Board Paper of Class 10 2008 Maths Abroad(SET 1) - Solutions

Find all the zeroes of the polynomial x⁴ + x³ − 34x² − 4x + 120, if two of its zeroes are 2 and −2.

A pair of dice is thrown once. Find the probability of getting the same number on each die.

If sec 4A = cosec (A − 20°), where 4A is an acute angle, then find the value of A.

OR

In a ΔABC, right-angled at C, if then find the value of sin A cos B + cos A sin B.

Find the value of k, if the points (k, 3), (6, −2), and (−3, 4) are collinear.

E is a point on the side AD produced of a ||^gm ABCD and BE intersects CD at F. Show that ΔABE ∼ ΔCFB.

Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or (3m + 1) for some integer m.

Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis:

x + 3y = 6

2x − 3y = 12

For what value of n are the n^th terms of two A.P.’s 63, 65, 67 … and 3, 10, 17 … equal?

OR

If m times the m^th term of an A.P. is equal to n times its n^th term, then find the (m + n)^th term of the A.P.

In an A.P., the first term is 8, n^th term is 33, and sum to first n terms is 123. Find n and d, the common difference.

Prove that:

(1+ cot A + tan A) (sin A − cos A) = sin A tan A − cot A cos A

OR

Without using trigonometric tables, evaluate the following:

If P divides the line joining the points A(−2, −2) and B(2, −4) such that , then find the coordinates of P.

The mid-points of the sides of a triangle are (3, 4), (4, 6), and (5, 7). Find the coordinates of the vertices of the triangle.

Draw a right triangle in which the sides containing the right angle are 5 cm and 4 cm. Construct a similar triangle whose sides are times the sides of the above triangle.

Prove that a parallelogram circumscribing a circle is a rhombus.

OR

In figure, AD ⊥ BC. Prove that AB² + CD² = BD² + AC².

In the figure, ABC is a quadrant of a circle of radius 14 cm and a semi-circle is drawn with BC as diameter. Find the area of the shaded region.

A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake, the peacock pounces on it. If their speeds are equal, then at what distance from the hole is the snake caught?

OR

The difference of two numbers is 4. If the difference of their reciprocals is, then find the two numbers.

The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a constant height of m, then find the speed, in km/hour, of the plane.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

Using the above, prove the following:

In figure, AB || DE and BC || EF. Prove that AC || DF.

OR

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

Using the above, prove the following:

ABC is an isosceles triangle in which AB = AC, circumscribed about a circle, as shown in figure. Prove that the base is bisected by the point of contact.

If the radii of the circular ends of a conical bucket, which is 16 cm high, are 20 cm and 8 cm, then find the capacity and total surface area of the bucket.

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

Class	Frequency
0 − 20	6
20 − 40	8
40 − 60	10
60 − 80	12
80 − 100	6
100 − 120	5
120 − 140	3