Chapter 5
2 ≤ 3x - 4 ≤ 5 2. 6 ≤ - 3 (2x - 4) < 12
(3x2 - 9x + 5) 9 2. sin3 x + cos 6 x
5 (2x - 7) - 3 (2x + 3) ≤ 0 , 2x + 19 ≤ 6x + 47 .
A solution is to be kept between 68° F and 77° F. What is the range in temperature in degree Celsius (C) if the Celsius / Fahrenheit (F) conversion formula is given by F = 5 C + 32 ?
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added? - v vv vv -
How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?
IQ of a person is given by the formula IQ = MA CA × 100, where MA is mental age and CA is chronological age. If 80 ≤ IQ ≤ 140 for a group of 12 years old children, find the range of their mental age.
(5x)3 cos 2x 4. sin- 1(x x ), 0 ≤ x ≤ 1 5. 1cos 2 2 7 x x − + , - 2 < x < 2 6. 1 1 sin 1 sincot 1 sin 1 sin x x x x − + + − + − − , 0 < x < 2 π 7. (log x)log x, x > 1 8. cos (a cos x + b sin x), for some constant a and b. 9. (sin x - cos x ) (sin x - cos x), 3 4 4xπ π< < 10. xx + xa + ax + aa, for some fixed a > 0 and x > 0 11. ( ) 22 3 3 xxx x− + − , for x > 3 12. Find dy dx , if y = 12 (1 - cos t), x = 10 (t - sin t), 2 2tπ π− < < 13. Find dy dx , if y = sin-1 x + sin-1 21 x− , 0 < x < 1 14. If 1 1 0x y y x+ + + = , for , - 1 < x < 1, prove that ( ) 2 1/1 / dy dx x = − + 15. If (x - a)2 + (y - b)2 = c2, for some c > 0, prove that 2 2 2/2 1 dy dx d y dx + is a constant independent of a and b. 16. If cos y = x cos (a + y), with cos a ≠ ± 1, prove that 2cos ( ) sin
| dy | a | y |
|---|---|---|
| dx a += . 17. If x = a (cos t + t sin t) and y = a (sin t - t cos t), find 2/2 d y dx . 18. If f(x) = | x | 3, show that f ″(x) exists for all real x and find it. 19. Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines. 20. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer. 21. If |
| ( ) ( ) ( )f | x | g |
| --- | --- | --- |
| y | l | m |
| a | b | c |
= , prove that
| ( ) ( ) ( )f | x | g | x | h | x |
|---|---|---|---|---|---|
| dy l m ndx | a | b | c |
′ ′ ′ = 22. If y = 1cosa xe − , - 1 ≤ x ≤ 1, show that ( ) 2 2 21 0d y dyx x a y dxdx − − − = . MATHEMATICS146
73 4 18 2 x- ≤ − ≤ 4. 3 215 0 5 ( x )−− < ≤
312 4 2 5 x− < − ≤ − 6. 3 117 112 ( x )+≤ ≤ . Solve the inequalities in Exercises 7 to 10 and represent the solution graphically on number line.
5x + 1 > - 24, 5x - 1 < 24
2 (x - 1) < x + 5, 3 ( x + 2) > 2 - x
3x - 7 > 2 ( x - 6) , 6 - x > 11 - 2x