Mathematics All India Set 3 2017-2018 English Medium Class 10 Question Paper Solution

What is the value of (cos² 67° – sin² 23°)?

In an AP, if the common difference (d) = –4, and the seventh term (a₇) is 4, then find the first term.

Given `triangle ABC ~ triangle PQR`, if `(AB)/(PQ) = 1/3`, then find `(ar  triangle ABC)/(ar triangle PQR)`

What is the HCF of the smallest prime number and the smallest composite number?

Find the distance of a point P(x, y) from the origin.

If x = 3 is one root of the quadratic equation x² – 2kx – 6 = 0, then find the value of k.

Two different dice are tossed together. Find the probability :

1) of getting a doublet

2) of getting a sum 10, of the numbers on the two dice.

Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find m.

An integer is chosen at random between 1 and 100. Find the probability that it is

1) divisible by 8.

2) not divisible by 8.

In Fig. 1, ABCD is a rectangle. Find the value of x and y.

Find the sum of first 8 multiples of 3

Given that `sqrt2` is irrational prove that `(5 + 3sqrt2)`  is an irrational number

If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides

If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD

Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)`  and `(2 - sqrt3)`

Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals

If the areas of two similar triangles are equal, prove that they are congruent.

A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed.

The table below shows the salaries of 280 persons :

Calculate the median salary of the data.

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in given figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

 [Use `pi = 22/7`]

A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much is canvas cloth required to just cover the heap?

Find the area of the shaded region in Fig. 3, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use π = 3.14]

If 4 tan θ = 3, evaluate `((4sin theta - cos theta + 1)/(4sin theta + cos theta - 1))`

If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A

As observed from the top of a 100 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. [Use `sqrt3` = 1.732]

The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm, find:

1) The area of the metal sheet used to make the bucket.

2) Why we should avoid the bucket made by ordinary plastic? [Use π = 3.14]

Prove that `(sin "A" - 2sin^3 "A")/(2cos^3 "A" - cos "A") = tan "A"`

The mean of the following distribution is 18. Find the frequency f of class 19 – 21.

The following distribution gives the daily income of 50 workers of a factory.

Convert the distribution above to a less than type cumulative frequency distribution, and draw its ogive.

A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original average speed?

The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7: 15. Find the numbers.

Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are `3/4` of the corresponding sides of the ∆ABC.

In an equilateral triangle ABC, D is a point on side BC such that BD = `1/3BC` . Prove that 9 AD² = 7 AB²

Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.