Miscallenous Test -01

Group A ( 10*1=10)

[ Q. No. 1 ] For what value of ‘a’ will the function f(x)=ax+b becomes a constant function ?
i. 0
ii. 1
iii. -3
iv. 1-3/x

[ Q. No. 2] A polynomial f(x) with f(b/a)=0 has one of its factor :
i. x-a
ii. ax-b
iii. ax+b
iv. bx-a

[ Q. No. 3 ] Which of the following is the correct expression for the sum of first n-natural numbers ?
i.n^2
ii. n(n+1)
iii. n/2*(n+1)
iv. n/2*(a+b)

[ Q. No. 4 ] If sinA=x & cosA=y, which of the following is incorrect for cos2A ?
i. 2xy
ii. y²-x²
iii. 1-2x²
iv. 2y²-1

[ Q. No. 6 ] The determinant of a 2×2 identity matrix is :
i. 0
ii. 1
iii. Doesn’t exist
iv. None of the above

[ Q. No. 7 ] Let A & B be any two non-empty sets with n(A n B)=0, then A & B are said to be :
i) Identity sets
ii) Disjoint sets
iii) Overlapping sets
iv) Both i) & iii)

[ Q. No. 8 ] The cardinality of the set of all even prime numbers is :
i. 1
ii. 2
iii.3
iv. 4

[ Q. No. 9 ] Ram takes a loan of Rs. 5,000 for 2 years & agrees to pay 0. 5 paisa per rupee per month as interest. The rate of interest he needs to pay annually is :
i. 5%
ii. 6%
iii. 10%
iv. 12%

[ Q. No. 10 ] Which of the following is the correct relation between simple interest & compound interest compounded annually on a sum of Rs. P for 1 year at 10% rate of interest ?
i. S. I. < C. I.
ii. S. I. > C. I.
iii. S. I. = C. I.
iv. None of the above

Group B ( 5×10=50 )

[ Q. No. 11 ]
a) Define composite function. [ 1 ]
b) If f(x)=4x & g(x)=2x-1, find the value of gof(x) & fog(x). [ 1+1 ]
c) If f = { (0,0) , (1,1) , (2,4) , (3,9)} and g = { (1,0) , (2,1) , (3,2) , (4,3) }, show the function fog in arrow diagram and find it in ordered-pair. [ 2 ]

[ Q. No. 12 ]
a) Differentiate between factor theorem & remainder theorem in one point. [ 1 ]
b) Use factor theorem to check whether (x+3) is a factor of the polynomial x³-8x+3. [ 2 ]
c) Use remainder theorem to find the remainder when 2x³-7x²+5x+4 is divided by (x-3). [ 2 ]

[ Q. No. 13 ]
a) Define arithmetic series with an example. [ 1 ]
b) Check whether the sequence 2, 5, 8, 11,…. is an arithmetic sequence or not. If yes, find its 20th term. If no, state why ? [ 1 + 1 ]
c) Which term of the series 84 + 78 + 72 + ……… is zero ? Find it. [ 2 ]

[ Q. No. 14 ]
a) Define determinant of a matrix A with an example. [ 1 ]
b) Under what condition, inverse of a matrix is possible ? State. Also find the inverse of a 2×2 identity matrix. [ 1 + 1 ]
c) Considering any 2×2 matrix, prove that AB = BA = I, such that A & B are two non-singular matrix & I is the identity matrix.

[ Q. No. 15 ]
a) Express tan2A in terms of sinA & cosA. [ 1 ]
b) Using the expression above, calculate the value of tan2A given that sinA = ¾. [ 2 ]
c) Prove that : [ 1 + 1 ]
i. cos2A = 2cos²A – 1
ii. sin3A = 3sinA – 4 sin³A

[ Q. No. 16 ]
a) Define scalar products of two vectors a and b. [ 1 ]
b) Under what condition, two vectors are said to be perpendicular to each other ? Clarify with necessary mathematical relations. [ 2 ]
c) For any two perpendicular vectors a & b , prove that (a+b)²=( a- b)².

[ Q. No. 17 ] A sum of Rs. 9,600 is invested for 2 years at 10% p.a at compound interest.
a) Find the sum due at the end of first year. [1.5]
b) Find the sum due at the end of second years. [ 1.5 ]
c) By what perfect is the sum due at the end of second year is more than that at the end of first year ? Find it. [ 2 ]

[ Q. No. 18 ] On a certain sum of money, the difference between the simple interest and the compound interest for 2 years is equal to Rs. 25, provided that the rate of interest charged for both is 5% p.a. ?
a) Find the sum. [ 1.5 ]
b) Calculate the simple interest and the compound interest. [ 1 + 1 ]
c) What would be the semi-annual compound interest for the same sum at same rate for the same time period ? Calculate. [ 1.5 ]

[ Q. No. 19 ] In a college, 200 students are randomly selected. 140 like tea, 120 like coffee and 80 like both tea and coffee. [ 1×5 = 5 ]
i. How many students like only tea?
ii. How many students like only coffee?
iii. How many students like neither tea nor coffee?
iv. How many students like only one of tea or coffee?
v. How many students like at least one of the beverages?

[ Q. No. 20 ] In a group of 150 people, 120 like to play volleyball, 85 like to play football and 25 like to play none of the games:
i) Show the above information in a Venn diagram. [ 1 ]
ii) How many people like to play both the games ? [ 2 ]
iii) How many people like to play volleyball only? [ 2 ]

Group C ( 5 × 3 = 15 )

[ Q. No. 21] In a survey of the students, it was found that 70% of the students studied BBA courses, 65% of the students studied BBS courses, 430 students studied both the course and 8% did not study both courses then:
a) Show the above information in a Venn-diagram. [ 1 ]
b) Find what percent students studied both the courses. [ 2 ]
c) Find the total number of students who take part in the survey. [ 2 ]

[ Q. No. 22 ] The compound amount of a certain sum of money in 2 years and 3 years become Rs 8820 and Rs 9261 respectively. Find the sum and the rate of interest. [ 3 + 2 ]

[ Q. No. 23 ] A businessman borrowed a sum of money at the rate of 5% simple interest for 2 years and immediately he lent it in compound interest compounded annually at the same rate for the same duration of time. In this transaction if he gained Rs 50, find the sum borrowed. [5]

* Best Of Luck *