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Which of the following is true if A and C are coefficient and augmented matrices respectively for a system of linear equation. Here n = number of unknowns

If $\frac{{\cos }^{4}A}{{\cos }^{2}B}+\frac{{\sin }^{4}A}{{\sin }^{2}B}=1$, then which of the following is/are correct?

A line passes through the centre of a sphere whose radius is 5 and one of the intercept points is $(1,-2,2)$. If the equation of the line is

$\frac{x}{1}=\frac{y}{-2}=\frac{z}{2}$

,then the equation of the sphere can be

Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in (A U B )

If the standard deviation of $X$ is $\sigma$, then s.d. of the variable $U=\frac{aX+b}{c}$ where $a,b,c$ are constants is

If $11+103+1005+\cdots n\text{terms}=\frac{{10}^{n+1}+x{n}^2+y}{9}$, then the value of $x-y$ is

For the given table choose the correct option

$\begin{array}{cccc}& & & \\ & \text{Column I}& & \text{Column II}\\ \left(a\right)& \text{The value of}\cot \left(\frac{41\pi }{4}\right)\text{is}& \left(p\right)& 1\\ \left(b\right)& \text{The value of}\sec (-{600}^{\circ })\text{is}& \left(q\right)& 2\\ \left(c\right)& \text{The value of}{\text{cosec}}^2\left(\frac{41\pi }{4}\right)\text{is}& \left(r\right)& -2\\ \left(d\right)& \text{The value of}\tan \left(\frac{19\pi }{4}\right)\text{is}& \left(s\right)& -1\end{array}$

If $(x-2{)}^{\circ }$ and $(2x+5{)}^{\circ }$ are supplementary angles, then the value of $x$ is

Which of the following function is a monotonic function?

The absolute value of $\frac{\underset{0}{\overset{\pi /2}{\int }}(x\cos x+1)e^{\sin x}dx}{\underset{0}{\overset{\pi /2}{\int }}(x\sin x-1)e^{\cos x}dx}$ is equal to

If $f\left(x\right)=\{\begin{array}{cc}{x}^3,& x<0\\ 3x-2,& 0\le x\le 2\\ x^2+1,& x>2\end{array}$

Then find the value(s) of x for which f(x)=2.

The first 3 terms in the expansion of $(1+ax{)}^n$ (n ≠ 0) are 1, 6x and 16$x^2$. Then the value of a and n are respectively

The distance between the foci of the hyperbola $9x^2-16y^2+18x+32y-151$ = 0 is

Which among the following point lie inside the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$

Solution set of $x(2^x-1)(3^x-9)(x-3)<0$ is

If $A$ is a square matrix of order $3$ and $||adj\left(A\right)|\cdot |A|\cdot A|=|A{|}^{\lambda },$ then the value of $\lambda$ is

The area (in sq. units) of the region bounded by the curve $x^2=4y$ and the straight line $x=4y-2$ is:

If $S$ be the sum, $P$ the product and $R$ the sum of reciprocals of $n$ terms in G.P., then the value of ${\left(\frac{S}{R}\right)}^n$ is

Let the foot of the perpendicular of $P(2,-3,1)$ on the line

$\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-2}{-1}$ be $Q.$

If direction ratios of the line segment joining P and Q be $l,m,n$, then which of the following relations are correct ?

The set of values of $m$ for which $f\left(x\right)=x^2-(m-3)x+m$ intersects the positive direction of $x-$axis atleast once, is

If the expansion of powers of x of the function $\frac{1}{(1-ax)(1-bx)}$ is $a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots ,$ then $a_n$ is?

The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If $e_1$ & $e_2$ are the eccentricities of ellipse and hyperbola respectively, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is

Which of the following function is an into function if all are defined on f: R $\to$ R

If P = $\left(\begin{array}{ccc}1& \propto & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right)$ is the adjoint of a 3x3 matrix A and det(A)=4,then α is equal to

The domain of $f\left(x\right)=\sqrt{\frac{1-5^x}{7^{-x}-7}}$ is

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

​​​​​​$\begin{array}{ccc}Column1& Column2& Column3\\ \left(I\right)x^2+y^2=a^2& \left(i\right)my=m^{2}x+a& \left(P\right)(\frac{a}{m^2},\frac{2a}{m})\\ \left(II\right)x^2+a^2{y}^2=a^2& \left(ii\right)y=mx+a\sqrt{m^2+1}& \left(Q\right)(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}})\\ \left(III\right)y^2=4ax& \left(iii\right)y=mx+\sqrt{a^2{m}^2-1}& \left(R\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}})\\ \left(IV\right)x^2-a^2{y}^2=a^2& \left(iv\right)y=mx+\sqrt{a^2{m}^2+1}& \left(S\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}})\end{array}$



If a tangent to a suitable conic (Column 1) is found to be $y=x+8$ and its point of contact is $(8,16)$, then which of the following options is the only CORRECT combination?

The value of $\underset{n\to \infty }{lim}\frac{1}{n^2}\{{\sin }^2\frac{\pi }{4n}+2{\sin }^2\frac{2\pi }{4n}+\cdots +n{\sin }^2\frac{4\pi }{4n}\}$ is

The equation of pair of straight lines joining the point of intersection of the curve $x^2+y^2=4$ and y - x = 2 to the origin, is

It is given that the events A and B are such that $P\left(A\right)=\frac{1}{4},P\left(\frac{A}{B}\right)=\frac{1}{2}andP\left(\frac{B}{A}\right)=\frac{2}{3}.$ Then P(B) is

If $z$ and $w$ be two complex numbers such that $|z|\le 1,|w|\le 1$ and $|z+iw|=|z-i\overline{w}|=2,$then

f(x) and f’(x) are differentiable at x = c. Which of the following is the condition for f(x) to have a local maximum at x = c, if f’(c) = 0

If the lines $(p-q)x^2+2(p+q)xy+(q-p)y^2=0$ are mutually perpendicular, then

Choose the correct pair of Equivalent Sets.

The range of the function $f\left(x\right)=\sqrt{x^2-3x+5}$ is

Find the sum upto first 11 terms of the series 1.4.7 + 4.7.10 + 7.10.13 + . . . . . is___

Angle between two planes $a_{1}x+b_{1}x+c_{1}x+d_1=0$ & $a_{2}x+b_{2}x+c_{2}x+d_2=0$ is given by-

$\int \frac{2x}{(1-x^2)\sqrt{x^4-1}}dx$ is equal to

The length of the major axis and the minor axis of the ellipse $2x^2+3y^2-4x-12y+13=0$ are and respectively.

If $A=\left\{\right(a,b):a^2+b^2=25\text{and}a,b\in N\}$ then $n\left(A\right)=$

The equation(s) of the circle passing through the points of intersection of the circles $x^2+y^2-2x-4y-4=0,$ $x^2+y^2-10x-12y+40=0$ and having radius $4$ units is/are

The domain of the function $f\left(x\right)=\sqrt{{0.625}^{4-3x}-{1.6}^{x(x+8)}}$ is

The set of the solutions for $\frac{(x+1)(x-3)}{(x+5)}<0$ is

The equation of the lines joining the vertex of the parabola $y^2=6x$ to the points on it whose abscissa is 24, is