MAT 2003 Question Paper

Let time taken by 12 men be $$t$$ minutes.

=> $$5\times60=12\times t$$

=> $$t=25$$ minutes

=> Ans - (A)

Framing equation for given data: 
$$\log\ \left(a^2+b^2\right)-\left(\log\ a+\log\ b\right)=C$$

Opening brackets and substituting a = b, we get
C = log 2

Principal = Rs. 10,000 and rate of interest = 6%

=> Interest = $$\frac{10,000\times6\times9}{12\times100}$$

= $$50\times9=Rs.$$ $$450$$

=> Ans - (A)

Prime factorization of :

385 = $$5\times7\times11$$

1001 = $$7\times11\times13$$

Thus, the 4 prime numbers are : 5,7,11,13

=> Smallest one = 5

=> Ans - (A)

Let speed of car = $$10x$$ km/hr

=> Speed of train = $$12x$$ km/hr

Time taken by both to travel 75 km = $$\frac{75}{10x}=\frac{75}{12x}+\frac{12.5}{60}$$

=> $$75(\frac{1}{60x})=\frac{12.5}{60}$$

=> $$x=\frac{75}{12.5}=6$$

$$\therefore$$ Speed of car = $$60$$ km/hr

=> Ans - (C)

Let number of cards with Pintoo = $$x$$ and number of cards laid aside = $$z$$

=> Number of cards with Mintoo = $$(52-x-z)$$

Let number of cards exchanged = $$k$$

Acc. to ques, => $$(x+k)=4(52-x-z-k)$$

=> $$x+k=208-4x-4k-4z$$

=> $$5x+5k+4z=208$$ ---------------(i)

Similarly, $$(x-k)=3(52-x-z+k)$$

=> $$x-k=156-3x+3k-3z$$

=> $$4x-4k+3z=156$$ ----------------(ii)

By applying 3(i)-4(ii), we get : $$-x+31k=0$$

=> $$x=31k$$

Now, we know that $$y$$ is a constant greater than 0, also since there are only 52 cards, we have $$k=1$$

$$\therefore$$ Number of cards with Pintoo = $$x=31$$

=> Ans - (A)

Normal time taken by express train (without stoppage) = $$\frac{600}{100}=6$$ hours

Number of time it stopped = $$\frac{600}{75}=8$$

Thus, it stopped 7 times for a total time of = $$3\times7=21$$ minutes

Now, distance travelled by local train in 6 hours and 21 minutes

= 300 km in 6 hours and 12 minutes (12 stoppages)

Thus, in the last 9 minutes, it will travel = $$\frac{50}{60}\times9=7.5$$ km

$$\therefore$$ Total distance travelled = 307.5 km

=> Ans - (A)

Let $$f(\theta)=sin\theta+cos\theta$$

=> $$f'(\theta)=cos\theta-sin\theta$$

Now, max value of $$f(\theta)$$ will be when $$f'(\theta)=0$$

=> $$cos\theta=sin\theta$$

$$\because \theta$$ is acute, => $$\theta=45^\circ$$

=> Ans - (B)

Let the third number be $$x$$. H.C.F. = 36. Prime factorization of :

3240 = $$2^3\times3^4\times5$$

3600 = $$2^4\times3^2\times5^2$$

=> $$x=2^2\times3^{n}\times k$$, where $$n\geq2$$ and $$k$$ is any prime number and $$k\neq5$$

Also, LCM is $$2^4 \times 3^5 \times 5^2 \times 7^2$$

=> $$n=5$$ and $$k=7^2$$

$$\therefore$$ $$x=2^2\times3^5\times7^2$$

=> Ans - (B)