Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where

In the given figure, the graph of the polynomial p(x) is shown. The number of zeros of p(x) is

In ΔABC, it is given that DE || BC. If AD = 3 cm, DB = 2 cm and DE = 6 cm, then BC = ?

If sin 3θ = cos (θ - 2°), where 3θ and (θ - 2°) are both acute angles, then θ = ?

The decimal expansion of 49/40 will terminate after how many places of decimal?

The pair of linear equations 6x - 3y + 10 = 0, 2x - y + 9 = 0 has

For a given data with 60 observations the 'less than ogive' and the 'more than ogive' intersect at (18.5, 30). The median of the data is

Is (7 x 5 x 3 X 2 + 3) a composite number? Justify your answer.

When a polynomial p(x) is divided by (2x + 1), is it possible to have (x - 1) as a remainder? Justify your answer.

If 3 cos²θ + 7sin²θ = 4, show that cotθ = √3

If tan θ = 8/15 evaluate

In the given figure, DE|| AC and DF || AE.

Prove that:

In the given figure, AD ⊥ BC and BD = 1/3 CD. Prove that: 2CA² = 2AB² + BC².

Find the mode of the following distribution of marks obtained by 80 students:

Show that any positive odd integer is of the form (4q + 1) or (4q + 3), where q is a positive integer.

Prove that (5 - √3) is irrational.

Prove that is irrational.

A man can row a boat at the rate of 4 km/hour in still water. He takes thrice as much time in going 30 km upstream as in going 30 km downstream. Find the speed of the stream.

In a competitive examination, 5 marks are awarded for each correct answer, while 2 marks are deducted for each wrong answer. Jayant answered 120 questions and got 348 marks. How many questions did he answer correctly?

If α and β are the zeros of the polynomial 2x² + x - 6, then form a quadratic equation whose zeros are 2α and 2β.

Prove that: (cosecθ - sinθ)(secθ - cosθ) =

If cosθ + sinθ = √2 cosθ, prove that cos θ - sinθ = √2 sinθ.

ΔABC and ΔDBC are on the same base BC and on opposite sides of BC. If O is the point of intersection of BC and AD, prove that:

In Δ ABC, the AD is a median and E is the midpoint of the AD. If BE is produced to meet 1 AC in F, show that AF = 1/3 AC.

Find the mean of the following frequency distribution using step deviation method:

The mean of the following frequency distribution is 24. Find the value of p.

Find the median of the following data:

Let p(x) = 2x⁴ - 3x³ - 5x² + 9x - 3 and two of its zeros are √3 and -√3. Find the other two zeros.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle. Prove it.

Prove that

Evaluate:

If secθ + tanθ = x, prove that sinθ =

Solve the following system of linear equations graphically:

2x - y = 1, x - y = -1.

Shade the region bounded by these lines and the y-axis.

The following table gives the yield per hectare of wheat of 100 farms of a village:

Change the above distribution to 'more than type' distribution and draw its ogive.

Solve for x and y:

ax + by-a + b = 0, bx-ay-a-b = O.

Prove that: = (cosecθ - cotθ)².

Δ ABC is right angled at B and D is the midpoint of BC.

Prove that: AC² = (4AD² - 3AB²).

Find the mean, mode and median of the following data: