Find the angle between (5i+3j) and (3i−5j)
180°
90°
45°
0°
Correct answer is B
a.b=|a||b|cosθ
cosθ=a.b|a||b|
= (5i+3j).(3i−5j)(√52+32)(√32+(−5)2)
= 034=0
θ=cos−10=90°
Given that AB=(43) and AC=(2−3), find |BC|.
4√2
6√2
2√10
4√10
Correct answer is C
BC=BA+AC
Given, AB, then BA=−AB
= AB=(43)⟹BA=(−4−3)
∴
= \begin{pmatrix} -2 \\ -6 \end{pmatrix}
|BC| = \sqrt{(-2)^{2} + (-6)^{2}} = \sqrt{40}
= 2\sqrt{10}
Integrate (x - \frac{1}{x})^{2} with respect to x.
\frac{1}{3}(x - \frac{1}{x})^{3} + c
\frac{x^{3}}{3} - x\sqrt{\frac{1}{x^{3}}} + c
\frac{x^{3}}{3} - 2x + \frac{1}{x^{3}} + c
\frac{x^3}{3} - 2x - \frac{1}{x} + c
Correct answer is D
(x - \frac{1}{x})^{2} = x^2 - 2 + \frac{1}{x^2}
\int (x^2 + \frac{1}{x^2} - 2) \mathrm {d} x
= \int (x^2 + x^{-2} - 2) \mathrm {d} x
= \frac{x^3}{3} - 2x - \frac{1}{x}
If Px^{2} + (P+1)x + P = 0 has equal roots, find the values of P.
\text{-1 and }\frac{-1}{3}
\text{1 and }\frac{-1}{3}
\text{-1 and }\frac{1}{3}
\text{1 and }\frac{1}{3}
Correct answer is B
For equal roots, b^{2} - 4ac = 0
From the equation, a = P, b = (P+1), c = P
(P+1)^{2} - 4(P)(P) = P^{2} + 2P + 1 - 4P^{2} = 0
-3P^{2} + 2P + 1 = 0 \implies 3P^{2} - 2P - 1 = 0
3P^{2} - 3P + P - 1 = 0
3P(P - 1) + 1(P - 1) = 0
P = \text{1 or }\frac{-1}{3}
| Age in years | 10 - 14 | 15 - 19 | 20 - 24 | 25 - 29 | 30 - 34 |
| Frequency | 6 | 8 | 14 | 10 | 12 |
Find the mean of the distribution.
23.4
23.6
24.3
24.6
Correct answer is B
| Age in years |
Classmark (x) |
Frequency (f) |
fx |
| 10 - 14 | 12 | 6 | 72 |
| 15 - 19 | 17 | 8 | 136 |
| 20 - 24 | 22 | 14 | 308 |
| 25 - 29 | 27 | 10 | 270 |
| 30 - 34 | 32 | 12 | 384 |
| Total | 50 | 1170 |
Mean = \frac{\sum fx}{\sum f}
= \frac{1170}{50} = 23.4