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

p varies directly as \(x^{2}+y^{2}+z^{2}\). 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 \( \sin^{2}\theta\,\cos^{2}\theta\,(\sec^{2}\theta+\cosec^{2}\theta) \) 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 \(A B=6 \mathrm{~cm}\) and \(B C=8 \mathrm{~cm}\), 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 kg, 2 kg, 4 kg, 5 kg respectively. If on an average he purchased \(x\) 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 \(\bigl(100^{97}+100^{54}+x+1\bigr)\) leaves a remainder 0 when divided by 9?