[Download PDF] PGDBA 2016 Paper with Solutions

Instructions

For the following questions answer them individually

Question 41

If [a] denotes the greatest integer less than or equal to a for $$a \in R$$, then the value of the integral
$$\int_{0}^{1.7} [x^2] dx$$ is equal to

Video Solution

Question 42

Which of the following functions is differentiable at x = 0 ?

Video Solution

Question 43

Let the function f be given by $$f(x) = \begin{cases}x + x^2 \sin(\frac{\pi}{x}); & x \neq 0 \\0; & x = 0\end{cases}$$

then $$\left( f'(1) - f'(0) \right)$$ is

Video Solution

Solution

$$f'\left(x\right)=1+2x\sin\ \frac{\pi}{x}+x^2\cos\ \frac{\pi}{x}\left(-\frac{\pi}{x^2}\ \right)$$ = $$f'\left(x\right)=1+2x\sin\ \frac{\pi}{x}-\pi\ \cos\ \frac{\pi}{x}$$

=> $$f'\left(1\right)=1+2\sin\ \pi-\pi\ \cos\ \pi=1+\pi\ $$

$$f'\left(0\right)=\lim_{h->0}\ \frac{\left(f\left(0+h\right)-f\left(0\right)\right)}{h}$$ = $$f'\left(0\right)=\lim_{h->0}\ \frac{f\left(h\right)-0}{h}$$ = $$f'\left(0\right)=\lim_{h->0}\ \frac{\left(h+h^2\sin\ \frac{\pi}{h}\right)}{h}$$

=$$f'\left(0\right)=\lim_{h->0}\ \left(1+h\sin\ \frac{\pi}{h}\right)=1+0\ =1$$

Hence $$f'\left(1\right)-f'\left(0\right)=1+\pi\ -1=\pi\ $$

Question 44

Let the function $$f$$ be given by

$$f(x) = \begin{cases}-(x -1)^4; & x \leq 2\\(x - 3)^3; & x > 2\end{cases}$$

Then local extrema of $$f$$ exist at

Video Solution

Question 45

The points in the xy-plane, which satisfy the equation
$$\sqrt{(x - 1)^2 + (y + 2)^2} = \sqrt{(x + 3)^2 + (y - 2)^2}$$

Video Solution

Question 46

Two pairs of straight lines $$x^2 — 7x + 6 = 0$$ and $$y^2 — 14y + 40 = 0$$ intersect to form a rectangle. Let the diagonals of the rectangle intersect at the point W. A circle with center W and with tangents as lines $$y^2 — 14y + 40 = 0$$ intersects lines $$x^2 — 7x + 6 = 0$$ at points P, Q, R, S. The area of the rectangle PQRS is

Video Solution

Solution

$$x^2 — 7x + 6 = 0$$ => x= 1, x=6$$PM\ =\sqrt{\ PW^2-MW^2}$$

$$y^2 — 14y + 40 = 0$$ => y= 4, y=10

$$W\ =\ \left(\frac{1+6}{2},\frac{4+10}{2}\right)\ =\ \left(\frac{7}{2},7\right)$$

QR = 6-1 =5

r = (10-4)/2 = 3

PW = r= 3

MW = QR/2 = 5/2

$$PM\ =\sqrt{\ PW^2-MW^2}$$ = $$\frac{\sqrt{\ 11}}{2}$$

PQ=2PM = $$\sqrt{\ 11}$$

Area of PQRS = QR x PQ = $$5\sqrt{\ 11}$$

Question 47

Normals to the parabola $$y^2 = 4x$$ are drawn at two points P and Q on it. These normals intersect the parabola at the point R (9, -6). Then PQ equals

Video Solution

Solution

$$y^2 = 4x$$ , a =1

Equation of normal is $$y=mx-2am-am^3$$

=> $$y=mx-2m-m^3$$

It passes through R(9,-6)

=> -6 = 9m -2m - $$m^3$$

=> $$\left(m+1\right)\left(m-3\right)\left(m+2\right)=0$$

=> m =-1,3,-2

$$y^2 = 4x$$

$$2yy'=4\ =>\ y'=\frac{2}{y}$$

Slope of the normal = -y/2

-y/2 = -1 => y=2 => x= 1 hence one point is P(1,2)

-y/2 = 3 => y=-6 => x=9 hence secoond point is R(9,-6)

-y/2=-2 => y =4 => x=4 hence third point is Q(4,4)

PQ = $$\sqrt{\ 13}$$

Question 48

If $$f : R \rightarrow R$$ be a continuous function satisfying $$f(x) + f(3 - x) = 4$$, then $$\int_{0}^{3} f(x) dx$$ is equal to

Video Solution

Solution

$$\int_0^3f(x)dx=\int_0^3f(3-x)dx=A\left(say\right)$$ (using property of definite integrals).

Given $$f(x) + f(3 - x) = 4$$

Integrating with limit 0 to 3, we have

$$\int_0^3f(x)dx+\int_0^3f(x)dx=\int_0^34dx$$

$$A+A=4\left[3-0\right]$$

I=6.

Question 49

Let PQRS be a cycle quadrilateral. Let O be the centre of the circumcircle of the quadrilateral. Then which of the following statements is NOT true?

Video Solution

Solution

The angle which an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circumference.

$$\angle\ POQ=2\angle\ PSQ$$

Angles in the same segment of a circle are equal to one another

$$\angle\ PRQ=\angle\ PSQ$$

OPS is isosceles ( OP =OS =radius)

$$\angle\ OPS=\angle\ OSP$$

$$\angle\ PRQ\ \ne\ \angle\ POQ$$

Question 50

Let $$P$$ and $$Q$$ be two distinct nonempty sets. Then $$(P \cup Q)^c \cup (P^c \cap Q)$$ equals

Video Solution

Solution

$$(P \cup Q)^c \cup (P^c \cap Q)$$ = $$\left(P^c\cap Q^c\right)\cup(P^c\cap Q)\ =\ P^c\cap\left(Q^cUQ\right)$$

= $$\ P^c\cap universal\ set$$

= $$\ P^c$$

CAT Content

PGDBA 2016

If [a] denotes the greatest integer less than or equal to a for $$a \in R$$, then the value of the integral
$$\int_{0}^{1.7} [x^2] dx$$ is equal to

Which of the following functions is differentiable at x = 0 ?

Let the function f be given by $$f(x) = \begin{cases}x + x^2 \sin(\frac{\pi}{x}); & x \neq 0 \\0; & x = 0\end{cases}$$

then $$\left( f'(1) - f'(0) \right)$$ is

Let the function $$f$$ be given by

$$f(x) = \begin{cases}-(x -1)^4; & x \leq 2\\(x - 3)^3; & x > 2\end{cases}$$

Then local extrema of $$f$$ exist at

The points in the xy-plane, which satisfy the equation
$$\sqrt{(x - 1)^2 + (y + 2)^2} = \sqrt{(x + 3)^2 + (y - 2)^2}$$

Normals to the parabola $$y^2 = 4x$$ are drawn at two points P and Q on it. These normals intersect the parabola at the point R (9, -6). Then PQ equals

If $$f : R \rightarrow R$$ be a continuous function satisfying $$f(x) + f(3 - x) = 4$$, then $$\int_{0}^{3} f(x) dx$$ is equal to

Let PQRS be a cycle quadrilateral. Let O be the centre of the circumcircle of the quadrilateral. Then which of the following statements is NOT true?

Let $$P$$ and $$Q$$ be two distinct nonempty sets. Then $$(P \cup Q)^c \cup (P^c \cap Q)$$ equals

CAT Quant Questions | CAT Quantitative Ability

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CAT Content

PGDBA 2016

If [a] denotes the greatest integer less than or equal to a for $$a \in R$$, then the value of the integral $$\int_{0}^{1.7} [x^2] dx$$ is equal to

Which of the following functions is differentiable at x = 0 ?

Let the function f be given by $$f(x) = \begin{cases}x + x^2 \sin(\frac{\pi}{x}); & x \neq 0 \\0; & x = 0\end{cases}$$ then $$\left( f'(1) - f'(0) \right)$$ is

Let the function $$f$$ be given by $$f(x) = \begin{cases}-(x -1)^4; & x \leq 2\\(x - 3)^3; & x > 2\end{cases}$$ Then local extrema of $$f$$ exist at

The points in the xy-plane, which satisfy the equation $$\sqrt{(x - 1)^2 + (y + 2)^2} = \sqrt{(x + 3)^2 + (y - 2)^2}$$

Normals to the parabola $$y^2 = 4x$$ are drawn at two points P and Q on it. These normals intersect the parabola at the point R (9, -6). Then PQ equals

If $$f : R \rightarrow R$$ be a continuous function satisfying $$f(x) + f(3 - x) = 4$$, then $$\int_{0}^{3} f(x) dx$$ is equal to

Let PQRS be a cycle quadrilateral. Let O be the centre of the circumcircle of the quadrilateral. Then which of the following statements is NOT true?

Let $$P$$ and $$Q$$ be two distinct nonempty sets. Then $$(P \cup Q)^c \cup (P^c \cap Q)$$ equals

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CAT Quant Questions | CAT Quantitative Ability

CAT DILR Questions | LRDI Questions For CAT

CAT Verbal Ability Questions | VARC Questions For CAT

If [a] denotes the greatest integer less than or equal to a for $$a \in R$$, then the value of the integral
$$\int_{0}^{1.7} [x^2] dx$$ is equal to

Let the function f be given by $$f(x) = \begin{cases}x + x^2 \sin(\frac{\pi}{x}); & x \neq 0 \\0; & x = 0\end{cases}$$

then $$\left( f'(1) - f'(0) \right)$$ is

Let the function $$f$$ be given by

$$f(x) = \begin{cases}-(x -1)^4; & x \leq 2\\(x - 3)^3; & x > 2\end{cases}$$

Then local extrema of $$f$$ exist at

The points in the xy-plane, which satisfy the equation
$$\sqrt{(x - 1)^2 + (y + 2)^2} = \sqrt{(x + 3)^2 + (y - 2)^2}$$