Exercise 3.1

Form the pair of linear equations in the following problems, and find their solutions graphically.

(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

(ii) 5 pencils and 7 pens together cost $\text{₹ } 50$, whereas 7 pencils and 5 pens together cost $\text{₹ } 46$. Find the cost of one pencil and that of one pen.

Find the radian measures corresponding to the following degree measures: (i) 25° (ii) - 47°30′ (iii) 240° (iv) 520°

5A 35 2 12 2 / 173 1 5 /  −    = −    −  , write: (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements a13, a21, a33, a24, a23.

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is: (A) 27 (B) 18 (C) 81 (D) 512

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Find the degree measures corresponding to the following radian measures (Use 22π 7= ). (i) 11 / 16 (ii) - 4 (iii) 5π / 3 (iv) 7π

On comparing the ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$, and $\dfrac{c_1}{c_2}$, 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) $6x - 3y + 10 = 0$, $2x - y + 9 = 0$

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

On comparing the ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$ and $\dfrac{c_1}{c_2}$, 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) $3x + 2y = 5$ ; $9x - 10y = 14$

(iv) $5x - 3y = 11$ ; $-10x + 6y = -22$

(v) $4x + 2y = 8$ ; $2x + 3y = 12$

A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Construct a 2 × 2 matrix, A = [aij], whose elements are given by: (i) 2( ) / ij i ja += (ii) ij ia j = (iii) 2( 2 ) ij i ja +=

Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

(i) $x + y = 5$, $2x + 2y = 10$

(ii) $x - y = 8$, $3x - 3y = 16$

(iii) $2x + y - 6 = 0$, $4x - 2y - 4 = 0$

(iv) $2x - 2y - 2 = 0$, $4x - 4y - 5 = 0$

Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (Use 22π 7= ).

In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

Half the perimeter of a rectangular garden, whose length is $4$ m more than its width, is $36$ m. Find the dimensions of the garden.

Construct a 3 × 4 matrix, whose elements are given by: (i) 1 | 3 |2 ija i j= − + (ii) 2ija i j= −

Given the linear equation $2x + 3y - 8 = 0$, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines (ii) parallel lines (iii) coincident lines

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

Find the values of x, y and z from the following equations: (i) 4 3 / 5 1 5 / y z / x     =       (ii) 2 6 2 / 5 5 8 / x y z xy +    =   +    (iii) 9 5/7

x z / y z + +       + =       +   

Find the angle in radian through which a pendulum swings if its length is 75 cm and th e tip describes an arc of length (i) 10 cm (ii) 15 cm (iii) 21 cm

Find the value of a, b, c and d from the equation: 2 1 5 2 3 0 13

− + −   =   − +   

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.

A = [aij]m × n\ is a square matrix, if (A) m < n (B) m > n (C) m = n (D) None of these

1 2 3 / x / y x +   + −  , 0 2 8 4 y −     (A) 1, 73x y −= = (B) Not possible to find (C) y = 7, 2 3x −= (D) 1 2,3 3x y − −= =