratio r and b, \[{{y}_{1}}\] and \[{{y}_{2}}\] are in G.P. with common ratio s, is given by