Mathematics Test 157

Result

Mathematics Test 157

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Weekly Quiz Competition

Question 1
4 / -1

If S denotes the sum of infinite and S_n denotes the sum of n terms of series then the least value of n is :-

A
8

B
9

C
10

D
11
Question 2
4 / -1

A and B are two square matrix such that A²B = BA and If (AB)¹⁰ = A^KB¹⁰ then K is :-

A
10

B
1001

C
1023

D
1042

Solution

(AB)² = (AB) (AB) = A(BA) B = A³B²

Now, (AB)³ = AB AB AB = A⁷B³

So, K = 2¹⁰ – 1 = 1023
Question 3
4 / -1

Coefficient of x⁶ in the expansion of (1 + x)¹⁰¹ (1 – x + x²)¹⁰⁰ is -

A
¹⁰⁰C₂

B
¹⁰⁰C₆

C
¹⁰⁰C₃

D
¹⁰⁰C₆ + ¹⁰⁰C₃

Solution

((1 + x)(1 – x + x²))¹⁰⁰(1 + x)

⇒ (1 + x) (1 + x³)¹⁰⁰

coefficient of x⁶ is ¹⁰⁰C₂
Question 4
4 / -1

If x is an odd integer and y, z are even integers, satisfying x + y + z = 47 and x, y, z ≥ 0, then number of solutions are

A
100

B
200

C
300

D
400

Solution

x = 2p + 1 ;           p ≥ 0

y = 2q              ;           q ≥ 0

z = 2r               ;           r ≥ 0

x + y + z = 47

p + q+ r = 23

25 × 12 = 300
Question 5
4 / -1

There are 100 tickets numbered 00, 01, 02, ...., 99. One ticked is selected at random. Suppose A & B are the sum & product of the digits of numbers found on the ticket then

A
1

B
0

C
1/2

D
3/4

Solution

Exhaustive cases (परिपूर्ण स्थितियाँ) = 26, 34, 43, 62 = 4

Favourable cases (अनुकूल स्थितियाँ) = 34, 43 = 2

∴ Required Probability = 1/2
Question 6
4 / -1

The existence of unique solution of the system of equations,

       x + y + z = β,

       5x – y + αz = 10,

       2x + 3y – z = 6

depends on

A
α only

B
β only

C
α and β both

D
neither α nor β

Solution

Here, Δ ≠ 0

As, Δ = (23 – α) ≠ 0
Question 7
4 / -1

Consider three statements

p: Suyash will come today evening

q: Shalini will be at his home

r: They will go to party.

Then the statement (~q ∧ ~r) ⇒ p is logically equivalent to

A
If Suyash will not come today then Shalini will go to party.

B
If Suyash will not come today then Shalini will not go to party.

C
Suyash will come today or Shalini will be at his home or they will go to party

D
Suyash will come today or Shalini will go to party

Solution

(~p ∧ ~r) ⇒ p ≡ ~ ( ~ q ∧ ~r) ∨ p

≡ (q ∨ r) ∨ p ≡ p ∨ q ∨ r
Question 8
4 / -1

Two data sets each of size 10 has the variance as 4 and k and the corresponding means as 2 and 4 respectively. If the variance of the combined data set is 5 · 5, then the value of k is equal to

A
5

B
6

C
4

D
3

Solution
Question 9
4 / -1

A
ln|sinx + cosx| + C

B
– ln|sinx + cosx| + C

C
ln|1 + sin²x| + C

D
ln|1 + cotx| + C

Solution
Question 10
4 / -1

A
4/3

B
1/3

C

D

Solution