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On comparing the ratio, (a₁/a₂) , (b₁/b₂) , (c₁/c₂) find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7 (ii) 2x – 3y = 8 ; 4x – 6y = 9 (iii) (3/2) x + (5/3) y = 7; 9x – 10y = 14 (iv) 5x – 3y = 11 ; – 10x + 6y = -22 (v) (4/3) x + 2y = 8 ; 2x + 3y = 12

(i) Given equations 3x + 2y = 5 or 3x + 2y -5 = 0 and 2x – 3y = 7 or 2x – 3y -7 = 0 Comparing these equations with a1x+b1y+c1 =... View Article

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 of their marks would have been 210. Find her marks in the two subjects.

Let us assume, The marks of Shefali in Maths be x. Then, the marks in English will be 30 – x. As per the given data, (x +... View Article

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: (i) t²-3, 2t⁴ + 3t³-2t²-9t-12 (ii) x² + 3x + 1 , 3x⁴ + 5x³-7x² + 2x + 2 (iii) x²-3x + 1, x⁵-4x³ + x² + 3x + 1

(i) t2-3, 2t4 +3t3-2t2-9t-12 S As per the given data First polynomial = t2-3 Second polynomial = 2t4 +3t3-2t2 -9t-12 We observe that, the... View Article

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 (see Fig.) . Find the sides AB and AC.

Let us consider the triangle ABC, We know that the length of any two tangents which are drawn from the same point to the circle is equal. So,... View Article

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.

In the quadrilateral OBPA, ∠B = 90°, ∠A = 90° Phe sum of the angles of a quadrilateral is 360° (Angle sum property of... View Article

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x²-2x -8 (ii) 4s²-4s + 1 (iii) 6×2-3-7x (iv) 4u² + 8u (v) t²-15 (vi) 3x²-x-4

(i) x2–2x –8 Let f(x) = x2 ˗ 2x ˗ 8 = x2 ˗ 4x + 2x ˗ 8 = x(x ˗ 4) + 2(x ˗ 4) = (x ˗ 4) (x + 2) To find the zeroes, set f(x) = 0, then... View Article

Fig.depicts 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.

Given Width of the track = 10 m Distance between two parallel lines = 60 m Length of parallel tracks = 106 m Find out We have to find the (i)... View Article

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following: (i) p(x) = x³-3x² + 5x-3 , g(x) = x²-2 (ii) p(x) = x²-3x² + 4x + 5 , g(x) = x² + 1-x (iii) p(x) = x²-5x + 6, g(x) = 2-x²

(i)p(x) = x3-3x2+5x–3 , g(x) = x2–2 As per given data Dividend = p(x) = x3-3x2+5x–3 Divisor = g(x) = x2– 2 ... View Article

ABCD is a rectangle formed by the points A (-1, – 1) , B (-1, 4) , C (5, 4) and D (5, -1) . P, Q, R and S are the midpoints of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

P id the mid-point of side AB, Coordinate of P = ( (-1 – 1)/2, (-1 + 4)/2 ) = (-1, 3/2) Similarly, Q, R and S are (As Q is mid-point of... View Article

A survey was made to find the type of music that a certain group of young people liked in a city. Adjoining pie chart shows the findings of this survey. From this pie chart, answer the following: (i) If 20 people liked classical music, how many young people were surveyed? (ii) Which type of music is liked by the maximum number of people? (iii) If a cassette company were to make 1000 CD’s, how many of each type would they make?

Solution (i) Given, 10% of x = 20 (10/100) * x = 20 x = (20 * 100) / 10 = 200 So, 200 young people were surveyed. (ii) Since 40% of... View Article

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

Given Height of cone (h) = 8 cm Radius of cone (r) = 5 cm Radius of sphere (R)= 0.5 cm Find out We have to find the number of lead shots... View Article

The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see Fig.) . The advertisements yield an earning of Rs. 5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?

Given The sides of the triangle ABC are 122 m, 22 m and 120 m respectively. AB = a = 120 m BC = b = 22 m AC = c = 122 m Find out We have... View Article

E and F are points on the sides PQ and PR respectively of a ΔPQR. For each of the following cases, state whether EF || QR. (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm (iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm

Given that in ΔPQR, E and F are two points on side PQ and PR respectively. The figure shows (i) As per the given data PE = 3.9... View Article

The value of π up to 50 decimal places is given below: 3.14159265358979323846264338327950288419716939937510 (i) Make a frequency distribution of the digits from 0 to 9 after the decimal point. (ii) What are the most and the least frequently occurring digits?

(i) The frequency distribution of the digits from 0 to 9 after the decimal point is tabulated below: Digits Frequency 0 2 1 5 2 5... View Article

A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

Solution Given The positions of Ankur, Syed and David are represented as A, B and C respectively. As they are sitting at equal... View Article

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see Fig.) . Show that: (i) ΔAMC ΔBMD (ii) DBC is a right angle. (iii) ΔDBC ΔACB (iv) CM = 1/2 AB

Solution Given M is the mid-point of AB So AM = BM ∠ ACB = 90° DM = CM To Prove (i) ∆ AMC ≅ ∆ BMD (ii) ∠ DBC is... View Article

On comparing the ratios a₁/a₂ , b₁/b₂ , c₁/c₂ find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0 7x + 6y – 9 = 0 (ii) 9x + 3y + 12 = 0 18x + 6y + 24 = 0 (iii) 6x – 3y + 10 = 0 2x – y + 9 = 0

(i) Given Equations 5x−4y+8 = 0 7x+6y−9 = 0 Comparing these equations with a1x+b1y+c1 = 0 And a2x+b2y+c2 = 0 We get, a1 = 5, b1 = -4,... View Article

In Fig., diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that: (i) ar (DOC) = ar (AOB) (ii) ar (DCB) = ar (ACB) (iii) DA || CB or ABCD is a parallelogram. [Hint : From D and B, draw perpendiculars to AC.]

Solution Given OB = OD and AB = CD Construction DE ⊥ AC and BF ⊥ AC are drawn. Proof: (i) In ΔDOE and ΔBOF,... View Article

One of the two digits of a two-digit number is three times the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 88. What is the original number?

Solution Let the digit at tens place be x The digit at ones place will be 3x. Original two-digit number = 10x + 3x After interchanging the... View Article

Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX) .

Given that A triangle with mid points To prove The line joining the mid-points of any two sides of a triangle is parallel to the third... View Article