If \[x\in \left( \frac{-\pi }{2},\,\,\frac{\pi }{2} \right),\] then the value of \[{{\tan }^{-1}}\left( \frac{\tan x}{4} \right)+{{\tan }^{-1}}\,\,\left( \frac{3\,\,\sin \,\,2x}{5+3\,\,\cos \,\,2x} \right)\]is ________.
If P and Q are square matrices of the same order such that \[(P+Q)\,\,(P-Q)={{P}^{2}}-{{Q}^{2}},\] then \[{{\left( PQ{{P}^{-1}} \right)}^{2}}\] is equal to.
If \[a\ne p,\] \[b\ne q,\]\[c\ne r\] and \[\left| \begin{matrix} p & b & c \\ a & q & c \\ a & b & r \\ \end{matrix} \right|=0,\] the value of \[\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\] is_________.
Let \[\hat{u}\] and \[\hat{v}\] be unit vectors such that \[\hat{u}.\,\hat{v}=0.\] If \[\overrightarrow{r}\] is any vector coplanar with \[\hat{u}\] and \[\hat{v}\], then the magnitude of the vector \[\overrightarrow{r}\times \left( \hat{u}\times \hat{v} \right)\] is _____.
The absolute value of parameter t for which the area of the triangle whose vertices are \[A\left( -1,1,2 \right),\] \[B\left( 1,2,3 \right)\] and \[C\left( t,1,1 \right)\] is minimum, is _____.
If \[\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}\]and \[\overrightarrow{r}.\overrightarrow{a}=0,\] where \[\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k},\] \[\overrightarrow{b}=3\hat{i}-\hat{j}+\hat{k},\] and \[\overrightarrow{c}=\hat{i}+\hat{j}+3\hat{k},\] then \[\overrightarrow{r}\] is equal to _______.
The angle between the line \[\overrightarrow{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda \,\,(2\hat{i}+3\hat{j}+4\hat{k})\] and the plane \[\overrightarrow{r}.(\hat{i}+2\hat{j}-2\hat{k})=0\] is_________.
Let Q be the foot of perpendicular from the origin to the plane \[4x-3y+z+13=0\] and P be a point \[\left( -1,1,-6 \right)\] on the plane. Then, the length PQ is _________.
A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of \[2cm/{{\sec }^{2}}\]and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/sec. Then, the lizard will catch the insect after _________.
If m is the minimum value of \[f\,\,(x,y)={{x}^{2}}-4x+{{y}^{2}}+6y,\] when x and y are subjected to the restrictions \[0\le x\le 1\] and \[0\le y\le 1,\] then the value of |m| is________.
The parabolas \[{{y}^{2}}=4x\] and \[{{x}^{2}}=4y\] divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If \[{{S}_{1}},{{S}_{2}},{{S}_{3}}\] are respectively the areas of these parts numbered from top to bottom, then \[{{S}_{1}}:{{S}_{2}}:{{S}_{3}}\] is _____.
For a linear programming problem, minimize \[Z=2x+y\] subject to constraints \[5x+10y\le 50,\] \[x+y\ge 1,\]\[y\le 4\]and \[x,\]\[y\ge 0,\] then Z is equal to _______.
A letter is known to have come either from TATANAGAR or from CALCUTTA. On the envelope, just two consecutive letters TA are visible. What is the probability that the letter has come from TATANAGAR?
Two events A and B have probability 0.25 and 0.50, respectively. The probability that both A and B occur simultaneously is 0.14. Then, the probability that neither A nor B occur is ________.
A car manufacturing factory has two plants, X and Y. Plant X manufactures 70% of cars and plant Y manufactures 30%. 80% of the cars at plant X and 90% of the cars at plant Y are rated of standard quality. A car is chosen at random and is found to be of standard quality. What is the probability that it has come from plant X?
In a certain code language, '3a, 2b, 7c' means 'Truth is Eternal', '7c, 9a, 8b, 3a' means 'Enmity is not Eternal' and '9a, 4b, 2b, 6b' means 'Truth does not perish'. Which of the following means 'enmity' in that language?
Direction: Each of the questions given below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question as per the options given below it.
How is M related to F?
Statements:
I. F is sister of N who is mother of R.
II. M has two brothers of which one is R.
A)
The data in statement I alone is sufficient to answer the question, while the data in statement II alone is not sufficient to answer the question.
doneclear
B)
The data in statement II alone is sufficient to answer the question, while the data in statement I alone is not sufficient to answer the question.
doneclear
C)
The data either in statement I alone or in statement II alone is sufficient to answer the question.
doneclear
D)
The data in both the statements I and II together are not sufficient to answer the question.
doneclear
E)
The data in both the statements I and II together are necessary to answer the question.
Direction: Each of the questions given below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question as per the options given below it.
Among M, N, T, R and D each having different age, who is the youngest?
Statements:
I. N is younger than only D among them.
II. T is older than R and younger than M.
A)
The data in statement I alone is sufficient to answer the question, while the data in statement II alone is not sufficient to answer the question.
doneclear
B)
The data in statement II alone is sufficient to answer the question, while the data in statement I alone is not sufficient to answer the question.
doneclear
C)
The data either in statement I alone or in statement II alone is sufficient to answer the question.
doneclear
D)
The data in both the statements I and II together are not sufficient to answer the question.
doneclear
E)
The data in both the statements I and II together are necessary to answer the question.
Direction: Each of the questions given below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question as per the options given below it.
Village D is in which direction of village H?
Statements:
I. Village H is to the South of village A which is to the South-East of village D.
II. Village M is to the East of village D and to the North-East of village H.
A)
The data in statement I alone is sufficient to answer the question, while the data in statement II alone is not sufficient to answer the question.
doneclear
B)
The data in statement II alone is sufficient to answer the question, while the data in statement I alone is not sufficient to answer the question.
doneclear
C)
The data either in statement I alone or in statement II alone is sufficient to answer the question.
doneclear
D)
The data in both the statements I and II together are not sufficient to answer the question.
doneclear
E)
The data in both the statements I and II together are necessary to answer the question.
Let X be the solution set of the equation \[{{A}^{x}}=I,\] where \[A=\left[ \begin{matrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \\ \end{matrix} \right]\] and I is the corresponding unit matrix and \[x\subseteq N,\] then the minimum value of \[\sum{({{\cos }^{x}}\theta +{{\sin }^{x}}\theta ),}\] \[\theta \in R\] is _________.
Consider a pyramid OPQRS located in the first octant \[(x\ge 0,\,\,y\ge 0,\,\,z\ge 0)\] with O as origin and OP and OR along the X-axis and the Y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ, such that TS = 3. Then, ________.
A)
the acute angle between OQ and OS is \[\frac{\pi }{3}\].
doneclear
B)
the equation of the plane containing the \[\Delta OQS\] is \[x-y=0.\]
doneclear
C)
the length of the perpendicular from p to the plane containing the \[\Delta OQS\] is \[\frac{1}{\sqrt{2}}\].
doneclear
D)
the perpendicular distance from O to the straight line containing RS is \[\sqrt{\frac{13}{2}}\].
If f(x) is a function such that \[f\,\,(x-1)+f\,\,(x+1)=\sqrt{3}\,\,f\,\,(x)\] and \[f\,\,(5)=10,\] then the value of \[\sum\limits_{r=0}^{19}{f\,\,(5+12\,\,r)}\] is ________.
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is\[100{{m}^{2}}\], the resulting box has maximum volume. The length of the longer side of the rectangular sheet (in m) is ________.
A curve passes through the point\[\left( 1,\,\,\frac{\pi }{6} \right)\]. If the slope of the curve at each point (x, y) is \[\frac{y}{x}+\sec \,\,\left( \frac{y}{x} \right),\]\[x>0\]. Then, the equation of the curve is ________.
Let F denotes the set of all onto functions from \[A=\{{{a}_{1}},\,\,{{a}_{2}},\,\,{{a}_{3}},\,\,{{a}_{4}}\}\] to \[B=\{x,\,\,y,\,\,z\}\]. A function f is chosen at random from F. The probability that \[{{f}^{-1}}(x)\] consists of exactly two elements is _____.
Let \[f:R\to R\] be a continuous odd function, which vanishes exactly at one point and \[f(1)=\frac{1}{2}\]. Suppose that \[F(x)=\int\limits_{-1}^{x}{f(t)}\,\,dt\] for all \[x\in [-1,\,\,2]\] and \[G\,\,(x)=\int\limits_{-1}^{x}{t|\{f\,\,(t)\}|}\,\,dt\] for all \[x\in [-1,\,\,2]\]. If \[\underset{x\to 1}{\mathop{\lim }}\,\frac{F(x)}{G(x)}=\frac{1}{14},\] then the value of \[f\,\,\left( \frac{1}{2} \right)\] is _______.
Match the statements of Column I with values of Column II:
Column I
Column II
a.
The number of polynomials f(x) with non-negative integer coefficients of degree \[\le \] 2, satisfying\[f(0)=0\,\,and\,\,\int\limits_{0}^{1}{f(x)\,\,dx=1}\]is _____.
p.
8
b.
The number of points in the interval \[\left[ -\sqrt{13},\sqrt{13} \right]\] at which \[f(x)=sin({{x}^{2}})+cos({{x}^{2}})\] attains, its maximum value is ________.
q.
2
c.
\[\int\limits_{-2}^{2}{\left( \frac{3{{x}^{2}}}{1+{{e}^{x}}} \right)}\,\,dx\] equals to ______.
Six faces of a die are marked with the numbers 1, \[-1,\] 0, \[-2,\] 2 and 3. The die is thrown thrice. The probability that the sum of the numbers thrown is six is ________.