CDS PYQ Test

A real number \(x\) is such that the sum of the number and four times its square is the least. What is that number?

The difference of the square of two natural numbers \(m\) and \(n\) (m>n) is 72. How many pairs of natural numbers will satisfy?

Let \( N \) be a 5-digit number. When \( N \) is divided by \( 6,12,15,24 \) it leaves respectively \( 2,8,11,20 \) as remainders. What is the greatest value of \( N \) ?

What is the remainder when \[111^{222}+222^{333}+333^{444}\] is divided by 5 ?

What are the last three digits in the multiplication of \( 4321012345 \times 98766789 \)?

\( p \) varies directly as \( \left(x^{2}+y^{2}+z^{2}\right) \). When \( x=1, y=2, z=3 \), then \( p=70 \). What is the value of \( p \) when \( x=-1, y=1 \), \( z=5 \)?

Let \( N \) be the least positive multiple of 11 that leaves a remainder of 5 when divided by \( 6,12,15,18 \). Which one of the following is correct?

What is \( \frac{1}{\sqrt{10}+\sqrt{9}}+\frac{1}{\sqrt{11}+\sqrt{10}}+\frac{1}{\sqrt{12}+\sqrt{11}}+\ldots+\frac{1}{\sqrt{196}+\sqrt{195}} \) equal to?

Train \(X\) crosses a man standing on the platform in 24 seconds and train \(Y\) crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of \(X\) to speed of \(Y\)?

Let \( p, q \) be the roots of the equation \( x^{2}+m x-n=0 \) and \( m, n \) be the roots of the equation \( x^{2}+p x-q=0\) (m, n, p, q are non-zero numbers). Which of the following statements is/are correct?
I. \( m(m+n)=-1 \)
II. \( p+q=1 \)

What is the maximum value of \( 8 \sin \theta - 4 \sin^2 \theta \)?

What is \( (1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2} \) equal to?

Consider the following statements:
I. \( \tan 50^{\circ}-\cot 50^{\circ} \) is positive
II. \( \cot 25^{\circ}-\tan 25^{\circ} \) is negative
Which of the statements is/are correct?

If \(0 \leqslant (\alpha-\beta) \leqslant (\alpha+\beta) \leqslant \frac{\pi}{2}\), \(\tan(\alpha+\beta)=\sqrt{3}\) and \(\tan(\alpha-\beta)=\frac{1}{\sqrt{3}}\), then what is \(\tan\alpha\cdot\cot2\beta\) equal to?

What is the value of \( \sin^{2} \theta \cos^{2} \theta\bigl(\sec^{2} \theta + \cosec^{2} \theta\bigr) \) equal to?

If \(64^{\sin^{2}\theta}+64^{\cos^{2}\theta}=16\) where \(0\leqslant\theta\leqslant\frac{\pi}{2}\), then what is the value of \(\tan\theta+\cot\theta\)?

If \( \operatorname{cosec} \theta - \cot \theta = m \) and \( \sec \theta - \tan \theta = n \), then what is \( \operatorname{cosec} \theta + \sec \theta \) equal to?

From a point \(X\) on a bridge across a river, the angles of depression of two points \(P\) and \(Q\) on the banks on opposite side of the river are \(\alpha\) and \(\beta\) respectively. If the point \(X\) is at a height \(h\) above the surface of the river, what is the width of the river if \(\alpha\) and \(\beta\) are complementary?

In a triangle \(A B C\), \(\angle A B C=60^{\circ}\) and \(A D\) is the altitude. If \(A B=6\) cm and \(B C=8\) cm, then what is the area of the triangle?

If \( p \) and \( q \) are the roots of the equation \( x^{2}-\sin^{2} \theta\,x-\cos^{2} \theta=0 \), then what is the minimum value of \( p^{2}+q^{2} \)?

The arithmetic mean of \( n \) numbers is \( M \). If the sum of first \( (n-1) \) terms is \( k \), then what is the \( n \)th number?

What is the geometric mean of \( 3,9,27,81,243,729,2187 \)?

A person purchases one kg of tea powder from each of the four places \( A, B, C, D \) at the rate of \( ₹1000 \) per \(1\mathrm{~kg}, 2\mathrm{~kg}, 4\mathrm{~kg}, 5\mathrm{~kg}\). If on an average he purchased \( x \mathrm{~kg} \) of tea powder per \( ₹1000 \), then what is the approximate value of \( x \)?

What is the sum of the largest and the smallest 4-digit numbers made by using single digit prime numbers (without repetition)?

What is the remainder when \( 3^{255} \) is divided by 28?

What is the value of \( x(0 \leqslant x \leqslant 8) \) if \( \left(100^{97}+100^{54}+x+1\right) \) leaves a remainder 0 when divided by 9 ?

In a triangle \( A B C, D \) is a point on \( B C \). If \( A B \cdot D C = A C \cdot B D, \angle B A D = \alpha \) and \( \angle C A D = \beta \) then which one of the following is correct?

Let \(N=12345678AB\) be a 10-digit number, where A, B are digits. If N is divisible by 9, then which of the following statements is/are correct?
I. \((A+B)\) is divisible by 9
II. If A is odd, then B is odd

If \( x^{3}+\frac{1}{x^{3}}=\frac{65}{8} \) and \( y^{3}+\frac{1}{y^{3}}=\frac{730}{27} \), then which one of the following is a value of \( x y \)?

If \( 11 x+5 y \) is a prime number where \( x, y \) are natural numbers then what is the minimum value of \( (x+y) \) ?

A 4-digit number \( N \) has exactly 15 distinct divisors. What is the total number of distinct divisors of \( N^{2} \)?

If \( p, q \) and \( r \) are the lengths (in cm) of the sides of a right-angled triangle, then \( (p-q-r)(q-r-p)(r-p-q) \) is always

What is the minimum value of \( \frac{\left(a^{8}+a^{4}+1\right)\left(b^{8}+b^{4}+1\right)}{a^{4} b^{4}} \) where \( a>0, b>0 \)?

In a class containing 200 students, \( n \) students prefer both tea and coffee; \( 2 n \) students prefer coffee, \( 3 n \) students prefer tea; \( 4 n \) students prefer neither tea nor coffee. What is the value of \( n \)?

Let \( A B C \) be a triangle with area 36 square cm. If \( A B = 9\,\mathrm{cm}, B C = 12\,\mathrm{cm} \) and \( \angle A B C = \theta \), then what is \( \cos \theta \) equal to?

Let \( n \) be a natural number. The HCF of \( n, n+10 \) is 10. If the LCM is \( x \) (a 2-digit number), then how many values of \( x \) are possible?

What is HCF of \( a^{4}+2 a^{3}+3 a^{2}+2 a+1 \) and \( a^{6}-2 a^{3}+1 ? \)

If the roots of the equation \( x^{2}-(k-2) x+(k+1)=0 \) are equal, then what are the values of \( k \) ?

What is \(\left(\frac{\cos \theta - \sin \theta + 1}{\cos \theta + \sin \theta - 1}\right)(\cot \theta - \operatorname{cosec} \theta)\) equal to?

What is \( \frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta} \) equal to ?