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Reduce the following equations into intercept form and find their
intercepts on the axes.

(i) 3x
+ 2y – 12 = 0 (ii) 4x – 3y = 6
(iii) 3y + 2 = 0.

The area bounded by the curve $y=\frac{1}{x^2}$ and its asymptote between the lines $x=1$ to $x=3$ is

A tangent is drawn to the parabola $y^2=6x$ which is perpendicular to the line $2x+y=1$. Which of the following points does $\text{NOT}$ lie on it ?

The number of integers $x$ satisfying
$-3x^4+\text{det}\left[\begin{array}{ccc}1& x& x^2\\ 1& x^2& x^4\\ 1& x^3& x^6\end{array}\right]=0$
is equal to

Find the coordinates of
the point where the line through (3, ­−4, −5) and (2,
− 3, 1) crosses the plane 2x + y + z = 7).

If P=$\left[\begin{array}{c}x00\\ 0y0\\ 00z\end{array}\right]andQ=\left[\begin{array}{c}a00\\ 0b0\\ 00c\end{array}\right]$ then prove that

$PQ=\left[\begin{array}{c}xa00\\ 0yb0\\ 00zc\end{array}\right]=QP$

The value of $\int {\cot }^{4}xdx$ is

Let $f:(-\frac{\pi }{2},\frac{\pi }{2})\to R$ be given by f(x) = $\left[log\right(secx+tanx){]}^3$. Then

The equation $(y–1)=m(x–2)$ represents a family of lines passing through $(2,1).$ But this equation does not include the line $x–2=0,$ since the slope of the line is not defined. A line $y=mx$ through $(0,0)$ cuts the lines $x+y=3$ and $x+y=5$ at two distinct points A and B respectively. $AB=d$

If $y=e^x+\sin x,$ then $\frac{d^{2}x}{d{y}^2}$ is equal to

The distance moved by the particle in time t is given by $x=t^3-12t^2+6t+8$. At the instant when its acceleration is zero, then the velocity is

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-zero vectors such that $|\overrightarrow{a}|=|\overrightarrow{b}|$ and $\overrightarrow{a}\cdot (2\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c})=\overrightarrow{b}\cdot (\overrightarrow{a}+2\overrightarrow{b}+\overrightarrow{c}),$ then which of the following is/are true ?

The shape factor of the section shown in figure is

1. 2

There are two bags, one of which contains 5 black and 4 white balls while the other contains 3 black and 6 white balls. A die is thrown. If it shows up 1 or 3, a ball is taken from the first bag. But if it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a white ball.

If the graph $y=g\left(x\right)$ has a minimum point at $(4,2),$ then minimum point of the graph $y=g(x-5)-3$ is

If $A=\{\alpha ,\beta ,\gamma \},B=\{1,2,3,4\}$, then the number of elements in set $A\times B\times B$ is

In a hyperbola e = $\frac{9}{4}$ and the distance between the directrices is 3. Then the length of Transverse axis is

If $\alpha ,\beta$ be roots $x^2+px+1=0$ and $\gamma ,\delta$ are the roots of $x^2+qx+1=0$, then $(a-\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )$ is

If ${\tan }^2{2}^{\circ }+{\tan }^2{4}^{\circ }+\cdots +{\tan }^2{88}^{\circ }=a,$ then the value of $\sum _{\theta =1^{\circ }}^{{89}^{\circ }}\frac{{\sin }^4\theta +{\cos }^4\theta }{{\sin }^2\theta {\cos }^2\theta }$ is

Using the fact that $\sin (A+B)=\sin A\cos B+\cos A\sin B$ and the differentiation, obtain the sum formula for $\text{cosines}$.

The value of $\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{n-r}{n^2+r^2}$ is

Integrate the following functions w.r.t. x.

$\int \frac{1}{cos(x+a)cos(x+b)}dx.$

If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right],\text{then}adj(3A^2+12A)$ is equal to

Let $\overrightarrow{a}=2\hat{i}+\hat{j}-2\hat{k}and\overrightarrow{b}=\hat{i}+\hat{j}.$ If $\overrightarrow{c}$ is a vector such that $\overrightarrow{a}.\overrightarrow{c}=|\overrightarrow{c}|,|\overrightarrow{c}-\overrightarrow{a}|=2\sqrt{2}$ and the angle between $(\overrightarrow{a}\times \overrightarrow{b})and\overrightarrow{c}is{30}^{\circ },then|(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}|$ is equal to

Find the equation of
the plane passing through the line of intersection of the planes
and
and
parallel to x-axis.

The following graph shows two straight lines passing through the origin. Find the slope of Line 2 from the given graph if the product of two slopes of the two lines is -1.

Let $A$ and $B$ be two $n\times n$ matrices such that $det\left(A\right)\ne 0,$ $A+B={\left(AB\right)}^2$ and $BAB=A+I,$ where $I$ is an identity matrix. Which of the following is/are CORRECT?

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