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Question 30

If on increasing a,b decreases in such a manner that___remains ___and positive, then a and b are said to vary inversely with each other.

Range of $f\left(X\right)=co{s}^{-1}\left(\frac{x^2+x+1}{x^4+1}\right)$ is

If $x=3(\cos t+\sin t);y=2(\cos t-\sin t)$ represents a conic, then its foci are

A survey shows that $76\%$ of the Indians like oranges, whereas $62\%$ like bananas. What percentage of the Insians like botth oranges and bananas ?

By using properties of definite integrals, evaluate the integrals
${\int }_0^{\frac{\pi }{2}}\frac{\sqrt{sin}x}{\sqrt{sinx}+\sqrt{cosx}}dx$

The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.

A linear function of several variables x and y is called ……...

The coefficient of x⁴⁹ in the product

(x - 1)(x - 3) ... (x - 99) is

(a) - 99² (b) 1 (c) - 2500 (d) - 50

If two sets $A$ and $B$ are such that $n\left(A\right)=20$, $n(A\cap B)=4$, and $n(A\cup B)=42$, then

The sum of the $n$ terms of the series $3+8+22+72+266+1036+\cdots$

The solution of the differential equation $y_1{y}_3=3y_2^2$ is

The angle $\frac{37\pi }{4}$ is equivalent to

Consider the ellipse $(3x-6{)}^2+(3y-9{)}^2=\frac{4}{169}(5x+12y+6{)}^2$

Column I contains the distances associated with this ellipse and Column II gives their value. Match the expressions/statements in column I with those in column II.

$\begin{array}{cccc}& \text{Column -I}& & \text{Column -II}\\ \left(a\right)& \text{The length of major axis}& \left(p\right)& \frac{72}{5}\\ \left(b\right)& \text{The length of minor axis}& \left(q\right)& \frac{16}{\sqrt{5}}\\ \left(c\right)& \text{The length of latus recturn}& \left(r\right)& \frac{16}{3}\\ \left(d\right)& \text{The distance between the directrices}& \left(s\right)& \frac{48}{5}\end{array}$

If the four roots of the equation $z^4+z^3+2z^2+z+1=0$ form a quadrilateral on the Argand plane, then the area of the quadrilateral is

If $cot\theta (1+sin\theta )=4mandcot\theta (1-sin\theta )=4n$, prove that $(m^2-n^2{)}^2=mn$

A circle intersect y axis at a distance b from origin and intercept on x-axis is 2a. The locus of the centre of the circle is?

${lim}_{x\to 0}\frac{8^x-2^x}{x}$

Rewrite each of the following statement in the form ' p if and only if q".

(i) p : If you watch television , then your mind is free and if your mind is free, then you watch television.

(ii) q : If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a you rectangle, then it is equiangular.

(iii) r : For you to get an A grade, it is necessary and sufficient that you do all the homework you regularly.

(iv) s : If atumbler is half empty, then it is half full and if a tumbler is half full, then it is ha;f empty.

If p, q and r are in GP and the equations $p{x}^2+2qx+r=0$ and $d{x}^2+2ex+f=0$ have a common root, show that $\frac{d}{p},\frac{e}{q}$ and $\frac{f}{r}$ are in AP.

Find the equations to the straight lines which pass through the origin and are inclined at an angle of ${75}^{\circ }$ to the straight line $x+y+\sqrt{3}(y-x)=a$.

If the equations $x^2+ax+12=0,x^2+bx+15=0$ and $x^2+(a+b)x+36=0$ have a common positive root, then the value of $a+b$ is equal to

If $a=2\hat{i}+2\hat{j}+3\hat{k}$, $b=-\hat{i}+2\hat{j}+\hat{k}$ and $c=3\hat{i}+\hat{j}$ such that $a+\lambda b$ is perpendicular to c, then find the value of $\lambda$.

If $(\frac{5}{3},3)$ is the centroid of a triangle and its two vertices are $(0,1)$ and $(2,3)$, then the coordinates of the third vertex are

The probabiltiy of a increasing G.P. being formed, if we select three numbers from the set $\{1,2,3,\cdots ,10\}$ is
(correct answer + 1, wrong answer - 0.25)

If $f\left(x\right)={\int }_0^{x}tsintdt$, then write the value of f'(x).

If ratio of the sum of $n$ terms of $2$ different A.P. is $\frac{3n+2}{4n+23}$, then the ratio of ${10}^{\text{th}}$ term is

Let $f:[-2,3]\to R$ and $g:[0,5]\to R$ be such that $f\left(x\right)=x^3-x,g\left(x\right)=x^3-2x^2.$ Then the domain of $\frac{f\left(x\right)}{g\left(x\right)}$ is

For $a>b>c>0$, the distance between (1, 1) and the point of intersection of the lines $ax+by+c=0$ and $bx+ay+c=0$ is less than $2\sqrt{2}$. Then,

If the arithmetic mean of two positive numbers a & b (a > b) is twice their G.M., then a:b is

Write the number of 5 digit numbers that can be formed using digits 0, 1 and 2.

Match the following

$$\begin{array}{cc}ColoumnI& ColoumnII\\ I.Zns& \left(P\right).Limitingratio=0.414\\ II.NaCl& \left(Q\right).Frenkeldefect\\ III.CsCl& \left(R\right).Coordinationnumber=4\\ IV.Ca{F}_2& \left(S\right).8nearestneighboursofthecation\end{array}$$

If the A.M. and G.M. of roots of a quadratic equations are 8 and 5 respectively, then the quadratic equation will be

The letters of the word COCHIN are arranged and all the words are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is:

If a straight line passing through the origin $O$ meets the lines $4x+2y=9$ and $2x+y+6=0$ at the points $P$ and $Q$ respectively, then the ratio in which $O$ devides the segment $PQ$ is

If the straight line, $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta )$ then $\beta$ equals:

Let a circle is touching exactly two sides of a square $ABCD$ and passes through exactly one of its vertices. If area of the square $ABCD$ is $1$ sq. units, then the radius of the circle (in units) is

Let $\alpha$ be a root of the equation $x^2+x+1=0$ and the matrix

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& \alpha & {\alpha }^2\\ 1& {\alpha }^2& {\alpha }^4\end{array}\right]$ then the matrix $A^{31}$ is equal to

In column 1 different setup for Young’s double slit experiment is given. Distance between two slits $S_1$ and $S_2$ is ‘d’ and between slits and screen is ‘D’ in every setup. In column 2 location of central maxima is given. In column 3 path difference $\Delta P$ is given at point P which is $\theta$ angle above MO line (d << D)

Mark the correct combination

If ${\cot }^{2}x=\cot (x-y)\cot (x-z)$, then the value of $\cot 2x$ is

Find the number of ways in which 8 distinct toys can be distributed among 5 children.

Two wires of the same material have diameter in the ratio 2:1 and length in the ratio 1:2. If they are stretched by the same force, their elongation will be in the ratio

If $\alpha ,\beta$ are the roots of the equation $x^2+px+1=0;\gamma ,\delta$ the roots of the equation $x^2+px+1=0;$ then $(\alpha -\gamma )(\alpha +\delta )(\beta -\gamma )(\beta +\delta )$

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ and $|\overrightarrow{a}|=3,|\overrightarrow{b}|=5,|\overrightarrow{c}|=7$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\alpha$, then $\alpha =$_____