Test

The sum of LCM and HCF of two numbers is 536 and the difference between LCM and HCF is 296. If one of the numbers is 104, then what is the other number?

Which one of the following is a factor of the polynomial \((x-1)(x-2)(x-4)-90\)?

20 men are supposed to complete a work in 10 days. After working for 5 days, they realise that only one-fourth of the work is done. How many more men do they need to employ to finish the work on time?

If \( x \) is a negative real number, then which of the following are not correct?
1. There is some natural number \( k \) such that \( k x>0 \)
2. \( x^{2}+x>0 \) always
3. \( 2 x4. \( x^{2} \) is always a rational number

What is the remainder after dividing the number \( 37^{1000} \) by 9 ?

What is the square root of \( 23-4 \sqrt{15} \) ?

What is the sum of the linear factors (in \( x \) and \( y \) ) of the expression \( 2 x^{2}+x y-3 y^{2} \)?

Which one of the following equations does not have real roots?

The sum and the product of the roots of a quadratic equation are 7 and 12 respectively. If the bigger root is halved and the smaller root is doubled, then what is the resulting quadratic equation?

5 pencils, 6 notebooks and 7 erasers cost ₹ 250; whereas 6 pencils, 4 notebooks and 2 erasers cost ₹ 180. What is the cost of 2 notebooks and 4 erasers?

For which values of \(k\) does the equation \(x^{2}-kx+2=0\) have real and distinct solutions?

Three runners are running in a circular track, and they complete one round in 20,30 and 35 minutes respectively. When will they next meet at the starting point?

If \( \log_{10} x + \log_{10} x^{2} = 2 \log_{10} x + 1 \), then what is the value of \( x \)?

The LCM of two prime numbers \( p \) and \( q \) is 2231, where \( p>q \). What is the value of \( p-q \)?

If \( \alpha+\beta+\gamma = \alpha\beta + \beta\gamma + \gamma\alpha \), then what is \( (1-\alpha)(1-\beta)(1-\gamma) \) equal to?

If \( p \) and \( q(p>q) \) are the roots of the equation \( x^{2}-60x+899=0 \), then which one of the following is correct?

How many zeros are there in the product \( 1^{50} \times 2^{49} \times 3^{48} \times \ldots \times 50^{1} \)?

If three positive numbers are in the ratio \( 2: 3: 5 \) and the sum of their squares is 1368, then what is sum of the numbers?

If \( \frac{x}{b+c-a}=\frac{y}{b-c-a}=\frac{z}{a+b-c}=k \), then what is \( x^{2}+y^{2}+z^{2}-2 x y-2 y z+2 z x \) equal to?

What is \( \frac{x^{4}}{\left(x^{2}-y^{2}\right)\left(x^{2}-z^{2}\right)} + \frac{y^{4}}{\left(y^{2}-x^{2}\right)\left(y^{2}-z^{2}\right)} + \frac{z^{4}}{\left(z^{2}-x^{2}\right)\left(z^{2}-y^{2}\right)} \) equal to?

If \( A+B=\frac{x^{2}-8}{x+2} \) and \( A-B=\frac{-x^{2}+2 x+4}{x+2} \), then what is \( B \) equal to?

If the roots of the equation \( x^{2} - 4x - \log_{10} N = 0 \) are real, then what is the minimum value of \( N \)?

If \(96-64 a^{3}+\frac{8}{a^{6}}-\frac{48}{a^{3}}-t^{3}=0\), then what is \(a^{2}t+4a^{3}\) equal to?

If \( (2ab - b^2):(6a^2 - ab) = 1:6 \), then what is \( (a+b):(a-b) \) equal to?

If \( 5^{x-1}=(2 \cdot 5)^{\log_{10}5}\), then what is the value of \(x\)?

If \( \frac{a y - b x}{c} = \frac{c x - a z}{b} = \frac{b z - c y}{a} \), then which of the following is/are correct?
1. \( \frac{x}{a} = \frac{y}{b} \)
2. \( \frac{x+y+z}{a+b+c} = \frac{z}{c} \)

If one of the roots of the equation \( a x^{2}-4 a x+15=0 \) is \( \frac{3}{2} \), then what is the sum of the squares of the roots?

Consider the following inequalities:
1. \( \frac{a^{2}-b^{2}}{a^{2}+b^{2}} > \frac{a-b}{a+b} \) where \( a>b>0 \)
2. \( \frac{a^{3}+b^{3}}{a^{2}+b^{2}} > \frac{a^{2}+b^{2}}{a+b} \) only when \( a>b>0 \)

Which of the above is/are correct?

Let work done by ( \(3n - 1\) ) men in ( \(2n + 1\) ) days be \(x\) and work done by ( \(3n + 1\) ) men in ( \(4n - 3\) ) days be \(y\). If \(x : y = 6 : 11\), then what is the value of \(n\)?

A two-digit number is such that the product of the digits is 8. If 63 is added to this number, the digits interchange their places. What is the sum of the digits in the number?

A person wishes to fence \( 375 \mathrm{~m}^{2} \) rectangular garden. He has 65 m of barbed wire and is able to fence only three sides of the garden. What is the perimeter of the garden?

A motor boat has speed \(30 \mathrm{~km} / \mathrm{hr}\) in still water. It goes 60 km downstream and comes back in \(\frac{9}{2}\) hours. What is the speed of the stream?

The sum of numerator and denominator of a fraction is 10. If the numerator is increased by 3 and denominator is decreased by 1, the fraction becomes 1. What is the difference between numerator and denominator of the fraction?

The present age of a father is equal to sum of the ages of his 4 children. After ten years the sum of the ages of the children will be 1.6 times the age of their father. What is the present age of father?

Sheela can stitch a suit in 2 days, while Meena can stitch a suit in \( 1 \frac{1}{2} \) days. How many days will both take in stitching 30 suits?

A sum of money compounded annually doubles itself in 5 years. In how many years will it become four times of itself?

If \( p \) varies directly as \( q \) and inversely as square of \( r \), what is the percentage increase in \( p \) due to an increase in \( q \) by \( 20 \% \) and a decrease in \( r \) by \( 20 \% \) ?

If \( 2 x-3 y-7=0 \), then what is the value of \( 8 x^{3}-36 x^{2} y+54 x y^{2}-27 y^{3}-340 ? \)

Between 3 and 4 O'clock, both hour hand and minute hand will coincide past 3 O'clock between

If the system of equations \( 7 x+k y=27 \) and \( k x+7 y=19 \) have unique solution, then which one of the following is correct?

A person agrees to work for 30 days, on a condition that for every day's work he should receive \( ₹ 500 \), and that for every day's absence from work he should forfeit \( ₹ 100 \). At the end of the time he received \( ₹ 11,400 \). How many days did he work?

A person bought a chair and a table for ₹ 750. He sold the chair at a gain of \(5\%\) and the table at a gain of \(20\%\). He gained \(16\%\) on the whole. What is the original cost of the table?

If \(14 \sin^{2} \theta + 10 \cos^{2} \theta = 11\) where \(0^{\circ} < \theta < 90^{\circ}\), then what is the value of \(\tan \theta + \cot \theta\)?

A person rode one-third of a journey at \( 60 \mathrm{~km} / \mathrm{hr} \), one-third at \( 50 \mathrm{~km} / \mathrm{hr} \) and the rest at \( 40 \mathrm{~km} / \mathrm{hr} \). Had the person ridden half of the journey at \( 60 \mathrm{~km} / \mathrm{hr} \) and the rest at \( 40 \mathrm{~km} / \mathrm{hr} \), he would have taken 4 minutes longer to complete the journey. What distance did the person ride?

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

A ladder 10 m long reaches a point 10 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is \(60^{\circ}\). What is the height of the flagstaff?

What is the minimum value of \(\cos^{3}\theta + \sec^{3}\theta\) where \(0^{\circ} \leqslant \theta < 90^{\circ}\)?

A person saves \( ₹ 1000 \) more than he did the previous year. If he saves ₹ 2000 in the first year, in how many years will he save \( ₹ 170000 \) ?

If \( \frac{\cos^{2}\theta - 3\cos\theta + 2}{\sin^{2}\theta} = 1\), where \(0^{\circ} < \theta < 90^{\circ}\), then what is \(\sin^{2}\theta + \cos\theta\) equal to?

If for some \( \theta \) lying between \( 0^{\circ} \) and \( 90^{\circ} \), \( \tan \theta = 1 \), then what is the value of \( \sin^{2} \theta - 2 \sin \theta \cos \theta \)?

If \( x=p \sin A \cos B, y=p \sin A \sin B \) and \( z=p \cos A \), then what is the value of \( x^{2}+y^{2}+z^{2} \)?

From an aeroplane flying above a river at an altitude of 1200 m, it is observed that the angles of depression of opposite points on the two banks of a river are \( 30^{\circ} \) and \( \theta \). If the width of the river is 3000 m, then which one of the following is correct?

What is the maximum value of \( 1+2 \sin^{2} \theta \cos^{2} \theta - \sin^{4} \theta - \cos^{4} \theta \) where \( 0^{\circ}<\theta<90^{\circ} \)?

If \( x=m \sec A+n \tan A \) and \( y=m \tan A+n \sec A \), then what is \( x^{2}-y^{2} \) equal to?

What is the value of \( \sin 24^{\circ} \sin 66^{\circ} - \cos 24^{\circ} \cos 66^{\circ} + \tan 24^{\circ} \tan 66^{\circ} - \cot 24^{\circ} \cot 66^{\circ} \)?

In a triangle \(A B C\), right-angled at \(B\), \(A B + B C = 10(1+\sqrt{3})\) cm and the length of the hypotenuse is 20 cm. What is the value of \(\tan A + \tan C\)?

Let \( p \) be the area of a square \( X \) and \( q \) be the area of the square formed on the diagonal of the square \( X \). What is the value of \( \frac{p}{q} \)?

Consider the following :
1. \( \sin^{4}\theta - \sin^{2}\theta = \cos^{4}\theta - \cos^{2}\theta \)
2. \( \sin^{4}\theta + \cos^{4}\theta = 1 + 2\sin^{2}\theta\cos^{2}\theta \)
3. \( \tan^{4}\theta + \tan^{2}\theta = \sec^{4}\theta - \sec^{2}\theta \)
Which of the above are identities?

Two circles touch externally. The sum of their areas is \( 41 \pi \) square cm. If the distance between their centres is 9 cm, then what is the difference between their diameters?

The area of a rhombus is 336 square cm. If the length of one of its diagonals is 48 cm, then what is the perimeter of the rhombus?

If \( A, B \) and \( C \) are interior angles of a triangle \( A B C \), then what is
\[
\begin{aligned}
\tan \left(\frac{B+C}{2}\right)+ & \sin \left(\frac{B+C}{2}\right)- \\
& \cot \left(\frac{A}{2}\right)-\cos \left(\frac{A}{2}\right)
\end{aligned}
\]
equal to?

The radius of circum-circle of a rightangled triangle is 10 cm and the altitude drawn to the hypotenuse is 8 cm. What is the area of the triangle?

What is \( \frac{2 \sin^{3} \theta - \sin \theta}{\cos \theta - 2 \cos^{3} \theta},\left(0^{\circ}<\theta<90^{\circ}\right) \) equal to?

How many silver coins, 3.5 cm in diameter and of thickness 4 mm, must be melted to form a cuboid of dimensions \(21\;\mathrm{cm} \times 11\;\mathrm{cm} \times 7\;\mathrm{cm}\)? (take \(\pi = \frac{22}{7}\))

A hemispherical bowl of internal radius 18 cm contains a liquid. The liquid is filled in small cylindrical bottles of internal radius 3 cm and internal height 4 cm. What is the number of bottles used to empty the bowl?

A cone of height 16 cm and diameter 14 cm is mounted on a hemisphere of same diameter. What is the volume of the solid thus formed? (take \(\pi=\frac{22}{7}\))

A cubical block of side 14 cm is surmounted by a hemisphere of radius 7 cm. What is the total surface area of the solid thus formed? (take \(\pi=\frac{22}{7}\) )

The length and breadth of a room are 21 m and 16 m respectively. If the length of the longest rod that can be placed in the room is 29 m, then what is the height of the room?

3 cubes each of volume \( 343 \mathrm{~cm}^{3} \) are joined end to end. What is the total surface area of the resulting cuboid ?

A hollow spherical shell is made of a metal of density \( 7 \mathrm{~g} / \mathrm{cm}^{3} \). If its internal and external radii are 3 cm and 6 cm respectively, then what is the mass of the shell? (take \( \pi=\frac{22}{7} \) )

The minute hand of a clock is 21 cm long. What is the area on the face of the clock described by the minute hand between \( 10 \cdot 10 \) a.m and \( 10 \cdot 30 \) a.m? (take \( \pi=\frac{22}{7} \))

A tub is in the shape of a frustum of a cone. The radii of two circular ends of the tub are 105 cm and 42 cm. If the vertical height of the tub is 16 cm, what is its slant height?

A solid rod consists of a cylinder of height 20 cm and radius 7 cm. It is surmounted by another solid cylinder of height 10 cm and radius 3.5 cm. If 1 cubic metre of rod weighs 10000 kg, what is the mass of the rod? (take \( \pi=\frac{22}{7} \))

\(ABC\) is a triangle right-angled at \(B\) with \(AB=8\mathrm{~cm}\) and \(BC=6\mathrm{~cm}\). It is made to revolve about its side \(BC\). What is the approximate total surface area of the cone so formed? (take \(\pi=\frac{22}{7}\))

Consider the following for the next three (03) items that follow:
A solid consisting of a right circular cone of radius \( x \) and height \( 2 x \) standing on a hemisphere of radius \( x \) (take \( \pi=\frac{22}{7} \) )
The solid is placed upright in a right circular cylinder full of water such that it touches the bottom. If the internal radius of cylinder is \( x \) and height is \( 3 x \), what is the approximate volume of water left in the cylinder?

Consider the following for the next three (03) items that follow:
A solid consisting of a right circular cone of radius \( x \) and height \( 2 x \) standing on a hemisphere of radius \( x \) (take \( \pi=\frac{22}{7} \) )
What is the approximate total surface area of the solid?

Consider the following for the next three (03) items that follow:
A solid consisting of a right circular cone of radius \( x \) and height \( 2x \) standing on a hemisphere of radius \( x \) (take \( \pi=\frac{22}{7} \))
The volume of the solid is equal to that of a

Consider the following for the next two (02) items that follow :
A chord of a circle of radius 2.1 cm subtends an angle of \( 120^{\circ} \) at the centre. (take \( \pi=\frac{22}{7} \) and \( \sqrt{3}=1.732 \) )
What is the approximate area of minor segment of the circle ?

Consider the following for the next three (03) items that follow :
ABC is a triangle with sides AB = 6 cm, BC = 10 cm and CA = 8 cm. With vertices A, B and C as centres, three circles are drawn each touching the other two externally.
If \( P, Q \) and \( R \) are the areas of sectors at \( A, B \) and \( C \) within the triangle respectively, then which of the following is/are correct?
1. \( \mathrm{P}=\pi\mathrm{cm}^{2} \)
2. \( 9Q + 4R = 36\pi\mathrm{cm}^{2} \)

Select the correct answer using the code given below:

Consider the following for the next two (02) items that follow :
\( A B C \) is triangle in which \( A B = A C \) and \( D \) is any point on \( B C \).
If \( A D = 5 \mathrm{~cm}, B D = 4 \mathrm{~cm} \) and \( C D = 6 \mathrm{~cm} \), then what is \( A B \) equal to?

What is the approximate area of major segment of the circle?

Consider the following for the next three (03) items that follow :
\( A B C \) is a triangle with sides \( A B=6 \, \mathrm{cm} \), \( B C=10 \, \mathrm{cm} \) and \( C A=8 \, \mathrm{cm} \). With vertices \( A, B \) and \( C \) as centres, three circles are drawn each touching the other two externally.
What is the length of the altitude of the triangle drawn from vertex \( A \) on \( B C \)?

Consider the following for the next three (03) items that follow :
\( A B C \) is a triangle with \( A B=1.6 \mathrm{~cm} \), \( B C=6.3 \mathrm{~cm} \) and \( C A=6.5 \mathrm{~cm} \). Let \( P \) and \( Q \) be the mid-points of \( A B \) and \( B C \) respectively.
What is \( A Q^{2}+C P^{2} \) equal to ?
86. What is \( A B^{2}+4 B Q^{2} \) equal to ?
88. What is \( 4\left(C P^{2}-A Q^{2}\right) \) equal to ?

Consider the following for the next three (03) items that follow :
\( A B C \) is a triangle with sides \( A B=6 \mathrm{~cm} \), \( B C=10 \mathrm{~cm} \) and \( C A=8 \mathrm{~cm} \). With vertices \( A, B \) and \( C \) as centres, three circles are drawn each touching the other two externally.
What is the sum of the radii of the circles?

Consider the following for the next two (02) items that follow :
\( A B C \) is triangle in which \( A B=A C \) and \( D \) is any point on \( B C \).
Which one of the following is correct ?

Consider the following for the next two (02) items that follow :

Collect all the sequences of five consecutive integers such that their product is equal to one of these integers. Let \( X \) be the collection of all possible such sequences. Let \( P \) be the smallest integer and \( Q \) be the largest integer occurring in these sequences.
How many such sequences of five consecutive integers are possible?
100. What is the arithmetic mean of \( P \) and \( Q \)?

Consider the sequence of numbers from 1 to 50. How many trailing zeros are there in the factorial of 50?

If you multiply all the integers from 1 to 50, what is the highest power of 5 that divides the product?

In the expression \( 1^{50} \times 2^{49} \times 3^{48} \times \ldots \times 50^{1} \), what is the highest power of 10 that divides the product?