If the point ( , +1) lies inside the region bounded by the curve … - Vidyadip

MCQ

If the point $(\lambda,\ \lambda+1)$ lies inside the region bounded by the curve $\text{x}=\sqrt{25-\text{y}^2}$ and y-axis, then $\lambda$ belongs to the interval:

✓
$(-1,\ 3)$
B
$(-4,\ 3)$
C
$(-\infty,\ -4)\cup(3,\ \infty)$
D
None of these

✓

Answer

Correct option: A.

$(-1,\ 3)$

$(-1,\ 3)$

Solution:
The given equation of the curve is $x^2 + y^2 = 25$
Since $(\lambda,\ \lambda+1)$ lies inside the region bounded by the curve $x^2 + y^2 = 25$ and the y-axis, we have:
$\lambda^2+(\lambda+1)^2<25,$ provided $\lambda+1>0$
$\Rightarrow\lambda^2+\lambda^2+12\lambda<25,\ \lambda>-1$
$\Rightarrow2\lambda^2+2\lambda-24<0,\ \lambda>-1$
$\Rightarrow\lambda^2+\lambda-12<0,\ \lambda>-1$
$\Rightarrow(\lambda-3)(\lambda+4)<0,\ \lambda>-1$
$\Rightarrow-4<\lambda<3,\ \lambda>-1$
$\Rightarrow\lambda\in(-1,\ 3)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from The Circle→The Circle — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Sum of $n$ terms of series $12 + 16 + 24 + 40 + .....$ will be

Two persons each make a single throw with a die. The probability they get equal value is ${p_1}$. Four persons each make a single throw and probability of three being equal is ${p_2}$, then

For $p\,>\,0$, a vector $\vec{v}_{2}=2 \hat{i}+(p+1) \hat{j}$ is obtained by rotating the vector $\vec{v}_{1}=\sqrt{3} p \hat{i}+\hat{j}$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta=\frac{(\alpha \sqrt{3}-2)}{4 \sqrt{3}+3}$, then the value of $\alpha$ is equal to $....$

The value of $\mathop {\lim }\limits_{x \to 0} \,\left[ {\frac{{\sqrt {a + x} - \sqrt {a - x} }}{x}} \right]$ is

Eccentricity of the conic $16{x^2} + 7{y^2} = 112$ is

The weight of a body, calculated as the average of seven different experiments is 53.735g.The average of the first three experiments is 54.005g. The fourth was greater than the fifth by 0.0040.004 g and the average of sixth and seventh was 0.010g less than the average of the first three. Find the weight of the body in the fourth experiment.

The sum of the infinite series $\frac{1}{9} + \frac{1}{{18}} + \frac{1}{{30}} + \frac{1}{{45}} + \frac{1}{{63}} + ..........\infty$ is equal to :-

The area of the curve ${x^2} + {y^2} = 2ax$ is

All the values of $m$ for which both roots of the equation $x^2-2 m x+m^2-1=0$ are greater than -2 but less than 4 lie in the interval:

If $\cos \,\alpha + \cos \,\beta = \frac{3}{2}$ and $\sin \,\alpha + \sin \,\beta = \frac{1}{2}$ and $\theta $ is the the arithmetic mean of $\alpha $ and $\beta $ , then $\sin \,2\theta + \cos \,2\theta $ is equal to