Mathematics Test 223

Result

Mathematics Test 223

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
4 / -1

If X = {8ⁿ - 7n - 1 : n ∈ N} and Y = {49(n - 1) : n ∈ N}, then

A
X ⊂ Y

B
Y ⊂ X

C
X = Y

D
X ≠ Y

Solution

We have: 8ⁿ - 7ⁿ - 1 = (7 + 1)ⁿ - 7n - 1 = (ⁿC₂ 7² + ⁿC₃ 7³ + … ⁿC_n 7ⁿ)

= 49(ⁿC₂ + ⁿC₃ 7 + … + ⁿC_n 7^{n - 2}) for n ≥ 2. For n = 1, 8ⁿ - 7ⁿ - 1 = 0.

Thus, 8ⁿ - 7n - 1 is a multiple of 49 for n ≥ 2 and 0 for n = 1.

Hence, X consists of all positive integral multiples of 49 of the form 49 K_n, where K_n = ⁿC₂ + ⁿC₃ 7 + … + ⁿC_n 7^{n - 2} together with zero.

Also, Y consists of all positive integral multiple of 49, including zero. Therefore, X ⊂ Y.
Question 2
4 / -1

If x² + b₁x + c₁ = 0 has two real roots α, β and x² + b²x + c² = 0 has two real roots α + 2 , β + 2 where b₁, b₂, c₁, c₂ are real numbers, then is equal to which of the following?

A
4c₁ - 4c₂

B
4c₁ + 4c₂

C
2c₁ - 2c₂

D
2c₁ + 2c₂

Solution
Question 3
4 / -1

The value of the determinant is equal to

A
2 (10! 11!)

B
2 (10! 13!)

C
2 (10! 11! 12!)

D
2 (11! 12! 13!)

Solution
Question 4
4 / -1

Coefficient of t²⁴ in (1 + t²)¹²(1 + t¹²)(1 + t²⁴) is

A
¹²C₆ + 3

B
¹²C₆ + 1

C
¹²C₆

D
¹²C₆ + 2

Solution

(1 + t²)¹²(1 + t¹²)(1 + t²⁴) = (1 + t¹² + t²⁴ + t³⁶)(1 + t²)¹²

Coefficient of t²⁴ = 1 × Coefficient of t²⁴ in (1 + t²)¹² + t¹² × Coefficient of t¹² in (1 + t²)¹² + t²⁴ × Constant term in (1 + t²)¹²

= ¹²C₁₂ + ¹²C₆ + ¹²C₀

= 1 + ¹²C₆ + 1

= ¹²C₆ + 2
Question 5
4 / -1

Let there exist a function f such that f'(a) = f²(a) = ... = f²ⁿ (a) = 0, where fn(x) stands for the n^th derivative of the function, f^{2n + 2}(a) = -ve and f has a local maximum value b at x = a. Find f(x).

A
(x - a)^{2n + 2}

B
(b - 1) - (x + 1 - a)^{2n + 1}

C
b - (x - a)^{2n + 2}

D
(x - a)^{2n + 2} - b

Solution

Since the derivatives of the function vanish at x = a, x - a should be a factor of derivatives of the function.

∴ f^{(2n + 1)}(a) = 0

f^{2n + 2}(a) = -ve

And f(a) = b

∴ f(x) = -(x - a)^{2n + 2} + b

= b - (x - a)^{2n + 2}
Question 6
4 / -1

A
1 + √5

B
-1 + √5

C
-1 + √2

D
1 + √2

Solution
Question 7
4 / -1

The three lines px + qy + r = 0, qx + ry + p = 0 and rx + py + q = 0 are concurrent, if

A
p + q + r = 1

B
p² + q² + r² = pr + rq

C
p³ + q³ + r³ = 3pqr

D
None of these

Solution

⇒ p(rq - p²) - q(q² - pr) + r(pq - r²) = 0

⇒ p³ + q³ + r³ = 3pqr
Question 8
4 / -1

If α and β are the roots of the quadratic equation ax² + bx + c = 0 and the plane βx + αy = z is passing through the line of intersection of the two planes x + αz = y and y + βz = x, then the value of b

A
is a negative number

B
is zero

C
is a positive number

D
cannot be determined

Solution
Question 9
4 / -1

A problem of mathematics is given to three students A, B and C, and their respective probabilities of solving the problem is 1/2, 1/3 and 1/4. The probability that the problem is solved is

A
3/4

B
1/2

C
2/3

D
1/3

Solution
Question 10
4 / -1

The trigonometric equation sin^-1 x = 2 sin^-1 a has a solution for

A

B

C

D

Solution