Match the following
DIRECTIONS (Q. 1 and 2) : Each question contains statements given in two columns, which have to be matched. The statements in ColumnI are labelled A, B, C and D, while the statements in ColumnII are labelled p, q, r, s and t. Any given statement in ColumnI can have correct matching with ONE OR MORE statement(s) in ColumnII. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example : If the correct matches are Ap, s and t; Bq and r; Cp and q; and Ds then the correct darkening of bubbles will look like the given.
Q. 1. In this questions there are entries in columns I and II. Each entry in column I is related to exactly one entry in column II. Write the correct letter from column II against the entry number in column I in your answer book.
Ans. (A)  p, (B)  r
Solution.
∴ Differentiable everywhere.
∴ (A) → (p)
(B) sin (p (x – [x])) = f (x)
We know that
It’s graph is, as shown in figure which is discontinuous at Clearly x – [x] and hence sin (p (x – [x])) is not differentiable
(B) → r
Q. 2. In the following [x] denotes the greatest integer less than or equal to x. Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
Ans. (A)  p, q, r ; (B)  p, s ; (C)  r, s ; (D)  p, q
Solution.
Graph is as follows :
From graph y = x  x  is continuous in (– 1, 1) (p)
differentiable in (– 1, 1) (q)
Strictly increasing in (–1, 1). (r)
(B)
{where y can take only + ve values}
and y^{2} = x, x > 0
∴ Graph is as follows :
From graph is continuous in (– 1, 1) (p) not differentiable at x = 0 (s)
(C) NOTE THIS STEP
∴ Graph of y = x + [x] is as follows :
From graph, y = x + [x] is neither con tin uous, nor differentiable at x = 0 and hence in ( – 1, 1). (s)
Also it is strictly increasing in (– 1, 1) (r)
Graph of function is as follows :
From graph, y = f (x) is continuous (p) and differentiable (q) in (– 1, 1) but not strictly increasing in (– 1, 1).
DIRECTIONS (Q. 3) : Following question has matching lists. The codes for the list have choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
Q. 3. Let f_{1} : R →R , f_{2} : [0, ∞)→R, f_{3} : R →R and f_{4} :R → [ 0,∞) be defined by
Ans. (d)
Solution:
From graph f is differentiable but not one one.
From graph f_{2 }of_{1} is neither continuous nor one one.
Integer Value Correct Type
Q. 1. Let f : [1, ∞) → [2, ∞) be a differentiable function such that for all x > 1 , then the value of f (2) is
Ans. 6
Solution.
Differentiating, we get 6f (x) = 3 f (x) + 3xf '(x) – 3x^{2}
∴ f (x) = x^{2} + cx
But f (1) = 2 ⇒ c = 1
∴ f (x) = x^{2} + x
Hence f (2) = 4 + 2 = 6
Note : Putting x = 1 in given integral equation, we get
∴ Data given in the question is inconsistent.
Q. 2. The largest value of nonnegative integer a for which
Ans. 2
Solution.
Q. 3. Let f : R →R and g : R → R be r espectively given by f (x) =  x  + 1 and g(x) = x^{2} + 1. Define h : R → R by
The number of points at which h(x) is not differentiable is
Ans. 3
Solution.
g(x) = x^{2} + 1
From graph there are 3 points at which h(x) is not differentiable.
Q. 4. Let m and n be two positive integers greater than 1. If then the value of
Ans. 2
Solution.
Q. 5. be such that Then 6 (α + β) equals.
Ans. 7
Solution.
For above to be possible, we should have
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