10th Standard - Mathematics

Unit Test Examination

Chapter 2 (Numbers & Sequences) & Chapter 3 (Ex: 3.15 to 3.19)

Time Allowed: 1.30 Hours

Subject: Mathematics

Maximum Marks: 50

PART - I (10 x 1 = 10 Marks)

Note: (i) Answer all the questions. (ii) Choose the correct answer from the given options.

1.Euclid's division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy:

(a) 1 < r < b	(b) 0 ≤ r < b
(c) 0 < r ≤ b	(d) 0 ≤ r ≤ b

2.Using Euclid's division lemma, if the cube of any positive integer is divided by 9 then the possible remainders are:

(a) 0, 1, 8	(b) 1, 4, 8
(c) 0, 1, 3	(d) 1, 3, 5

3.The sum of the exponents of the prime factors in the prime factorization of 1729 is:

(a) 1	(b) 2
(c) 3	(d) 4

4.If t₁, t₂, t₃, ... are in A.P., then t₆, t₁₂, t₁₈, ... is:

(a) Geometric Progression	(b) Arithmetic Progression
(c) Neither A.P. nor G.P.	(d) Constant Sequence

5.The value of (1³ + 2³ + 3³ + ... + 15³) - (1 + 2 + 3 + ... + 15) is:

(a) 14280	(b) 14400
(c) 14200	(d) 14220

6.If a matrix has 16 elements, then the number of possible orders it can have is:

(a) 3	(b) 4
(c) 5	(d) 6

7.If A is a 2 x 3 matrix and B is a 3 x 4 matrix, how many columns does the matrix AB have?

(a) 3	(b) 4
(c) 2	(d) 5

8.If A = [1 2 3], then the order of its transpose matrix A^T is:

(a) 1 x 3	(b) 3 x 1
(c) 1 x 1	(d) 3 x 3

9.Given F₁ = 1, F₂ = 3 and F_n = F_n-1 + F_n-2, then F₅ is:

(a) 3	(b) 5
(c) 8	(d) 11

10.A matrix A = [a_ij]_{m x n} is said to be a square matrix if:

(a) m < n	(b) m > n
(c) m = n	(d) m ≠ n

PART - II (10 x 2 = 20 Marks)

Note: Answer any 10 questions. Question No. 22 is compulsory.

11.Find the HCF of 396, 504, 636 using Euclid's division algorithm.

12.If 13824 = 2^a x 3^b then find the values of a and b.

13.Find the general term (n^th term) of the A.P.: 3, 15, 27, 39, ...

14.Find the 10^th term of a G.P. whose 8^th term is 768 and common ratio is 2.

15.Find the sum of the series: 1 + 2 + 3 + ... + 50.

16.Find the sum of cubes: 1³ + 2³ + 3³ + ... + 20³.

17.Construct a 3 x 3 matrix whose elements are given by a_ij = |i - 2j|.

18.If A = [[7, 8, 6], [1, 3, 9], [-4, 3, -1]], find the transpose matrix A^T.

19.Find the values of x, y, and z from the equation: [x+y, y+z, z+x]^T = [9, 5, 7]^T.

20.If A = [[2, 1], [1, 3]] and B = [[2, 0], [1, 3]], find AB.

21.Show that matrix A = [[1, 2], [3, 1]] satisfies the equation A² - 2A - 5I₂ = 0.

22.[Compulsory] If 3 + k, 18 - k, 5k + 1 are in A.P., find the value of k.

PART - III (4 x 5 = 20 Marks)

Note: Answer any 4 questions. (Question No. 27 is compulsory/Graph question)

23.The sum of three consecutive terms of an Arithmetic Progression is 27 and their product is 288. Find the three terms.

24.Find the sum to n terms of the series: 5 + 55 + 555 + ...

25.If A = [[1, 1], [-1, 3]], B = [[1, 2], [-4, 2]], C = [[-7, 6], [3, 2]], verify that A(B + C) = AB + AC.

26.If A = [[3, 1], [-1, 2]], show that A² - 5A + 7I₂ = 0.

27.[Compulsory Graph] Draw the graph of y = x² - 4x + 3 and use it to solve x² - 6x + 9 = 0.