Let \[P=\left[ \begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \\ \end{matrix} \right]\] and I be the identity matrix of order 3. If \[Q=[{{q}_{ij}}]\] is a matrix such that \[{{P}^{50}}-Q=I,\] then \[\frac{{{q}_{31}}+{{q}_{32}}}{{{q}_{21}}}\]equals
let \[\overrightarrow{a},\]\[\overrightarrow{b}\]and \[\overrightarrow{c}\] be three unit vectors such that \[\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{\sqrt{3}}{2}(\overrightarrow{b}+\overrightarrow{c}).\] If \[\overrightarrow{b}\] is not parallel to \[\overrightarrow{c}\]then the angles between \[\overrightarrow{a}\]and \[\overrightarrow{b}\] is
Let p and q be the position vectors of P and Q respectively with respect to 0 and | p | = p, |q| = q. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If \[\overrightarrow{OR}\]and \[\overrightarrow{OS}\]are perpendicular, then
The equation of the plane through the points \[(2,-1,-3)\] and parallel to the lines \[\frac{x-1}{3}=\frac{y+2}{2}=\frac{z}{-4}\] and \[\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}\] is
The solution of primitive integral equation \[({{x}^{2}}+{{y}^{2}})\,dy=xy\,\,dx\]is \[y=y(x).\] If \[y(1)=1\] and \[({{x}_{0}})=e,\] then \[{{x}_{0}}\]is equal to
Let \[f:\left[ -\frac{1}{2},2 \right]\to R\] and \[g:\left[ -\frac{1}{2},2 \right]\to R\] functions defined by \[f(x)=[{{x}^{2}}-3]\] and \[g(x)=|x|f(x)+|4x-7|f(x),\] where [y] denotes the greatest integer less than or equal to y for \[y\in R.\]Then
A)
f is discontinuous exactly at three points in \[\left[ -\frac{1}{2},2 \right]\]
doneclear
B)
f is discontinuous exactly at four points in \[\left[ -\frac{1}{2},2 \right]\]
doneclear
C)
g is NOT differentiable exactly at four points in \[\left( -\frac{1}{2},2 \right)\]
The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is:
In a triangle ABC, D and E are points on BC and AC respectively, such that BD = 2 DC and AE = 3EC. Let P be the point of intersection of AD and BE. Find BP/PE, using the vector method.
Consider\[f(x)={{\tan }^{-1}}\left( \sqrt{\frac{1+\sin x}{1-\sin x}} \right),\]\[x\in \left( 0,\frac{\pi }{2} \right)\]. A normal to y = f(x) at \[x=\frac{\pi }{6}\]also passes through the point:
If \[a=\sum\limits_{n=0}^{\infty }{\frac{{{x}^{3n}}}{(3\,\,n)!}},\] \[b=\sum\limits_{n=1}^{\infty }{\frac{{{x}^{3n-2}}}{(3n-2)!}}\] and \[c=\sum\limits_{n=1}^{\infty }{\frac{{{x}^{3n-1}}}{(3n-1)!}},\] then the value of \[{{a}^{3}}+{{b}^{3+}}{{c}^{3}}-3abc\]=
Internal bisector of ZA of triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F. If a, b, c represent sides of A ABC then
If \[{{\tan }^{-1}}x+{{\cos }^{-1}}\frac{y}{\sqrt{(1+{{y}^{2}})}}={{\sin }^{-1}}\frac{3}{\sqrt{10}}\]and both x and y are positive and integral, then x and Y=
Let the vectors \[\overrightarrow{PQ},\]\[\overrightarrow{QR,}\]\[\overrightarrow{RS,}\]\[\overrightarrow{ST,}\]\[\overrightarrow{TU,}\]and \[\overrightarrow{UP}\] represent the sides of a regular hexagon.
Statement 1: \[\overrightarrow{PQ}\times (\overrightarrow{RS}+\overrightarrow{ST})\ne \overrightarrow{0}\] because
Statement 2: \[\overrightarrow{PQ}\times \overrightarrow{RS}=\overrightarrow{0}\] and \[\overrightarrow{PQ}\times \overrightarrow{ST}\ne \overrightarrow{0}\]
A)
Statement-1 is True, Statement-2 is True; Statement-2 is the correct explanation for Statement-1
doneclear
B)
Statement-1 is True, Statement-2 is True; Statement-2 is not the correct explanation for Statement-1
Let function \[f(x)={{x}^{2}}+x+\sin x-\cos x+\log \,(1+|x|)\] be defined over the interval [0, 1]. The odd extensions of f(x) to interval [-1, 1] will be
If \[\int{\frac{(2{{x}^{2}}+1)dx}{({{x}^{2}}-4)({{x}^{2}}-1)}=\log \left[ {{\left( \frac{x+1}{x-1} \right)}^{a}}{{\left( \frac{x-2}{x+2} \right)}^{b}} \right]}\] then the values of a and b are respectively
If \[{{I}_{1}}=\int\limits_{0}^{\pi /2}{x\sin x\,\,dx}\] and \[{{I}_{2}}=\int\limits_{0}^{\pi /2}{x\cos x\,\,dx},\] then which one of the following is true?
The incident ray on a surface is along the unit vector \[\hat{v},\]the reflected ray is along the unit vector \[\hat{w}\]and the normal is along unit vector \[\hat{a}\] outwards. Express \[\hat{w}\]in terms of \[\hat{a}\]and \[\hat{v}\].
Directions: Study the following information to answer the given questions.
A test with five sections namely Quant, Reasoning, English, Computer and GA has to be proofread in 6 hours where one hour needs to be spent on each section. A break of one hour has to be taken in the third or the fourth hour. Proofreading cannot start with Quant and has to end with English. Computer has to immediately follow Reasoning with no break in-between. Quant cannot be done immediately after Computers. Quant has to immediately follow GA with no break in-between.
Directions: Study the following information to answer the given questions.
A test with five sections namely Quant, Reasoning, English, Computer and GA has to be proofread in 6 hours where one hour needs to be spent on each section. A break of one hour has to be taken in the third or the fourth hour. Proofreading cannot start with Quant and has to end with English. Computer has to immediately follow Reasoning with no break in-between. Quant cannot be done immediately after Computers. Quant has to immediately follow GA with no break in-between.
Which section is to be proofread immediately after GA?
Directions: Study the following information carefully to answer following questions.
A building has seven floors numbered one to seven, in such a way that the ground floor is numbered one, the floor above it is numbered two and so on such that the topmost floor is numbered seven. One out of seven people viz. Koyal, Naina, Dimple, Shruti, Payal, Arti and Anjali lives on each floor. Koyal lives on fourth floor. Payal lives on the floor immediately below Arti's floor. Arti does not live on the second or seventh floor. Dimple does not live on an odd numbered floor. Naina does not live on a floor immediately above or below Dimple's floor. Shruti does not live on the topmost floor. Anjali does not lie on any floor below Payal's floor.
Directions: Study the following information carefully to answer following questions.
A building has seven floors numbered one to seven, in such a way that the ground floor is numbered one, the floor above it is numbered two and so on such that the topmost floor is numbered seven. One out of seven people viz. Koyal, Naina, Dimple, Shruti, Payal, Arti and Anjali lives on each floor. Koyal lives on fourth floor. Payal lives on the floor immediately below Arti's floor. Arti does not live on the second or seventh floor. Dimple does not live on an odd numbered floor. Naina does not live on a floor immediately above or below Dimple's floor. Shruti does not live on the topmost floor. Anjali does not lie on any floor below Payal's floor.
The life of a farmer is very tough. With no other revenue source, he has to totally depend upon the year's crop. And if it happens to fail, an entire year is lost," says, a farmer's son who comes from a very humble background. The education of the farmer's children is often the first casualty of a failed crop. A large number of students is reported to be dropping out of schools in villages as their parents want their children to help them on farms.
Courses of action:
I. The government should immediately launch a programme to create awareness among the farmers about the value of education.
II. The government should offer incentives to those farmers whose children remain in schools.
III. Education should be made compulsory for all the children up to the age of 14 and their employment should be banned. You have to assume everything in the statement to be true, and then decide which of the following three given suggested courses of action logically follows for pursuing.
If for \[AX=B,\,\,B=\left[ \begin{matrix} 9 \\ 52 \\ 0 \\ \end{matrix} \right]\] and \[{{A}^{-1}}=\left[ \begin{matrix} 3-\frac{1}{2}-\frac{1}{2} \\ -\,4\,\,\,\,\frac{3}{4}\,\,\,\,\frac{5}{4} \\ 2-\frac{1}{4}-\frac{3}{4} \\ \end{matrix} \right],\] then X is equal to
A curve passing through the point (1, 1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x-axis. Determine the equation of the curve.
The length of a tangent at any point on the curve y = f(x) is intercepted between the point and the x-axis is of length 1. Find the equation of the curve.
Let \[\hat{u}={{u}_{1}}\hat{i}+{{u}_{2}}\hat{j}+{{u}_{3}}\hat{k}\] be a unit vector in \[{{R}^{3}}\] and \[\hat{w}=\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+2\hat{k}).\] Given that there exists a vector \[\vec{v}\] in \[{{R}^{3}}\] such that \[|\hat{u}\times \vec{v}|\,=1\] and \[\hat{w}.|\hat{u}\times \vec{v}|\,=1.\] Which of the following statements(s) is (are) correct?
A)
There are infinitely many choices for such \[\hat{v}\]
doneclear
B)
If \[\hat{u}\]lies in the xy-plane then \[|{{u}_{1}}|\,=\,|{{u}_{2}}|\]
doneclear
C)
If \[\hat{u}\] lies in the xz-plane then \[|{{u}_{1}}|\,=2|{{u}_{3}}|\]
Let \[\overrightarrow{a},\] \[\overrightarrow{b}\] and \[\overrightarrow{c}\] be non-coplanar unit vectors, equally inclined to one another at an angle\[\theta \]. If \[\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+r\overrightarrow{c},\] find the scalars p, q and r in terms of\[\theta \].
Let\[S=\left\{ x\in (-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2} \right\}\]. The sum of all distinct solutions for the equation \[\sqrt{3}\sec \,\,x+\cos ec\,\,x+2\,\,(\tan \,\,x-\cot \,\,x)=0\] in the set S is equal to
If the gradient of the tangent at any point (x, y) of a curve which passes through the point \[\left( 1,\frac{\pi }{4} \right)\] is \[\left\{ \frac{y}{x}-{{\sin }^{2}}\left( \frac{y}{x} \right) \right\},\]the equation of the curve will be
A straight rod AB of length 1ft balances about a point 5 inches from A when masses of 9 lbs and 6 lbs are suspended from A and B respectively. It balances about a point 3- inches from B when the mass of 6 lbs is replaced by one of 23 lbs. The distance of C.G. of the rod from the end B is
Cards are drawn one by one at random from a well shuffled full pack of 52 cards until two aces are obtained for the first time. If N is the number of cards required to be drawn, then \[P\{N=n\},\]where \[2\le n\le 50,\]is