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Show that any positive even integer can be written in the form 6q,6q+2 and 6q+4 where q is an integer

The number of solutions of the pair of linear equations x + 2y - 8 = 0 and 2x + 4y = 16 are:

A circular garden of radius 10m has a straight line fence between the two points on the boundary of the garden. The fence lies inside the garden arena. This fence separates the walking area which is a small region from the plants. The fence is at a distance of 6 m from the centre of the garden. Find the angle subtended by the fence with the center of the garden. It is given that cos(${53}^{\circ }$) = $\frac{3}{5}$.

Define : 'Zero of the Polynomial' With examples.

Then, Zero of the Polynomial 1 + 2x + 3 + 4x + 5 + 6x + 7 is _______

1) -3/4

2) 4/3

3) -4/3

4) 3/4

If the vectors $3\hat{i}+\hat{j}-2\hat{k},\hat{i}-3\hat{j}+4\hat{k}$ are diagonals of quadrilateral, then vector area is

Calculation (i) Arithmetic Mean (ii) Standard Deviation and (iii) Co-efficient of variation for the following frequency distribution.

Question 16(d)

The present age of the father is 42 years and that of his son is 14 years. Find the ratio of:

Age of father to the age of son when father was 30 years old.

I have thin matchsticks each of thickness 1 mm. The length of each matchstick is ‘L ’ mm. There are a few matchsticks kept side by side on a table such that the heads of all matchsticks are in one line and the tail ends of all matchsticks are together in another different line and the lengths of the matchsticks are parallel. The number of matchsticks kept is ‘B’. What is the area of the plane formed by the matchsticks (in ${mm}^2$)?

Question 3

Recall, $\pi$ is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, $\pi =\frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?

If $1+si{n}^2\theta =3sin\theta cos\theta$ then prove that $tan\theta =1or\frac{1}{2}$.

Find the mean of the following data, using assumed-mean method:
$\begin{array}{ccccccc}\text{Class}& 0-20& 20-40& 40-60& 60-80& 80-100& 100-120\\ \text{Frequency}& 20& 35& 52& 44& 38& 31\end{array}$

Refer to the figure shown:

AB = 6 and BC = 4. BC is extended to E such that C is the midpoint of BE.

Join AE. It cuts CD in P. Now consider the two points:

i) P is the midpoint of AE.

ii) P is the midpoint of CD.

(a) x² + 3x − 5 = 0



(b) −x² + 3x + 3 = 0

(c)



(d) 3x² − 3x − 3 = 0

A bucket contains 10 brown balls, 8 green balls, and 12 red balls and you pick one randomly without looking. What is the probability that the ball will be brown?

If the $t^{\text{th}}$ term of an AP is $s$ and $s^{\text{th}}$ term of the same AP is $t,$ then $a_n$ is ___.

Pick a common word among the pictures given below.

The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.[4 MARKS]

In the given figure, O is the centre of the circle. Chord AB is parallel to chord CD and CB is a diameter. Prove that arc AC = arc BD. [4 MARKS]

Area of a trapezium is 160 sq cm and the height is 10 cm . Find the sum of its two parallel sides.

Find the sum of the first 15 multiples of 8.

Assume $X,Y,Z,W$ and $P$ are matrices of order $2\times n,3\times k,2\times p,n\times 3$, and $p\times k$ respectively.

If $n=p$, then the order of the matrix $7X-5Z$ is

The quadratic equation whose roots are 3 ± $\sqrt{5}$ is

In a flag race, a pole is placed at the starting point, which is 10m from the first flag and the other flags are placed 6m apart in a straight line. There are 10 flags in the line. Each competitor starts from the pole, picks up the nearest flag, comes back to the pole and continues the same way until all the flags are on the pole. Find the total distance covered.

In the figure given below, if BD = 2.4 cm, AC = 3.6 cm, PD = 4 cm, PB = 3.2 cm, and AC is parallel to BD, then find the lengths of PA & PC.