11. If a fair coin is tossed 3 times, what is the probability of getting at least two heads?
- A.\(\frac{2}{3}\)
- B.\(\frac{4}{5}\)
- C. \(\frac{2}{5}\)
- D. \(\frac{}{2}\)
Correct Option: Answer is D
Solution:
The outcomes are {HHH, HHT, HTT, HTH, THH, THT, TTH, TTT}
P(at least two heads) = 4/8
= ½
12. In how many ways can the word MATHEMATICIAN be arranged?
- A. 6,794,800 ways
- B. 2,664,910 ways
- C. 6,227,020,800 ways
- D. 129,729,600 ways
Correct Option: Answer is D
Solution:
MATHEMATICIAN = 13 letters with 2M, 3A, 2T, 2I.
Hence, the word MATHEMATICIAN can be arranged in 13!/2!3!2!2!
= 129,729,600 ways
13. Given matrix M = \(\begin{vmatrix} -2 ~~~ 0 ~~~ 4 \\ 0 ~~~ -1 ~~~ 6 \\ 5 ~~~ 6 ~~~ 3 \end{vmatrix}\), find \(M^{T} + 2M\)
- A.
\(\begin{vmatrix} -4 & 2 & 1\\ 6 & 0 & 5 \\ 0 & 6 & 2 \end{vmatrix}\) - B.
\(\begin{vmatrix} -6 & 0 & 13\\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{vmatrix}\) - C.
\(\begin{vmatrix} 5 & 2 & 6 \\ 0 & 1 & 1\\ 3 & 4 & -7 \end{vmatrix}\) - D.
\(\begin{vmatrix} -4 & 0 & 8 \\ 0 & -2 & -16 \\ 10 & 12 & 6 \end{vmatrix}\)
Correct Option: Answer is B
solution:
M = \(\begin{vmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{vmatrix}\)
M\(^{T}\) = \(\begin{vmatrix} -2 & 0 & 5 \\ 0 & -1 & 6\\ 4 & 6 & 3 \end{vmatrix}\)
2M = \(\begin{vmatrix} -4 & 0 & 8\\ 0 & -2 & 12\\ 10 & 12 & 6\end{vmatrix}\)
M\(^T\) + 2M = \(\begin{vmatrix} -6 & 0 & 13 \\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{vmatrix}\)
14.
Score (x) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Freq (f) | 5 | 7 | 3 | 7 | 11 | 6 | 7 |
Find the mean of the data.
- A. 3.26
- B. 4.91
- C. 6.57
- D. 3.0
Correct Option: Answer is A
Mean = Σfx/Σx
= 150/46
= 3.26
15.
Score (x) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Freq (f) | 5 | 7 | 3 | 7 | 11 | 6 | 7 |
Find the variance
- A. 42
- B. 20
- C. 70
- D. 54
Correct Option: Answer is D
Variance= Σf(x-x)/Σf
=170.888/46
=3.72
16. The locus of a point which moves so that it is equidistant from two intersecting straight lines is the
- A. bisector of the two lines
- B. line parallel to the two lines
- C. angle bisector of the two lines
- D. perpendicular bisector of the two lines
Correct Option: Answer is A
17.
From the cyclic quadrilateral MNOP above, find the value of x.
- A. 16°
- B. 25°
- C. 42°
- D. 39°
Correct Option: Answer is D
The sum of two opposite angles of a cyclic quadrilateral = 180°
∴ (2x + 18)° + 84° = 180°
2x + 102° = 180° ⟹ 2x = 78°
x = 39°
18. If 4sin2x−3=0, find the value of x, when 0° ≤ x ≤ 90°
- A. 90°
- B. 45°
- C. 60°
- D. 30°
Correct Option: Answer is C
\(4\sin^2 x – 3 = 0\)
\(4 \sin^2 x = 3 \implies \sin^2 x = \frac{3}{4}\)
\(\sin x = \frac{\sqrt{3}}{2}\)
\(\therefore x = \sin^{-1} (\frac{\sqrt{3}}{2})\)
x = 60°
19.
In the figure above, |CD| is the base of the triangle CDE. Find the area of the figure to the nearest whole number.
- A. 56 cm2
- B. 24 cm2
- C. 42 cm2
- D. 34 cm2
Correct Option: Answer is D
Area of rectangle ABCD = length x breadth
= 7 x 4
= 28 cm\(^2\)
Area of triangle CDE = \(\frac{1}{2}\) base x height
= \(\frac{1}{2} \times 3 \times 4\)
= 6 cm\(^2\)
Area of the figure = 28 cm\(^2\) + 6 cm\(^2\)
= 34 cm\(^2\)
20. The marks scored by 30 students in a Mathematics test are recorded in the table below:
Scores (Mark) | 0 | 1 | 2 | 3 | 4 | 5 |
No of students | 4 | 3 | 7 | 8 | 6 | 2 |
What is the total number of marks scored by the children?
- A. 82
- B. 15
- C. 63
- D. 75
Correct Option: Answer is D
Scores (Mark) | 0 | 1 | 2 | 3 | 4 | 5 | |
No of students | 4 | 3 | 7 | 8 | 6 | 2 | |
fx | 0 | 3 | 14 | 24 | 24 | 10 | 75 |