Manipur Class 10 Sample Paper Higher Maths

Manipur

SAMPLE PAPER
Class 10

HIGHER MATHEMATICS

Full Marks: 80

Pass Marks: 20

Time: Three hours

Attempt all questions

The figures in the right hand margin indicate full marks for the questions

1. The identity element for the algebraic structure ( Q ,  ) where
xy
x y = , x, y  Q is 1
3

1 1
(A) 3 (B) (C) -3 (D) −
3 3

2. The nth term of a G.P. with first term a and common ratio r is 1

(A) a + ( n − 1) r (B) a + nr
(C) ar n−1 (D) ar n

The middle term in the expansion of ( x + y ) is
8
3. 1

(A) T4 (B) T5 (C) T6 (D) T7

4. If A is a 2x2 matrix and B, a 2x3 matrix, then the order of the matrix AB is
1

(A) 2x2 (B) 3x3 (C) 2x3 (D) 3x2

5. Which of the following angles is coterminus with −600 ? 1

(A) −3000 (B) 3000 (C) −2400 (D) 4200

6. If P(n) is the statement “n2+ 2 is divisible by 3”, is P(2) true ? 1

What is the number of terms in the expansion of (1 − 2x ) ?
15
7. 1

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8. When are two matrices said to be equal? 1

2
9. Compute the product  −2  1 2 3 . 1
 
 1 

10. If a matrix A has 7 elements, find all possible orders A may have. 1

11. When are two angles said to be allied to each other? 1

4
12. Find the value of tan . 1
3

13. Define principal solutions of a trigonometric equation. 1

14. Prove that the identity element for an algebraic structure, if it exists, is
unique. 2
12
 1 
15. Find the term independent of x in the expansion of  x + 2  . 2
 x 

16. For any matrix A, show that ( A ) = A . 2

17. Simplify: sin 4050 cos3000 − cos4200 sin 2250 . 2

A+ B C
18. If A, B, C denote the angles of a triangle, show that tan = cot .
2 2
2

19. Show that the binary operation  on N defined by a  b = b is associative
but not commutative. 3

20. Find the sum of first n terms of a G.P. whose first term and common
ratio are respectively a and r. 3

2
21. Insert three harmonic means between 2 and 12. 3
5

22. What are the basic steps of the method of mathematical induction in

proving a mathematical proposition? 3

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2 1
23. If A =   , show that A2 − 6 A + 5I = 0, where I is the unit matrix of
3 4
order 2. 3

24. Solve for  ( 00    3600 ) : cos + 3 sin  = 2. 3

25. Form the composition table for the set S = {1, 2, 3, 4, 5, 6} with respect
to the binary operation of multiplication modulo 7. From the table, find
the identity element and the inverse of each element of S. 4

26. Prove by using the principle of mathematical induction that n  N ,
2 + 4 + 6 + . . . + 2n = n ( n + 1) . 4

OR

32 n − 1 is divisible by 4.

1 1 3 4 
27. If A =   and B =   , show that A2 − B 2  ( A + B )( A − B ) .
 −1 −2 1 −1

28. If S be the sum, P, the product and R, the sum of the reciprocals of n
n
S
terms of a G.P., prove that P =   . 2
5
R

Or

Sum the series 1.2.3 + 2.3.4 + 3.4.5 + … … … to n terms.

29. Prove that every square matrix can be expressed in one and only way, as
a sum of a symmetric matrix and a skew-symmetric matrix. 5

30. Find the trigonometric ratios of ( 900 +  ) in terms of those of  . 5

31. 150 workers were engaged to finish a piece of work in a certain number
of days. Four workers dropped the second day, four more workers
dropped the third day and so on. It takes 8 more days to finish the work
now. Find the number of days in which the work was completed. 6

Or

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The digits in the one’s place, ten’s place and hundred’s place of a
positive integer having three digits are in A.P. and their sum is 15. The
number obtained by reversing the digits is 594 less than the original
number. Find the number.

32. Without using principle of mathematical induction, prove that
23n − 7 n ( n  N ) always leaves the remainder 1 when divided by 49. 6

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