Pythagoras Theorem

Q. 32. SOLUTION OF TRIANGLES : In a triangle ABC, the angles A, B, C are in A.P. Show that :2cos [(A-C)/2] = (a+c)/sqrt(a²-ac+c²)

Q.

The perpendicular from the focus on either asymptote meets it in the same points as the corresponding directrix & common points of intersection lie on the auxiliary circle.

1. False

2. True

Q.

The combination of algebra and geometry is called:

1. Calculus

2. Co-ordinate geometry

3. Algebra

4. Geometry

Q. If the normal at any point on the standard ellipse x^2/a^2 +y^2/b^2 =1, meets its auxillary circle at Q and R such that angle QOR equals to 90 degree , where O is the centre of the ellipse, then prove that (a^4)+(2 b^4)>=(5a^2 b^2)+(a^3 b).

Q. x={(t+5)}^{-1} find d^2x/dt^2

Q.

Consider the two statements

Statement 1: To each point P in the space there corresponds an ordered triplet (x, y, z) of real numbers.

Statement 2: Given any triplet (x, y, z), we would first fix the point P in the space to which it corresponds

The two statements are

1. Only Statement 1 but not Statement 2 is true

2. Only Statement 2 but not Statement 1 is true

3. Both statements are not true

4. Both true

Q.

WHAT IS TRIGNOMETRIC FUNCTION

Q. The combined equation of three sides of a triangle is (x^2 - y^2)(2x +3y-6)=0. If (secθ, 1) is an Interior point of the triangle then find the value of θ.

Q.

In a right triangle, let the base be x, height is y and hypotenuse be z. Which of the following is correct representation of Pythagoras theorem?

1. 

2. 

3. 

4. 

Q. The angle of elevation of the top of a tower from a point $A$ in east of it is ${45}^{\circ }$. The angle of elevation of the top of the same tower from point $B$ which is in south of $A$ is ${30}^{\circ }$. If the distance between $A$ and $B$ is $30\sqrt{2}\text{m}$, then height (in metre) of the tower is

Q.

The line x + y = 1 meets x - axis at A and y - axis at B. P is the mid - point of AB $P_1$ is the foot of the perpendicular from P to OA; $M_1$ is that from $P_1$ to OP; $P_2$ is that from $M_1$ to OA; $M_2$ is that from $P_2$ to OP; $P_3$ is that from $M_2$ to OA and so on. If $P_n$ denotes the nth foot of the perpendicular on OA from $M_{n-1}$, then $O{P}_n$ =

1. $\frac{1}{2^{\frac{n}{2}}}$

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

3. $\frac{1}{2^n}$

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

Q.

Given below are pairs of statements. In each case, combine them using 'if and only if'.

(i) p: In $\Delta ABC,\angle B=\angle C.$

q: In $\Delta ABC,AC=AB.$

(ii) p: A and B are two sets such that $A\subseteq B$ and $B\subseteq A.$

q: $A=B.$

(iii) p: $\Delta ABC$ is equilateral.

q: $\Delta ABC$ is equiangular.

(iv) p: $\{a\in Rsuchthat\left|a\right|<2\}.$

q: $\{a\in Rsuchthat(a>-2anda<2\left)\right\}.$

Q. The points of intersection of the two ellipse $x^2+2y^2-6x-12y+23=0,4x^2+2y^2-20x-12y+35=0$

1. Lie on a circle centered at and of radius
2. Lie on a circle centered at and of radius
3. Are not concyclic
4. Lie on a circle centered at (8, 9) and of radius

Q. 3. cotx- , x lies in third quadrant.

Q. the value of∫_{50}^{40}(1/θ-θ^°)\operatorname dθ where θ^°=30 (a cons†an t ) is

Q. The sides of a triangle are a = 4, b = 6 and c = 8. Show that $8\cos A+16\cos B+4\cos C=17$.