Bhaskara Contest (NMTC JUNIOR LEVEL-IX and X Grades) 2023 - Problems and Solutions

Problem 1

If $a, b, c$ are real numbers such that the polynomial $x^3+6 x^2+a x+b$ is the cube of $(x+c)$ then
a) $(a+b+c)$ is divisible by 13
b) $a+b=11 c$
c) $a>b$ and $b<c$ .
(4) $(a+b+c)$ is divisible by 11 .

Problem 2

In the adjoining figure, $A B=9 \mathrm{~cm}, A C=7 \mathrm{~cm}$ , $B C=8 \mathrm{~cm}, A D$ is the median and $\angle C=40^{\circ}$ . Then measure of $\angle A D B$ (in degrees) is

a) 100
b) 140
c) 45
d) 120

Problem 3

If $x^2+6 x+1=0$ and $\frac{x^3+k x^2+1}{3 x^3+k x^2+3 x}=2$ then the value of $k$ is

a) 68
b) 72
c) 65
d) 70

Problem 4

If $x=\sqrt[3]{49}+\sqrt[3]{42}+\sqrt{36}$ , then the value of $x-\frac{1}{x^2}$ is

a) $2 \sqrt[3]{42}$
b) $3 \sqrt[4]{42}$
c) $\sqrt[4]{42}$
d) $4 \sqrt[3]{42}$

Problem 5

In the adjoining figure,
AB = BC = CD

$P$ is the midpoint of $AQ$ . If $CR = 4, QC=12$ , then $PQ$ is equal to

a) $4 \sqrt{3}$
b) $6 \sqrt{3}$
c) $8 \sqrt{3}$
d) $2 \sqrt{3}$

Problem 6

In the adjoining figure, $A$ is the midpoint of the are $B A C$ .
Given that $A B=15$ and $A D=10$ .

Then the value of $A B$ is

a) 22
b) 23
c) 22.5
d) 23.5

Problem 7

The number of real numbers $x$ which satisfy the equation $\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$ is

a) 1
b) 2
c) 0
d) 4

Problem 8

$a, b$ are real numbers such that $2 a^2+5 b^2=20$ . Then the maximum value of $a^4 b^6$ is

a) 256
b) 1024
c) 1262
d) 16

Problem 9

The number of ordered pairs $(x, y)$ of integers such that $x-y^2=4$ and $x^2+y^4=26$ is

a) 4
b) 3
c) 2
d) 1

Problem 10

In the adjoining figure, three equal squares are placed. The squares are unit squares. The area of the shaded region $\left(\mathrm{in} \mathrm{cm}^2\right)$ is

a) $\frac{5}{4}$
b) $\frac{4}{5}$
c) $\frac{3}{2}$
d) $\frac{3}{4}$

Problem 11

In the adjoining figure, $A B$ is a diameter of the circle. Given $\angle B A C=20^{\circ}, \angle A E B=56^{\circ}$ , Then the measure (in degrees) of $\angle B C D$ is

a) 12
b) 10
c) 14
d) 16

Problem 12

The number of ordered pairs $((m, n)$ of integers such that $1 \leq m, n \leq 100$ and $m^n n^*$ leaves a remainder 1 when divided by 4 is

a) 2250
b) 1000
c) 1125
d) 1250

Problem 13

The number of ordered pairs of positive integers $(x, y)$ satisfying the equation $x^2+4 y=3 x+16$ is

a) 1
b) 2
c) 3
d) 4

Problem 14

The algebraic expression $(a+b+a b+2)^2+(a-a b+2-b)^2-2 b^2\left(1+a^2\right)$ reduces to

a) $4(a+2)^{2}$
b) $2(a+2)^2+4 a b^2$
c) $(a-2)^2$
d) $2(a-2)^2+4 a b^2$

Problem 15

The sum of $(1 \times 4)+(2 \times 7)+(3 \times 10)+(4 \times 13)+\ldots 49$ terms is equal to

a) 122500
b) 116800
c) 11800
d) 117600

Problem 16

If the equations $x^3+\alpha x+1=0$ and $x^4-\alpha x^2+1=0$ have a common root, then the value of $a^2$ is $\rule{1cm}{0.15mm}$ .

Problem 17

If $a, b, c, d$ are positive reals such that abcd=1 then the maximum value of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$ is $\rule{1cm}{0.15mm}$ .

Problem 18

The sum of all natural numbers ' $n$ ' for which $n(n+1)$ is a perfect square is $\rule{1cm}{0.15mm}$ .

Problem 19

$P$ is a point inside the square $A B C D$ such that $P A=P B=$ Distance of $P$ from $C D$ .
The ratio of the areas of the triangle $P A B$ to the area of the square $A B C D$ is $\frac{m}{n}$ where $m, n$ are relatively prime integers. Then the value of $m+n=$ $\rule{1cm}{0.15mm}$ .

Problem 20

The sum of roots of the simultuneous equations
$\sqrt[y]{4^x}=32 \sqrt[x]{8^y} , \sqrt[y]{3^x}=3 \sqrt[y]{9^{1-y}}$ is $\rule{1cm}{0.15mm}$ .

Problem 21

If $2 \sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}=\sqrt{a}+\sqrt{b}$ where $a, b$ are natural numbers, then the value of $a+b$ is $\rule{1cm}{0.15mm}$ .

Problem 22

In the adjoining figure, $\angle A C D=38^{\circ}$ .
Then the measure (in degrees) of angle $x$ is $\rule{1cm}{0.15mm}$ .

Problem 23

If $\frac{a}{b+c}+\frac{c}{a+b}=\frac{2 b}{c+a}$ (where $a+b, b+c, c+a, a+b+c$ are all not zero), then the numerical value of $\frac{a^2+c^2}{b^2}$ is $\rule{1cm}{0.15mm}$ .

Problem 24

The geometric and arithmetic means of two positive numbers are respectively 8 and 17 . The larger among the two numbers is $\rule{1cm}{0.15mm}$ .

Problem 25

The number of two-digit numbers in which the tens and the units digit are different and odd is $\rule{1cm}{0.15mm}$ .

Problem 26

The value of $(5 \sqrt[3]{4}-3 \sqrt[3]{\frac{1}{2}})(12 \sqrt[3]{2}+\sqrt[3]{16}-2 \sqrt[3]{2})$ is equal to $\rule{1cm}{0.15mm}$ .

Problem 27

If $\frac{x y}{x+y}=1, \frac{y z}{y+z}=2, \frac{z x}{z+x}=3$ , then the numerical value of $15 x-7 y-z$ is $\rule{1cm}{0.15mm}$ .

Problem 28

The sum of all natural numbers which satisfy the simultaneous inequations $x+3<4+2 x$ and $5 x-3<4 x-1$ is $\rule{1cm}{0.15mm}$ .

Problem 29

In an increasing geometric progression (with $1^{\text {st }}$ term $a$ and $n^{\text {th }}$ term $t_n$ ). the difference between the fourth and the first term is 52 and the sum of the first three terms is 26. Then the numerical value of $\frac{t_{2024}}{t_{2023}}+\frac{a^{2024}}{a^{2023}}$ is $\rule{1cm}{0.15mm}$ .

Problem 30

The base of a triangle is 4 units less than the altitude drawn to it. The area of the triangle is $96\left(\right.$ unit $\left.^2\right)$ . The ratio of the base to height is $\frac{p}{q}$ where $p , q$ are relatively prime to each other. Then the value of $p+q$ is $\rule{1cm}{0.15mm}$ .