Circum Radius

Q.

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3+4x+2=0$, then ${\alpha }^3+{\beta }^3+{\gamma }^3$ is equal to

1. $2$

2. $6$

3. $-2$

4. $-6$

Q.

The value of $\cos \left(\frac{\pi }{15}\right)·\cos \left(\frac{2\pi }{15}\right)·\cos \left(\frac{4\pi }{15}\right)·\cos \left(\frac{8\pi }{15}\right)$ is

1. $\frac{1}{16}$

2. $-\frac{1}{16}$

3. $1$

4. $0$

Q.

The expression $2\cos \frac{\mathrm{\pi }}{13}\cos \frac{9\mathrm{\pi }}{13}+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13}$ is equal to

1. $-1$

2. $0$

3. $1$

4. None of these

Q. Relation between circumRadius and Sine Rule is given by

1. $R=\frac{a}{SinA}=\frac{b}{SinB}=\frac{c}{SinC}$
2. $R=\frac{a}{2SinA}=\frac{b}{2SinB}=\frac{c}{2SinC}$
3. $R=\frac{ab}{SinASinB}=\frac{bc}{SinBSinC}=\frac{ca}{SinCSinA}$
4. $R=\frac{abc}{SinASinBSinC}$

Q.

If $\cos \theta =\cos \alpha \cos \beta$, then $\tan \frac{(\theta +\alpha )}{2}\tan \frac{(\theta -\alpha )}{2}$ is equal to

1. ${\tan }^2\frac{\alpha }{2}$

2. ${\tan }^2\frac{\beta }{2}$

3. ${\tan }^2\frac{\theta }{2}$

4. $co{t}^2\frac{\beta }{2}$

Q. The ratio of area of a regular polygon of $n$ sides inscribed in a circle to that of the polygon of same number of sides circumscribing the same circle is $3:4$. Then the value of $n$ is

1. $6$
2. $4$
3. $8$
4. $1$

Q. Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE ?

1. If $n=8$, then $(\sqrt{2}–1)r<R$
2. If $n=4$, then $(\sqrt{2}–1)r<R$
3. If $n=12$, then $\sqrt{2}(\sqrt{3}+1)r>R$
4. If $n=5$, then $r<R$

Q. $\mathrm{In}a\Delta \mathrm{ABC},\mathrm{if}a=18,b=24\mathrm{and}c=30,\mathrm{then}R=$ units.

1. 18
2. 25
3. 12
4. 15

Q. If two sides of a triangle are roots of the equation $x^2-7x+8=0$ and the angle between these sides is ${60}^{\circ },$ then the product of in-radius and circum-radius of triangle is

1. $\frac{8}{7}$
2. $\frac{5}{3}$
3. $\frac{5\sqrt{2}}{3}$
4. $8$

Q.

Find the value of $x$ if: $8^{255}={32}^x$

Q. 9. Let f(x)=log(cos2ix) {base of logarithm is (cos3x)}. If x is not equal to zero and f(0)=k (i=-1) is continuous at x=0 then find the value of k.

Q. O is the circumcenter of $\Delta \mathrm{ABC}\mathrm{and}{R}_1,R_2\mathrm{and}{R}_3$are the radii of the circumcircles of the triangles OBC, OCA and OAB. Then $\frac{a}{R_1}+\frac{b}{R_2}+\frac{c}{R_3}=$ .

1. $\frac{4\mathrm{abc}}{R^3}$
2. $\frac{8\mathrm{abc}}{R^3}$
3. $\frac{\mathrm{abc}}{R^3}$

Q. For a $△ABC;\angle B={60}^{\circ },b=3,c=4.$ Then possible number of triangle(s) is
(where $a,b,c$ are sides(in $\text{units}$) of $△ABC$ opposite to $\angle A,\angle B$ and $\angle C$ respectively)

1. Zero
2. One
3. Two
4. Infinite

Q. In $△ABC,R,r,r_1,r_2,r_3$ denote the circumradius, inradius, the exradii opposite to the vertices $A,B,C$ respectively. Given that $r_1:r_2:r_3=1:2:3$.
The value of $R:r$ is

1. $5:2$
2. $5:4$
3. $5:3$
4. $3:2$

Q. Relation between circumRadius and Sine Rule is given by

1. $R=\frac{a}{SinA}=\frac{b}{SinB}=\frac{c}{SinC}$
2. $R=\frac{a}{2SinA}=\frac{b}{2SinB}=\frac{c}{2SinC}$
3. $R=\frac{ab}{SinASinB}=\frac{bc}{SinBSinC}=\frac{ca}{SinCSinA}$
4. $R=\frac{abc}{SinASinBSinC}$

Q. In a $△ABC,a=5,c=2\sqrt{6},\angle C={60}^0.$ If $b_1,b_2$ are two possible values of third side, then the value of $\frac{b_1+b_2}{b_1{b}_2}=$
(In $△ABC,$ usual notations are given.)

1. $3$
2. $5$
3. $7$
4. $9$

Q. If A + B + C = $\pi ,$ the maximum value of cos A + cos B + cos C can have is --

1. $\frac{3}{2}$
2. 2
3. 3
4. 1

Q. The area of a regular polygon of $2n$ sides inscribed in a circle is given by?

1. The arithmetic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
   
2. The harmonic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
   
3. The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
   
4. None of the above

Q. The ratio of area of a regular polygon of $n$ sides inscribed in a circle to that of the polygon of same number of sides circumscribing the same circle is $3:4$. Then the value of $n$ is

1. $6$
2. $4$
3. $8$
4. $1$

Q.

Fill in the blanks in following table:

	P(A)	P(B)	P(A ∩ B)	P(A ∪ B)
(i)				…
(ii)	0.35	…	0.25	0.6
(iii)	0.5	0.35	…	0.7

Q. If two sides of a triangle are roots of the equation $x^2-7x+8=0$ and the angle between these sides is ${60}^{\circ }$ then the product of inradius and circumradius of the triangle is

1. $\frac{8}{7}$
2. $\frac{5}{3}$
3. $\frac{5\sqrt{2}}{3}$
4. $8$

Q.

1. 4/3 sq units

2. 5/3 sq units

3. 2/3 sq units

4. 1/3 sq units

Q.

$A.M$ of the three numbers which are in $G.P$. is $\frac{14}{3}$. If adding $1$ in the first and second number and subtracting $1$ from the third number, the resulting numbers are in $A.P$. then the sum of the squares of the original three numbers is

1. $91$

2. $80$

3. $84$

4. $88$

Q. In a triangle $\Delta ABC$, the sum of the lengths of two sides is represented by $p$ and the product of the lengths of the same two sides is $q$. Let $c$ be the third side. If $p^2=c^2+2q$, then the area of the circumcircle of $\Delta ABC$ is

1. $\frac{\pi c^2}{2}$
2. $\pi c^2$
3. $2\pi c^2$
4. $\frac{\pi c^2}{4}$

Q. Find the value of 19.5 X 0.3 X 12 / 3.6 X5.7 X 0.07 using logarithms.

Q. Let a right angled triangle with maximum possible area is drawn keeping the sum of lengths of the hypotenuse and a side constant. If a square is drawn taking the hypotenuse as the side and the ratio of the area of triangle to square is $r$, then which of the following is (are) CORRECT?

1. $r=\frac{\sqrt{3}}{8}$
2. The other angles of the triangle are $30°,60°.$
3. The other angles of the triangle are $45°,45°$
4. $r=\frac{3}{8}$

Q. If A + B + C = $\pi ,$ the maximum value of cos A + cos B + cos C can have is --

1. 1
2. 2
3. 3
4. $\frac{3}{2}$

Q. If two sides of a triangle are roots of the equation $x^2-7x+8=0$ and the angle between these sides is ${60}^{\circ }$ then the product of inradius and circumradius of the triangle is

1. $\frac{8}{7}$
2. $\frac{5}{3}$
3. $\frac{5\sqrt{2}}{3}$
4. $8$

Q.

Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:

(i) {2, 3, 4} … {1, 2, 3, 4, 5}

(ii) {a, b, c} … {b, c, d}

(iii) {x: x is a student of Class XI of your school} … {x: x student of your school}

(iv) {x: x is a circle in the plane} … {x: x is a circle in the same plane with radius 1 unit}

(v) {x: x is a triangle in a plane}…{x: x is a rectangle in the plane}

(vi) {x: x is an equilateral triangle in a plane}… {x: x is a triangle in the same plane}

(vii) {x: x is an even natural number} … {x: x is an integer}

Q. Which of the following relations hold true ?

1. $r=\frac{\Delta }{s},r=2Rsin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
2. $r=\frac{\Delta }{s},r=Rsin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
3. $r=\frac{\Delta }{s},r=2Rsin\frac{A}{3}sin\frac{B}{3}sin\frac{C}{3}$
4. $r=\frac{\Delta }{s},r=2Rsin\frac{A}{3}sin\frac{B}{2}sin\frac{C}{2}$