MAT 2009 Question Paper

Let edge of each cube be $$a=1$$ cm and length, width and height of cuboid be $$l=3$$ cm, $$b=1$$ cm and $$h=1$$ cm

Total surface area of cuboid = $$2(lb+bh+hl)$$

= $$2\times(3+1+3)=14$$ $$cm^2$$

Total surface area of each cube = $$3\times6a^2 = 18\times(1)^2=18$$ $$cm^2$$

$$\therefore$$ Required ratio = $$\frac{14}{18}=7:9$$

=> Ans - (D)

Ratio of their heights = Ratio of length of shadow (since angle will be same)

Let height of Qutab Minar = $$h$$ m

=> $$\frac{h}{80}=\frac{150}{120}$$

=> $$h=\frac{150}{1.5}=100$$ m

=> Ans - (A)

Ram's salary for each month = $$\frac{90}{100}\times8000=Rs.$$ $$7200$$

Expenses were increased at 5% each month

Thus, from the month of April, he wont have any savings.

=> Ans - (A)

Case 1 : When we refine oil for 1 hour

Input = 1000 L

=> Output = $$\frac{90}{100}\times1000=900$$ L

Profit = $$30\times900=Rs.$$ $$27,000$$

Case 2 : When we refine oil for 1/2 hour

Input = 900 L

=> Output = $$\frac{90}{100}\times900=810$$ L

Profit = $$50\times810=Rs.$$ $$40,500$$

$$\therefore$$ Clearly, maximum profit earned is Rs. 40,500

=> Ans - (D)

Radius of cistern $$R=10m$$ and height = $$H=15$$ m

Radius of cylindrical pipe $$r=0.25m$$ and height = $$h=5$$ m (for 1 sec)

=> Volume of cylindrical cistern = $$\pi r^2h$$

$$\therefore$$ Time taken to fill the cistern (in minutes) = $$\frac{\pi (10)^2\times15}{\pi\times(0.25)^2\times5}\times\frac{1}{60}$$

= $$\frac{100\times10000}{625\times5\times4}$$

= $$5\times16=80$$ minutes

=> Ans - (D)

Expression : $$2^m - 2^n = 960$$ -------------(i)

=> $$2^m>960$$

=> $$m\geq10$$

If, $$m=10$$

Substituting in equation (i), we get : $$1024-2^n=960$$

=> $$2^n=64=2^6$$

=> $$n=6$$

Since, both are natural numbers, => $$m=10$$

=> Ans - (A)

Let the 2 sides be $$a=b=2$$ cm each, and the angle between them be $$\theta$$

Now, area of triangle = $$\frac{1}{2} ab sin\theta$$

=> $$A=\frac{1}{2}\times2\times2\times sin\theta=2sin\theta$$ $$cm^2$$

When the sides are doubled, => $$a'=b'=4$$ cm

=> $$A'=\frac{1}{2}\times4\times4\times sin\theta=8sin\theta$$ $$cm^2$$

$$\therefore$$ $$\frac{A'}{A}=4$$

=> Ans - (C)

The number when divided by 765 leaves remainder 42, => Number = $$N=765k+42$$, where $$k$$ is a whole number

Let $$k=0$$, => $$N=42$$

Now, when 42 is divided by 17, remainder = $$42\%17=8$$

=> Ans - (A)

If we add 1 to our dividend, it becomes = 11,109,999 + 1 = 11,110,000 which is clearly a multiple of 1111 ($$\because 1111\times10000=11110000$$)

Now, since we added 1 to the number, we need to subtract 1 from 1111 to get the remainder.

Thus, remainder = 1111 - 1 = 1110

=> Ans - (D)

Quantity of coniac in the mixture = 30 $$l$$ and quantity of water = 10 $$l$$

Let $$x$$ $$l$$ of water is added.

=> $$\frac{30}{10+x}=\frac{5}{2}$$

=> $$12=10+x$$

=> $$x=2$$

$$\therefore$$ 2 L of additional water is required.

=> Ans - (D)