MAT 2003 Question Paper

Let the A.P. be $$A_1,A_2,A_3,....$$ and so on, with first term = $$a$$ and common difference = $$d$$

Also, $$n^{th}$$ term of an A.P. = $$A_n=a+(n-1)d$$

Acc to ques,

=> $$A_6+A_{15}=A_7+A_{10}+A_{12}$$

=> $$(a+5d)+(a+14d)=(a+6d)+(a+9d)+(a+11d)$$

=> $$a+7d=0$$

Thus, $$A_8=0$$

=> Ans - (B)

Since, we do not the tax on the initial salary, we cannot determine the increase in tax liability.

=> Ans - (D)

Let number of pages types per hour by Rohit, Harsha and Sanjeev respectively be $$x,y,z$$

=> $$4(x+y+z)=216$$

=> $$x+y+z=54$$ -----------(i)

Also, $$z-y=y-x$$

=> $$x+z-2y=0$$----------(ii)

Subtracting equation (ii) from (i), we get : $$y=18$$

=> $$x+z=36$$ --------------(iii)

Also, $$5z=7x$$ ------------(iv)

Now, solving equations (iii) and (iv), we get : $$x=15$$ and $$z=21$$

$$\therefore$$ Pages typed per hour by Rohit, Harsha and Sanjeev respectively are : 15,18,21

=> Ans - (D)

Total units of bulbs used by the worker = 17+13 = 30

Number of bulbs in each box = 24

$$\therefore\ $$Number of boxes used by the worker = $$\frac{30}{24}$$ = $$\frac{5}{4}$$ = $$1\frac{1}{4}$$

Hence, the correct answer is Option B

Number of roads from Delhi to Chennai via Mumbai

= $$3\times4=12$$

=> Ans - (B)

Principal = Rs. 1000 and rate of interest = 5% and time period = 3 years

=> Amount when compounded annually = $$P(1+\frac{r}{100})^t$$

= $$1000(1+\frac{5}{100})^3$$

= $$1000\times(\frac{21}{20})^3$$

$$\approx Rs.$$ $$1157$$

=> Ans - (C)

There are 2 red, 3 green and 2 blue balls

Probability of drawing blue balls = $$\frac{2}{7}$$

=> Probability that the balls drawn contain no blue ball = $$1-\frac{2}{7}$$

= $$\frac{5}{7}$$

=> Ans - (A)

Baskets made in $$\frac{2}{3}$$ hours = 1 basket

=> Baskets made in $$\frac{15}{2}$$ hours = $$\frac{3}{2}\times\frac{15}{2}$$

= $$\frac{45}{4}=11\frac{1}{4}$$

=> Ans - (B)

The slope of the function $$y = x^3 + kx$$  at x = 2 can be calculated as shown below,

Slope = $$\dfrac{dy}{dx}\ =\ 3x^2\ +\ k$$

Slope at x = 2 is $$\ 3\left(2\right)^2\ +\ k\ =\ 12\ +\ k$$

The area under the curve $$z = a^2 + a$$ between points a = 0 and a = 3 can be calculated as,

Area = $$\int\ _0^3\left(a^2\ +\ a\right)da\ =\ \left[\dfrac{a^3}{3}\ +\ \dfrac{a^2}{2}\right]_{_0}^{^{^3}}\ =\ \dfrac{3^3}{3}\ +\ \dfrac{3^2}{2}\ -\ 0\ =\ 9\ +\ 4.5\ =\ 13.5$$

Given that the area is equal to the slope. Equating them, we get,

13.5 = 12 + k

k = 1.5

Hence, the correct answer is option A.