CAT 2023 QA Questions and Solutions
Question 1
The number of all natural numbers up to 1000 with non-repeating digits is:
A. 648
B. 585
C. 738
D. 504
Answer: C. 738
Detailed Solution:
– **1-digit numbers:** 9
– **2-digit numbers:** 9 choices for the tens place, 9 remaining choices for the units place = 81.
– **3-digit numbers:** 9 choices for the hundreds place, 9 remaining choices for the tens place, and 8 choices for the units place = 648.
Total = 9 + 81 + 648 = **738**.
Question 2
Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km, where x is a palindromic number. At the end of the trip, the car had travelled a total of 26862 km. If Brishti never drove at more than 110 kmph, then the greatest possible average speed at which she drove during the trip, in kmph, was:
A. 110
B. 80
C. 90
D. 100
Answer: D. 100
Detailed Solution:
– Before the trip: least palindromic number greater than 25982 is 26062.
– Distance during the trip = 26862 – 26062 = 800 km.
– Average speed = 800 km / 8 hours = **100 kmph**.
Question 3
If \(\frac{\sqrt{5x + 9}}{\sqrt{5x – 9}} = 3(\sqrt{2} + 1)\), then find the value of \( \sqrt{10x + 9} \):
A. 3√7
B. 3√5
C. 4√3
D. 7√3
Answer: A. 3√7
Detailed Solution:
Given, \(\frac{\sqrt{5x + 9}}{\sqrt{5x – 9}} = 3(\sqrt{2} + 1)\)
– Simplifying, we get \( \sqrt{5x + 9} = 3(\sqrt{2} + 1) \sqrt{5x – 9} \).
– Solving, we find that \( x = 9 \).
– Therefore, \( \sqrt{10x + 9} = 3√7 \).
Question 4
A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals:
A. 2 : 1
B. 5 : 8
C. 1 : 2
D. 8 : 5
Answer: D. 8 : 5
Detailed Solution:
– Using properties of similar triangles and cyclic quadrilaterals:
– The given ratio implies that the required ratio of AE : CE is **8 : 5**.
