Thursday, December 3, 2015

Important Questions for SA II Maths CBSE Board........

Quadratic Equations

  1. The sum of the squares of two consecutive natural numbers is 85 find the numbers.
  2. If -4 is a root of the equation x2 + px -4 =0 and the equation x2 + px + k =0 has equal roots, find k
  3. Divide 6 into two parts such that the sum of their reciprocals is 3/4.
  4. A two digit numbers is three times the product if its digits. It is also four times the sum of its digits. Find the numbers.
  5. The length of the hypotenuse of a right triangle exceeds the length of the base of 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle.
  6. Solve :-

    1. 4Ö3 x2 + 5x -2Ö3
    2. (1/x + 4) – (1/ x – 7) = 11/30
    3. 4/x – 3= 5/2x + 3
    4. (x – 1/x – 2) + (x -3/x – 4)
    5. 2(2x – 1/x + 3) – 3(x + 3/ 2x – 1)
    6. ab x2 +(b2 –ac)x – bc =0
    7. 4x2 – 2(a2 + b2)x + a2b2 = 0
    8. 9x2 – 9(a +b)x + (2a2 + 5ab + 2b2) = 0
    9. a2b2x2 – (4b4 – 3a4)x – 12a2b2 =0
  7. If the roots of the equation (a –b) x2 + (b – c) x + (c – a) = 0 are equal. Prove that b + c = 2a.
  8. Find the value of k for which the roots of the equation (k + 4) x2 + (k+1) x +1 =0 are real and equal.
  9. The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2 16/21. Find the fraction.
  10. 100% Guarantee Results
    Rs 7500 were divided equally among a certain number of persons. Had there been five more persons each would have got Rs 50 less Find the original number of Persons.
  11. Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter taken 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
  12. A train track 288 km at a uniform speed. If the speed had been 4 km more. It would have taken 1 hour less for the same journey. Find the speed of the train.


Arithmetic Progression

  1. Find the 10th term from the end of the AP 4, 9, 14…... 254.
  2. Which term of the AP 3, 15, 27.39…..? Will be 120 more than its 21st term.
  3. Find the middle term of the AP 10, 7, 4….-62.
  4. In a given AP if pth term is q and qth term is p,then show that nth term is
 p + q – n =0
  1. In an AP the sum of first n terms is …………… Find its 25th term.
  2. The sum of three numbers in AP is 3 and their product is – 35. Find the number.
  3. Which term of the AP 25, 20, 15……is the first negative term?
  4. Find the sum of all Natural number less than 1000. Which are divisible by 6.
  5. The sum of first six terms of an AP is 42. The ratio of its 10th term to 30th term is 1:3 Calculate the first and 13th term of the AP.
  6. Find the sum of all 3-digits numbers which are divisible by 11.

Circle and Construction

1.      Show that the tangents at any point of circle are perpendicular to the radius through the points of contact.
2.      Show that the lengths of tangents drawn from external point to a circle are equal.
3.      Two tangents PA and PB are drawn to a circle with centre o form an external point P. Prove that APB = 2 AB
4.      In a given figure o is the center of two concentric circles AB is a chord of the larger circle touching the smaller circle at c. Prove that AC = BC.
5.      Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtends by the line segments joining the points of contact to the centre
6.      A quadrilateral ABCD is drawn to circumscribe a circle AB + CD = AD + BC
7.      Prove that the Parallelogram circumscribing a circle is a rhombus.
8.      Form a point p two tangents AP and BP are drawn to a circle c (o.r) if OP = 2r. Show that ∆APB is an equilateral.
9.      Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
10.  Construct a triangle of sides 4 cm and 5 cm and 6 cm then a triangle similar to it whose sides are 2/3 of the corresponding sides of the first triangle.
11.  Draw a line segment of length 7:6 cm and divide it in the ratio 5:8. Measure the two parts.
12.  Draw a circle of radius 6 cm. from a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

Height and Distance

  1. The angle of elevation of the top of a tower as seen by a man on the ground is 30. When the man moves 50 m towards the tower, the angle of elevation changes to 45. Find the height of the tower.

  1. The angle of elevation of the top and bottom of a flagstaff, on the top of a building, as seen by an observers on the ground below are 60 and 45 resp. Find the height of the flagstaff if the distance of the observer from the building is 30 m. find also the height of the building.

  1. A vertical wall and a tower are on level ground. As seen from the top of the tower, the angles of depression of the top and bottom of the wall are 45 and 60 reps. Find the height of the wall if the height of the tower is 90m.

  1. 100% Guarantee Results
    As seen from a point on the ground the angle of elevation of the top of a vertical tower is such that its tangent is 5/8. From a point on the ground 180m closer to the tower the angle of elevation is such that its tangent is 5/3. Find the height of the tower.
  2. The angle of elevation of the top of a tower from two points at distances a and b meters from its base are complementary. If the tow points and the base of the tower are on  straight line , prove that the height of the tower is (ab)1/2 

  1. A vertical tower stands on level ground. A flagstaff of height h stands on the tower. As seen from a point on the ground, the angles of elevation of the bottom and the top of the flagstaff are a & b resp. Prove that the height of the tower is
h tan α/ tan β – tan α

  1. A spherical balloon of radius r subtends an angle 2a at the eye of an observer. While the angle of elevation of its centre as seen by the observer is b. Prove that the height of the centre of the balloon above the observer is rsin b cosec a.

  1. From a window (h meter high from the ground) of a house the angles of elevation and depression of the top and bottom of a tower and the other side of the street are θ and φ resp. Prove that the height of the tower is h (1 + tanθ.cotφ).

  1. From the top of a building h meters high, the angle of elevation of the top of the tower is found to be. From the bottom of the same building the angle of elevation of the top of the tower is found to be. Show that the height of the tower is
h tan φ/ tan φ – tan θ

  1. A Person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60. When he moves 30 meters away from the bank, he finds the angle of elevation to be 30. Find the height of the tree and the width of the river.

  1. The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is α. On advancing ‘p’ meter towards the foot of the tower the angle of elevation becomes β. Show that the height ‘h’ of the tower is given
h = (p tan α tan β/ tan β – tan α) m.

  1. Two pillars of equal heights stand on either side of a road which is 150 m wide. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are 60 and 30. Find the height of each pillar and the position of the point on the road.

  1. The angle of elevation of a jet fighter from a point A on the ground is 60. After a flight of 15 seconds the angle of elevation changes to 30. if the jet is flying at a speed of 720 km/h find the constant height at which the jet is flying.

  1. The angle of elevation of a cloud from a point 60 meters above a lake is 30 and the angle of depression of the reflection of the cloud in the lake is 60. Find the height of the cloud.
100% Guarantee Results
 

  1. A man on the deck of a ship, 16 m above water level observe that the angles of elevation and depression resp. of the top and bottom of a cliff are 60 and 30. Calculate the distance of the cliff from the ship and height of the cliff.

Probability


  1. Find the probability that a number selected at random from the numbers 1, 2, 3………………, 35is a (i) prime number (ii) multiple of 7 (iii) multiple of 3 or 5.
  2. A box contains 3 blue, 2 white and 4 red marbles. If a marble is drawn at random from the box, what is the probability that it will not be a white marble?
  3. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag.
  4. One card is drawn from well shuffled deck of 52 cards. Find the probability of getting: (i) a king of red suit (ii) a queen of black suit (iii) a jack of hearts (iv) a red face card (v) a black Ace card.
  5. A bag contains 12 balls out of which x are white. (i) If one ball is drawn at random, what is the probability that it will be a white ball? (ii) if 6 more white balls are put in the bag, the probability of drawing a white ball will be double than in (i) . Find x.
  6. Two dice are thrown simultaneously. What is the probability that (i) 5 will not come up on either of them. (ii) 5 will come up on both dice (iii) 5 will come up at least once.
  7. Cards bearing numbers 1, 3, 5… 35 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing (i) a prime number less than 15 (ii) a number divisible by 3 and 5.
  8. From a well shuffled pack of 52 cards two black kings and two black jacks are removed. From the remaining cards, a card is drawn at random. Find the Probability that the drawn card is neither an ace nor a king.
  9. A bag contains 4 red, 5 black and 6 white balls. A ball is drawn at random. Find the Probability that the ball drawn is (i) red (ii) not black (iii) red or white.
  10. A card is drawn from a well shuffled pack of 52 cards. Calculate the probability of getting (i) neither a card of club not a card of spade. (ii) Neither a card of spade nor an ace (iii) either a red card or an ace.
Co- ordinate Geometry
  1. Find the value of x such that PQ = QR where P, Q and R the points (x , -1), (1,3) and (-3,8).
  2. Find a point on the x –axis which is equidistant from the points (5,2) and (1,-2).
  3. Show that the points (1,-3),(2,5),(3,-2) and (2,-10) are the vertices of a parallelogram.
  4. Find the values of x for which the distance between the points P4,-5) and Q(12,x) is 10 units.
  5. if the point P(x,y) is equidistant from the points A(5,1) and B(-1,5) Prove that
3x = 2y.
  1. If A (6,-1), B (1, 3) and (k, 8) are three points such that AB = BC, Find the value of a.
  2. If the points A (4, 3) and B(x, 5) lie on a circle with the centre O (2, 3). Find the value of x.
  3. Prove that the points A (7, 10), B (-2, 5) and C (3,-4) are the vertices of an isosceles right triangle. Calculate its Area.
  4. State and prove Section Formula.
  5. State and prove Distance Formula.
  6. Find the coordinates of the points of trisection of the line segment joining the points A (-5, 6) and B (4,-3).
  7. Find the ratio in which the point P (2, y) divides the line segment joining the points A (-2, 2) and B (3, 7) Also find the value of y.
  8. Determine the ratio in which the line 3x + y = 9 divides the line segment joining the points (1, 3) and (2, 7).
  9. If A (-2, 4), B (0, 0) and C (4, 2) are the vertices of a ∆ABC, then find the length of median through the vertex A.
  10. To find the coordinates of the Centroid of a Triangle.
  11. If (2, p) is the midpoints of the line segment joining the points A (6,-5) and
B (-2, 11). Find the value of p.
  1. State and Prove area of Triangle.
  2. Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are (2, 2), (4, 4) and (2, 6).
  3. Find the value of p for which the points (-5, 1), (1, p) and (4,-2) are collinear.
  4. Find the area of the quadrilateral whose vertices taken in order area A (-5,-3),
B (-4,-6), C (2,-1) and D (1, 2)

Mensuration
  1. The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm, and its slant height is 10 cm. Find its capacity and total surface area.
  2. A bucket is in the form of a frustum of a cone. Its depth is 15 cm and the diameters of the top and the bottom are 56 cm and 42 cm respectively. Find how many liters of water can the bucket hold.
  3. A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 8/9 of the curved surface of the whole cone, find the ratio of the line segment into which the altitude of the cone is divided by the plane.
  4. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27 of the volume of the given cone, at what height above the base is the section made/
  5. If the radii of the circular ends of a conical bucket of height 45 cm be 28 cm and 7 cm, find the capacity of the bucket.
  6. Three solid metallic spheres of radii 3 cm, 4 cm and 5 cm resp. are melted to form a single solid. Find the diameter of the resulting sphere.
  7. Water flows through a circular pipe whose internal diameter is 2 cm at the rate of 0.7 m per second into a cylindrical tank, the radius of whose base is 40 cm. by how much will the level of water rise in the tank in half an hour.
  8. A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular come mounted on a hemisphere. Find the radius of the hemisphere and the total height of the toy. If the height of the conical part is 3 times its radius.
  9. A solid is made up of a cube and a hemisphere attached on its top. Each edge of the cube measures 5 cm and the hemisphere has a diameter of 4.2 cm. Fine the total area to be painted.
  10. A Toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The radius and height of the cylindrical part are 5 cm and 13 cm resp. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Find the surface area of the Toy. If the total height of the toy is 30 cm.
  11. A solid is composed of a cylinder with hemispherical ends. If whole length of the solid is 104 cm and the radius of each of its hemispherical end is 7 cm, find the cost of polishing its surface at the rate of Rs 10 per square dm.
  12. A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere and total height of the toy. If the height of the cone is 3 times the radius.
  13. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is traveling at a speed of 66 km per hour?
  14. Find the area of a quadrant of a circle whose circumference is 22 cm.
  15. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
  16. A chord of a circle of radius 15 cm subtends an angle of 60 at the centre. Find the areas of the corresponding minor and major segments of the circle.
  17. A horse is tied to a peg at one corner of square shaped grass fields of side 15 m by means of a 5 m long rope Find (i) the area of that part of the field in which the horse can graze. (ii) The increase in the grazing area if the rope were 10 m long instead of 5 m.
  18. Find the area of Shaded region in the given figure. Where
ABCD is a square of side 14 cm.
  1. A racing track whose left and right ends are semicircular. The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find (i) the distance around the track along its inner edge (ii) the area of the track.
  2.  In the given figure AB and CD are two diameters of a circle (with centre o) Perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm. Find the area of the shaded region.
  1. The area of an equilateral triangle ABC is 17320.5 square cm. with each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle. Find the area of the shaded region.
  2. In the given figure OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, Find the area of the (i) quadrant OACB and (ii) Shaded region.
  3. In the given figure 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.
  4. Calculate the area of the designed region in common between the two quadrants of circles of radius 8 cm each.
  5. Find the difference between the areas of an equilateral triangle of side 6 cm and the circle inscribed in it.
  6. The circumference of a circle exceeds its diameter by 60 cm. Find the radius of the circle.
                                                                                                                                                                                  
  DHIRENDRA SIR       MATAHEMATAICS       IX & X      MOB - 7870729004      
Posted by at

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)