Mathematics Test 115

Result

Mathematics Test 115

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Weekly Quiz Competition

Question 1
4 / -1

The negation of the boolean expression ~ s ∨ (~ r ∧ s) is equivalent to :

A
r

B
s ∧ r

C
s ∨ r

D
~s ∧ ~r

Solution

~ (~ s ∨ (~ r ∧ s))

s ∧ (r ∨ ~s)

(s ∧ r) ∨ (s ∧ ~s)

(s ∧ r) ∨ (c)

(s ∧ r)
Question 2
4 / -1

The angle of elevation of the top of vertical tower standing on a horizontal plane is observed to be 45° from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the top of the tower from B be 30°, then the distance (in m) of the foot of the tower from the point A is:

A
15 (3 - √3)

B
15 (3 + √3)

C
15 (1+ √3)

D
15 (5 - √3)

Solution
Question 3
4 / -1

If a ∈ z, and the equation (x – a) (x – 10) + 1 = 0 has integral roots, then the values of a are :-

A
8, 10

B
10, 12

C
12, 8

D
None of these

Solution

(x – a) (x – 10) = –1 (x, a ∈ I)

so x – a = 1 and x – 10 = –1

or x – a = –1 and x – 10 = 1
Question 4
4 / -1

Consider the series

1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4...... such that value r repeats upto r² times, then,

A
sum of 1000 terms will be 10815

B
sum of 1000 terms will be 14400

C
sum of 1000 terms will be 10810

D
sum of 1000 terms will be 11825
Question 5
4 / -1

If then locus of z is :-

A
a straight line

B
a circle having centre on x-axis

C
a circle having centre on y-axis

D
a parabola

Solution
Question 6
4 / -1

The number of 5 digit numbers which are divisible by 3, using digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 when repetition of digit is allowed is :-

A
3⁹

B
4.3⁸

C
5.3⁸

D
7.3⁸

Solution

If the sum of first 4 blanks is in form 3n then last blank can fill by 3, 6 or 9, so that sum of digit is divisible by 3.

If sum is in form 3n+1, than last blank can fill by 2, 5 or 8. If sum is in form 3n+2, than last blank can fill by 1,4 or 7 therefore in any case, last blank can be filled in 3 ways only.

∴ Number of five digit numbers

= 9 × 9 × 9 × 9 ×3 = 3⁹.
Question 7
4 / -1

If ¹⁷C₁ 2¹⁶ + ¹⁷C₂ 2¹⁶ + 3.¹⁷C₃ 2¹⁴ + 4.¹⁷C₄.2¹³ +... ....... + 17.¹⁷C₁₇ = x.3^y then x + y is equal to –

A
30

B
76

C
66

D
33
Question 8
4 / -1

A key bunch contains 100 keys exactly one of which open the door. A man tries to open the door by choosing a key random discarding the wrong key after each trial. The probability then door open in 25^th trial is –

A
1/100

B
1/76

C
1/25

D
1/24
Question 9
4 / -1

Let P(h, k) be a point on the curve y = x² + 7x + 2, nearest to the line y = 3x – 3. Then the equation of the normal to the curve at P is

A
x – 3y – 11 = 0

B
x – 3y + 22 = 0

C
x + 3y – 62 = 0

D
x + 3y + 26 = 0

Solution

As the point P(H, k) is the nearest point on the curve y = x² + 7x + 2, to the line y = 3x – 3, so the tangent to the parabola y = x² + 7x + 7 at point P(h, k) is parallel to the line y = 3x – 3
Question 10
4 / -1

Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T = {x ∈ R} f(x) = f(0), then the sum of squares of all the elements of T is

A
2

B
4

C
8

D
6

Solution

Let a polynomial f(x) of degree four is

f(x) = ax⁴ + bx³ + cx² + dx + e

∴ f'(x) = 4ax³ + 3bx² + 2cx + d = 0

⇒ x = – 1, 0, 1
∴ Sum of roots = 0 ⇒ –  3b/4a = 0

⇒ b = 0

and product of roots = 0 ⇒ – d/4a = 0

⇒ d = 0 and product of roots taking two at a time = –1

⇒ 2c/4a = –1   ⇒    c = –2a

∴ f(x) = ax⁴ – 2ax² + e

Now,   f(x) = f(0) [given]

⇒ ax⁴ – 2ax² + e = e

⇒ 2x²(x² – 2) = 0

⇒ x = 0, 0,– √2, √2 ∈ T = {x ∈ R | f(x) = f(0)}

∴ Sum of squares of all the element of

T = 2 + 2 = 4