Area of Polygon Using Coordinates

Q.

The area enclosed by 2|x| + 3|y| $\le$ 6 is

1. 4 sq units

2. 24 sq units

3. 3 sq units

4. 12 sq unit

Q. Area bounded by curve $y=x^3$, x-axis and ordinates $x=1$ and $x=4$, is

1. 64 sq. unit
2. 27 sq. unit
3. $\frac{127}{4}$ sq. unit
4. $\frac{255}{4}$

Q.

Find the area of the quadrilateral whose vertices, taken in order, are $(-4,-2),(-3,-5),(3,-2)$ and $(2,3)$ .

Q. The area bounded by $y=x+1$ and $y=\cos x$ and the $x$-axis, is

1. $1\text{sq. unit}$
2. $\frac{3}{2}\text{sq. unit}$
3. $\frac{1}{4}\text{sq. unit}$
4. $\frac{1}{8}\text{sq. unit}$

Q.

Prove that the area of the parallelogram formed by the lines.

$a_{1}x+b_{1}y+c_1=0,a_{1}x+b_{1}y+d_1=0,a_{2}x+b_{2}y+c_2=0,a_{2}x+b_{2}y+d_2=0$ is $\left|\frac{(d_1-c_1)(d_2-c_2)}{a_1{b}_2-a_2{b}_1}\right|$ sq. units.

Deduce the condition for these lines to form a rhombus.

Q. The area (in sq. units) of the quadrilateral whose vertices are $A(1,1),$ $B(3,4),$ $C(5,-2)$ and $D(4,-7)$ is

1. $41$
2. $64$
3. $\frac{37}{4}$
4. $\frac{41}{2}$

Q. A system of line is given as $y=m_{i}x+c_i,$ where $m_i$ can take any value out of 0, 1, -1 and when $m_i$ is positive then $c_i$ can be 1 or -1 when $m_i$ equal 0, $c_i$ can be 0 or 1 and when $m_i$ equals -1, $c_i$ can take 0 or 2. Then the area enclosed by all these straight lines is

1. $\frac{3}{\sqrt{2}}(\sqrt{2}-1)$ sq. units
2. $\frac{3}{\sqrt{2}}$ sq. units
3. $\frac{3}{2}$ sq. units
4. $\frac{3}{4}$ sq. units

Q. A system of line is given as $y=m_{i}x+c_i,$ where $m_i$ can take any value out of 0, 1, -1 and when $m_i$ is positive then $c_i$ can be 1 or -1 when $m_i$ equal 0, $c_i$ can be 0 or 1 and when $m_i$ equals -1, $c_i$ can take 0 or 2. Then the area enclosed by all these straight lines is

1. sq. units
2. sq. units
3. sq. untis
4. sq. units

Q. If Y = SX, Z = tX all the variables being differentiable functions of x and lower suffices denote the derivative with respect to x and $\left|\begin{array}{ccc}X& Y& Z\\ X_1& Y_1& Z_1\\ X_2& Y_2& Z_2\end{array}\right|÷\left|\begin{array}{cc}{S}_1& t_1\\ S_2& t_2\end{array}\right|=X^n,thenn=$

1. 3
2. 4
3. 1
4. 2

Q. Area bounded by the lines y=x, x=-1, x=2 and x- axis is

1. None of these
2. $\frac{5}{2}$ sq. unit
3. $\frac{3}{2}$ sq. unit
4. $\frac{1}{2}$ sq. unit

Q.

If $A=(-3,4),B=(-1,-2),C=(5,6),D(x,-4)$are vertices of a quadrilateral such that$ar(∆ABD)=2ar(∆ACD)$ then $x$ is equal to

1. $6$

2. $9$

3. $69$

4. $96$

Q. If the vertices of a quadrilateral are $A(-3,1),$ $B(-1,4),$ $C(3,2)$ and $D(t_1,-2)$ and the area of the quadrilateral is $19\text{sq. units}$, then which of the following is/are correct?

1. $t_1=1$
2. $t_1=-75$
3. $t_1=75$
4. $t_1=-1$

Q. $\frac{10}{x+y}+\frac{2}{x-y}=4;\frac{15}{x+y}-\frac{5}{x-y}=-2$

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

Let the area bounded by x = –1, x-axis, y-axis and f(x) = 1 + x² be A₁ and the area bounded by x = 1, x-axis, y-axis and f(x) = 1 + x² be A₂ then

माना x = –1, x-अक्ष, y-अक्ष तथा f(x) = 1 + x² द्वारा परिबद्ध क्षेत्रफल A₁ है तथा x = 1, x-अक्ष, y-अक्ष व f(x) = 1 + x² द्वारा परिबद्ध क्षेत्रफल A₂ है, तब

1. A₁ = A₂
2. A₁ + A₂ = $\frac{4}{3}$
3. A₁A₂ = $\frac{16}{9}$
4. $A_1+A_2=\frac{10}{3}$

Q.

Prove that the area of the parallelogram formed by the lines $3x-4y+a=0$, $3x-4y+3a=0$, $4x-3y-a=0$ and $4x-3y-2a=0$ is $\frac{2a^2}{7}$ sq. units.

Q. Consider the experiment of throwing a die, if a multiple of $3$ comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event 'the coin shows a tail', given that 'at least one die shows a $3$'.

Q. If the function f and g are defined from the set of real numbers R to R such that $f\left(x\right)=e^x,g\left(x\right)=3x-2$ then find functions fog and gof. Also find the domains of the functions $(fog{)}^{-1}$ and $(gof{)}^{-1}$

1. $(2,\infty )$
2. $(-\infty ,-2)$
3. $(-2,\infty )$
4. $(-2,2)$

Q. The equation of the line passing through origin and making an angle ${30}^{\circ }$ with xaxis is

1. $y=\sqrt{3}x$
2. $x=3y$
3. $y=3x$
4. $x=\sqrt{3}y$

Q. Prove that the locus of the vertices of all parabolas that can be drawn touching a given circle of radius a and having a fixed point on the circumference as focus is $r=2a{\cos }^3\frac{\theta }{3},$ the fixed point being the pole and the diameter through it the initial line.

Q. If for a sequence, $t_n=\frac{2^{n-2}}{5^{n-3}}$, show that the sequence is a G.P . Find its first term and common ratio.

Q. $2\times 2$ matrix where elements come from $\{p,q,r,s\}$ without repeating satisfying following conditions :-
$a_{11}\ne p$
$a_{12}\ne q$
$a_{21}\ne r$
$a_{22}\ne s$

Q. Find the Cartesian equation of the line which passes through the point $(-2,4,-5)$ and parallel to the line given by $\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$

Q. Using the method of integraton find the area of the region bounded by lines: $2x+y=4,3x-2y=6$ and $x-3y+5=0$

Q. Let $m_1$ be the slope of the line containing the points $(3,5)$ and $(6,10)$. Let there be another line with slope $m_2$ such that $m_1\times m_2=-1$. Find the equation of the other line.

1. $x+5y=15$
2. $3x+5y=15$
3. $3x+y=5$
4. $-5x+y=\frac{1}{3}$

Q. How do you find the inverse of $y=-\log x$?

Q. A straight line cuts intercepts from the axis of coordinates the sum of the reciprocals of which is a constant $K$. Then it always passes through a fixed point :

1. $(K,K)$
2. $(-K,-K)$
3. $(\frac{1}{K},\frac{1}{K})$
4. $(K-1,K-1)$

Q. The vertex A of a triangle ABC is the point $(-2,3)$ whereas the line perpendicular to the sides AB and AC are $x-y-4=0$ and $2x-y-5=0$ respectively. The right bisectors of sides meet at P$(3/2,5/2)$ . Then the equation of side BC is

1. $5x-2y=16$
2. $5x+2y=10$
3. $2x-5y=10$
4. none of these

Q. Solve the equation $2x+1=x-3$, and represent the solution (s) on (i)the number line (ii) the Cartesian plane.

Q. The area (in sq. units) of the quadrilateral whose vertices are $A(1,1),$ $B(3,4),$ $C(5,-2)$ and $D(4,-7)$ is

1. $41$
2. $64$
3. $\frac{37}{4}$
4. $\frac{41}{2}$

Q. If $81=3^n$, then $n=5$

1. True
2. False