Practice Test-I
Q:1. Equation of normal to the curve where it cuts x – axis is
(A) x + y = 1
(B) x – y = 1
(C) x + y = 0
(D) None of these
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Q:2. If y = a ln |x| + bx2 + x has its extreme values at x = -1 and x = 2 then P ≡ (a , b) is
(A) (2 , -1)
(B) (2 , -1/2)
(C) (-2 , 1/2)
(D) none of these
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Q:3. f(x) = tan-1 (sinx + cosx) is an increasing function in
(A) (–π/2, 0)
(B) (0, π/2)
(C) (-π/4, π/4)
(D) none of these
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Q:4. If the parabola y2 = 4x meets a circle with centre at (6,5) orthogonally, then possible point (s) of intersection can be;
(A) (4, 4)
(B) (9, 4)
(C) (2, √8)
(D) (3, 2)
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Q:5. If f (x) = xa log x and f(0) = 0, then the value of ‘ a ‘ for which Rolle’s theorem can be applied in [0, 1] is
(A) -2
(B) -1
(C) 0
(D) 1/2
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Q:6. In [0, 1] Lagrange Mean Value theorem is NOT applicable to
(A)$ \large f(x) = \left\{\begin{array}{ll} \frac{1}{2}-x \; , x < 1/2 \\ (\frac{1}{2}-x)^2 \; , x \ge 1/2 \end{array} \right. $
(B) $ \large f(x) = \left\{\begin{array}{ll} \frac{sinx}{x} \; , x \ne 0 \\ 1 \; , x = 0 \end{array} \right. $
(C) f(x) = x|x|
(D) f(x) = |x|
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Q:7. For what value of ‘a’ does the curve f(x) = (a2 -2a -2 ) + cosx is always strictly monotonic for all x ∈ R
(A) a ∈ R
(B) a > 0
(C) 1-√2 < a < 1 + √2
(D) None of these
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Q:8. If the curves a x2 + b y2 = 1 and a1 x2 + b1 y2 = 1 intersect orthogonally then
(A)1/a – 1/b = 1/a1 + 1/b1
(B)1/a + 1/b = 1/a1 + 1/b1
(C)1/a + 1/b = 1/a1 + 1/b1
(D) none of these
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Q:9. If the equation x5 – 10a3 x2 + b4 x + c5 = 0 has three equal roots, then
(A) 2b2 –10 a3b2+c5 = 0
(B) 6a5 + c5 = 0
(C) 2c5 –10 a3b2+ b4c5 = 0
(D) b4 = 15 a5
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Q:10. The function f is a differentiable function and satisfies the functional equation f(x) + f(y) = f(x + y) – xy – 1for every pair x, y of real numbers. If f(1) = 1, then the number of integers n ≠ 1 for which f(n) = n is
(A) 0
(B) 1
(C) 2
(D) 3
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Q:11. Let f(x) = x – sinx and g(x) = x – tanx where x∈ (0, π/2) . Then for these values of x ;
(A) f(x). g(x)>0
(B) f(x) g(x) <0
(C)f(x)/ g(x)>0
(D) none of these
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Q:12. If f (x) = x3 + bx2 + cx + d and 0 < b2 < c, then in (-∞ , ∞)
(A) f (x) is a strictly increasing function
(B) f(x) has a local maxima
(C) f(x) is a strictly decreasing function
(D) f(x) is bounded
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Q:13. If f(x) = 1 + x/1! + x2/2! + x3/3! …………+ xn/n! , then f(x) = 0 (n is odd , n≥3)
(A) can’t have any real root
(B) can’t have any repeated root
(C) has one positive root
(D) none of these
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Q:14. If f (x) = sin x + a2x + b is an increasing function for all values of x, then
(A) a ∈ (-∞,-1)
(B) a ∈ R
(C) a ∈ (-1 , 1)
(D) none of these
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Q:15. If f(x)= x2e-x2/a2 is an increasing function then (for a > 0), x lies in the interval
(A) [a , 2a]
(B) (-∞, -a] ∪ [0 , a]
(C) (-a, 0)
(D) None of these
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Q:16. Let f(x) = |x-1| + |x-2| + |x-3| + |x-4| ∀x ∈ R . Then
(A) x = 2 is the point of local minima
(B) x = 3 is the point of local minima
(C) x = 1 is the point of local minima
(D) none of these
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Q:17. If a, b are real numbers such that x3-ax2 + bx – 6 = 0 has its roots real and positive then minimum value of b is
(A) 1
(B) 2
(C) 3(36)1/3
(D) None of these
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Q:18. If f'(x) exists for all x∈R and g(x) = f(x) – (f(x))2 + (f(x))3 ∀ x ∈R , then
(A) g(x) is increasing whenever ‘ f ’ is increasing
(B)g(x) is increasing whenever ‘ f ’ is decreasing
(C)g(x) is decreasing whenever ‘ f ’ is increasing
(D)none of these
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Q:19. Cosine of the angle of intersection of curves y = 3x–1 logx and y = xx -1 is
(A) 1
(B) 1/2
(C) 0
(D) 1/3
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Q:20. The equation 8x3 – ax2 + bx –1 = 0 has three real roots in G.P. If λ1 ≤ a ≤ λ2 , then ordered pair (λ1, λ2) can be
(A) (-2, 2)
(B) (18, 12)
(C) (-10, -8)
(D) none of these
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