Class 11 Mathematics Limits And Derivatives MCQs Set 05

Question. Let \( f(x) = \frac{\tan^n x}{\sum_{r=0}^{2n} \tan^r x}, n \in N \), where \( x \in [0, \pi/2) \)
(a) f(x) is bounded and it takes both of it's bounds and the range of f(x) contains exactly one integral point.
(b) f(x) is bounded and it takes both of it's bounds and the range of f(x) contains more than one integral point.
(c) f(x) is bounded but minimum and maximum does not exists.
(d) f(x) is not bounded as the upper bound does not exist.
Answer: (a) f(x) is bounded and it takes both of it's bounds and the range of f(x) contains exactly one integral point.

Question. Two curves \( C_1 : y = x^2 - 3 \) and \( C_2 : y = kx^2 \), \( k \in R \) intersect each other at two different points. The tangent drawn to \( C_2 \) at one of the points of intersection \( A \equiv (a, y_1) \), \( (a > 0) \) meets \( C_1 \) again at \( B(1, y_2) \), \( y_1 \neq y_2 \). The value of 'a' is
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (b) 3

Question. A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines \( y = 0 \), \( y = 3x \), and \( y = 30 - 2x \). The largest area of such a rectangle is
(a) \( \frac{135}{8} \)
(b) 45
(c) \( \frac{135}{2} \)
(d) 90
Answer: (c) \( \frac{135}{2} \)

Question. Which of the following statement is true for the function \( f(x) = \begin{cases} \sqrt{x} & x \ge 1 \\ x^3 & 0 \le x \le 1 \\ \frac{x^3}{3} - 4x & x < 0 \end{cases} \)
(a) It is monotonic increasing \( \forall x \in R \)
(b) \( f'(x) \) fails to exist for 3 distinct real values of x
(c) \( f'(x) \) changes its sign twice as x varies from \( (-\infty, \infty) \)
(d) function attains its extreme values at \( x_1 \) & \( x_2 \), such that \( x_1, x_2 > 0 \)
Answer: (c) \( f'(x) \) changes its sign twice as x varies from \( (-\infty, \infty) \)

Question. Coffee is draining from a conical filter, height and diameter both 15 cms into a cylindrical coffee pot diameter 15 cm. The rate at which coffee drains from the filter into the pot is 100 cu cm/min. The rate in cms/min at which the level in the pot is rising at the instant when the coffee in the pot is 10 cm, is
(a) \( \frac{9}{16\pi} \)
(b) \( \frac{25}{9\pi} \)
(c) \( \frac{5}{3\pi} \)
(d) \( \frac{16}{9\pi} \)
Answer: (d) \( \frac{16}{9\pi} \)

Question. Let f(x) and g(x) be two differentiable function in R and f(2) = 8, g(2) = 0, f(4) = 10 and g(4) = 8 then
(a) \( g'(x) > 4f'(x) \forall x \in (2, 4) \)
(b) \( 3g'(x) = 4f'(x) \) for at least one \( x \in (2, 4) \)
(c) \( g(x) > f(x) \forall x \in (2, 4) \)
(d) \( g'(x) = 4f'(x) \) for at least one \( x \in (2, 4) \)
Answer: (d) \( g'(x) = 4f'(x) \) for at least one \( x \in (2, 4) \)

Question. A horse runs along a circle with a speed of 20 km/hr. A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse move along the fence at the moment when it covers 1/8 of the circle in km/hr is
(a) 20
(b) 40
(c) 30
(d) 60
Answer: (b) 40

Question. Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false.
Statement-1: If \( f : R \to R \) and \( c \in R \) is such that f is increasing in \( (c - \delta, c) \) and f is decreasing in \( (c, c + \delta) \) then f has a local maximum at c. Where \( \delta \) is a sufficiently small positive quantity.
Statement-2 : Let \( f : (a, b) \to R, c \in (a, b) \). Then f can not have both a local maximum and a point of inflection at \( x = c \).
Statement-3 : The function \( f(x) = x^2 |x| \) is twice differentiable at \( x = 0 \).
Statement-4 : Let \( f : [c - 1, c + 1] \to [a, b] \) be bijective map such that f is differentiable at c then \( f^{-1} \) is also differentiable at f(c).
(a) FFTF
(b) TTFT
(c) FTTF
(d) TTTF
Answer: (a) FFTF

Question. Let \( f : [-1, 2] \to R \) be differentiable such that \( 0 \le f'(t) \le 1 \) for \( t \in [-1, 0] \) and \( -1 \le f'(t) \le 0 \) for \( t \in [0, 2] \). Then
(a) \( -2 \le f(2) - f(-1) \le 1 \)
(b) \( 1 \le f(2) - f(-1) \le 2 \)
(c) \( -3 \le f(2) - f(-1) \le 0 \)
(d) \( -2 \le f(2) - f(-1) \le 0 \)
Answer: (a) \( -2 \le f(2) - f(-1) \le 1 \)

Question. If the function \( f(x) = \frac{t + 3x - x^2}{x - 4} \), where 't' is a parameter has a minimum and a maximum then the range of values of 't' is
(a) (0, 4)
(b) \( (0, \infty) \)
(c) \( (-\infty, 4) \)
(d) \( (4, \infty) \)
Answer: (c) \( (-\infty, 4) \)

Question. The function \( S(x) = \int_0^x \sin \left( \frac{\pi t^2}{2} \right) dt \) has two critical points in the interval [1, 2.4]. One of the critical points is a local minimum and the other is a local maximum. The local minimum occurs at x =
(a) 1
(b) \( \sqrt{2} \)
(c) 2
(d) \( \frac{\pi}{2} \)
Answer: (c) 2

Question. Read the following mathematical statements carefully:
I. A differentiable function 'f' with maximum at \( x = c \implies f''(c) < 0 \).
II. Antiderivative of a periodic function is also a periodic function.
III. If f has a period T then for any \( a \in R \), \( \int_0^T f(x) dx = \int_0^T f(x+a) dx \)
IV. If f(x) has a maxima at \( x = c \), then 'f' is increasing in \( (c - h, c) \) and decreasing in \( (c, c + h) \) as \( h \to 0 \) for \( h > 0 \).
Now indicate the correct alternative.
(a) exactly one statement is correct.
(b) exactly two statements are correct.
(c) exactly three statements are correct.
(d) All the four statements are correct.
Answer: (a) exactly one statement is correct.

Question. If the point of minima of the function, \( f(x) = 1 + a^2x - x^3 \) satisfy the inequality \( \frac{x^2 + x + 2}{x^2 + 5x + 6} < 0 \), then 'a' must lie in the interval
(a) \( (-3\sqrt{3}, 3\sqrt{3}) \)
(b) \( (-2\sqrt{3}, -3\sqrt{3}) \)
(c) \( (2\sqrt{3}, 3\sqrt{3}) \)
(d) \( (-3\sqrt{3}, -2\sqrt{3}) \cup (2\sqrt{3}, 3\sqrt{3}) \)
Answer: (b) \( (-2\sqrt{3}, -3\sqrt{3}) \)

Question. The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is 1cm the altitude is 6 cm. When the radius is 6cm, the volume is increasing at the rate of 1Cu cm/sec. When the radius is 36cm, the volume is increasing at a rate of n cu. cm/sec. The value of 'n' is equal to
(a) 12
(b) 22
(c) 30
(d) 33
Answer: (d) 33

Question. Consider \( f(x) = |1 - x|, 1 \le x \le 2 \) and \( g(x) = f(x) + b \sin \frac{\pi}{2}x, 1 \le x \le 2 \) then which of the following is correct?
(a) Rolles theorem is applicable to both f, g and \( b = \frac{3}{2} \)
(b) LMVT is not applicable to f and Rolles theorem if applicable to g with \( b = \frac{1}{2} \)
(c) LMVT is applicable to f and Rolles theorem is applicable to g with b = 1
(d) Rolles theorem is not applicable to both f, g for any real b.
Answer: (c) LMVT is applicable to f and Rolles theorem is applicable to g with b = 1

Question. Given that f(x) is continuously differentiable on \( a \le x \le b \) where \( a < b, f(a) < 0 \) and \( f(b) > 0 \), which of the following are always true?
(i) f(x) is bounded on \( a \le x \le b \).
(ii) The equation \( f(x) = 0 \) has at least one solution in \( a < x < b \).
(iii) The maximum and minimum values of f(x) on \( a \le x \le b \) occur at points where \( f'(c) = 0 \).
(iv) There is at least one point c with \( a < c < b \) where \( f'(c) > 0 \).
(v) There is at least one point d with \( a < d < b \) where \( f'(c) < 0 \).
(a) only (ii) and (iv) are true
(b) all but (iii) are true
(c) all but (v) are true
(d) only (i), (ii) and (iv) are true
Answer: (d) only (i), (ii) and (iv) are true

Question. Suppose that f is a polynomial of degree 3 and that \( f''(x) \neq 0 \) at any of the stationary point. Then
(a) f has exactly one stationary point.
(b) f must have no stationary point.
(c) f must have exactly 2 stationary points.
(d) f has either 0 or 2 stationary points.
Answer: (d) f has either 0 or 2 stationary points.