(English Medium) ICSE Class 10 - CISCE Question Bank Solutions

If a, b, c, d are in continued proportion, prove that (b − c)² + (c − a)² + (d − b)² = (d − a)².

If q is the mean proportional between p and r, prove that
`p^2 - q^2 + r^2 = q^4[(1)/p^2 - (1)/q^2 + (1)/r^2]`.

If `a/b = c/d` Show that a + b : c + d = `sqrt(a^2 + b^2) : sqrt(c^2 + d^2)`.

If a : b = c : d, show that (a - c) b² : (b - d) cd = (a² - b² - ab) : (c² - d² - cd).

If `a/b = c/d = e/f`, prove that `(ab + cd + ef)^2 = (a^2 + c^2 + e^2) (b^2 + d^2 + f^2)`.

If `a/b = c/d = r/f`, prove that `((a^2b^2 + c^2d^2 + e^2f^2)/(ab^3 + cd^3 + ef^3))^(3/2) = sqrt((ace)/(bdf)`

If a, b, c, d are in continued proportion, prove that:
`sqrt(ab) - sqrt(bc) + sqrt(cd) = sqrt((a - b + c) (b - c + d)`

If a, b, c, d are in continued proportion, prove that:
(a² + b² + c²) (b² + c² + d²) = (ab + bc + cd)².

If `x/a = y/b = z/c`, show that `x^3/a^3 - y^3/b^3 = z^3/c^3 = (xyz)/(zbc).`

If `x/a = y/b = z/c`, prove that `x^3/a^2 + y^2/b^2 + z^3/c^2 = ((x + y + z)^3)/((a + b ++ c)^2)`.

If `a = (b + c)/(2), c = (a + b)/(2)` and b is mean proportional between a and c, prove that `(1)/a + (1)/c = (1)/b`.

Using a graph paper, drawn an Ogive for the following distribution which shows a record of the weight in kilograms of 200 students.

Use your ogive to estimate the following:
(i) The percentage of students weighing 55kg or more.
(ii) The weight above which the heaviest 30% of the students fall.
(iii) The number of students who are:
(1) under-weight and
(2) over-weight, if 55·70 kg is considered as standard weight.

The frequency distribution of scores obtained by 230 candidates in a medical entrance test is as ahead:

Draw a cummulative polygon (ogive) to represent the above data.

Use graph paper for this question. The following table shows the weights in gm of a sample of 100 potatoes taken from a large consignment:

(i) Calculate the cumulative frequencies.
(ii) Draw the cumulative frequency curve and form it determine the median weights of the potatoes.

Find the value of x in the following proportions : 10 : 35 = x : 42

Find the value of x in the following proportions : 3 : x = 24 : 2

Find the value of x in the following proportions : 2.5 : 1.5 = x : 3

Find the value of x in the following proportions : x : 50 :: 3 : 2

Find the fourth proportional to 3, 12, 15

Find the fourth proportional to `(1)/(3), (1)/(4), (1)/(5)`