MATHS Important Questions -CLASS-X-2017-18,

KV NO. 4, DLW-CAMPUS, VARANASI

1.       Check whether 6ⁿ  can end with the digit 0 for any natural number n.

2.       Prove that √3 is irrational.

3.       Use Euclid’s algorithm to find the HCF of 4052 and 12576.

4.       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.

5.       Find the zeroes of the quadratic polynomial  x² + 7x + 10, and verify the relationship between the zeroes and the coefficients.

6.       Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.

7.       Obtain all other zeroes of 3x⁴  + 6x³  – 2x² – 10x – 5, if two of its zeroes are  and –  .

8.       Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

9.       For what values of k will the following pair of linear equations have infinitely many solutions?

kx + 3y – (k – 3) = 0,                            12x + ky – k = 0

10.    A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.

11.    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.

12.    A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

13.    Find the values of k for the quadratic equations, so that they have two equal roots. kx (x – 2) + 6 = 0

14.    How many three-digit numbers are divisible by 7?

15.    The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.

16.    If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

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

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

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

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

21.    Prove that the tangent to a circle is perpendicular to the radius through the point of contact.

22.    ABC is an isosceles triangle with AC = BC. If AB² = 2 AC², prove that ABC is a right triangle.

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

24.    Find the point on the x-axis which is equidistant from (2, –5) and (–2, 9).

25.    Find the ratio in which the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4). Also find the point of intersection.

26.    If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

27.    Find the area of the quadrilateral whose vertices, taken in order, are (– 4, – 2), (– 3, – 5), (3, – 2) and (2, 3).

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

29.    If tan (A – B) = and tan (A + B) = √3 , then find the value of A and B.

30.    If sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A.

31.    Prove that (cos A – sin A + 1) / (cos A + sin A – 1) = cosec A + cot A.

32.    A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30° (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

33.    A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

34.    Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠ ∠PTQ = 2 ∠ ∠OPQ.

35.    PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T . Find the length TP.

36.    A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC

37.    XY and X´Y´ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X´Y´ at B. Prove that ∠∠AOB = 90°.

38.    Prove that the parallelogram circumscribing a circle is a rhombus.

39.    A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC.

40.    Draw a triangle ABC with side BC = 7 cm, ∠ B = 45°, ∠ A = 105°. Then, construct a triangle whose sides are 4/3 times the corresponding sides of ∆ABC.

41.    Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.

42.    Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.

43.    In below left figure, AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O. If ∠ AOB = 60°, find the area of the shaded region.

                                       

44.    In above right Fig., ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

45.    Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

46.    A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can completely fill the container, at the rate of Rs. 20 per litre. Also find the cost of metal sheet used to make the container, if it costs Rs. 8 per 100 cm². (Take = 3.14)

47.    If the median of the distribution given below is 28.5, find the values of x and y.

C. I.	0-10	10-20	20-30	30-40	40-50	50-60	Total
F	5	X	20	15	Y	5	100

48.    The frequency distribution table of agriculture holdings in a village is given below:

Area of land(in ha)	1-3	3-5	5-7	7-9	9-11	11-13
No. of families	20	45	80	55	40	12

          Find the modal agriculture holdings of the village.

49.    Draw less than ogive for the following frequency distribution:

Marks	0-10	10-20	20-30	30-40	40-50	50-60
Number of students	5	8	6	10	6	6

Also find the median from the graph and verify that by using the formula.

50.    A card is drawn at random from a well-shuffled pack of 52 playing cards. Find the probability of getting (i) neither a red card nor a queen (ii) a face card or a spade card.

51.    Find the probability that in a leap year there will be 53 Tuesdays.

52.    A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5.