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If ∝, β, γ ∈ R, then the determinant

Which of the following functions are homogeneous?

Find the minors and cofactors of elements$a_{23},a_{32}\mathrm{and}{a}_{13}$of matrix $A=\left(a_{\mathrm{ij}}\right]=\left(\begin{array}{ccc}5& 6& 7\\ 5& 2& 3\\ 4& 8& 9\end{array}\right].$

The equation(s) of tangents drawn from the point $(1,4)$ to the parabola $y^2=12x$ is/are

The value of $5^{{\log }_{\frac{1}{5}}\left(\frac{1}{2}\right)}+{\log }_{\sqrt{2}}\left(\frac{4}{\sqrt{3}+\sqrt{7}}\right)+{\log }_{\frac{1}{2}}\left(\frac{1}{10+2\sqrt{21}}\right)$ is

Which among the following are classifications of triangular matrices?

The value of $lo{g}_6$ $\left(216\sqrt{6}\right)$ is equal to

then x equal to

If A = $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ and B = $\left[\begin{array}{ccc}4& 5& 6\\ 7& 8& 9\\ 1& 2& 3\end{array}\right]$ then the order of A+B will be nxn where n= -----

The greatest positive integer $k$, for which ${49}^k+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\cdots +{49}^2+49+1,$ is

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is :

A value of $\theta \in (0,\frac{\pi }{3})$, for which $\left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & 1+{\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & {\sin }^2\theta & 1+4\cos 6\theta \end{array}\right|=0\text{is}:$

If a variable circle having fixed radius $a$, passes through origin and meets the coordinates axes at point $A$ and $B$ respectively, then the locus of centroid of $△OAB$, where $O$ is the origin, is

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 $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\cdots +C_n{x}^n,$ then the value of $\underset{0\le r<s\le n}{\sum \sum }(r\cdot s)C_r{C}_s$ is

If $A=x:x\text{is a natural number}B=x:x\text{is even natural number}C=x:x\text{is prime number}$

Then, $(A\cup B)\cap C=$

Find the value of i.$i^i$ is _______.

If $5^{th}$ and $6^{th}$ term of an $A.P.$ are $6$ and $5$ respectively, then the ${11}^{th}$ term is

Suppose $y=f\left(x\right)$ and $y=g\left(x\right)$ are two functions whose graphs intersect at the three points $(0,4),(2,2)$ and $(4,0)$. And also $f\left(x\right)>g\left(x\right)$ for $x\in (0,2),f\left(x\right)<g\left(x\right)$ for $x\in (2,4)$. If $\underset{0}{\overset{4}{\int }}\left(f\right(x)-g(x\left)\right)dx=10$ and $\underset{2}{\overset{4}{\int }}\left(g\right(x)-f(x\left)\right)dx=5$, then the area between the two curves for $x\in (0,2)$ is

The expression $\tan \left(i{\log }_e\left(\frac{2-3i}{2+3i}\right)\right)$ is equal to

Range of the rational expression $y=\frac{x+3}{2x^2+3x+9},x\in R$ is

If $\sin \alpha =\frac{-4}{5}$, $\alpha \in \left[\frac{3\pi }{2},2\pi \right]$, then the value of $\cos \frac{\alpha }{2}$ is equal to



यदि $\sin \alpha =\frac{-4}{5}$, $\alpha \in \left[\frac{3\pi }{2},2\pi \right]$, तब $\cos \frac{\alpha }{2}$ का मान बराबर है

Which of the following set of points are non- collinear

[MP PET 1990]

If both roots of the equation $x^2+ax+2=0$ lie in the interval $(0,3),$ then the range of values of $a$ is

If $f\left(x\right)=max(x^3,x^2,\frac{1}{64})\forall x\in [0,\infty )$, then

The equation of the circle in the first quadrant which touches each axis at a distance 5 from the origin is

Sum upto $100$ term of the series $i+2i^2+3i^3+\cdots$ is

Let $n_1$ and $n_2$ be the number of red and black balls, respectively in box I. Let $n_3$ and $n_4$ be the number of red and black balls, respectively in box II.

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac{1}{3}$, then the correct option(s) with the possible values of $n_1$ and $n_2$ is/are

In the expansion of $(1+x{)}^2(1+y{)}^3(1+z{)}^4(1+w{)}^5,$ the sum of coefficients of the term of degree $12$ is $k.$ Then the value of $\frac{k}{13}$ is

If tan $\theta$ = $\frac{-4}{3}$ then sin$\theta$ is

If $z$ is a complex number satisfying $\overline{z^2}=1$, where $\overline{z}$ is the conjugate of $z$, then

Consider the two sets:

$A=\{m\in R:\text{both the roots of}{x}^2-(m+1)x+m+4=0\text{are real}\}$ and $B=[-3,5).$

Which of the following is not true ?

The range of the function $f\left(x\right)={\log }_2(3-2x-x^2)$ is

If ${\log }_{10}7=0.8451$, then the position of the first significant figure of $7^{-20}$, is

The mid point of line joining the common points of the line $2x-3y+8=0$ and $y^2=8x,$ is

Using quadratic formula solve the following quadratic equation:



$p^2{x}^2+(p^2-q^2)x-q^2=0,p\ne 0$

The number of $7$-digit numbers formed by the digits $1,$ $2$ and $3$ only whose sum of the digits equals $10,$ is

$\overrightarrow{a}$ and $\overrightarrow{c}$ are unit vectors and $|\overrightarrow{b}|=4.$ The angle between $\overrightarrow{a}and\overrightarrow{c}isco{s}^{-1}\left(\frac{1}{4}\right).$

If $\overrightarrow{b}-2\overrightarrow{c}=\lambda \overrightarrow{a}$, then $\lambda$ is

Let $A$ and $B$ (where $A>B$), be acute angles. If $\sin (A+B)=\frac{12}{13}$ and $\cos (A-B)=\frac{3}{5}$, then the value(s) of $\sin \left(2A\right)$ is/are

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System of vectors $a_1,a_2,.......a_n$ is said to be linearly dependent if there exists a system of scalars $(c_1,c_2---c_{n}suchthat{c}_1\overline{a}+c_2\overline{a_2}+...c_n\overline{a_n}=\overline{0}$

Which of the following gives a description about standard and mean deviation.

The coordinates of the point(s) which divide(s) the line segment joining the point $(5,-2)$ and $(9,6)$ in the ratio $3:1,$ are

The expression $\frac{36}{4\sin (x+\frac{\pi }{3})-4\sqrt{3}\cos x+8}$ lies in the interval

What is the value of determinant given if x, y, z are in AP with common difference d.

$\left|\begin{array}{ccc}x+y& y& 1\\ y+z& z& 1\\ z+x& x& 1\end{array}\right|$

The value of $\underset{n\to \infty }{lim}\underset{k=1}{\sum ^n}(\frac{n-k}{n^2})\cos \frac{4k}{n}=\frac{1}{a}(1-\cos c),$ then

If the ${10}^{th}$ term of an $A.P.$ is $35$ and the $5^{th}$ term is $20$, then