__Important Questions__

__Class X__

__2008-2009__

__Mathematics__

__UNIT : 1 Number System__

**Q. 1.**** ** State Euclid’s Division Lemma.

**Q. 2.**** ** What is the HCF of 84 and 270 ?

**Q. 3.**** ** Why the number 4n, where n is a natural number, cannot end with 0?

**Q. 4.**** ** Why is is a composite number?

**Q. . 5. **Why 40/9 is is a non – terminating decimal?

**Q. 6. **Using Euclid’s division algorithm, find the HCF of 2160 and 3520.

**Q. 7. **Use Euclid’s division algorithm to show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

**Q. 8. **Four ribbons measuring 14 cm, 18 cm, 22 cm and 26 cm respectively are to be cut into least number of pieces of equal length. What is the length of each piece?

**Q. 9. **Write the condition to be satisfied by q so that a rational number p/q has a terminating expression.

**Q. 10. **Prove that 3 – is an irrational number.

**Q. 11. **Show that any one of the numbers (n + 2), n and (n + 4) is divisible by 3.

**Q. 12. **Prove that Ön is an irrational number.

**Q. 13. **If , find the value of m and n.

**Q. 14. **Explain why is a composite number while is not a composite number.

**Q. 15. **Prove that no number of the type 4K + 2 be a perfect square.

__UNIT: 2 Algebra__

**Q. 1.**** ** Find the number of zeroes in each of the following:

**Q. 2.**** ** Solve for x and y: x – 4y = 13 and 3x + 2y = – 3.

**Q. 3.**** ** Find the discriminant of the quadratic equation 3x^{2} – 5x – 11 = 0.

**Q. 4.**** ** Find AP, if first term – 1 and common difference is 2.

**Q. 5.**** ** Write the formula for sum of first n terms of an AP whose first term is a and the last term is *l*.

**Q. 6.**** ** Find a quadratic polynomial such that sum of zeroes is 0 and product of zeroes is –.

**Q. 7.**** ** Find the value of k, so that the pair of linear equations will have infinite number of solutions: x + (2k – 1)y = 4 and kx + 6y = k + 6.

**Q. 8.**** ** Draw the graphs of the equations 4x – y – 8 = 0 and 2x – 3y + 6 = 0. Also determine the vertices of the triangle formed by the lines and x – axis.

**Q. 9.**** ** In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product would have been 210, find her marks in the two subjects.

**Q. 10.**** ** If the p^{th}term of an AP is *q *and the q^{th}term is *p*. Find the r^{th}term.

**Q. 11.**** ** Find the zeroes of the quadratic polynomial 8x^{2} – 4 and verify the relationship between zeroes and their coefficients.

**Q. 12.**** ** Solve the following system of linear equations: 2(ax – by) + (a + 4b) = 0 and 2(bx + ay) + (b – 4a) = 0.

**Q. 13.**** ** Solve for x: 9x^{2} – 9(a + b)x + (2a^{2} + 5ab + 2b^{2}) = 0

**Q. 14.**** ** If m times the mth term of an AP is equal to n times its nth term, find its (m + n)^{th} term.

**Q. 15.**** ** In an AP, prove that t _{m + n} + t _{m – n} = 2t _{m}.

__UNIT : 3 TRIGONOMETRY __

**Q. 1.**** ** Find the value of 9 tan^{2} A – 9 sec^{2} A

**Q. 2.**** ** Express sin 81^{0}+ cos 81^{0} in terms of trigonometric ratios of angles lying between 0° and 45°.

**Q. 3.**** ** Evaluate: (cos 0^{0} + sin 45^{0} + sin 30^{0})(sin 90^{0} – cos 45^{0} + cos 60^{0})

**Q. 4.**** ** The string of a kite is 150 m long and it makes an angle of 60° with the horizontal. Find the height of the kite from the ground.

**Q. 5.**** ** If tan A = ¾ , find all other trigonometric ratios.

**Q. 6.**** ** If x = a cos A – b sin A and y = a sin A + b cos A, then prove that a^{2} + b^{2} = x2 + y^{2}.

**Q. 7.**** ** Evaluate:

**Q. 8.**** ** Prove that

**Q. 9.**** ** If A , B and C are the interior angles of a triangle ABC, show that

**Q. 10.**** ** A tree is broken by the wind. The top struck the ground at an angle of 30° at a distance of 30 m from the root. Find the whole height of the tree.

**Q. 11.**** ** If sec q = x + 1/4x, then prove that sec q + tan q = 2x or 1/ 2x.

**Q. 12.**** ** If 7 sin2 q + 3 cos2 q = 4, show that tan q = 1/Ö3.

**Q. 13.**** ** Prove that

**Q. 14.**** ** The angle of elevation of a cloud from a point *h* metres above a lake is b and the angle of depression of its reflection in the lake is a. Prove that the height of the cloud above the lake is

**Q. 15.**** ** If tan

__UNIT: 4 coordinate geometry__

**Q. 1.** Find the mid – point of the line segment joining the points (4, 3) and (2, 1).

**Q. 2.**** ** Find the coordinates of the point which divides the line segment joining the points (1, 3) and (2, 7) in the ratio 3 : 4.

**Q. 3.**** ** Show that the points (1, 1); (3, – 2) and (– 1, 4) are collinear.

**Q. 4.**** ** Find the centroid of the triangle whose vertices are (3, – 5); (– 7, 4) and (10, – 2).

**Q. 5.**** ** Find the area of a triangle whose vertices are A (1, 2); B (3, 5) and C (– 4, – 7)

**Q. 6.**** ** If the distance of the point P(x, y) from the point A (5, 1) and (– 1, 5) are equal, show that 3x = 2y.

**Q. 7.**** ** In what ratio does the point (– 4, 6) divide the line segment joining the points A (– 6, 10) and B (3, – 8).

**Q. 9.**** ** For what value of m, the points (4, 3); (m, 1) and (1, 9) are collinear.

**Q. 10.**** ** Prove that the coordinates of the centroid of a triangle ABC with vertices A(x1, y1); B(x2, y2); C(x3, y3) are given by (x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3.

**Q. 11.**** ** Prove that the diagonals of a rectangle bisect each other and are of equal length.Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.

**Q. 12.**** ** In what ratio does the line 4x + y = 11 divide the line segment joining the points (1, 3) and (2, 7).

**Q. 13.**** ** PQRS is a square of side ‘b’ units. If P lies at the origin, sides PQ and PS lie along x – axis and y – axis respectively, find the coordinates of the vertices of the square PQRS.

**Q. 14.**** ** If the points (5, 4) and (x, y) are equidistant from the point (4, 5); then show that x^{2} + y^{2} – 8x – 10y + 39 = 0

**Q. 15.**** ** The line segment joining the points (3, – 4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, – 2) and (5/3, q) respectively, find the value of p and q.

__UNIT: 5 Geometry__

**Q. 1.** State Pythagoras theorem.

**Q. 2.** State Thale’s theorem.

**Q. 3.** P and Q are the points on the sides AB and AC respectively of triangle ABC. If AP = 3 cm, PB = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ.

**Q. 4.** Find the length of the tangent drawn from a point P whose distance from the centre of the circle is 25 cm. It is given that the radius of the circle is 7 cm.

**Q. 5.** Prove that the lengths of the two tangents drawn from an external point to a circle are equal.

**Q. 6.** Draw a circle of diameter 4 cm. Take a point P, 5 cm away from the centre of the circle. From P, draw a pair of tangents to the circle.

**Q. 7.** Prove converse of Pythagoras theorem.

**Q. 8.** Prove that the ratio of the corresponding altitudes of two similar triangles is equal to the ratio of their corresponding sides.

**Q. 9.** In an equilateral triangle, prove that the three times the square on one side is equal to four times the square of its altitude.

**Q. 10.** Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

**Q. 11.** Construct a triangle ABC with and altitude through A is 2.5 cm. Draw another triangle similar to this triangle with scale factor ½.

**Q. 12.** Prove that the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

**Q. 13.** In a right triangle ABC, right angled at A with AB = 6 cm and AC = 8 cm. A circle with centre O is inscribed in it. Find r, the radius of the circle.

**Q. 14.** PA and PB are the two tangents to a circle with centre O in which OP is equal to the diameter of the circle. Prove that APB is an equilateral triangle.

**Q. 15.** If ‘a’ is the area of the of a right triangle and ‘b’ is one of the side sides containing the right angle, prove that the length of the altitude on the hypotenuse is

__UNIT: 6 MENSURATION__

**Q. 1.** The sum of circumferences of two circles is 132 cm. If the radius of one circle is 14 cm, find the radius of the second circle.

**Q. 2.** Find the area of the quadrant of a circle whose circumference is 22 cm.

**Q. 3.** What is the surface area of the solid hemisphere of radius r cm?

**Q. 4.** How many spherical bullets can be made out of a solid cube of lead whose edge measures 44 cm, each bullet being 4 cm in diameter.

**Q. 5.** Two cubes each with 12 cm edge are joined end to end. Find the surface area of the resulting cuboid.

**Q. 6.** If the perimeter of a semi-circular protractor is 36 cm. Find the diameter of the protractor.

**Q. 7.** A drain cover is made from a square metal plate of side 40 cm by having 441 holes of diameter 1 cm, each drilled in it. Find the area of the remaining square plate.

**Q. 8.** Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 9 cm.

**Q. 9.** If the radius of the base of a right circular cylinder is halved, keeping the height same, find the ratio of the volume of the reduced cylinder to that of the original cylinder.

**Q. 10.** Write all the formulae used in the chapters: -> Areas related to Circles and Surface area & Volume.

**Q. 11.** Two circles touch internally, the sum of their areas is 116p sq. cm. and the distance between their centres is 6 cm. Find the radii of the circles.

**Q. 12.** A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1 : 2 : 3.

**Q. 13.** A shuttle cock used for playing badminton has the shape of a frustum of a cone surmounted on a hemisphere. The external diameters of the frustum are 5 cm and 2 cm, the height of the entire shuttle cock is 7 cm. Find the external surface area.

**Q. 14.** A cone is divided into two parts by drawing a plane through the mid – point of its axis, parallel to its base. Compare the volumes of the two parts.

**Q. 15.** Solid spheres of diameter 6 cm are dropped into a cylindrical beaker containing some water and are fully submerged. If the diameter of the beaker is 18 cm and the water rises by 40 cm, find the number of solid spheres dropped in the water.

__UNIT: 7 STATISTIC & PROBABILITY__

**Q. 1.** For a given data ‘less than’ ogive and ‘more than’ ogive intersect at a point P(x, y). Then what does abscissa of the point represent.

**Q. 2.** For the given data ‘less than’ ogive and ‘more than’ ogive intersect at the point P(25.7, 42). Find the median of the data.

**Q. 3.** Find the mode of the data whose mean and median are respectively 20 and 20.

**Q. 4.** A coin is tossed once; find the probability of getting a tail.

**Q. 5.** There are 48 students in a class of which 18 are girls, find the probability of selecting a boy.

**Q. 6.** A group of 10 items has arithmetic mean 6. If the arithmetic mean of 4 of these items is 7.5, find the mean of the remaining items.

**Q. 7.** Name the three measures of central tendency you have studied in your class and name those measures of central tendency which can be found graphically.

**Q. 8.** If a variable takes discrete values, then find the median.

**Q. 9.** A die is thrown; find the probability of getting a number other than 3.

**Q. 10.** A card is drawn from a well shuffled pack of 52 cardss. Find the probability of getting neither a club nor a jack.

**Q. 11.** A coin is tossed once. If it results in a tail, a dice is thrown and if it results in a head, a coin is tossed again. Write the possible outcomes.

**Q. 12.** Find the probability of getting 53 Fridays in a leap year.

**Q. 13.** The Arithmetic Mean of the following frequency distribution is 47. Determine the value of p.

| 0 – 20 | 20 – 40 | 40 – 60 | 60 – 80 | 80 – 100 |

| 8 | 15 | 20 | P | 5 |

**Q. 14. **Find median and mode of the data:

| 1 – 3 | 3 – 5 | 5 – 7 | 7 – 9 | 9 – 11 |

| 7 | 8 | 2 | 2 | 1 |

**Q. 15. ** Draw a less than ogive for the given data and hence obtain median:

| 145 – 150 | 150 – 155 | 155 – 160 | 160 – 165 | 165 – 170 |

| 8 | 10 | 9 | 15 | 10 |