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-3 is a root of the quadratic equation 2x² +px - 15 = 0. For what value of q , the equation p(x² + x ) + q= 0 has equal roots?

Two circles touch each other externally at a point C and P is a point on the common tangent at C. If PA and PB are tangents to the two circles, prove that PA = PB.

D is a point on the side BC of $△$ ABC such that $\angle ADC=\angle BAC$. Prove that : $C{A}^2=CB\times CD$.

A restaurant POQR is being constructed in the middle of a circular lake. The circle, which has its center at O is touching the mid-points of the boundary ABCD as shown in the figure. AB = CD and AD = BC. If BC = 8 cm, then the area of AQOP is

In three line segments OA, OB and OC, points L,M,N respectively are so chosen that LM is parellel to AB and MN is parellel to BC. But neither of L, M, N nor A, B, C are collinear. Show that LN is parellel to AC.

In solving a problem, one student made a mistake in the coefficient of the first degree term and obtained -9 and -1 as the roots. Another student made a mistake in the constant term of the equation and obtained 8 and 2 for the roots. Assuming the coefficient of x² as 1, the correct equation was-

Given that the LCM of 306 and 657 is 22338, what is the LCM of 102, 306 and 657

State the SAS-similarity criterion.

If a square with side ‘a’ is inserted within a circle such that the corners coincide with the circumference of the circle with diameter ‘d’. Find the relation between ‘a’ and ‘d’.

For a given value of n, the total number of possible l values is

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio will be equal to

Match the equations A, B, C, D with the lines L₁, L₂, L₃, L₄, L₅ whose graphs are roughly drawn in the adjoining figure.

A: y = 2

B: y – 2 + 2 = 0

C: 3 + 2y = 6

D: y = -2

E : = 2

The value of $\sqrt{2+\sqrt{2+\sqrt{2+.........}}}$ is

Using quadratic formula, find the value of x.
$p^2{x}^2+(p^2-q^2)x-q^2=0,p\ne 0$

Match the following:

$$\begin{array}{cc}1.sin({90}^{\circ }-A)& a)cosecA\\ 2.tan({90}^{\circ }-A)& b)cotA\\ 3.sec({90}^{\circ }-A)& c)cosA\end{array}$$

Calculate the mean of the following distribution:
$\begin{array}{cc}ClassInterval& Frequency\\ 0-10& 8\\ 10-20& 5\\ 20-30& 12\\ 30-40& 35\\ 40-50& 24\\ 50-60& 16\end{array}$

The monthly income of a group of 320 employees in a company is given below:

Calculate:

(i) Median wage

(ii) Number of employees whose income is below Rs. 8500

(iii) Number of senior employees in the company

(iv) Upper quartile

[4 MARKS]

In parallelogram ABCD, A(3,2), B(8,4) and D(5,7). Then the coordinates of C is

The number of times the graph of y = p(x) cuts the X axis is ‘a’. The degree of the polynomial p(x) is ‘b’ (> 0). Then $\frac{b}{a}$ ___ 1

If $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\dots =\frac{\pi }{4},$ then the value of $\frac{1}{1\times 3}+\frac{1}{5\times 7}+\frac{1}{9\times 11}+\dots$ is

How many terms of the G.P. $3,\frac{3}{2},\frac{3}{4}.....$ are needed to give the sum $\frac{3069}{512}$ ?

Arjun and Karna had an archery competition where they had to shoot a target board with 7 concurrent circles dividing the board into 8 concurrent circular strips. Score for each outcome decreases by unity with the centermost circle having a score of 8. Dronacharya had difficulty in deciding who the winner was after the scores were recorded. Calculate the mean deviation of the score of the innermost circle so as to decide who the winner is assuming the winner is less deviated from the bulls eye.

$$\begin{array}{ccccccccc}Points& 8& 7& 6& 5& 4& 3& 2& 1\\ Arjun\left(f_i\right)& 5& 7& 14& 23& 11& 10& 4& 3\\ karana\left(f_i\right)& 5& 8& 11& 24& 14& 8& 5& 3\end{array}$$

The sum of first n terms of an AP is $(4n^2+2n)$ The nth term of this AP is
(a) (6n-2) (b) (7n-3) (c) (8 n - 2) (d) (8n + 2)

In the adjoing figure, a circle touches all the four sides fo a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length side AD.

Find the angle subtended at the centre of a circle of radius 'a' by an arc of length $\left(\frac{a\pi }{4}\right)cm$

The sum of first q terms of an A.P. is 162. The ratio of its 6th term to its 13th term is 1.: 2. Find the first and 15th term of the A.P

A metallic cone of radius 12 cm and height 24 cm is melted and made into spheres of radius 2 cm each. How many spheres are formed?

A shuttlecock used for playing badminton is the combination of

(a) cylinder and a hemisphere

(b) frustum of a cone and a hemisphere

(c) a cone and a hemisphere

(d) a cylinder and a sphere

If {x} denotes the fractional part of x, then $\left\{\frac{3^{10001}}{82}\right\}$ is:

If S is the set of distinct values of b for which the folowing system of linear equations
$x+y+z=1$,
$x+ay+z=1$
and $ax+by+z=0$ has no solution, then S is

Match the expressions on the left side with their HCF given on the right side.

The number of real roots in the equation $x^2-3x+5=0$ is:

Cosec^4 A(1-cos^4)-2cot^2 A=1

Find the circumradius of the triangle whose vertices are (8, 6), (8, – 2) and (2, – 2).

In fig. XP and XQ are tangents from X to the circle with center O. R is a point on the circle. Prove that, XA + AR = XB + BR. [2 MARKS]

If A = {2, 4, 6, 8} , B = {4, 8, 12, 16} Find A $\Delta$ B

P (3, 4), Q (7, –2) & R (–2, –1) are the vertices of Δ PQR. Write the equation of altitude of the Δ through R.

The HCF of $21x(x^2-1)and3x^3(x+1)$ is:

In ________ the circumcenter always lies outside the triangle.

Chord AM is drawn in a circle of diameter 30cm. Calculate the distance of chord from the centre, if AM = 18cm.

A manufacturer sells an article to a wholesaler for ₹ 5000 . The wholesaler sells it to a trader at a profit of ₹ 1000 and trader , in turn sells it to a customer at a profit of ₹ 1500. If the rate of VAT is 8 % find :

i) the amount of VAT received by the State Goverment on the sale of this article

ii) the amount that customer pays for the article

If $3sin\theta +4cos\theta =5$, then the value of $sin\theta$ is

Number of solutions for the system of linear equations, will be,

x + y + z =6

x + 2y + 3z = 14

x + 4y + 2z =30