Maharashtra SSC Algebra (Mathematics Part 1) Practice Paper 1 with Solutions | Class 10

Q.1 (A)(i) The face value of a share is ₹ 10 and its market value is ₹ 35. Then the share is .................... .

Q.1 (A)(ii) The value of the discriminant of the quadratic equation \(x^2 + 4x + 4 = 0\) is .................... .

Q.1 (A)(iii) If \(3x + 2y = 11\) and \(2x + 3y = 4\), then \(x + y =\) .................... .

Q.1 (A)(iv) The common difference of the A.P. 5, 2, −1, −4, .................... is .................... .

Q.1 (B)(i) In a pie diagram, the central angle of a sector is 72°. What percentage of the total data does this sector show?

Q.1 (B)(ii) Find the value of the determinant \(\begin{vmatrix} 5 & 3 \\ -2 & 4 \end{vmatrix}\).

Q.1 (B)(iii) Two-digit numbers are formed using the digits 2, 3 and 5 without repeating a digit. Write the sample space S and n(S).

Q.1 (B)(iv) Find the sum of the roots of the quadratic equation \(3x^2 - 7x + 2 = 0\).

Q.2 (A)(i) Complete the following activity to solve the simultaneous equations by Cramer's rule :

\[ 2x + 3y = 12 \]

\[ x - y = 1 \]

Solution :

\(D = \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2 \times -1) - (3 \times 1) =\) ____
\(D_x = \begin{vmatrix} \text{____} & 3 \\ 1 & -1 \end{vmatrix} = -12 - 3 = -15\)
\(D_y = \begin{vmatrix} 2 & 12 \\ 1 & \text{____} \end{vmatrix} = 2 - 12 = -10\)
\(\therefore x = \dfrac{D_x}{D} =\) ____ and \(y = \dfrac{D_y}{D} =\) ____

Q.2 (A)(ii) Complete the following activity to find the 15th term of the A.P. 4, 9, 14, 19, ...

Solution :

Here \(a =\) ____, \(d = t_2 - t_1 = 9 - 4 =\) ____
\(t_n = a + (n - 1)d\)
\(\therefore t_{15} = 4 + (15 - 1) \times\) ____
\(= 4 +\) ____
\(\therefore t_{15} =\) ____

Q.2 (A)(iii) The monthly expenditure of a family is ₹ 24,000. Out of this, ₹ 6,000 is spent on food and ₹ 8,000 on house rent. Complete the following activity to find the central angles for these two items in a pie diagram.

Solution :

Central angle \(= \dfrac{\text{Expenditure on the item}}{\text{Total expenditure}} \times\) ____
Central angle for food \(= \dfrac{6000}{24000} \times 360^\circ =\) ____
Central angle for house rent \(= \dfrac{\text{____}}{24000} \times 360^\circ =\) ____

Q.2 (B)(i) One card is drawn at random from a well-shuffled pack of 52 playing cards. Find the probability that the card drawn is a face card.

Q.2 (B)(ii) Solve the following simultaneous equations :

\[ 3x + 4y = 18 \]

\[ 5x - 2y = 4 \]

Q.2 (B)(iii) Solve the following quadratic equation using the formula :

\[ x^2 - 6x + 7 = 0 \]

Q.2 (B)(iv) Find the sum of the first 20 terms of the A.P. 3, 7, 11, 15, .....

Q.2 (B)(v) Find the mean of the following data by the direct method.

Q.3 (A)(i) Ms. Rutuja bought 120 shares of face value ₹ 10 when the market value was ₹ 55. She paid brokerage at 0.5% and GST at 18% on the brokerage. Complete the following activity to find the total amount she paid.

Solution :

Value of shares \(= 120 \times\) ____ \(=\) ₹ 6,600
Brokerage \(= 0.5\%\) of 6,600 \(= \dfrac{0.5}{100} \times 6600 =\) ₹ ____
GST on brokerage \(= \dfrac{18}{100} \times\) ____ \(=\) ₹ ____
Total amount paid \(= 6600 + 33 +\) ____ \(=\) ₹ ____

Q.3 (A)(ii) The roots of the quadratic equation \(x^2 + kx + 16 = 0\) are real and equal. Complete the following activity to find the value of \(k\).

Solution :

Comparing with \(ax^2 + bx + c = 0\), \(a = 1,\ b =\) ____, \(c = 16\)
Since the roots are real and equal, \(b^2 - 4ac =\) ____
\(\therefore k^2 - 4 \times 1 \times\) ____ \(= 0\)
\(\therefore k^2 =\) ____
\(\therefore k =\) ____ or \(k =\) ____

Q.3 (B)(i) The sum of a two-digit number and the number obtained by interchanging its digits is 99. The digit in the ten's place is greater than the digit in the unit's place by 3. Find the number.

Q.3 (B)(ii) In an auditorium there are 20 seats in the first row, 22 seats in the second row, 24 seats in the third row, and so on. If there are 25 rows, find the number of seats in the last row and the total number of seats.

Q.3 (B)(iii) Two dice are rolled simultaneously. Find the probability of each of the following events :

(1) Event A : the sum of the numbers on the upper faces is 7.
(2) Event B : the sum of the numbers on the upper faces is a multiple of 5.
(3) Event C : both dice show the same number.

Q.4 (i) A train travels a distance of 360 km at a uniform speed. If the speed had been 5 km/hr more, the train would have taken 1 hour less for the same journey. Find the speed of the train.

Q.4 (ii) Mr. Desai bought 200 shares of face value ₹ 10 when the market value was ₹ 120. Later he sold all of them when the market value was ₹ 150. He paid brokerage at 0.3% and GST at 18% on brokerage on both the purchase and the sale. Find :

(1) Total amount paid for buying the shares.
(2) Amount received on selling the shares.
(3) His profit or loss.

Q.4 (iii) The following table shows the marks obtained by students in a test. Find the mode of the data.

Q.5 (i) In an A.P., the 4th term is 11 and the sum of the 5th and 7th terms is 34. Find the first term and the common difference, write the first four terms, and check whether 150 is a term of this A.P.

Q.5 (ii) A bag contains 3 red balls, 5 blue balls and some green balls. One ball is drawn at random. If the probability of drawing a green ball is \(\dfrac{1}{3}\), find the number of green balls in the bag. Also find the probability of drawing a red ball.