Mathematics Test 233

Result

Mathematics Test 233

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Question 1
4 / -1

If tanA + tan B + tan C = tan A. tan B. tan C, then

A
A, B, C must be angles of a triangle

B
The sum of any two of A, B, C is qual to the third

C
A + B + C must be an integral multiple of π

D
none of these

Solution
Question 2
4 / -1

The mean of 5 observations is 4.4 and their variance is 8.24. If three observations are 1, 2 and 6. Then the other two observations are

A
4 and 8

B
4 and 9

C
5 and 7

D
5 and 9

Solution

Solving (i) and (iii), we get:

2x = 18

⇒ x = 9

⇒ y = 4

∴ The other 2 observations are 9 and 4.

The correct answer is Option 2.
Question 3
4 / -1

The probability that a man can hit a target is (3/4). He tries 5 times. The probability that he will hit the target at least three times is

A

B

C

D

Solution
Question 4
4 / -1

Four numbers are multiplied together. Then the probability that the product will be divisible by 5 or 10 is

A

B

C

D

Solution

Calculation:

The divisibility of the product of four numbers depends upon the value of the last digit of each number.

The last digit of a number can be any of the 10 digits

⇒ The total number of ways of selecting last digits of four numbers is 10 × 10 × 10 × 10 = 10⁴

Now, if the product of the 4 numbers is not divisible by 5 or 10

⇒ Then the number of choices for the last digit of each number is 8 (excluding 0 or 5).

⇒ Favourable number of ways = 8 × 8 × 8 × 8 = 8⁴
Question 5
4 / -1

The general solution of the equation sin¹⁰⁰ x – cos¹⁰⁰x = 1, is

A

B

C

D

Solution
Question 6
4 / -1

The number of common tangents that can be drawn to the circle x² + y² – 4x – 6y – 3 = 0 and x² + y² + 2x + 2y + 1 = 0 is

A
1

B
2

C
3

D
4

Solution

Also, r₁ + r₂ = 4 + 1 = 5 = C₁C₂

Hence, there are 3 common tangents

∴ The number of common tangents that can be drawn to the circle x² + y² – 4x – 6y – 3 = 0 and x² + y² + 2x + 2y + 1 = 0 is 3.

The correct answer is Option 3.
Question 7
4 / -1

If the locus of the point of intersection of perpendicular tangents to the ellipse is a circle with centre at (0, 0), then the radius of the circle would be

A
a + b

B
ab

C
b/a

D

Solution

Calculation:
Question 8
4 / -1

If f (–3) = f (2) = 0 and f ´(–3) < 0, for the function f(x) = x³ + ax² + bx + c then the largest integral value of c is

A
- 18

B
- 19

C
- 12

D
- 6

Solution
Question 9
4 / -1

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than then

A
a + b – c > 0

B
a – b + c < 0

C
a – b + c > 0

D
a + b – c < 0

Solution
Question 10
4 / -1

The variable straight line 3x (a + 1) + 4y(a – 1) – 3(a –1) = 0 for different values of ‘a’ passes through a fixed point whose coordinates are

A
(0, 1)

B
(1, 2)

C

D
(2, 1)

Solution

Calculation:

Given, 3x (a + 1) + 4y(a – 1) – 3(a –1) = 0

⇒ 3ax + 3x + 4ay – 4y – 3a + 3 = 0

⇒ a(3x + 4y – 3) + (3x – 4y + 3) = 0

It represents a family of lines.

∴ 3x + 4y – 3 = 0 … (1)

3x – 4y + 3 = 0 … (2)

Adding (1) and (2), we get:

6x = 0

⇒ x = 0

Putting the value of x in (1), we get: y = 3/4