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Q.1. The larger of cos (ln θ) and ln (cos θ) if .......... (1983  1 Mark)
Ans. cos(ln θ)
Solution.
⇒ cos (–π/2) < cos (ln θ) < cos (ln π/2)
⇒ cos (ln θ) > 0 ....... (1)
⇒ ln (cos q) < 0 ....... (2)
From (1) and (2) we get, cos (ln θ) > ln (cos θ)
∴ cos (ln θ) is larger.
Q.2. The function y = 2x^{2} ln x is monotonically increasing for values of x(≠ 0) satisfying the inequalities ....... and monotonically decreasing for values of x satisfying the inequalities .................. (1983  2 Marks)
Ans.
Solution.
Critical points are 0, 1/2, –1/2
Clearly f (x) is increasing on
f (x) is decreasing on
Q.3. The set of all x for which ln(1 + x) < x is equal to ............... (1987  2 Marks)
Ans. x > 0
Solution. Let f (x) = log (1 + x) – x for x > – 1
We observe that,
f ' (x) >0 if – 1 < x < 0 and f '(x) <0 if x > 0
Therefore f increases in (– 1, 0) and decreases in (0, ∞).
Q.4. Let P be a variable point on the ellipse with foci F_{1} and F_{2} . If A is the area of the triangle PF_{1}F_{2} then the maximum value of A is .............. (1994  2 Marks)
Ans. abe
Solution. Let P(a cos θ,b sinθ) be any point on the ellipse with foci F_{1} (ae, 0) and F_{2} (–ae, 0)
Then area of DPF_{1}F_{2} is given by
∵  sin θ  < 1
∴ A_{max} = abe
Q.5. Let C be the curve y^{3} – 3xy + 2 = 0. If H is the set of points on the curve C where the tangent is horizontal and V is the set of the point on the curve C where the tangent is vertical then H =............ and V = ............. (1994  2 Marks)
Ans. φ, {(1, 1)}
Solution. The given curve is C : y^{3} – 3xy + 2 = 0
Differentiating it with respect to x, we get
∴ Slope of tangent to C at point (x_{1}, y_{1}) is
For horizontal tangent, dy/dx = 0 ⇒ y_{1} = 0
For y_{1} = 0 in C, we get no value of x_{1}
∴ There is no point on C at which tangent is horizontal
∴ H = φ
For vertical tangent
∴ There is only one point (1, 1) at which vertical tangent can be drawn
∴ V = {(1, 1)}
True / False
Q. 1. If x – r is a factor of the polynomial f (x) = a_{n}x^{4} + .... + a_{0 }, repeated m times (1 < m < n), then r is a root of f '(x) = 0 repeated m times. (1983  1 Mark)
Ans. F
Solution. If (x – r) is a factor of f (x) repeated m times then f ' (x) is a polynomial with (x – r) as factor repeated (m – 1) times.
∴ Statement is false.
Q. 2. For 0 < a < x, th e minimum value of the function log_{a} x + log_{x}a is 2. (1984  1 Mark)
Ans. F
Solution. Given that 0 < a < x.
But equality holds for log_{a} x = 1
⇒ x = a which is not possible.
∴ f (x) > 2
∴ f_{min} cannot be 2.
∴ Statement is false.
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