Test 01

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

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 squares of two positive integers \( m \) and \( n (m > n) \) is 72. How many pairs of positive integers satisfy this?

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

Use the code given below to choose the answer :

Select the answer using the code given below :

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

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 is the remainder when \( 111^{222} + 222^{333} + 333^{444} \) is divided by 5?

What are the last three digits in the multiplication of \(432101 2345 \times 98766789\ )?

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

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

\( p \) 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 \( x=-1, y=1, z=5 \)?

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

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?

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

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?

A train \( X \) takes 24 seconds to pass a person standing on the platform , and train \( Y \) takes 18 seconds to pass a person standing on the platform . They pass each other in 20 seconds while moving in opposite directions . What is the ratio of the speed of \( X \) to the speed of \( Y \)?

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?

Assume \(p , q\) are the roots of the equation \( x^{2} + m x - n = 0\) and \(m, n \) are 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 true ?
I. \( m(m+n) = - 1\)
II. \(p + q = 1\)

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?

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

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?

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 these statements is/are correct ?

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 \le qslant(\alpha -\beta) \le qslant(\alpha +\beta) \le qslant \frac {\pi}{2} \ ), \( \tan (\alpha+\ beta)=\sqrt {3} \) and \( \tan (\ alpha-\beta )=\frac{1}{ \sqrt{3}} \ ), then what is the value of \( \tan \alpha \cd ot \cot 2 \ beta \)?

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 \( \tan \alpha \cdot \cot 2 \beta \) equal to?

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

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 \( 0 \leq slant \theta \leqslant \frac{\pi }{2} \), what is the value of \( \tan \theta + \ cot \theta \)?

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

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 over a river, located at a height \(h\) above the river surface, the angles of depression to two points \(P\) and \(Q\) on the opposite bank are \(\alpha \) and \(\ beta\) respectively . If \(\alpha + \beta = 90^\circ\ ), what is the width of the river ?

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

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?